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class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Teoria dell'apprendimento statistico</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="Vai a una voce in un'altra lingua. 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interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Statistical_learning_theory" title="Statistical learning theory - inglese" lang="en" hreflang="en" data-title="Statistical learning theory" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_del_aprendizaje_estad%C3%ADstico" title="Teoría del aprendizaje estadístico - spagnolo" lang="es" hreflang="es" data-title="Teoría del aprendizaje estadístico" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DB%8C%D8%A7%D8%AF%DA%AF%DB%8C%D8%B1%DB%8C_%D8%A2%D9%85%D8%A7%D8%B1%DB%8C" title="نظریه یادگیری آماری - persiano" lang="fa" hreflang="fa" data-title="نظریه یادگیری آماری" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_l%27apprentissage_statistique" title="Théorie de l'apprentissage statistique - francese" lang="fr" hreflang="fr" data-title="Théorie de l'apprentissage statistique" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Aprendizaxe_estat%C3%ADstica" title="Aprendizaxe estatística - galiziano" lang="gl" hreflang="gl" data-title="Aprendizaxe estatística" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a 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title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g"><span>Elemento Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>La <b>teoria dell'apprendimento statistico</b> è il fondamento teorico su cui si basa l'<a href="/wiki/Apprendimento_automatico" title="Apprendimento automatico">apprendimento automatico</a>. </p><p>Attingendo ai campi della <a href="/wiki/Statistica" title="Statistica">statistica</a> e dell'<a href="/wiki/Analisi_funzionale" title="Analisi funzionale">analisi funzionale</a>,<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> la teoria dell'apprendimento statistico cerca di risolvere il generico problema di trovare una funzione capace di effettuare previsioni basandosi sui dati. Questo campo di studio ha portato ad applicazioni pratiche in campi come la <a href="/wiki/Visione_artificiale" title="Visione artificiale">visione artificiale</a>, <a href="/wiki/Riconoscimento_vocale" title="Riconoscimento vocale">il riconoscimento vocale</a> e la <a href="/wiki/Bioinformatica" title="Bioinformatica">bioinformatica</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduzione">Introduzione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&veaction=edit&section=1" title="Modifica la sezione Introduzione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&action=edit&section=1" title="Edit section's source code: Introduzione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gli obiettivi dell'apprendimento sono la comprensione dei dati presenti e la previsione dei dati futuri. L'apprendimento si divide in molte categorie, tra cui l'<a href="/wiki/Apprendimento_supervisionato" title="Apprendimento supervisionato">apprendimento supervisionato</a>, l'<a href="/wiki/Apprendimento_non_supervisionato" title="Apprendimento non supervisionato">apprendimento non supervisionato</a>, l'apprendimento online e l'<a href="/wiki/Apprendimento_per_rinforzo" title="Apprendimento per rinforzo">apprendimento per rinforzo</a>. L'apprendimento supervisionato riguarda l'osservazione di dati contenuti in un <a href="/wiki/Insieme_di_addestramento" class="mw-redirect" title="Insieme di addestramento">insieme di addestramento</a> (<i>training set</i>). Ogni punto nel <i>training set</i> è una coppia di valori input-output, in cui l'input viene mappato a un output. Il problema di apprendimento consiste nell'inferire la funzione che mappa l'input all'output, in modo tale che la funzione appresa possa essere utilizzata per prevedere l'output associato ad input del futuro. </p><p>La funzione stimata che associa un input ad un output è detta ipotesi, stimatore o predittore (nella letteratura inglese <i>hypothesis, estimator, predictor)</i>, e si usa <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 {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"></span> come notazione. </p><p>A seconda del tipo di output, i problemi di apprendimento supervisionato sono problemi di <a href="/wiki/Analisi_della_regressione" title="Analisi della regressione">regressione</a> o problemi di <a href="/wiki/Classificazione_statistica" title="Classificazione statistica">classificazione</a>. Se l'output appartiene ad un intervallo continuo di valori, si tratta