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

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<p>In <a href="Machine_learning" title="Machine learning">machine learning</a>, <b>supervised learning</b> (<b>SL</b>) is a type of <a href="Machine_learning#Paradigms" title="Machine learning">machine learning paradigm</a> where an algorithm learns to map input data to a specific output based on example input-output pairs. This process involves training a statistical model using labeled data, meaning each piece of input data is provided with the correct output. For instance, if you want a model to identify cats in images, supervised learning would involve feeding it many images of cats (inputs) that are explicitly labeled "cat" (outputs).

</p><p>The goal of supervised learning is for the trained model to accurately predict the output for new, unseen data.<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> This requires the algorithm to effectively <a href="Generalization_(learning)" title="Generalization (learning)">generalize</a> from the training examples, a quality measured by its <i><a href="Generalization_error" title="Generalization error">generalization error</a></i>. Supervised learning is commonly used for tasks like <a href="Classification" title="Classification">classification</a> (predicting a category, e.g., spam or not spam) and <a href="Regression_analysis" title="Regression analysis">regression</a> (predicting a continuous value, e.g., house prices).

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<div class="mw-heading mw-heading2"><h2 id="Steps_to_follow">Steps to follow</h2></div>

<p>To solve a given problem of supervised learning, the following steps must be performed:

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<ol><li>Determine the type of training samples. Before doing anything else, the user should decide what kind of data is to be used as a <a href="Training%2C_validation%2C_and_test_data_sets" title="Training, validation, and test data sets">training set</a>. In the case of <a href="Handwriting_analysis" class="mw-redirect" title="Handwriting analysis">handwriting analysis</a>, for example, this might be a single handwritten character, an entire handwritten word, an entire sentence of handwriting, or a full paragraph of handwriting.</li>

<li>Gather a training set. The training set needs to be representative of the real-world use of the function. Thus, a set of input objects is gathered together with corresponding outputs, either from <a href="Subject-matter_expert" title="Subject-matter expert">human experts</a> or from measurements.</li>

<li>Determine the input <a href="Feature_(machine_learning)" title="Feature (machine learning)">feature</a> representation of the learned function. The accuracy of the learned function depends strongly on how the input object is represented. Typically, the input object is transformed into a <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vector</a>, which contains a number of features that are descriptive of the object. The number of features should not be too large, because of the <a href="Curse_of_dimensionality" title="Curse of dimensionality">curse of dimensionality</a>; but should contain enough information to accurately predict the output.</li>

<li>Determine the structure of the learned function and corresponding learning algorithm. For example, one may choose to use <a href="Support-vector_machine" class="mw-redirect" title="Support-vector machine">support-vector machines</a> or <a href="Decision_tree_learning" title="Decision tree learning">decision trees</a>.</li>

<li>Complete the design. Run the learning algorithm on the gathered training set. Some supervised learning algorithms require the user to determine certain <a href="Hyperparameter_(machine_learning)" title="Hyperparameter (machine learning)">control parameters</a>. These parameters may be adjusted by optimizing performance on a subset (called a <i><a href="Validation_set" class="mw-redirect" title="Validation set">validation set</a></i>) of the training set, or via <a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">cross-validation</a>.</li>

<li>Evaluate the accuracy of the learned function. After parameter adjustment and learning, the performance of the resulting function should be measured on a <a href="Test_set" class="mw-redirect" title="Test set">test set</a> that is separate from the training set.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Algorithm_choice">Algorithm choice</h2></div>

<p>A wide range of supervised learning algorithms are available, each with its strengths and weaknesses. There is no single learning algorithm that works best on all supervised learning problems (see the <a href="No_free_lunch_in_search_and_optimization" title="No free lunch in search and optimization">No free lunch theorem</a>).

</p><p>There are four major issues to consider in supervised learning:

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<div class="mw-heading mw-heading3"><h3 id="Bias–variance_tradeoff">Bias–variance tradeoff</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></div>

<p>A first issue is the tradeoff between <i>bias</i> and <i>variance</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> Imagine that we have available several different, but equally good, training data sets. A learning algorithm is biased for a particular input <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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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> if, when trained on each of these data sets, it is systematically incorrect when predicting the correct output for <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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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>. A learning algorithm has high variance for a particular input <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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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> if it predicts different output values when trained on different training sets. The prediction error of a learned classifier is related to the sum of the bias and the variance of the learning algorithm.<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> Generally, there is a tradeoff between bias and variance. A learning algorithm with low bias must be "flexible" so that it can fit the data well. But if the learning algorithm is too flexible, it will fit each training data set differently, and hence have high variance. A key aspect of many supervised learning methods is that they are able to adjust this tradeoff between bias and variance (either automatically or by providing a bias/variance parameter that the user can adjust).

