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<title>Preference learning</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Preference learning</span></span>

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<p><b>Preference learning</b> is a subfield of <a href="Machine_learning" title="Machine learning">machine learning</a> that focuses on modeling and predicting preferences based on observed preference information.<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> Preference learning typically involves <a href="Supervised_learning" title="Supervised learning">supervised learning</a> using datasets of pairwise preference comparisons, rankings, or other preference information.

</p>

<div class="mw-heading mw-heading2"><h2 id="Tasks">Tasks</h2></div>

<p>The main task in preference learning concerns problems in "<a href="Learning_to_rank" title="Learning to rank">learning to rank</a>". According to different types of preference information observed, the tasks are categorized as three main problems in the book <i>Preference Learning</i>:<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>

<div class="mw-heading mw-heading3"><h3 id="Label_ranking">Label ranking</h3></div>

<p>In label ranking, the model has an instance space <span class="mwe-math-element mwe-math-element-inline"><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=\{x_{i}\}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle X=\{x_{i}\}\,\!}</annotation>

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</math></span><img src="./20cbffee19d83be657109e348357ee38e9d91c84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.92ex; height:2.843ex;" alt="{\displaystyle X=\{x_{i}\}\,\!}" loading="lazy"></span> and a finite set of labels <span class="mwe-math-element mwe-math-element-inline"><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=\{y_{i}|i=1,2,\cdots ,k\}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle Y=\{y_{i}|i=1,2,\cdots ,k\}\,\!}</annotation>

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</math></span><img src="./ea110f3ec12166b80bf7e9bc02145aeca7d2c5d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:23.819ex; height:2.843ex;" alt="{\displaystyle Y=\{y_{i}|i=1,2,\cdots ,k\}\,\!}" loading="lazy"></span>. The preference information is given in the form <span class="mwe-math-element mwe-math-element-inline"><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}\succ _{x}y_{j}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}\succ _{x}y_{j}\,\!}</annotation>

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</math></span><img src="./f5e27eb54ab8388f4ddc30298366245f88ad247c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:8.646ex; height:2.509ex;" alt="{\displaystyle y_{i}\succ _{x}y_{j}\,\!}" loading="lazy"></span> indicating instance <span class="mwe-math-element mwe-math-element-inline"><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\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle x\,\!}</annotation>

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</math></span><img src="./ec21bb206c6c9f458130ab7ffddfe3fd8d0fa6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,\!}" loading="lazy"></span> shows preference in <span class="mwe-math-element mwe-math-element-inline"><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}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}\,\!}</annotation>

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</math></span><img src="./af6d96f2a38af431eb1835d53c6d3bba618f16c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.326ex; height:2.009ex;" alt="{\displaystyle y_{i}\,\!}" loading="lazy"></span> rather than <span class="mwe-math-element mwe-math-element-inline"><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_{j}\,\!}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle y_{j}\,\!}</annotation>

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</math></span><img src="./04f853c8e793c2f60e7ad940588d751916bb59fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:2.436ex; height:2.343ex;" alt="{\displaystyle y_{j}\,\!}" loading="lazy"></span>. A set of preference information is used as training data in the model. The task of this model is to find a preference ranking among the labels for any instance.

</p><p>It was observed that some conventional <a href="Classification_in_machine_learning" class="mw-redirect" title="Classification in machine learning">classification</a> problems can be generalized in the framework of label ranking problem:<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> if a training instance <span class="mwe-math-element mwe-math-element-inline"><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\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle x\,\!}</annotation>

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</math></span><img src="./ec21bb206c6c9f458130ab7ffddfe3fd8d0fa6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,\!}" loading="lazy"></span> is labeled as class <span class="mwe-math-element mwe-math-element-inline"><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}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}\,\!}</annotation>

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</math></span><img src="./af6d96f2a38af431eb1835d53c6d3bba618f16c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.326ex; height:2.009ex;" alt="{\displaystyle y_{i}\,\!}" loading="lazy"></span>, it implies that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall j\neq i,y_{i}\succ _{x}y_{j}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle \forall j\neq i,y_{i}\succ _{x}y_{j}\,\!}</annotation>

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</math></span><img src="./fda5dacef331e10ad55c68b5fb68f66cfb7f35d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:15.831ex; height:2.843ex;" alt="{\displaystyle \forall j\neq i,y_{i}\succ _{x}y_{j}\,\!}" loading="lazy"></span>. In the <a href="Multi-label_classification" title="Multi-label classification">multi-label</a> case, <span class="mwe-math-element mwe-math-element-inline"><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\,\!}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\,\!}</annotation>