di un problema di regressione. </p><p>I problemi di classificazione sono quelli per i quali l'output apparterrà ad un elemento di un insieme discreto di etichette. La classificazione è molto comune per le applicazioni di intelligenza artificiale. Nel riconoscimento facciale, ad esempio, l'immagine del volto di una persona sarebbe l'input e l'etichetta di output sarebbe il nome di quella persona. L'immagine input sarebbe rappresentata da un grande vettore multidimensionale i cui elementi rappresentano i pixel nell'immagine. </p> <div class="mw-heading mw-heading2"><h2 id="Algoritmo">Algoritmo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&veaction=edit&section=2" title="Modifica la sezione Algoritmo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&action=edit&section=2" title="Edit section's source code: Algoritmo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo scopo di un algoritmo di apprendimento è osservare i dati del training set e generare una funzione capace di predire l'output associato ad un input. Tale funzione viene convalidata su un test set, contenente dati che non sono presenti nel training set </p> <div class="mw-heading mw-heading2"><h2 id="Formalismo">Formalismo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&veaction=edit&section=3" title="Modifica la sezione Formalismo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&action=edit&section=3" title="Edit section's source code: Formalismo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I valori di input <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 {\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}"></span> vivono in uno <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a> multidimensionale <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 X\subset \mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\subset \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13a16fd233a972e276f5f215b63f05ecdcfe8bbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.849ex; height:2.676ex;" alt="{\displaystyle X\subset \mathbb {R} ^{d}}"></span>, mentre gli output <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}"></span> sono scalari reali, appartenenti 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 Y\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198307a9ef3ba7869c232066445b5dd23fc256ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.55ex; height:2.176ex;" alt="{\displaystyle Y\subset \mathbb {R} }"></span>. La coppia di valori <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 ({\vec {x}},y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {x}},y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d258cf90194bd99a40645db9457b000d90bf5de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle ({\vec {x}},y)}"></span> è detta <i>punto</i> o <i>campione</i> e compone l''insieme di addestramento, o <i>training set</i>, spesso denotato con <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, che si scrive </p> <dl><dd><i><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 S=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c585881f07effd0b556d05e2c9bafc890222095a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.271ex; height:2.843ex;" alt="{\displaystyle S=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}}"></span></i></dd></dl> <p>L'assunzione di base del procedimento è l'esistenza di una <a href="/wiki/Distribuzione_di_probabilit%C3%A0" class="mw-redirect" title="Distribuzione di probabilità">distribuzione di probabilità</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 p({\vec {x}},y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\vec {x}},y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1595e98c5caecb110c42a3b688b2e2c294220fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle p({\vec {x}},y)}"></span>, definita sullo spazio del prodotto <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 X\times Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle X\times Y}"></span>, che lega gli input e gli output; tale distribuzione è fissa ma sconosciuta. In questo formalismo, il problema di inferenza consiste nel trovare una funzione <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 f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> tale che <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 f({\vec {x}})\sim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f({\vec {x}})\sim y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f6d2201a45b4626332d3ec00e7bcd3abdeac4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle f({\vec {x}})\sim y}"></span>. L'algoritmo cerchera