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<div class="mw-heading mw-heading3"><h3 id="Function_complexity_and_amount_of_training_data">Function complexity and amount of training data</h3></div>

<p>The second issue is of the amount of training data available relative to the complexity of the "true" function (classifier or regression function). If the true function is simple, then an "inflexible" learning algorithm with high bias and low variance will be able to learn it from a small amount of data. But if the true function is highly complex (e.g., because it involves complex interactions among many different input features and behaves differently in different parts of the input space), then the function will only be able to learn with a large amount of training data paired with a "flexible" learning algorithm with low bias and high variance.

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<div class="mw-heading mw-heading3"><h3 id="Dimensionality_of_the_input_space">Dimensionality of the input space</h3></div>

<p>A third issue is the dimensionality of the input space. If the input feature vectors have large dimensions, learning the function can be difficult even if the true function only depends on a small number of those features. This is because the many "extra" dimensions can confuse the learning algorithm and cause it to have high variance. Hence, input data of large dimensions typically requires tuning the classifier to have low variance and high bias. In practice, if the engineer can manually remove irrelevant features from the input data, it will likely improve the accuracy of the learned function. In addition, there are many algorithms for <a href="Feature_selection" title="Feature selection">feature selection</a> that seek to identify the relevant features and discard the irrelevant ones. This is an instance of the more general strategy of <a href="Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a>, which seeks to map the input data into a lower-dimensional space prior to running the supervised learning algorithm.

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<div class="mw-heading mw-heading3"><h3 id="Noise_in_the_output_values">Noise in the output values</h3></div>

<p>A fourth issue is the degree of noise in the desired output values (the supervisory <a href="Target_variable" class="mw-redirect" title="Target variable">target variables</a>). If the desired output values are often incorrect (because of human error or sensor errors), then the learning algorithm should not attempt to find a function that exactly matches the training examples. Attempting to fit the data too carefully leads to <a href="Overfitting" title="Overfitting">overfitting</a>. You can overfit even when there are no measurement errors (stochastic noise) if the function you are trying to learn is too complex for your learning model. In such a situation, the part of the target function that cannot be modeled "corrupts" your training data – this phenomenon has been called <a href="Deterministic_noise" title="Deterministic noise">deterministic noise</a>. When either type of noise is present, it is better to go with a higher bias, lower variance estimator.

</p><p>In practice, there are several approaches to alleviate noise in the output values such as <a href="Early_stopping" title="Early stopping">early stopping</a> to prevent overfitting as well as <a href="Anomaly_detection" title="Anomaly detection">detecting</a> and removing the noisy training examples prior to training the supervised learning algorithm. There are several algorithms that identify noisy training examples and removing the suspected noisy training examples prior to training has decreased <a href="Generalization_error" title="Generalization error">generalization error</a> with <a href="Statistical_significance" title="Statistical significance">statistical significance</a>.<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><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>

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<div class="mw-heading mw-heading3"><h3 id="Other_factors_to_consider">Other factors to consider</h3></div>

<p>Other factors to consider when choosing and applying a learning algorithm include the following:

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<ul><li>Heterogeneity of the data. If the feature vectors include features of many different kinds (discrete, discrete ordered, counts, continuous values), some algorithms are easier to apply than others. Many algorithms, including <a href="Support_Vector_Machines" class="mw-redirect" title="Support Vector Machines">support-vector machines</a>, <a href="Linear_regression" title="Linear regression">linear regression</a>, <a href="Logistic_regression" title="Logistic regression">logistic regression</a>, <a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">neural networks</a>, and <a href="K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm">nearest neighbor methods</a>, require that the input features be numerical and scaled to similar ranges (e.g., to the [-1,1] interval). Methods that employ a distance function, such as nearest neighbor methods and <a href="Support_Vector_Machines" class="mw-redirect" title="Support Vector Machines">support-vector machines with Gaussian kernels</a>, are particularly sensitive to this. An advantage of <a href="Decision_tree_learning" title="Decision tree learning">decision trees</a> is that they easily handle heterogeneous data.</li>

