Hinge loss - Wikipedia

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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hinge loss</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Hinge_loss_vs_zero_one_loss.svg/220px-Hinge_loss_vs_zero_one_loss.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Hinge_loss_vs_zero_one_loss.svg/330px-Hinge_loss_vs_zero_one_loss.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Hinge_loss_vs_zero_one_loss.svg/440px-Hinge_loss_vs_zero_one_loss.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption>The vertical axis represents the value of the Hinge loss (in blue) and zero-one loss (in green) for fixed <span class="texhtml"><i>t</i> = 1</span>, while the horizontal axis represents the value of the prediction <span class="texhtml mvar" style="font-style:italic;">y</span>. The plot shows that the Hinge loss penalizes predictions <span class="texhtml"><i>y</i> < 1</span>, corresponding to the notion of a margin in a support vector machine.</figcaption></figure> <p>In <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, the <b>hinge loss</b> is a <a href="/wiki/Loss_function" title="Loss function">loss function</a> used for training <a href="/wiki/Statistical_classification" title="Statistical classification">classifiers</a>. The hinge loss is used for "maximum-margin" classification, most notably for <a href="/wiki/Support_vector_machine" title="Support vector machine">support vector machines</a> (SVMs).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For an intended output <span class="texhtml"><i>t</i> = ±1</span> and a classifier score <span class="texhtml mvar" style="font-style:italic;">y</span>, the hinge loss of the prediction <span class="texhtml mvar" style="font-style:italic;">y</span> is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (y)=\max(0,1-t\cdot y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (y)=\max(0,1-t\cdot y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f42d461f1a28b27438e8f1641e042ff2e40102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.042ex; height:2.843ex;" alt="{\displaystyle \ell (y)=\max(0,1-t\cdot y)}" /></span></dd></dl> <p>Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span> should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c430c17f27e7d7dd52e5a12741eceacc355d87d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.113ex; height:2.509ex;" alt="{\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {w} ,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {w} ,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11219e8065d442b9e95cc1f161ad4ff9d2c273e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.772ex; height:2.843ex;" alt="{\displaystyle (\mathbf {w} ,b)}" /></span> are the parameters of the <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" /></span> is the input variable(s). </p><p>When <span class="texhtml mvar" style="font-style:italic;">t</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> have the same sign (meaning <span class="texhtml mvar" style="font-style:italic;">y</span> predicts the right class) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y|\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |y|\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e949835f150dfe886776296b912457c704c17697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle |y|\geq 1}" /></span>, the hinge loss <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd23c92c905a4fd0e2ec0beb6c21e32e4f119e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.195ex; height:2.843ex;" alt="{\displaystyle \ell (y)=0}" /></span>. When they have opposite signs, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a247d45484db98651f830fd511a6927ed6caa5bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.934ex; height:2.843ex;" alt="{\displaystyle \ell (y)}" /></span> increases linearly with <span class="texhtml mvar" style="font-style:italic;">y</span>, and similarly if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |y|<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7a1cfbe0bc926e313771f06b0dda3cdbee363c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle |y|<1}" /></span>, even if it has the same sign (correct prediction, but not by enough margin). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hinge_loss&action=edit&section=1" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While binary SVMs are commonly extended to <a href="/wiki/Multiclass_classification" title="Multiclass classification">multiclass classification</a> in a one-vs.-all or one-vs.-one fashion,<sup id="cite_ref-duan2005_2-0" class="reference"><a href="#cite_note-duan2005-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed.<sup id="cite_ref-unifiedview_3-0" class="reference"><a href="#cite_note-unifiedview-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For example, Crammer and Singer<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> defined it for a linear classifier as<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (y)=\max(0,1+\max _{y\neq t}\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>+</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>≠</mo> <mi>t</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (y)=\max(0,1+\max _{y\neq t}\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d2483d6d6511dc0fce3afb627f0c58c84a205d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.48ex; height:4.509ex;" alt="{\displaystyle \ell (y)=\max(0,1+\max _{y\neq t}\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )}" /></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span> is the target label, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0afbe09786834d6f4736c0a909028be24ca395e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.757ex; height:2.009ex;" alt="{\displaystyle \mathbf {w} _{t}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f846639551748f166fc4ac42c3b7366342b30d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.981ex; height:2.343ex;" alt="{\displaystyle \mathbf {w} _{y}}" /></span> are the model parameters. </p><p>Weston and Watkins provided a similar definition, but with a sum rather than a max:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-unifiedview_3-1" class="reference"><a href="#cite_note-unifiedview-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (y)=\sum _{y\neq t}\max(0,1+\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>≠</mo> <mi>t</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (y)=\sum _{y\neq t}\max(0,1+\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ecfffb75eba0c8368d3542b91778009fb2513c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.509ex; height:6.009ex;" alt="{\displaystyle \ell (y)=\sum _{y\neq t}\max(0,1+\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )}" /></span>.