la migliore funzione <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> in un sottospazio denotato con <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 {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span> e chiamato <i>spazio delle ipotesi</i>. </p><p>Sia <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 V(f({\vec {x}}),y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(f({\vec {x}}),y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6f4858385026efb76bd83a1a5d640cecaee985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.204ex; height:2.843ex;" alt="{\displaystyle V(f({\vec {x}}),y)}"></span> la <a href="/wiki/Funzione_obiettivo" title="Funzione obiettivo">funzione di perdita</a>, uno strumento per misurare la differenza tra il valore previsto <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 f({\vec {x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f({\vec {x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546ff1db2de71c7abcb09ef533f42dad5e89bf19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f({\vec {x}})}"></span> e il valore vero <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>. Il rischio atteso (o errore atteso) è il <a href="/wiki/Valore_atteso" title="Valore atteso">valore atteso</a> della funzione di perdita, ed è definito come </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 I[f]=\displaystyle \int _{X\times Y}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </msub> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I[f]=\displaystyle \int _{X\times Y}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c97de659352fa60d6624a1f54f4d3f0b32c73af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.466ex; height:5.676ex;" alt="{\displaystyle I[f]=\displaystyle \int _{X\times Y}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy}"></span></dd></dl> <p>La migliore funzione possibile <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> che può essere scelta, soddisfa la condizione </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 f=\inf _{h\in {\mathcal {H}}}I[h]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> </munder> <mi>I</mi> <mo stretchy="false">[</mo> <mi>h</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\inf _{h\in {\mathcal {H}}}I[h]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c982d6049c9ce70930b327589b5ff00915406f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12ex; height:4.176ex;" alt="{\displaystyle f=\inf _{h\in {\mathcal {H}}}I[h]}"></span></dd></dl> <p>Poiché la distribuzione di probabilità <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 p({\vec {x}},y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\vec {x}},y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1595e98c5caecb110c42a3b688b2e2c294220fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle p({\vec {x}},y)}"></span> non è nota, deve essere utilizzata una stima per il valore atteso della funzione di perdita. Questa misura si basa sul training set, un campione di questa distribuzione di probabilità sconosciuta. Si chiama rischio empirico </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 I_{S}[f]={\frac {1}{n}}\displaystyle \sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{S}[f]={\frac {1}{n}}\displaystyle \sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0418e949bbea67e3cdbe4455429c3af2c3f09781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.133ex; height:6.843ex;" alt="{\displaystyle I_{S}[f]={\frac {1}{n}}\displaystyle \sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})}"></span></dd></dl> <p>Un algoritmo di apprendimento che sceglie la funzione <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 f_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44c97f83aebb50c3fd26b567ff9b005dc7b82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle f_{S}}"></span> che minimizza il rischio empirico si chiama minimizzazione del rischio empirico. </p> <div class="mw-heading mw-heading2"><h2 id="Funzioni_di_perdita">Funzioni di perdita</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&veaction=edit&section=4" title="Modifica la sezione Funzioni di perdita" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&action=edit&section=4" title="Edit section's source code: Funzioni di perdita"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La scelta della funzione di perdita è un fattore determinante sulla funzione <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 f_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44c97f83aebb50c3fd26b567ff9b005dc7b82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle f_{S}}"></span> che sarà scelto dall'algoritmo di apprendimento. La funzione di perdita influenza anche il tasso di convergenza per un algoritmo. È importante che la funzione di perdita sia convessa.