<li>Redundancy in the data. If the input features contain redundant information (e.g., highly correlated features), some learning algorithms (e.g., <a href="Linear_regression" title="Linear regression">linear regression</a>, <a href="Logistic_regression" title="Logistic regression">logistic regression</a>, and <a href="K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"> distance-based methods</a>) will perform poorly because of numerical instabilities. These problems can often be solved by imposing some form of <a href="Regularization_(mathematics)" title="Regularization (mathematics)">regularization</a>.</li>

<li>Presence of interactions and non-linearities. If each of the features makes an independent contribution to the output, then algorithms based on linear functions (e.g., <a href="Linear_regression" title="Linear regression">linear regression</a>, <a href="Logistic_regression" title="Logistic regression">logistic regression</a>, <a href="Support-vector_machine" class="mw-redirect" title="Support-vector machine">support-vector machines</a>, <a href="Naive_Bayes_classifier" title="Naive Bayes classifier">naive Bayes</a>) and distance functions (e.g., nearest neighbor methods, <a href="Support_Vector_Machines" class="mw-redirect" title="Support Vector Machines">support-vector machines with Gaussian kernels</a>) generally perform well. However, if there are complex interactions among features, then algorithms such as <a href="Decision_tree_learning" title="Decision tree learning">decision trees</a> and neural networks work better, because they are specifically designed to discover these interactions. Linear methods can also be applied, but the engineer must manually specify the interactions when using them.</li></ul>

<p>When considering a new application, the engineer can compare multiple learning algorithms and experimentally determine which one works best on the problem at hand (see <a href="Cross-validation_(statistics)" title="Cross-validation (statistics)"> cross-validation</a>). Tuning the performance of a learning algorithm can be very time-consuming. Given fixed resources, it is often better to spend more time collecting additional training data and more informative features than it is to spend extra time tuning the learning algorithms.

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<div class="mw-heading mw-heading3"><h3 id="Algorithms">Algorithms</h3></div>

<p>The most widely used learning algorithms are:

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<ul><li><a href="Support-vector_machine" class="mw-redirect" title="Support-vector machine">Support-vector machines</a></li>

<li><a href="Linear_regression" title="Linear regression">Linear regression</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a></li>

<li><a href="Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Linear discriminant analysis</a></li>

<li><a href="Decision_tree_learning" title="Decision tree learning">Decision trees</a></li>

<li><a href="K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-nearest neighbors algorithm</a></li>

<li><a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">Neural networks</a> (e.g., <a href="Multilayer_perceptron" title="Multilayer perceptron">Multilayer perceptron</a>)</li>

<li><a href="Similarity_learning" title="Similarity learning">Similarity learning</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="How_supervised_learning_algorithms_work">How supervised learning algorithms work</h2></div>

<p>Given a set of <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 N}">

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

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> training examples of 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_{1},y_{1}),...,(x_{N},\;y_{N})\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{(x_{1},y_{1}),...,(x_{N},\;y_{N})\}}</annotation>

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</math></span><img src="./3688abd5ec58027a9d879916c83b6890744ce4dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.255ex; height:2.843ex;" alt="{\displaystyle \{(x_{1},y_{1}),...,(x_{N},\;y_{N})\}}" 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 x_{i}}">

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

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</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> is the <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vector</a> of the <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 i}">

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

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-th example 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 y_{i}}">

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

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</math></span><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" 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}}" loading="lazy"></span> is its label (i.e., class), a learning algorithm seeks 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 g:X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle g:X\to Y}</annotation>

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</math></span><img src="./5c825617c180ee9cbba1d56f8514978bf7c33b7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.421ex; height:2.509ex;" alt="{\displaystyle g:X\to Y}" loading="lazy"></span>, where <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="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is the input space 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 Y}">

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

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</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> is the output space. The 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 g}">

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

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is an element of some space of possible functions <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 G}">

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

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</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>, usually called the <i>hypothesis space</i>. It is sometimes convenient to represent <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 g}">

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

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> using a <a href="Scoring_function" class="mw-redirect" title="Scoring function">scoring function</a> <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\to \mathbb {R} }">

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

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</math></span><img src="./e2c7bc3330b76d58cdb3bff67e7b0ec60a2509c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.102ex; height:2.509ex;" alt="{\displaystyle f:X\times Y\to \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 g}">

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

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is defined as returning the <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>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" 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}" loading="lazy"></span> value that gives the highest score: <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 g(x)={\underset {y}{\arg \max }}\;f(x,y)}">

<mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)={\underset {y}{\arg \max }}\;f(x,y)}</annotation>

</semantics>

</math></span><img src="./cba4bc7532dc87fa4312abad85326b21e99f96d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.555ex; height:4.676ex;" alt="{\displaystyle g(x)={\underset {y}{\arg \max }}\;f(x,y)}" loading="lazy"></span>. Let <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> denote the space of scoring functions.

