"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { a: "a", annotation: "annotation", blockquote: "blockquote", details: "details", div: "div", em: "em", h1: "h1", h2: "h2", h3: "h3", h4: "h4", img: "img", li: "li", math: "math", mfrac: "mfrac", mi: "mi", mn: "mn", mo: "mo", mover: "mover", mpadded: "mpadded", mrow: "mrow", mspace: "mspace", msqrt: "msqrt", mstyle: "mstyle", msub: "msub", msubsup: "msubsup", msup: "msup", mtable: "mtable", mtd: "mtd", mtext: "mtext", mtr: "mtr", munder: "munder", munderover: "munderover", ol: "ol", p: "p", path: "path", semantics: "semantics", span: "span", strong: "strong", summary: "summary", svg: "svg", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsxs(_components.h1, { id: "the-many-faces-of-linear-regression", children: [_jsx(_components.a, { className: "heading-link", href: "#the-many-faces-of-linear-regression", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "The Many Faces of Linear Regression"] }), "\n", _jsxs(_components.p, { children: ["In this article I'll review some basics on linear regression (or to use a better term, ", _jsx(_components.em, { children: "ordinary least squares" }), " (OLS)), show several approaches to solving an OLS problem, and how these connect to (", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "L" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "L_{2}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), ") regularization techniques."] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["What you already know:", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Linear algebra: matrix operations, inner products, vector norm and distance.", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "The section on the convergence condition for gradient descent requires some more theory; in particular, symmetric and orthogonal matrices, diagonalization, and eigendecomposition." }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["How this article might interest you", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "You already have some familiarity with linear regression but want a deeper, unified understanding of how this topic is introduced in textbooks belonging to different fields (linear algebra, statistics, etc.)" }), "\n", _jsx(_components.li, { children: "You have seen some rather intimidating formulas on linear regression and wonder where they come from!" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "preliminaries", children: [_jsx(_components.a, { className: "heading-link", href: "#preliminaries", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Preliminaries"] }), "\n", _jsxs(_components.p, { children: ["Basically, we have an ", _jsx(_components.em, { children: "inconsistent" }), " system of equations ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { mathvariant: "normal", children: "≠" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X\\mathbf{w}\\neq\\mathbf{y}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mrel", children: [_jsx(_components.span, { className: "mrel", children: _jsx(_components.span, { className: "mord vbox", children: _jsx(_components.span, { className: "thinbox", children: _jsxs(_components.span, { className: "rlap", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "inner", children: _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mrel", children: "" }) }) }), _jsx(_components.span, { className: "fix" })] }) }) }) }), _jsx(_components.span, { className: "mrel", children: "=" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }), ". 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The model can be formulated as"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mo, { children: "^" })] }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "m" })] }), _jsxs(_components.msub, { children: 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[_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }) })] }), " already contains a column of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "1" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) })] }), "s, which, when multiplied by it corresponding parameter ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "w" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "w" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02691em" }, children: "w" })] }) })] }), " in ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { mathvariant: "bold", children: "w" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4444em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" })] }) })] }), ", is equivalent to adding a bias term ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }), " (", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}^T\\mathbf{x}+b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9247em", verticalAlign: "-0.0833em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8413em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: 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This greatly simplifies the formulas we'll see later."] }), "\n"] }), "\n", _jsxs(_components.details, { "data-callout": true, "data-callout-type": "disclaimer", children: ["\n", _jsx(_components.summary, { "data-callout-title": true, children: "Disclaimer on Notation" }), "\n", _jsxs(_components.div, { "data-callout-body": true, children: ["\n", _jsx(_components.p, { children: "There's a variety of letters used to describe the above problem:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Linear algebra books introduce systems of equations as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "A" }), _jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { mathvariant: "bold", children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A\\mathbf{x}=\\mathbf{b}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mord mathbf", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathbf", children: "b" })] })] })] }), " and least-squares problems as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "A" }), _jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mo, { mathvariant: "normal", children: "≠" }), _jsx(_components.mi, { mathvariant: "bold", children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A\\mathbf{x} \\neq \\mathbf{b}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mord mathbf", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mrel", children: [_jsx(_components.span, { className: "mrel", children: _jsx(_components.span, { className: "mord vbox", children: _jsx(_components.span, { className: "thinbox", children: _jsxs(_components.span, { className: "rlap", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "inner", children: _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mrel", children: "" }) }) }), _jsx(_components.span, { className: "fix" })] }) }) }) }), _jsx(_components.span, { className: "mrel", children: "=" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: 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children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "Xβ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mrel", children: [_jsx(_components.span, { className: "mrel", children: _jsx(_components.span, { className: "mord vbox", children: _jsx(_components.span, { className: "thinbox", children: _jsxs(_components.span, { className: "rlap", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "inner", children: _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mrel", children: "" }) }) }), _jsx(_components.span, { className: "fix" })] }) }) }) }), _jsx(_components.span, { className: "mrel", children: "=" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }), " and the solution as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "X" }), _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mo, { children: "^" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { children: "^" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X \\hat{\\beta}=\\hat{\\mathbf{y}}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1523em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9579em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: 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verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.7079em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] }), _jsxs(_components.span, { style: { top: "-3.0134em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1944em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] }), ", where ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\beta}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1523em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9579em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] }), _jsxs(_components.span, { style: { top: "-3.2634em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }), " is the optimal set of parameters and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\mathbf{y}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9023em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.7079em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] }), _jsxs(_components.span, { style: { top: "-3.0134em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1944em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }), " is the vector of predictions made by the optimal model; also, ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "X" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }) })] }), " is known as the ", _jsx(_components.em, { children: "design matrix" }), "."] }), "\n", _jsxs(_components.li, { children: ["Machine learning literature denotes the parameters of the model as ", _jsx(_components.em, { children: "weights" }), ", therefore the problem is written as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { mathvariant: "normal", children: "≠" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X \\mathbf{w} \\neq \\mathbf{y}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mrel", children: [_jsx(_components.span, { className: "mrel", children: _jsx(_components.span, { className: "mord vbox", children: _jsx(_components.span, { className: "thinbox", children: _jsxs(_components.span, { className: "rlap", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "inner", children: _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mrel", children: "" }) }) }), _jsx(_components.span, { className: "fix" })] }) }) }) }), _jsx(_components.span, { className: "mrel", children: "=" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }), "."] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["Since we can't find a vector ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { mathvariant: "bold", children: "w" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4444em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" })] }) })] }), " that satisfies ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X\\mathbf{w}=\\mathbf{y}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }), ", we're instead in minimizing the (Euclidean) distance between ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X\\mathbf{w}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" })] }) })] }), " and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: 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children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { mathvariant: "bold", children: "T" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "−" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.mi, { children: "m" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{{\\sum_{i=1}^m (\\mathbf{w^T}\\mathbf{x}_{i}-y_{i})^2}}{m}" })] }) }) }), 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"vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.143em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3281em" }, children: _jsxs(_components.span, { style: { top: "-2.357em", marginLeft: "-0.0359em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.143em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mclose mtight", children: [_jsx(_components.span, { className: "mclose mtight", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) })] }), ", which is just the ", _jsx(_components.em, { children: "mean square error" }), ", a well-known performance metric for regression models.\nFormally, we're interesting the following:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mo, { children: _jsx(_components.mtext, { children: "argmin" }) }), _jsx(_components.mi, { mathvariant: "bold", children: "w" })] }), _jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathop{\\text{argmin}}_{\\mathbf{w}}\\lVert X\\mathbf{w}-\\mathbf{y} \\rVert^2 " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop", children: _jsx(_components.span, { className: "mord text", children: _jsx(_components.span, { className: "mord", children: "argmin" }) }) }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.067em" }, children: _jsxs(_components.span, { style: { top: "-2.4559em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathbf mtight", style: { marginRight: "0.01597em" }, children: "w" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2441em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "∥" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1141em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "∥" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Going forward, I'll be assuming the columns (features) in ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "X" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }) })] }), " are linearly independent. This assumption guarantees the ", _jsx(_components.em, { children: "uniqueness" }), " of the optimal solution to the problem, denoted as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\mathbf{w}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7079em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.7079em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" })] }), _jsxs(_components.span, { style: { top: "-3.0134em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }) }) }) })] }) })] }), ", and also the invertibility of the Gram matrix ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { children: "X" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X^TX" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8413em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8413em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }) })] }), "."] }), "\n", _jsxs(_components.li, { children: ["In machine learning terms, we can consider ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lVert X\\mathbf{w}-\\mathbf{y} \\rVert^2" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "∥" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: 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children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f(x)=\\lVert X \\mathbf{w}-\\mathbf{y} \\rVert^2=\\sum_{i=1}^m \\left( \\sum_{j=1}^n X_{ij}w_{j}-y_{i} \\right)^2 " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10764em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: 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className: "strut", style: { height: "3.3678em", verticalAlign: "-1.4138em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8723em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: 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{ style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.4138em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: 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className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: ")" }) })] }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.954em" }, children: _jsxs(_components.span, { style: { top: "-4.