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Vengono utilizzate diverse funzioni di perdita a seconda che il problema sia di regressione o di classificazione. </p><p>La funzione di perdita più comune per la regressione è la funzione di perdita quadrata (nota anche come <a href="/wiki/Norma_(matematica)" title="Norma (matematica)">norma L2</a>). Questa familiare funzione di perdita viene utilizzata nella regressione dei minimi quadrati ordinari. Il modulo è: </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 V(f({\vec {x}}),y)=(y-f({\vec {x}}))^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(f({\vec {x}}),y)=(y-f({\vec {x}}))^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c971ef7591ca3bc4c5d7642d8f90369b29992a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.579ex; height:3.176ex;" alt="{\displaystyle V(f({\vec {x}}),y)=(y-f({\vec {x}}))^{2}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Classificazione">Classificazione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&veaction=edit&section=5" title="Modifica la sezione Classificazione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&action=edit&section=5" title="Edit section's source code: Classificazione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In un certo senso la <a href="/wiki/Funzione_indicatrice" title="Funzione indicatrice">funzione indicatrice</a> 0-1 è la funzione di perdita più naturale per la classificazione. Prende il valore 0 se l'output previsto è lo stesso dell'output effettivo e assume il valore 1 se l'output previsto è diverso dall'output effettivo. Per la classificazione binaria con <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=\{-1,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\{-1,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4916f8d7711119d44385e1d2f628cc139de2c953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.364ex; height:2.843ex;" alt="{\displaystyle Y=\{-1,1\}}"></span>, questo è: </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 V(f({\vec {x}}),y)=\theta (-yf({\vec {x}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(f({\vec {x}}),y)=\theta (-yf({\vec {x}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aed24bedecfd114ed5bc77e5fdb1f36537d9330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.583ex; height:2.843ex;" alt="{\displaystyle V(f({\vec {x}}),y)=\theta (-yf({\vec {x}}))}"></span></dd></dl> <p>dove <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> è la <a href="/wiki/Funzione_gradino_di_Heaviside" title="Funzione gradino di Heaviside">funzione gradino di Heaviside</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Regolarizzazione">Regolarizzazione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&veaction=edit&section=6" title="Modifica la sezione Regolarizzazione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&action=edit&section=6" title="Edit section's source code: Regolarizzazione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Overfitting_on_Training_Set_Data.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Overfitting_on_Training_Set_Data.pdf/page1-220px-Overfitting_on_Training_Set_Data.pdf.jpg" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Overfitting_on_Training_Set_Data.pdf/page1-330px-Overfitting_on_Training_Set_Data.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Overfitting_on_Training_Set_Data.pdf/page1-440px-Overfitting_on_Training_Set_Data.pdf.jpg 2x" data-file-width="760" data-file-height="743" /></a><figcaption> Questa immagine rappresenta un esempio di overfitting nell'apprendimento automatico. I punti rossi rappresentano i dati del training set. La linea verde rappresenta la vera relazione funzionale, mentre la linea blu mostra la funzione appresa, che è stata sovraadattata ai dati del training set.</figcaption></figure> <p>Nei problemi di apprendimento automatico, un grosso problema che si pone è quello del <a href="/wiki/Overfitting" title="Overfitting">sovradattamento</a>. Poiché l'apprendimento è un problema di previsione, l'obiettivo non è trovare una funzione che si adatti maggiormente ai dati (osservati in precedenza), ma trovarne una che preveda in modo più accurato l'output dall'input futuro. La minimizzazione del rischio empirico corre questo rischio di sovradattamento: trovare una funzione che corrisponda esattamente ai dati ma non preveda bene l'output futuro. </p><p>Il sovradattamento è sintomatico di soluzioni instabili; una piccola perturbazione nei dati del training set causerebbe una grande variazione nella funzione appresa. Si può dimostrare che se può essere garantita la stabilità per la soluzione, sono garantite anche la generalizzazione e la consistenza.