</dd></dl> <p>In <a href="/wiki/Structured_prediction" title="Structured prediction">structured prediction</a>, the hinge loss can be further extended to structured output spaces. <a href="/wiki/Structured_support_vector_machine" title="Structured support vector machine">Structured SVMs</a> with margin rescaling use the following variant, where <span class="texhtml"><b>w</b></span> denotes the SVM's parameters, <span class="texhtml"><b>y</b></span> the SVM's predictions, <span class="texhtml mvar" style="font-style:italic;">φ</span> the joint feature function, and <span class="texhtml">Δ</span> the <a href="/wiki/Hamming_loss" class="mw-redirect" title="Hamming loss">Hamming loss</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\max _{y\in {\mathcal {Y}}}\left(\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle \right)-\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Y</mi> </mrow> </mrow> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\max _{y\in {\mathcal {Y}}}\left(\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle \right)-\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d33faf87d61823bedfb567943aa105462bf1743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.082ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\max _{y\in {\mathcal {Y}}}\left(\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle \right)-\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\end{aligned}}}" /></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Optimization">Optimization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hinge_loss&action=edit&section=2" title="Edit section: Optimization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The hinge loss is a <a href="/wiki/Convex_function" title="Convex function">convex function</a>, so many of the usual convex optimizers used in machine learning can work with it. It is not <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a>, but has a <a href="/wiki/Subderivative#The_subgradient" title="Subderivative">subgradient</a> with respect to model parameters <span class="texhtml"><b>w</b></span> of a linear SVM with score function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\mathbf {w} \cdot \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\mathbf {w} \cdot \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f49b13bd63db9eae74000efb1024d67af5c0e71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.275ex; height:2.009ex;" alt="{\displaystyle y=\mathbf {w} \cdot \mathbf {x} }" /></span> that is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1,\\0&{\text{otherwise}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ℓ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mi>t</mi> <mo>⋅</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>t</mi> <mo>⋅</mo> <mi>y</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1,\\0&{\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e40c2baa2c4848f880de007411ad01b6b9c18c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.511ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1,\\0&{\text{otherwise}}.\end{cases}}}" /></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hinge_loss_variants.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Hinge_loss_variants.svg/220px-Hinge_loss_variants.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Hinge_loss_variants.svg/330px-Hinge_loss_variants.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Hinge_loss_variants.svg/440px-Hinge_loss_variants.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption>Plot of three variants of the hinge loss as a function of <span class="texhtml"><i>z</i> = <i>ty</i></span>: the "ordinary" variant (blue), its square (green), and the piece-wise smooth version by Rennie and Srebro (red). The y-axis is the <span class="texhtml"><i>l(y)</i></span> hinge loss, and the x-axis is the parameter <span class="texhtml mvar" style="font-style:italic;">t</span></figcaption></figure> <p>However, since the derivative of the hinge loss at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ty=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ty=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0e5e69a71b34eac222a4d842a103824d5e0f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.256ex; height:2.509ex;" alt="{\displaystyle ty=1}" /></span> is undefined, <a href="/wiki/Smoothness" title="Smoothness">smoothed</a> versions may be preferred for optimization, such as Rennie and Srebro's<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0<ty<1,\\0&{\text{if}}~~1\leq ty\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>t</mi> <mi>y</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>t</mi> <mi>y</mi> <mo>≤</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> <mo><</mo> <mi>t</mi> <mi>y</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> <mo>≤</mo> <mi>t</mi> <mi>y</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0<ty<1,\\0&{\text{if}}~~1\leq ty\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05817f4bf8e27ff4369c363a3d1af00fa9faf833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:36.376ex; height:9.843ex;" alt="{\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0<ty<1,\\0&{\text{if}}~~1\leq ty\end{cases}}}" /></span></dd></dl> <p>or the quadratically smoothed </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{\gamma }(y)={\begin{cases}{\frac {1}{2\gamma }}\max(0,1-ty)^{2}&{\text{if}}~~ty\geq 1-\gamma ,\\1-{\frac {\gamma }{2}}-ty&{\text{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>t</mi> <mi>y</mi> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <mi>γ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>t</mi> <mi>y</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\gamma }(y)={\begin{cases}{\frac {1}{2\gamma }}\max(0,1-ty)^{2}&{\text{if}}~~ty\geq 1-\gamma ,\\1-{\frac {\gamma }{2}}-ty&{\text{otherwise}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b06681f6df8930073fa395242047323f8c662b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.952ex; height:7.509ex;" alt="{\displaystyle \ell _{\gamma }(y)={\begin{cases}{\frac {1}{2\gamma }}\max(0,1-ty)^{2}&{\text{if}}~~ty\geq 1-\gamma ,\\1-{\frac {\gamma }{2}}-ty&{\text{otherwise}}\end{cases}}}" /></span></dd></dl> <p>suggested by Zhang.<sup id="cite_ref-zhang_8-0" class="reference"><a href="#cite_note-zhang-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Huber_loss#Variant_for_classification" title="Huber loss">modified Huber loss</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span> is a special case of this loss function with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a00d4b82f21339bad9eb70a85c5c4fa847f8413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =2}" /></span>, specifically <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(t,y)=4\ell _{2}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(t,y)=4\ell _{2}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088f275bce6ce83da8a88cf2c574d7de4084b390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.671ex; height:2.843ex;" alt="{\displaystyle L(t,y)=4\ell _{2}(y)}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hinge_loss&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Multivariate_adaptive_regression_spline#Hinge_functions" title="Multivariate adaptive regression spline">Multivariate adaptive regression spline § Hinge functions</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hinge_loss&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRosascoDe_VitoCaponnettoPiana2004" class="citation journal cs1">Rosasco, L.; 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Hinge loss - Wikipedia