</p><p>Although <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 G}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" 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 F}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> can be any space of functions, many learning algorithms are probabilistic models where <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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> takes the form of a <a href="Conditional_probability" title="Conditional probability">conditional probability</a> model <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 g(x)={\underset {y}{\arg \max }}\;P(y|x)}">

<mrow>

</mrow>

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)={\underset {y}{\arg \max }}\;P(y|x)}</annotation>

</semantics>

</math></span><img src="./9416212f1466c7ce91578b8f87a42a15c9d831bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.635ex; height:4.676ex;" alt="{\displaystyle g(x)={\underset {y}{\arg \max }}\;P(y|x)}" 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 f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> takes the form of a <a href="Joint_probability" class="mw-redirect" title="Joint probability">joint probability</a> model <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,y)=P(x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x,y)=P(x,y)}</annotation>

</semantics>

</math></span><img src="./497c7a6614b31a86270a3a7bed89410bcd1e001c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.779ex; height:2.843ex;" alt="{\displaystyle f(x,y)=P(x,y)}" loading="lazy"></span>. For example, <a href="Naive_Bayes_classifier" title="Naive Bayes classifier">naive Bayes</a> and <a href="Linear_discriminant_analysis" title="Linear discriminant analysis">linear discriminant analysis</a> are joint probability models, whereas <a href="Logistic_regression" title="Logistic regression">logistic regression</a> is a conditional probability model.

</p><p>There are two basic approaches to choosing <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" 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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>: <a href="Empirical_risk_minimization" title="Empirical risk minimization">empirical risk minimization</a> and <a href="Structural_risk_minimization" title="Structural risk minimization">structural risk minimization</a>.<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> Empirical risk minimization seeks the function that best fits the training data. Structural risk minimization includes a <i>penalty function</i> that controls the bias/variance tradeoff.

</p><p>In both cases, it is assumed that the training set consists of a sample of <a href="Independent_and_identically-distributed_random_variables" class="mw-redirect" title="Independent and identically-distributed random variables">independent and identically distributed pairs</a>, <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},\;y_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{i},\;y_{i})}</annotation>

</semantics>

</math></span><img src="./413bed828a4750e458455095965e7082bece1e98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.557ex; height:2.843ex;" alt="{\displaystyle (x_{i},\;y_{i})}" loading="lazy"></span>. In order to measure how well a function fits the training data, a <a href="Loss_function" title="Loss function">loss function</a> <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:Y\times Y\to \mathbb {R} ^{\geq 0}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L:Y\times Y\to \mathbb {R} ^{\geq 0}}</annotation>

</semantics>

</math></span><img src="./881dc7b502a2b6a4970fb06609e170cf79475a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.532ex; height:2.676ex;" alt="{\displaystyle L:Y\times Y\to \mathbb {R} ^{\geq 0}}" loading="lazy"></span> is defined. For training example <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},\;y_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{i},\;y_{i})}</annotation>

</semantics>

</math></span><img src="./413bed828a4750e458455095965e7082bece1e98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.557ex; height:2.843ex;" alt="{\displaystyle (x_{i},\;y_{i})}" loading="lazy"></span>, the loss of predicting the 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 {\hat {y}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}}</annotation>

</semantics>

</math></span><img src="./3dc8de3d8ea01304329ef9518fad7a6d196c4c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}}" loading="lazy"></span> is <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(y_{i},{\hat {y}})}">

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(y_{i},{\hat {y}})}</annotation>

</semantics>

</math></span><img src="./6ac9025f28e57a00e0b20d64598436fe7a7477e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.667ex; height:2.843ex;" alt="{\displaystyle L(y_{i},{\hat {y}})}" loading="lazy"></span>.

</p><p>The <i>risk</i> <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 R(g)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(g)}</annotation>

</semantics>

</math></span><img src="./ae075d527038ed7d449f85ced07e487b5807446c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle R(g)}" loading="lazy"></span> of 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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is defined as the expected loss of <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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>. This can be estimated from the training data as

</p>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{emp}(g)={\frac {1}{N}}\sum _{i}L(y_{i},g(x_{i}))}</annotation>

</semantics>

</math></span><img src="./87fa2fe2f731e8e1b02e2710d36fc16076e449bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.504ex; height:6.343ex;" alt="{\displaystyle R_{emp}(g)={\frac {1}{N}}\sum _{i}L(y_{i},g(x_{i}))}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Empirical_risk_minimization">Empirical risk minimization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></div>

<p>In empirical risk minimization, the supervised learning algorithm seeks the 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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> that minimizes <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 R(g)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(g)}</annotation>

</semantics>

</math></span><img src="./ae075d527038ed7d449f85ced07e487b5807446c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle R(g)}" loading="lazy"></span>. Hence, a supervised learning algorithm can be constructed by applying an <a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization algorithm</a> to find <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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>.