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> La <a href="/wiki/Regolarizzazione_(matematica)" title="Regolarizzazione (matematica)">regolarizzazione</a> può risolvere il problema del sovradattamento e dare stabilità al problema. </p><p>La regolarizzazione può essere ottenuta restringendo lo spazio delle ipotesi <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 {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span>. Un esempio comune sarebbe la restrizione <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 {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span> alle funzioni lineari: questo può essere visto come una riduzione al problema standard della <a href="/wiki/Regressione_lineare" title="Regressione lineare">regressione lineare</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 {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span> potrebbe anche essere limitato al polinomio di grado <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, esponenziali o funzioni limitate su <a href="/wiki/Spazio_Lp" title="Spazio Lp">L1</a>. La restrizione dello spazio delle ipotesi evita l'overfitting perché la forma delle funzioni potenziali è limitata, e quindi non consente la scelta di una funzione che dia un rischio empirico arbitrariamente vicino allo zero. </p><p>Un esempio di regolarizzazione è la <a href="/wiki/Regolarizzazione_di_Tichonov" title="Regolarizzazione di Tichonov">regolarizzazione di Tichonov</a>. Consiste nel minimizzare </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 {1}{n}}\displaystyle \sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})+\gamma \|f\|_{\mathcal {H}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>γ</mi> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\displaystyle \sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})+\gamma \|f\|_{\mathcal {H}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d281a9698fde8d2015871615891a677457d0a4e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.474ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{n}}\displaystyle \sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})+\gamma \|f\|_{\mathcal {H}}^{2}}"></span></dd></dl> <p>dove <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> è un parametro fisso e positivo, il parametro di regolarizzazione. La regolarizzazione di Tikhonov garantisce l'esistenza, l'unicità e la stabilità della soluzione.<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> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&veaction=edit&section=7" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_dell%27apprendimento_statistico&action=edit&section=7" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><a href="/wiki/Vladimir_Vapnik" title="Vladimir Vapnik">Vladimir Vapnik</a> (1995) <i>The Nature of Statistical Learning Theory</i>, Springer New York <a href="/wiki/Speciale:RicercaISBN/978-1-475-72440-0" title="Speciale:RicercaISBN/978-1-475-72440-0">ISBN 978-1-475-72440-0</a>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text">Rosasco, L., Vito, E.D., Caponnetto, A., Fiana, M., and Verri A. 2004. <i>Neural computation</i> Vol 16, pp 1063-1076</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text">Vapnik, V.N. and Chervonenkis, A.Y. 1971. <a rel="nofollow" class="external text" href="http://ai2-s2-pdfs.s3.amazonaws.com/a36b/028d024bf358c4af1a5e1dc3ca0aed23b553.pdf">On the uniform convergence of relative frequencies of events to their probabilities</a>. <i>Theory of Probability and Its Applications</i> Vol 16, pp 264-280.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4"><b>^</b></a> <span class="reference-text">Mukherjee, S., Niyogi, P. Poggio, T., and Rifkin, R. 2006. <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s10444-004-7634-z">Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization</a>. <i>Advances in Computational Mathematics</i>. Vol 25, pp 161-193.