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</math></span><img src="./ec21bb206c6c9f458130ab7ffddfe3fd8d0fa6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,\!}" loading="lazy"></span> is associated with a set of labels <span class="mwe-math-element mwe-math-element-inline"><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\subseteq Y\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle L\subseteq Y\,\!}</annotation>

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</math></span><img src="./8e0d218e7306fa02d25960f151ab5dd1105cbd62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:6.842ex; height:2.343ex;" alt="{\displaystyle L\subseteq Y\,\!}" loading="lazy"></span> and thus the model can extract a set of preference information <span class="mwe-math-element mwe-math-element-inline"><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}\succ _{x}y_{j}|y_{i}\in L,y_{j}\in Y\backslash L\}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle \{y_{i}\succ _{x}y_{j}|y_{i}\in L,y_{j}\in Y\backslash L\}\,\!}</annotation>

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</math></span><img src="./d270ddb926591b532338685f0f2fe21698ae3878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:28.422ex; height:3.009ex;" alt="{\displaystyle \{y_{i}\succ _{x}y_{j}|y_{i}\in L,y_{j}\in Y\backslash L\}\,\!}" loading="lazy"></span>. Training a preference model on this preference information and the classification result of an instance is just the corresponding top ranking label.

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<div class="mw-heading mw-heading3"><h3 id="Instance_ranking">Instance ranking</h3></div>

<p>Instance ranking also has the instance space <span class="mwe-math-element mwe-math-element-inline"><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\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle X\,\!}</annotation>

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</math></span><img src="./98c6c570e3a2c130fc8d968160962b5e0fe4b0e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.367ex; height:2.176ex;" alt="{\displaystyle X\,\!}" loading="lazy"></span> and label set <span class="mwe-math-element mwe-math-element-inline"><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\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle Y\,\!}</annotation>

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</math></span><img src="./82e65066132416db15d76f148ea4436daf49c863.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; margin-right: -0.387ex; width:2.16ex; height:2.009ex;" alt="{\displaystyle Y\,\!}" loading="lazy"></span>. In this task, labels are defined to have a fixed order <span class="mwe-math-element mwe-math-element-inline"><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}\succ y_{2}\succ \cdots \succ y_{k}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle y_{1}\succ y_{2}\succ \cdots \succ y_{k}\,\!}</annotation>

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</math></span><img src="./bc54b0fa46229805d71bbbae7ceef52d7b60e618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:19.021ex; height:2.176ex;" alt="{\displaystyle y_{1}\succ y_{2}\succ \cdots \succ y_{k}\,\!}" loading="lazy"></span> and each instance <span class="mwe-math-element mwe-math-element-inline"><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_{l}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle x_{l}\,\!}</annotation>

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</math></span><img src="./640798b4c49797777a2367b5c41a020c5ada21cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.439ex; height:2.009ex;" alt="{\displaystyle x_{l}\,\!}" loading="lazy"></span> is associated with a label <span class="mwe-math-element mwe-math-element-inline"><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_{l}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle y_{l}\,\!}</annotation>

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</math></span><img src="./d16e64cc40dec862adc4d199743c5e000c32f279.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.249ex; height:2.009ex;" alt="{\displaystyle y_{l}\,\!}" loading="lazy"></span>. Giving a set of instances as training data, the goal of this task is to find the ranking order for a new set of instances.

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<div class="mw-heading mw-heading3"><h3 id="Object_ranking">Object ranking</h3></div>

<p>Object ranking is similar to instance ranking except that no labels are associated with instances. Given a set of pairwise preference information in the form <span class="mwe-math-element mwe-math-element-inline"><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_{i}\succ x_{j}\,\!}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x_{i}\succ x_{j}\,\!}</annotation>

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</math></span><img src="./e1e80ef53124759cea37706a8130f9763b447fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:7.854ex; height:2.509ex;" alt="{\displaystyle x_{i}\succ x_{j}\,\!}" loading="lazy"></span> and the model should find out a ranking order among instances.