</p><p>When <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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is a conditional probability distribution <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 P(y|x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(y|x)}</annotation>

</semantics>

</math></span><img src="./5d08508dff9e465cc317804ff19999c4ffbf7d94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.687ex; height:2.843ex;" alt="{\displaystyle P(y|x)}" loading="lazy"></span> and the loss function is the negative log likelihood: <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(y,{\hat {y}})=-\log P(y|x)}">

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(y,{\hat {y}})=-\log P(y|x)}</annotation>

</semantics>

</math></span><img src="./eaee3db78a569f2bb35e3dced4a1953ace44665b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.223ex; height:2.843ex;" alt="{\displaystyle L(y,{\hat {y}})=-\log P(y|x)}" loading="lazy"></span>, then empirical risk minimization is equivalent to <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood estimation</a>.

</p><p>When <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 G}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> contains many candidate functions or the training set is not sufficiently large, empirical risk minimization leads to high variance and poor generalization. The learning algorithm is able to memorize the training examples without generalizing well (overfitting).

</p>

<div class="mw-heading mw-heading3"><h3 id="Structural_risk_minimization">Structural risk minimization</h3></div>

<p><a href="Structural_risk_minimization" title="Structural risk minimization">Structural risk minimization</a> seeks to prevent overfitting by incorporating a <a href="Regularization_(mathematics)" title="Regularization (mathematics)">regularization penalty</a> into the optimization. The regularization penalty can be viewed as implementing a form of <a href="Occam's_razor" title="Occam's razor">Occam's razor</a> that prefers simpler functions over more complex ones.

</p><p>A wide variety of penalties have been employed that correspond to different definitions of complexity. For example, consider the case where the 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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is a linear function of the form

</p>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)=\sum _{j=1}^{d}\beta _{j}x_{j}}</annotation>

</semantics>

</math></span><img src="./86f23781682d33c596d0cda976b013e2b6fdf939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.56ex; height:7.676ex;" alt="{\displaystyle g(x)=\sum _{j=1}^{d}\beta _{j}x_{j}}" loading="lazy"></span>.</dd></dl>

<p>A popular regularization penalty is <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 \sum _{j}\beta _{j}^{2}}">

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j}\beta _{j}^{2}}</annotation>

</semantics>

</math></span><img src="./f7df304919e1cdf3dfc49b9db648afce80a189ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:6.133ex; height:5.843ex;" alt="{\displaystyle \sum _{j}\beta _{j}^{2}}" loading="lazy"></span>, which is the squared <a href="Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a> of the weights, also known as the <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_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation>

</semantics>

</math></span><img src="./c6a952cfe42c86b7741f55a817da0e251793a358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}" loading="lazy"></span> norm. Other norms include the <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_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation>

</semantics>

</math></span><img src="./0e79dc1b001f8b923df475ed14de023cbc456013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" loading="lazy"></span> norm, <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 \sum _{j}|\beta _{j}|}">

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j}|\beta _{j}|}</annotation>

</semantics>

</math></span><img src="./4f7c345923255a0cf76b3e2080b915eeb02f4126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:7.261ex; height:5.843ex;" alt="{\displaystyle \sum _{j}|\beta _{j}|}" loading="lazy"></span>, and the <a href="L0_%22norm%22" class="mw-redirect" title="L0 "norm""><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_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

</math></span><img src="./db742b8c210fc611329a4c2dcc3af4b4e1a110cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{0}}" loading="lazy"></span> "norm"</a>, which is the number of non-zero <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 \beta _{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{j}}</annotation>

</semantics>

</math></span><img src="./83edf0558c67ad56ca5c05096b550bd733d62c4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.225ex; height:2.843ex;" alt="{\displaystyle \beta _{j}}" loading="lazy"></span>s. The penalty will be denoted by <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 C(g)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(g)}</annotation>

</semantics>

</math></span><img src="./e60538c5c17aadb0567923ff496ba4da02bcd40c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.692ex; height:2.843ex;" alt="{\displaystyle C(g)}" loading="lazy"></span>.