</span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text">Tomaso Poggio, Lorenzo Rosasco, et al. <i>Statistical Learning Theory and Applications</i>, 2012, <a rel="nofollow" class="external text" href="https://www.mit.edu/~9.520/spring12/slides/class02/class02.pdf">Class 2</a></span> </li> </ol></div> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Apprendimento_automatico"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Apprendimento_automatico" title="Template:Apprendimento automatico"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Apprendimento_automatico&action=edit&redlink=1" class="new" title="Discussioni template:Apprendimento automatico (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Apprendimento_automatico&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Apprendimento_automatico" title="Apprendimento automatico">Apprendimento automatico</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;">Problemi</th><td colspan="1" class="navbox_list navbox_odd"><a class="mw-selflink selflink">Teoria dell'apprendimento statistico</a><b> ·</b> <a href="/wiki/Classificazione" title="Classificazione">Classificazione</a><b> ·</b> <a href="/wiki/Analisi_della_regressione" title="Analisi della regressione">Regressione</a><b> ·</b> <a href="/wiki/Regole_di_associazione" title="Regole di associazione">Regole di associazione</a><b> ·</b> <a href="/wiki/Apprendimento_non_supervisionato" title="Apprendimento non supervisionato">Apprendimento non supervisionato</a><b> ·</b> <a href="/wiki/Apprendimento_supervisionato" title="Apprendimento supervisionato">Apprendimento supervisionato</a><b> ·</b> <a href="/wiki/Apprendimento_per_rinforzo" title="Apprendimento per rinforzo">Apprendimento per rinforzo</a><b> ·</b> <a href="/wiki/Apprendimento_profondo" title="Apprendimento profondo">Apprendimento profondo</a></td><td rowspan="8" class="navbox_image"><span typeof="mw:File"><a href="/wiki/File:Kernel_Machine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/150px-Kernel_Machine.svg.png" decoding="async" width="150" height="68" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/225px-Kernel_Machine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/300px-Kernel_Machine.svg.png 2x" data-file-width="512" data-file-height="233" /></a></span><br /><br /><span typeof="mw:File"><a href="/wiki/File:Random_forest_model_space.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Random_forest_model_space.png/70px-Random_forest_model_space.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Random_forest_model_space.png/105px-Random_forest_model_space.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Random_forest_model_space.png/140px-Random_forest_model_space.png 2x" data-file-width="512" data-file-height="512" /></a></span></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Apprendimento_non_supervisionato" title="Apprendimento non supervisionato">Apprendimento non supervisionato</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Clustering" title="Clustering">Clustering</a><b> ·</b> <a href="/wiki/Clustering_gerarchico" title="Clustering gerarchico">Clustering gerarchico</a><b> ·</b> <a href="/wiki/K-means" title="K-means">K-means</a><b> ·</b> <a href="/wiki/Algoritmo_EM" title="Algoritmo EM">Algoritmo EM</a><b> ·</b> <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a><b> ·</b> <a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a><b> ·</b> <a href="/wiki/Rete_generativa_avversaria" title="Rete generativa avversaria">Rete generativa avversaria</a> (cGAN<b> ·</b> VAE-GAN<b> ·</b> cycleGAN)</td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Apprendimento_supervisionato" title="Apprendimento supervisionato">Apprendimento supervisionato</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Albero_di_decisione" title="Albero di decisione">Albero di decisione</a><b> ·</b> <a href="/wiki/Foresta_casuale" title="Foresta casuale">Foresta casuale</a><b> ·</b> <a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random fields</a> CRF<b> ·</b> <a href="/wiki/Modello_di_Markov_nascosto" title="Modello di Markov nascosto">Modello di Markov nascosto</a><b> ·</b> <a href="/wiki/K-nearest_neighbors" title="K-nearest neighbors">K-nearest neighbors</a><b> ·</b> <a href="/wiki/Classificatore_bayesiano" title="Classificatore bayesiano">Classificatore bayesiano</a><b> ·</b> <a href="/wiki/Rete_neurale_artificiale" title="Rete neurale artificiale">Rete neurale artificiale</a><b> ·</b> <a href="/wiki/Regressione_lineare" title="Regressione lineare">Regressione lineare</a><b> ·</b> <a href="/wiki/Modello_logit" title="Modello logit">Regressione logistica</a><b> ·</b> <a href="/wiki/Modello_grafico" title="Modello grafico">Modelli grafici</a><b> ·</b> <a href="/wiki/Macchine_a_vettori_di_supporto" title="Macchine a vettori di supporto">Macchine a vettori di supporto</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Apprendimento_per_rinforzo" title="Apprendimento per rinforzo">Apprendimento per rinforzo</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a><b> ·</b> <a