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<div class="mw-heading mw-heading2"><h2 id="Techniques">Techniques</h2></div>

<p>There are two practical representations of the preference information <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\succ B\,\!}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A\succ B\,\!}</annotation>

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</math></span><img src="./163ba6337ca37cf6a0152ceac6d4c0b41c8857f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:6.993ex; height:2.176ex;" alt="{\displaystyle A\succ B\,\!}" loading="lazy"></span>. One is assigning <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation>

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</math></span><img src="./b041434092d1f0042c2b4c7ab32ea84d462cb53e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation>

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</math></span><img src="./cf2670cb2bfb471dca31dd3da10251997439eed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}" loading="lazy"></span> with two real numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation>

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</math></span><img src="./99a94f96d2455b9d7faf3cec3eb02ab3c455aec1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle b\,\!}</annotation>

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</math></span><img src="./cb1dfcc2fcd791aacf48286147257b507b009e78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.385ex; height:2.176ex;" alt="{\displaystyle b\,\!}" loading="lazy"></span> respectively such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle a>b\,\!}</annotation>

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</math></span><img src="./6ccd39ed8239edfa3d343caaa7c4e77996b9b2d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.713ex; height:2.176ex;" alt="{\displaystyle a>b\,\!}" loading="lazy"></span>. Another one is assigning a binary value <span class="mwe-math-element mwe-math-element-inline"><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(A,B)\in \{0,1\}\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle V(A,B)\in \{0,1\}\,\!}</annotation>

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</math></span><img src="./aa8d177a2c86315bbda1508756b5e28d5b726d1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:17.049ex; height:2.843ex;" alt="{\displaystyle V(A,B)\in \{0,1\}\,\!}" loading="lazy"></span> for all pairs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle (A,B)\,\!}</annotation>

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</math></span><img src="./f02198cd4e497bffa5ffeabdfaf47617db7730e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:6.737ex; height:2.843ex;" alt="{\displaystyle (A,B)\,\!}" loading="lazy"></span> denoting whether <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\succ B\,\!}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A\succ B\,\!}</annotation>

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</math></span><img src="./163ba6337ca37cf6a0152ceac6d4c0b41c8857f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:6.993ex; height:2.176ex;" alt="{\displaystyle A\succ B\,\!}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\succ A\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle B\succ A\,\!}</annotation>

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</math></span><img src="./1a9fc40172404eceaf9ebd83149c9b9fdae7af0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:6.993ex; height:2.176ex;" alt="{\displaystyle B\succ A\,\!}" loading="lazy"></span>. Corresponding to these two different representations, there are two different techniques applied to the learning process.

</p>

<div class="mw-heading mw-heading3"><h3 id="Utility_function">Utility function</h3></div>

<p>If we can find a mapping from data to real numbers, ranking the data can be solved by ranking the real numbers. This mapping is called <a href="Utility_function" class="mw-redirect" title="Utility function">utility function</a>. For label ranking the mapping is a function <span class="mwe-math-element mwe-math-element-inline"><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\times Y\rightarrow \mathbb {R} \,\!}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\times Y\rightarrow \mathbb {R} \,\!}</annotation>

</semantics>

</math></span><img src="./8c25389918f337b5a94da7f9ea1f8ab7c121ea66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:15.489ex; height:2.509ex;" alt="{\displaystyle f:X\times Y\rightarrow \mathbb {R} \,\!}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><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}\succ _{x}y_{j}\Rightarrow f(x,y_{i})>f(x,y_{j})\,\!}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}\succ _{x}y_{j}\Rightarrow f(x,y_{i})>f(x,y_{j})\,\!}</annotation>

</semantics>

</math></span><img src="./3fffc156b22eaef053a25ce4e284f959a7b46ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:30.249ex; height:3.009ex;" alt="{\displaystyle y_{i}\succ _{x}y_{j}\Rightarrow f(x,y_{i})>f(x,y_{j})\,\!}" loading="lazy"></span>. For instance ranking and object ranking, the mapping is a function <span class="mwe-math-element mwe-math-element-inline"><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\rightarrow \mathbb {R} \,\!}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\rightarrow \mathbb {R} \,\!}</annotation>

</semantics>

</math></span><img src="./3cbf2a14e87e5852063751a7ee5e321d4adeb805.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:10.875ex; height:2.509ex;" alt="{\displaystyle f:X\rightarrow \mathbb {R} \,\!}" loading="lazy"></span>.

</p><p>Finding the utility function is a <a href="Regression_analysis" title="Regression analysis">regression</a> learning problem which is well developed in machine learning.