</p><p>The supervised learning optimization problem is to find the 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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> that minimizes

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(g)=R_{emp}(g)+\lambda C(g).}</annotation>

</semantics>

</math></span><img src="./f66b4b813c97f6598ed08791352e014fc7e16514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.987ex; height:3.009ex;" alt="{\displaystyle J(g)=R_{emp}(g)+\lambda C(g).}" loading="lazy"></span></dd></dl>

<p>The parameter <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 \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> controls the bias-variance tradeoff. When <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 \lambda =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda =0}</annotation>

</semantics>

</math></span><img src="./00c4bba30544017fe76932de5a4e25adb5512d95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =0}" loading="lazy"></span>, this gives empirical risk minimization with low bias and high variance. When <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 \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> is large, the learning algorithm will have high bias and low variance. The value of <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 \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> can be chosen empirically via <a href="Cross-validation_(statistics)" title="Cross-validation (statistics)"> cross-validation</a>.

</p><p>The complexity penalty has a Bayesian interpretation as the negative log prior probability of <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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>, <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 -\log P(g)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\log P(g)}</annotation>

</semantics>

</math></span><img src="./140f8e655a53130f7068d9341b32891142e3a80c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.225ex; height:2.843ex;" alt="{\displaystyle -\log P(g)}" loading="lazy"></span>, in which 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 J(g)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(g)}</annotation>

</semantics>

</math></span><img src="./81eae208007bd0ddec50b7f177e991b914a11508.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.397ex; height:2.843ex;" alt="{\displaystyle J(g)}" loading="lazy"></span> is the <a href="Posterior_probability" title="Posterior probability">posterior probability</a> of <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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Generative_training">Generative training</h2></div>

<p>The training methods described above are <i>discriminative training</i> methods, because they seek to find 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 g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> that discriminates well between the different output values (see <a href="Discriminative_model" title="Discriminative model">discriminative model</a>). For the special case where <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,y)=P(x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x,y)=P(x,y)}</annotation>

</semantics>

</math></span><img src="./497c7a6614b31a86270a3a7bed89410bcd1e001c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.779ex; height:2.843ex;" alt="{\displaystyle f(x,y)=P(x,y)}" loading="lazy"></span> is a <a href="Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> and the loss function is the negative log likelihood <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 -\sum _{i}\log P(x_{i},y_{i}),}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\sum _{i}\log P(x_{i},y_{i}),}</annotation>

</semantics>

</math></span><img src="./89be2acbbec6725f4ac254167f2d9cd7e97d2c7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.6ex; height:5.509ex;" alt="{\displaystyle -\sum _{i}\log P(x_{i},y_{i}),}" loading="lazy"></span> a risk minimization algorithm is said to perform <i>generative training</i>, because <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> can be regarded as a <a href="Generative_model" title="Generative model">generative model</a> that explains how the data were generated. Generative training algorithms are often simpler and more computationally efficient than discriminative training algorithms. In some cases, the solution can be computed in closed form as in <a href="Naive_Bayes_classifier" title="Naive Bayes classifier">naive Bayes</a> and <a href="Linear_discriminant_analysis" title="Linear discriminant analysis">linear discriminant analysis</a>.

</p>

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

<p>There are several ways in which the standard supervised learning problem can be generalized:

</p><ul><li><a href="Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a> or <a href="Weak_supervision" title="Weak supervision">weak supervision</a>: the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled.</li>

<li><a href="Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a>: Instead of assuming that all of the training examples are given at the start, active learning algorithms interactively collect new examples, typically by making queries to a human user. Often, the queries are based on unlabeled data, which is a scenario that combines semi-supervised learning with active learning.</li>

<li><a href="Structured_prediction" title="Structured prediction">Structured prediction</a>: When the desired output value is a complex object, such as a <a href="Parse_tree" title="Parse tree">parse tree</a> or a labeled graph, then standard methods must be extended.</li>

<li><a href="Learning_to_rank" title="Learning to rank">Learning to rank</a>: When the input is a set of objects and the desired output is a ranking of those objects, then again the standard methods must be extended.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Approaches_and_algorithms">Approaches and algorithms</h2></div>

<ul><li>Analytical learning</li>

<li><a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a></li>

<li><a href="Backpropagation" title="Backpropagation">Backpropagation</a></li>

<li><a href="Boosting_(meta-algorithm)" class="mw-redirect" title="Boosting (meta-algorithm)">Boosting (meta-algorithm)</a></li>

<li><a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a></li>

<li><a href="Case-based_reasoning" title="Case-based reasoning">Case-based reasoning</a></li>