href="/wiki/SARSA" title="SARSA">SARSA</a><b> ·</b> <a href="/wiki/Temporal_difference_learning" title="Temporal difference learning">TD</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Riduzione_della_dimensionalit%C3%A0" title="Riduzione della dimensionalità">Riduzione della dimensionalità</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Analisi_fattoriale" title="Analisi fattoriale">Analisi fattoriale</a><b> ·</b> <a href="/wiki/Analisi_della_correlazione_canonica" title="Analisi della correlazione canonica">Analisi della correlazione canonica</a> (CCA)<b> ·</b> <a href="/wiki/Analisi_delle_componenti_indipendenti" title="Analisi delle componenti indipendenti">Analisi delle componenti indipendenti</a> (ICA)<b> ·</b> <a href="/wiki/Analisi_discriminante_lineare" title="Analisi discriminante lineare">Analisi discriminante lineare</a> (LDA)<b> ·</b> <a href="/wiki/Analisi_delle_componenti_principali" title="Analisi delle componenti principali">Analisi delle componenti principali</a> (PCA)<b> ·</b> <a href="/wiki/Selezione_delle_caratteristiche" title="Selezione delle caratteristiche">Selezione delle caratteristiche</a><b> ·</b> <a href="/wiki/Estrazione_di_caratteristiche" title="Estrazione di caratteristiche">Estrazione di caratteristiche</a><b> ·</b> <a href="/wiki/T-distributed_stochastic_neighbor_embedding" title="T-distributed stochastic neighbor embedding">t-distributed stochastic neighbor embedding</a> (t-SNE)</td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Rete_neurale_artificiale" title="Rete neurale artificiale">Reti neurali artificiali</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Percettrone" title="Percettrone">Percettrone</a><b> ·</b> <a href="/wiki/Rete_neurale_a_base_radiale" title="Rete neurale a base radiale">Rete neurale a base radiale</a><b> ·</b> <a href="/wiki/Rete_bayesiana" title="Rete bayesiana">Rete bayesiana</a><b> ·</b> <a href="/wiki/Rete_neurale_feed-forward" title="Rete neurale feed-forward">Rete neurale feed-forward</a><b> ·</b> <a href="/wiki/Rete_di_Hopfield" title="Rete di Hopfield">Rete di Hopfield</a><b> ·</b> <a href="/wiki/Percettrone_multistrato" title="Percettrone multistrato">Percettrone multistrato</a><b> ·</b> <a href="/wiki/Rete_neurale_ricorrente" title="Rete neurale ricorrente">Rete neurale ricorrente</a> (<a href="/w/index.php?title=Long_short-term_memory&action=edit&redlink=1" class="new" title="Long short-term memory (la pagina non esiste)">LSTM</a>)<b> ·</b> <a href="/wiki/Macchina_di_Boltzmann_ristretta" title="Macchina di Boltzmann ristretta">Macchina di Boltzmann ristretta</a><b> ·</b> <a href="/wiki/Mappa_auto-organizzata" title="Mappa auto-organizzata">Mappa auto-organizzata</a><b> ·</b> <a href="/wiki/Rete_neurale_convoluzionale" title="Rete neurale convoluzionale">Rete neurale convoluzionale</a><b> ·</b> <a href="/wiki/Rete_neurale_a_ritardo" title="Rete neurale a ritardo">Rete neurale a ritardo</a><b> ·</b> <a href="/wiki/Rete_neurale_spiking" title="Rete neurale spiking">Rete neurale spiking</a><b> ·</b> <a href="/wiki/Trasformatore_(informatica)" title="Trasformatore (informatica)">Trasformatore</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;">Software</th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Keras" title="Keras">Keras</a><b> ·</b> <a href="/wiki/Microsoft_Cognitive_Toolkit" title="Microsoft Cognitive Toolkit">Microsoft Cognitive Toolkit</a><b> ·</b> <a href="/wiki/Scikit-learn" title="Scikit-learn">Scikit-learn</a><b> ·</b> <a href="/wiki/TensorFlow" title="TensorFlow">TensorFlow</a><b> ·</b> <a href="/wiki/Theano" title="Theano">Theano</a><b> ·</b> <a href="/w/index.php?title=Torch_(libreria_software)&action=edit&redlink=1" class="new" title="Torch (libreria software) (la pagina non esiste)">Torch</a><b> ·</b> <a href="/wiki/Weka_(software)" title="Weka (software)">Weka</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;">Altro</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Algoritmo_genetico" title="Algoritmo genetico">Algoritmo genetico</a><b> ·</b> <a href="/wiki/Particle_Swarm_Optimization" title="Particle Swarm Optimization">Particle Swarm Optimization</a><b> ·</b> <a href="/wiki/Caratteristica_(apprendimento_automatico)" title="Caratteristica (apprendimento automatico)">Caratteristica</a><b> ·</b> <a href="/wiki/Compromesso_bias-varianza" title="Compromesso bias-varianza">Compromesso bias-varianza</a><b> ·</b> <a href="/wiki/Minimizzazione_del_rischio_empirico" title="Minimizzazione del rischio empirico">Minimizzazione del rischio empirico</a></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: 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