</p>

<div class="mw-heading mw-heading3"><h3 id="Preference_relations">Preference relations</h3></div>

<p>The binary representation of preference information is called preference relation. For each pair of alternatives (instances or labels), a binary predicate can be learned by conventional supervised learning approach. Fürnkranz and Hüllermeier proposed this approach in label ranking problem.<sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For object ranking, there is an early approach by Cohen et al.<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><p>Using preference relations to predict the ranking will not be so intuitive. Since observed preference relations may not always be transitive due to inconsistencies in the data, finding a ranking that satisfies all the preference relations may not be possible or may result in multiple possible solutions. A more common approach is to find a ranking solution which is maximally consistent with the preference relations. This approach is a natural extension of pairwise classification.<sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<p>Preference learning can be used in ranking search results according to feedback of user preference. Given a query and a set of documents, a learning model is used to find the ranking of documents corresponding to the <a href="Relevance_(information_retrieval)" title="Relevance (information retrieval)">relevance</a> with this query. More discussions on research in this field can be found in <a href="Tie-Yan_Liu" title="Tie-Yan Liu">Tie-Yan Liu</a>'s survey paper.<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>

</p><p>Another application of preference learning is <a href="Recommender_systems" class="mw-redirect" title="Recommender systems">recommender systems</a>.<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> Online store may analyze customer's purchase record to learn a preference model and then recommend similar products to customers. Internet content providers can make use of user's ratings to provide more user preferred contents.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMohriRostamizadehTalwalkar2012" class="citation book cs1"><a href="Mehryar_Mohri" title="Mehryar Mohri">Mohri, Mehryar</a>; Rostamizadeh, Afshin; Talwalkar, Ameet (2012). <i>Foundations of Machine Learning</i>. US, Massachusetts: MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262018258</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFFürnkranzHüllermeier2010" class="citation book cs1">Fürnkranz, Johannes; Hüllermeier, Eyke, eds. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nc3XcH9XSgYC&pg=PA4"><i>Preference learning</i></a>. Springer. pp. <span class="nowrap">3–</span>8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-14124-9</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://proceedings.neurips.cc/paper_files/paper/2002/file/16026d60ff9b54410b3435b403afd226-Paper.pdf">"Constraint classification for multiclass classification and ranking"</a> <span class="cs1-format">(PDF)</span>. <i>NeurIPS</i>. 2002.</cite></span>

</li>

<li id="cite_note-:0-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFFürnkranzHüllermeier2003" class="citation book cs1">Fürnkranz, Johannes; Hüllermeier, Eyke (2003). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-540-39857-8_15">"Pairwise Preference Learning and Ranking"</a>. In Lavrač, Nada; Gamberger, Dragan; Blockeel, Hendrik; Todorovski, Ljupčo (eds.). <i>Machine Learning: ECML 2003</i>. Lecture Notes in Computer Science. Vol. 2837. Berlin, Heidelberg: Springer. pp. <span class="nowrap">145–</span>156. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-39857-8_15">10.1007/978-3-540-39857-8_15</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-39857-8</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFCohenSchapireSinger1998" class="citation journal cs1">Cohen, William W.; Schapire, Robert E.; Singer, Yoram (1998-07-31). <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/10.5555/302528.302736">"Learning to order things"</a>. <i>NeurIPS</i>. Cambridge, MA, USA: MIT Press: <span class="nowrap">451–</span>457. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-10076-2</bdi>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFLiu2007" class="citation journal cs1">Liu, Tie-Yan (2007). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://www.nowpublishers.com/article/Details/INR-016">"Learning to Rank for Information Retrieval"</a></span>. <i>Foundations and Trends in Information Retrieval</i>. <b>3</b> (3): <span class="nowrap">225–</span>331. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1561%2F1500000016">10.1561/1500000016</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1554-0669">1554-0669</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFGemmisIaquintaLopsMusto2010" class="citation cs2">Gemmis, Marco de; Iaquinta, Leo; Lops, Pasquale; Musto, Cataldo; Narducci, Fedelucio; Semeraro, Giovanni (2010), Fürnkranz, Johannes; Hüllermeier, Eyke (eds.), <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/978-3-642-14125-6_18">"Learning Preference Models in Recommender Systems"</a></span>, <i>Preference Learning</i>, Springer, pp. <span class="nowrap">387–</span>407, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-14125-6_18">10.1007/978-3-642-14125-6_18</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-14124-9</bdi><span class="reference-accessdate">, retrieved <span class="nowrap">2024-11-05</span></span></cite></span>

</li>

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