<li><a href="Decision_tree_learning" title="Decision tree learning">Decision tree learning</a></li>

<li><a href="Inductive_logic_programming" title="Inductive logic programming">Inductive logic programming</a></li>

<li><a href="Gaussian_process_regression" class="mw-redirect" title="Gaussian process regression">Gaussian process regression</a></li>

<li><a href="Genetic_programming" title="Genetic programming">Genetic programming</a></li>

<li><a href="Group_method_of_data_handling" title="Group method of data handling">Group method of data handling</a></li>

<li><a href="Variable_kernel_density_estimation#Use_for_statistical_classification" title="Variable kernel density estimation">Kernel estimators</a></li>

<li><a href="Learning_automaton" title="Learning automaton">Learning automata</a></li>

<li><a href="Learning_classifier_system" title="Learning classifier system">Learning classifier systems</a></li>

<li><a href="Learning_vector_quantization" title="Learning vector quantization">Learning vector quantization</a></li>

<li><a href="Minimum_message_length" title="Minimum message length">Minimum message length</a> (<a href="Decision_tree" title="Decision tree">decision trees</a>, decision graphs, etc.)</li>

<li><a href="Multilinear_subspace_learning" title="Multilinear subspace learning">Multilinear subspace learning</a></li>

<li><a href="Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes classifier</a></li>

<li><a href="Maximum_entropy_classifier" class="mw-redirect" title="Maximum entropy classifier">Maximum entropy classifier</a></li>

<li><a href="Conditional_random_field" title="Conditional random field">Conditional random field</a></li>

<li><a href="Nearest_neighbor_(pattern_recognition)" class="mw-redirect" title="Nearest neighbor (pattern recognition)">Nearest neighbor algorithm</a></li>

<li><a href="Probably_approximately_correct_learning" title="Probably approximately correct learning">Probably approximately correct learning</a> (PAC) learning</li>

<li><a href="Ripple_down_rules" class="mw-redirect" title="Ripple down rules">Ripple down rules</a>, a knowledge acquisition methodology</li>

<li>Symbolic machine learning algorithms</li>

<li>Subsymbolic machine learning algorithms</li>

<li><a href="Support_vector_machine" title="Support vector machine">Support vector machines</a></li>

<li>Minimum complexity machines (MCM)</li>

<li><a href="Random_forest" title="Random forest">Random forests</a></li>

<li><a href="Ensembles_of_classifiers" class="mw-redirect" title="Ensembles of classifiers">Ensembles of classifiers</a></li>

<li><a href="Ordinal_classification" class="mw-redirect" title="Ordinal classification">Ordinal classification</a></li>

<li><a href="Data_pre-processing" class="mw-redirect" title="Data pre-processing">Data pre-processing</a></li>

<li>Handling imbalanced datasets</li>

<li><a href="Statistical_relational_learning" title="Statistical relational learning">Statistical relational learning</a></li>

<li><a href="Proaftn" title="Proaftn">Proaftn</a>, a multicriteria classification algorithm</li></ul>

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

<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Cheminformatics" title="Cheminformatics">Cheminformatics</a>

<ul><li><a href="Quantitative_structure%E2%80%93activity_relationship" title="Quantitative structure–activity relationship">Quantitative structure–activity relationship</a></li></ul></li>

<li><a href="Database_marketing" title="Database marketing">Database marketing</a></li>

<li><a href="Handwriting_recognition" title="Handwriting recognition">Handwriting recognition</a></li>

<li><a href="Information_retrieval" title="Information retrieval">Information retrieval</a>

<ul><li><a href="Learning_to_rank" title="Learning to rank">Learning to rank</a></li></ul></li>

<li><a href="Information_extraction" title="Information extraction">Information extraction</a></li>

<li>Object recognition in <a href="Computer_vision" title="Computer vision">computer vision</a></li>

<li><a href="Optical_character_recognition" title="Optical character recognition">Optical character recognition</a></li>

<li><a href="Spamming" title="Spamming">Spam detection</a></li>

<li><a href="Pattern_recognition" title="Pattern recognition">Pattern recognition</a></li>

<li><a href="Speech_recognition" title="Speech recognition">Speech recognition</a></li>

<li>Supervised learning is a special case of <a href="Downward_causation" title="Downward causation">downward causation</a> in biological systems</li>

<li>Landform classification using <a href="Satellite_imagery" title="Satellite imagery">satellite imagery</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></li>