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["Let's inspect the ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "m" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" })] }) })] }), " individual (no. subjects) error terms, denoted as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "r_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "r" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0278em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), ":"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: ":" })] })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "w" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "−" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { fence: "true", children: ")" })] }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mspace, { width: "2em" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: "…" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "r_{i}^2=\\left( X_{i,:}w_{i}-b_{i} \\right)^2, \\qquad i=1,\\dots,m" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1111em", verticalAlign: "-0.247em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "r" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: [_jsxs(_components.span, { style: { top: "-2.453em", marginLeft: "-0.0278em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: 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className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2401em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "minner", children: [_jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { 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[_jsx(_components.span, { className: "strut", style: { height: "0.7944em", verticalAlign: "-0.15em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "r" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0278em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "ij" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "This is the partial derivative, for one subject, w.r.t a single parameter. Let's start summing up to get back to the gradient:" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsx(_components.mi, { children: "f" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { children: "j" })] })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mn, { children: "2" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "j" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{ \\partial f }{ \\partial x_{j} } =\\sum_{i=1}^m 2r_{i}X_{ij}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.3435em", verticalAlign: "-0.9721em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10764em" }, children: "f" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.9721em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9291em", verticalAlign: "-1.2777em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8723em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2777em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "r" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0278em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "ij" }) }) })] }) }), _jsx(_components.span, { className: 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_jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "[" }), _jsxs(_components.mtable, { rowspacing: "0.16em", columnalign: "center", columnspacing: "1em", children: [_jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mn, { children: "2" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "1" })] })] })] }) }) }) }), _jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "⋮" }), _jsx(_components.mpadded, { height: "0em", voffset: "0em", children: _jsx(_components.mspace, { mathbackground: "black", width: "0em", height: "1.5em" }) })] }) }) }) }), _jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mn, { children: "2" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "n" })] })] })] }) }) }) })] }), _jsx(_components.mo, { fence: "true", children: "]" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "2" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { children: "r" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "2" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\nabla f(x)=\\begin{bmatrix}\n\\sum_{i=1}^m 2r_{i}X_{i,1} \\\\ \\vdots \\\\ \\sum_{i=1}^m 2r_{i}X_{in}\n\\end{bmatrix}=2X^Tr=2X^T(X\\mathbf{w}-\\mathbf{y})" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∇" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10764em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { 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_jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "∥" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["Notice how each objective has its own ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "X" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }) })] }), " matrix and its own ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { mathvariant: "bold", children: "y" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{y}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] }) })] }), " vector, but the parameters vector ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { mathvariant: "bold", children: "w" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: 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A convenient approach is to frame the overall objective function as a weighted sum of the individual objective function:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mtable, { rowspacing: "0.25em", columnalign: "right left", columnspacing: "0em", children: [_jsxs(_components.mtr, { children: [_jsx(_components.mtd, { className: "mtr-glue" }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "true", children: _jsx(_components.mi, { children: "J" }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "true", children: _jsxs(_components.mrow, { children: [_jsx(_components.mrow, {}), 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\n\\end{bmatrix} \\right\\Vert ^2 \\\\\n & = \\left\\lVert \\begin{bmatrix}\n\\sqrt{ \\lambda_{1} }X_{1} \\\\ \\vdots \\\\ \\sqrt{ \\lambda_{k} }X_{k}\n\\end{bmatrix}\\mathbf{w}- \\begin{bmatrix}\n\\sqrt{ \\lambda_{1} }\\mathbf{y}_{1} \\\\ \\vdots \\\\ \\sqrt{ \\lambda_{k} }\\mathbf{y}_{k}\n\\end{bmatrix} \\right\\rVert ^2 \\\\\n & = \\lVert \\tilde{X}\\mathbf{w}-\\tilde{\\mathbf{y}} \\rVert^2 \n\\end{align}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "14.2012em", verticalAlign: "-6.8506em" } }), _jsxs(_components.span, { className: "mtable", children: [_jsx(_components.span, { className: "col-align-r", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "7.3506em" }, 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_jsxs(_components.span, { className: "vlist", style: { height: "0.9202em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }), _jsxs(_components.span, { style: { top: "-3.6023em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "~" }) })] })] }) }) }) }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6813em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] }), _jsxs(_components.span, { style: { top: "-3.3634em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1944em" }, children: _jsx(_components.span, { className: "mord", children: "~" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "∥" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "6.8506em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }), _jsx(_components.span, { className: "tag", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "7.3506em" }, children: [_jsxs(_components.span, { style: { top: "-11.1118em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "4.6012em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "-9.5877em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "4.6012em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "-6.3265em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "4.6012em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "-1.528em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "4.6012em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "1.5894em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "4.6012em" } }), _jsx(_components.span, { className: "eqn-num" })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "6.8506em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) }), "\n", _jsxs(_components.p, { children: ["Interesting! We can write a multi-objective least-squares problem as an equivalent single objective problem (with appropriately stacked ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mo, { children: "~" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\tilde{X}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9202em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.9202em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }), _jsxs(_components.span, { style: { top: "-3.6023em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "~" }) })] })] }) }) }) })] }) })] }), " and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { children: "~" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\tilde{\\mathbf{y}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8757em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6813em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] }), _jsxs(_components.span, { style: { top: "-3.3634em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1944em" }, children: _jsx(_components.span, { className: "mord", children: "~" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }), ")."] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["In practice, all the ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "λ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lambda" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "λ" })] }) })] }), " weights are divided by ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "λ" }), _jsx(_components.mn, { children: "1" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lambda_{1}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "λ" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " so that the weight for the first objective is ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "1" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) })] }), ".\nGiven what we know about ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mo, { children: "~" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\tilde{X}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9202em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.9202em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }), _jsxs(_components.span, { style: { top: "-3.6023em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "~" }) })] })] }) }) }) })] }) })] }), " and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { children: "~" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\tilde{\\mathbf{y}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8757em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6813em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] }), _jsxs(_components.span, { style: { top: "-3.3634em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1944em" }, children: _jsx(_components.span, { className: "mord", children: "~" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }), ", we can derive the closed form optimal solution:"] }), "\n"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: 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_jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.247em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.3158em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }), _jsx(_components.span, { className: "tag", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.8158em" }, children: [_jsxs(_components.span, { style: { top: "-3.8158em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.9202em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "-2.2644em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.9202em" } }), _jsx(_components.span, { className: "eqn-num" })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.3158em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) }), "\n", _jsx(_components.p, { children: "At this point you might be wondering about the scenarios that give rise to a multi-objective least squares problem. Fortunately, you probably already know about a well-known example: regularized least squares!" }), "\n", _jsxs(_components.h3, { id: "regularized-least-squares", children: [_jsx(_components.a, { className: "heading-link", href: "#regularized-least-squares", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Regularized Least Squares"] }), "\n", _jsxs(_components.p, { children: ["In this section we'll show how many regularization methods can in fact be framed as a multi-objective least squares problem, and also how this view can let us easily derive the closed form solutions. In particular, we'll focus on ridge regression and regularization based on ", _jsx(_components.em, { children: "prior" }), " assumptions on ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { mathvariant: "bold", children: "x" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{x}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4444em" } }), _jsx(_components.span, { className: "mord mathbf", children: "x" })] }) })] }), "."] }), "\n", _jsxs(_components.h4, { id: "ridge-regression", children: [_jsx(_components.a, { className: "heading-link", href: "#ridge-regression", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Ridge Regression"] }), "\n", _jsxs(_components.p, { children: ["Let's start with the objective function for ridge regression, conventionally written as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "λ" }), _jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lVert X \\mathbf{w}-\\mathbf{y} \\rVert^2+ \\lambda \\lVert \\mathbf{w} \\rVert^2" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "∥" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "∥" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "λ" }), _jsx(_components.span, { className: "mopen", children: "∥" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "∥" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing 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"1" })] }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }) }), "\n", _jsxs(_components.h4, { id: "regularization-using-mathbfx_textprior", children: [_jsx(_components.a, { className: "heading-link", href: "#regularization-using-mathbfx_textprior", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Regularization using ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mtext, { children: "prior" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{x}_{\\text{prior}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7305em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3175em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord text mtight", children: _jsx(_components.span, { className: "mord mtight", children: "prior" }) }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] })] }), "\n", _jsxs(_components.p, { children: ["In many applications, we don't approach the data without any prior assumptions about ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\mathbf{w}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7079em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.7079em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" })] }), _jsxs(_components.span, { style: { top: "-3.0134em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }) }) }) })] }) })] }), ". In fact, domain knowledge or a previously developed model could easily equip us with a ", _jsx(_components.em, { children: "prior" }), " estimate of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\mathbf{w}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7079em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.7079em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" })] }), _jsxs(_components.span, { style: { top: "-3.0134em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }) }) }) })] }) })] }), ", denoted as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mtext, { children: "prior" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}_{\\text{prior}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7305em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3175em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord text mtight", children: _jsx(_components.span, { className: "mord mtight", children: "prior" }) }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), ". The cost function is similar to ridge regression:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "λ" }), _jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mtext, { children: "prior" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.msqrt, { children: _jsx(_components.mi, { children: "λ" }) }), _jsx(_components.mi, { children: "I" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.msqrt, { children: _jsx(_components.mi, { children: "λ" }) }), _jsx(_components.mi, { children: "I" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mtext, { children: "prior" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "∥" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "[" }), _jsxs(_components.mtable, { rowspacing: "0.16em", columnalign: "center", columnspacing: "1em", children: [_jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsx(_components.mi, { children: "X" }) }) }) }), _jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.msqrt, { children: _jsx(_components.mi, { children: "λ" }) }), _jsx(_components.mi, { children: "I" })] }) }) }) })] }), _jsx(_components.mo, { fence: "true", children: "]" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "[" }), _jsxs(_components.mtable, { rowspacing: "0.16em", columnalign: "center", columnspacing: "1em", children: [_jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsx(_components.mi, { mathvariant: "bold", children: "y" }) }) }) }), _jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.msqrt, { children: _jsx(_components.mi, { children: "λ" }) }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mtext, { children: "prior" })] })] }) }) }) })] }), _jsx(_components.mo, { fence: "true", children: "]" })] }), _jsx(_components.mo, { fence: "true", children: "∥" })] }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lVert X\\mathbf{w}- \\mathbf{y} \\rVert^2+ \\lambda \\lVert \\mathbf{w} - \\mathbf{w}_{\\text{prior}} \\rVert^2=\\lVert X\\mathbf{w}-\\mathbf{y} \\rVert^2 + \\lVert \\sqrt{ \\lambda }I \\mathbf{w}-\\sqrt{ \\lambda }I\\mathbf{w}_{\\text{prior}} \\rVert^2=\\left\\lVert \\begin{bmatrix}\nX \\\\ \\sqrt{ \\lambda }I\n\\end{bmatrix}\\mathbf{w}-\\begin{bmatrix}\n\\mathbf{y} \\\\ \\sqrt{ \\lambda }\\mathbf{w}_{\\text{prior}}\n\\end{bmatrix} \\right\\rVert^2" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "∥" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: 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"vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "∥" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", 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height: "1.7001em" }, children: _jsxs(_components.span, { style: { top: "-3.949em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["This second regularization method clearly shows how ridge regression uses ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { mathvariant: "bold", children: "0" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{0}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord mathbf", children: "0" })] }) })] }), " as a ", _jsx(_components.em, { children: "prior" }), " assumption on ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\mathbf{w}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7079em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.7079em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" })] }), _jsxs(_components.span, { style: { top: "-3.0134em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }) }) }) })] }) })] }), ".\nAt this point you should be able to compute the closed form solution below:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "^" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "λ" }), _jsx(_components.mi, { children: "I" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "λ" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mtext, { children: "prior" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\mathbf{w}}=(X^TX+\\lambda I)^{-1}(X^T \\mathbf{y}+\\lambda \\mathbf{w}_{\\text{prior}})" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { 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_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0361em", verticalAlign: "-0.2861em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "λ" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3175em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord text mtight", children: _jsx(_components.span, { className: "mord mtight", children: "prior" }) }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["The use of the word ", _jsx(_components.em, { children: "prior" }), " in the notation and text for this section is quite intentional: the idea introduced here can act as a gateway into the Bayesian perspective on regression."] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "perspective-2-gradient-descent-for-ols", children: [_jsx(_components.a, { className: "heading-link", href: "#perspective-2-gradient-descent-for-ols", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Perspective 2: Gradient descent for OLS"] }), "\n", _jsxs(_components.p, { children: ["We can instead start ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { mathvariant: "bold", children: "x" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{x}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4444em" } }), _jsx(_components.span, { className: "mord mathbf", children: "x" })] }) })] }), " at random ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mn, { children: "0" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{x}_{0}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5944em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: 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[_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "μ" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "X" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}_{t+1}=\\mathbf{w}_{t}-\\mu \\nabla \\lVert X\\mathbf{w}_{t}-\\mathbf{y} \\rVert^2=\\mathbf{w}_{t}-2\\mu X^T(X\\mathbf{w}_{t}-\\mathbf{y}) " })] }) }) 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_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["Where ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "μ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mu" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) })] }), " is the ", _jsx(_components.em, { children: "learning rate" }), " hyperparameter.