<li>Spend classification in <a href="Procurement" title="Procurement">procurement</a> processes<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="General_issues">General issues</h2></div>

<ul><li><a href="Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li>

<li><a href="Inductive_bias" title="Inductive bias">Inductive bias</a></li>

<li><a href="Overfitting" title="Overfitting">Overfitting</a></li>

<li>(Uncalibrated) <a href="Class_membership_probabilities" class="mw-redirect" title="Class membership probabilities">class membership probabilities</a></li>

<li><a href="Version_space" class="mw-redirect" title="Version space">Version spaces</a></li></ul>

<ul><li><a href="List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a></li>

<li><a href="Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li></ul>

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

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</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Mehryar_Mohri" title="Mehryar Mohri">Mehryar Mohri</a>, Afshin Rostamizadeh, Ameet Talwalkar (2012) <i>Foundations of Machine Learning</i>, The MIT Press <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262018258</bdi>.</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">S. Geman, E. Bienenstock, and R. Doursat (1992). <a rel="nofollow" class="external text" href="http://delta-apache-vm.cs.tau.ac.il/~nin/Courses/NC06/VarbiasBiasGeman.pdf">Neural networks and the bias/variance dilemma</a>. Neural Computation 4, 1–58.</span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">G. James (2003) Variance and Bias for General Loss Functions, Machine Learning 51, 115–135. (<a rel="nofollow" class="external free" href="http://www-bcf.usc.edu/~gareth/research/bv.pdf">http://www-bcf.usc.edu/~gareth/research/bv.pdf</a>)</span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">C.E. Brodely and M.A. Friedl (1999). Identifying and Eliminating Mislabeled Training Instances, Journal of Artificial Intelligence Research 11, 131–167. (<a rel="nofollow" class="external free" href="http://jair.org/media/606/live-606-1803-jair.pdf">http://jair.org/media/606/live-606-1803-jair.pdf</a>)</span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Vapnik, V. N. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EqgACAAAQBAJ&q=%22empirical+risk+minimization%22+OR+%22structural+risk+minimization%22">The Nature of Statistical Learning Theory</a> (2nd Ed.), Springer Verlag, 2000.</span>

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<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="CITEREFA._Maity2016" class="citation arxiv cs1">A. Maity (2016). "Supervised Classification of RADARSAT-2 Polarimetric Data for Different Land Features". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1608.00501">1608.00501</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.CV">cs.CV</a>].</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://publication.sipmm.edu.sg/key-technologies-agile-procurement/">"Key Technologies for Agile Procurement | SIPMM Publications"</a>. <i>publication.sipmm.edu.sg</i>. 2020-10-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-06-16</span></span>.</cite></span>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.mloss.org/">Machine Learning Open Source Software (MLOSS)</a></li></ul>

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</style><div id="Differentiable_computing254" style="font-size:114%;margin:0 4em">Differentiable computing</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Differentiable_function" title="Differentiable function">General</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><b><a href="Differentiable_programming" title="Differentiable programming">Differentiable programming</a></b></li>

<li><a href="Information_geometry" title="Information geometry">Information geometry</a></li>

<li><a href="Statistical_manifold" title="Statistical manifold">Statistical manifold</a></li>

<li><a href="Automatic_differentiation" title="Automatic differentiation">Automatic differentiation</a></li>

<li><a href="Neuromorphic_computing" title="Neuromorphic computing">Neuromorphic computing</a></li>

<li><a href="Pattern_recognition" title="Pattern recognition">Pattern recognition</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li>

<li><a href="Inductive_bias" title="Inductive bias">Inductive bias</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Hardware</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Memristor" title="Memristor">Memristor</a></li>

<li><a href="SpiNNaker" title="SpiNNaker">SpiNNaker</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software libraries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="TensorFlow" title="TensorFlow">TensorFlow</a></li>

<li><a href="PyTorch" title="PyTorch">PyTorch</a></li>

<li><a href="Keras" title="Keras">Keras</a></li>

<li><a href="Scikit-learn" title="Scikit-learn">scikit-learn</a></li>

<li><a href="Theano_(software)" title="Theano (software)">Theano</a></li>

<li><a href="MindSpore" title="MindSpore">MindSpore</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"></span> Portals

<ul><li><a href="Portal%3AComputer_programming" title="Portal:Computer programming">Computer programming</a></li>

<li><a href="Portal%3ATechnology" title="Portal:Technology">Technology</a></li></ul></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q334384#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata644" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q334384#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata644" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh94008290">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007561023305171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/cf583102-236b-4653-a870-fc87150b4484">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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