\nFor simplicity we can change our objective function from ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: 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_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "∥" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }), " to ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: "∥" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lVert X \\mathbf{w}-\\mathbf{y} \\rVert^2 /2" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "∥" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: 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This does ", _jsx(_components.em, { children: "not" }), " change the optimal solution, and allows us to write gradient descent as the following:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.mo, { children: 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_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1413em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["The series ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "{" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { stretchy: "false", children: "}" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\{ \\mathbf{w}_{k} \\}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: "}" })] }) })] }), " does ", _jsx(_components.em, { children: "not" }), " always converge to the optimal solution. We'll see the condition for convergence in the next section."] }), "\n", _jsxs(_components.h2, { id: "perspective-3-gradient-descent-as-a-discrete-linear-dynamical-system-lds", children: [_jsx(_components.a, { className: "heading-link", href: "#perspective-3-gradient-descent-as-a-discrete-linear-dynamical-system-lds", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Perspective 3: Gradient descent as a discrete linear dynamical system (LDS)"] }), "\n", _jsxs(_components.p, { children: ["Some rearrangement leads to a pretty fascinating result: gradient descent for OLS can be considered as a LDS with fixed input (i.e., ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "F" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { mathvariant: "bold", children: "g" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}_{k+1}=F\\mathbf{w}_{k}+\\mathbf{g}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6528em", verticalAlign: "-0.2083em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: 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_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "F" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { 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children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "F" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "F" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "F" })] }) })] }), " is the ", _jsx(_components.em, { children: "transition matrix" }), " and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: 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"0.01597em" }, children: "y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "⟹" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7387em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 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className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "2.077em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }), _jsx(_components.span, { className: "tag", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "2.577em" }, children: [_jsxs(_components.span, { style: { top: "-4.577em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.8913em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "-3.0257em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.8913em" } }), _jsx(_components.span, { className: "eqn-num" })] }), _jsxs(_components.span, { style: { top: "-1.4743em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.8913em" } }), _jsx(_components.span, { className: "eqn-num" })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "2.077em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) }), "\n", _jsxs(_components.p, { children: ["We have arrived at the same solution for OLS using the definition of a fixed point (", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "∗" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "F" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "∗" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { mathvariant: "bold", children: "g" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}^*=F\\mathbf{w}^*+\\mathbf{g}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6887em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.772em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "F" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "g" })] })] })] }), ")."] }), "\n", _jsxs(_components.h3, { id: "condition-for-convergence", children: [_jsx(_components.a, { className: "heading-link", href: "#condition-for-convergence", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Condition for Convergence"] }), "\n", _jsxs(_components.p, { children: ["Equipped with the dynamical system view and the definition of a fixed point, we can now find out the the condition for convergence (i.e., ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "∗" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lim_{ k \\to \\infty } \\mathbf{w}_{k}=\\mathbf{w^*}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop", children: "lim" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6887em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] })] })] }), "). First we define the ", _jsx(_components.strong, { children: "error" }), " ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "e" }), _jsx(_components.mi, { children: "k" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{e}_{k}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5944em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " as the difference between ", 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children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6887em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), ":"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsx(_components.mtable, { rowspacing: "0.25em", columnalign: "right", columnspacing: "", children: _jsxs(_components.mtr, { children: [_jsx(_components.mtd, { className: "mtr-glue" }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "true", children: _jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "e" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.mo, { children: ":" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.mo, { children: "−" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "∗" })] })] }) }) }), _jsx(_components.mtd, { className: "mtr-glue" }), _jsx(_components.mtd, { className: "mml-eqn-num" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\begin{align}\n\\mathbf{e}_{t}:=\\mathbf{w}_{t}-\\mathbf{w}^*\n\\end{align}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.5em", verticalAlign: "-0.5em" } }), _jsx(_components.span, { className: "mtable", children: _jsx(_components.span, { className: "col-align-r", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "1em" }, children: _jsxs(_components.span, { style: { top: "-3.16em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ":=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7387em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5em" }, children: _jsx(_components.span, {}) }) })] }) }) })] }), _jsx(_components.span, { className: "tag", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "1em" }, children: _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.84em" } }), _jsx(_components.span, { className: "eqn-num" })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) }), "\n", _jsxs(_components.p, { children: ["We reframe the problem as checking the condition for ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "e" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { mathvariant: "bold", children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lim_{ t \\to \\infty} \\mathbf{e}_{t}=\\mathbf{0}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop", children: "lim" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { 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As usual for dynamical systems, we're interested in the long-term behavior:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "e" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsxs(_components.munder, { children: [_jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "I" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "μ" }), _jsxs(_components.msup, { 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The worst case scenario is for the largest eigenvalue of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { children: "X" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X^TX" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8413em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), 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}), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8413em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), ". We can finally clearly state the condition for convergence, which simply depends on avoiding inappropriate values for ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "μ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mu" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) })] }), ": ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: 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className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6835em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0913em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "2/" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "λ" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord text mtight", children: _jsx(_components.span, { className: "mord mtight", children: "max" }) }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8413em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), ". To be fair, the ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "μ" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0 < \\mu" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6835em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] })] })] }), " part is obvious; you can't call it 'descent' if the descent rate is negative."] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["You might remember related warnings on avoiding learning rates that are too large from machine learning courses, since this would lead to ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mi, { children: "t" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}_{t}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5944em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.016em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " jumping from non-minimum point (w.r.t the loss function) to non-minimum point instead of descending to ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "w" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{w}^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6887em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "w" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), ". Now, you know the rigorous proof of this!\n", _jsx(_components.img, { src: "https://my.flowershow.app/@AmirV97/Blog/LR_fig%20(1)%201.svg", alt: "LR_fig (1) 1", className: "internal" }), "\n", _jsx(_components.strong, { children: "Figure 2" }), ". Geometric interpretation of the condition for convergence, visualized for single variable OLS problem."] }), "\n"] }), "\n", _jsxs(_components.h2, { id: "perspective-4-attention", children: [_jsx(_components.a, { className: "heading-link", href: "#perspective-4-attention", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Perspective 4: Attention?"] }), "\n", _jsxs(_components.p, { children: ["Let's frame OLS as a retrieval problem: Any new (test) subject ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "t" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x_{test}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "es" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " can be considered as a ", _jsx(_components.em, { children: "query" }), " ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" })] }) })] }), ", and our prediction ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{y}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1944em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }), " is a function of the individual rows in our training dataset ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: "…" }), 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The features vectors ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{x}_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5944em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " can be considered as ", _jsx(_components.em, { children: "keys" }), " and their labels can be considered as ", _jsx(_components.em, { children: "values" }), ". In other words, the prediction for ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "t" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{x}_{test}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5944em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "es" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "t" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " is a function of how much 'attention' our model pays to each of the ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " values in the training data, and the amount of attention depends on the keys (", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "x" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf{x}_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5944em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), ").\nLet's formulate this in more precise terms. We need a function ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "f" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "q" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f_{attn}(q,k_{i})" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10764em" }, children: "f" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.2806em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1076em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "a" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "tt" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0315em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " that computes ", _jsx(_components.em, { children: "attention scores" }), " ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "α" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\alpha_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0037em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " which will be used as the weights in the weighted sum ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "m" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "α" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{i}^m\\alpha_{i}y_{i}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0037em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), ":"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mo, { children: "^" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "t" })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "m" })] }), _jsxs(_components.munder, { children: 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