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| {{machine learning bar}}
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| '''Statistical learning theory''' is a framework for [[machine learning]]
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| drawing from the fields of [[statistics]] and [[functional analysis]].<ref>[[Mehryar Mohri]], Afshin Rostamizadeh, Ameet Talwalkar (2012) ''Foundations of Machine Learning'', The
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| MIT Press ISBN 9780262018258.</ref>
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| Statistical learning theory deals with the problem of finding a
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| predictive function based on data. Statistical learning
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| theory has led to successful applications in fields such as [[computer vision]], [[speech recognition]], [[bioinformatics]], and [[baseball]].<ref>Gagan Sidhu, Brian Caffo. Exploiting pitcher decision-making using Reinforcement Learning. ''Annals of Applied Statistics''</ref> It is the theoretical
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| framework underlying [[support vector machines]].
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| ==Introduction==
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| The goal of learning is prediction. Learning falls into many
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| categories, including [[supervised learning]], [[unsupervised learning]],
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| [[Online machine learning|online learning]], and [[reinforcement learning]]. From the perspective of
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| statistical learning theory, supervised learning is best understood.<ref>Tomaso Poggio, Lorenzo Rosasco, et al. ''Statistical Learning Theory and Applications'', 2012, Class 1 [http://www.mit.edu/~9.520/spring12/slides/class01/class01.pdf]</ref>
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| Supervised learning involves learning from a [[training set]] of data.
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| Every point in the training is an input-output pair, where the input
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| maps to an output. The learning problem consists of inferring the
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| function that maps between the input and the output in a predictive fashion,
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| such that the learned function can be used to predict output from
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| future input.
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| Depending of the type of output, supervised learning problems are
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| either problems of [[regression analysis|regression]] or problems of [[Statistical classification|classification]]. If the
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| output takes a continuous range of values, it is a regression problem.
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| Using [[Ohm's Law]] as an example, a regression could be performed with
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| voltage as input and current as output. The regression would find the
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| functional relationship between voltage and current to be
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| {{nowrap|<math>\frac{1}{R}</math>}}, such that
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| :<math>
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| I = \frac{1}{R} V
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| </math>
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| Classification problems are those for which the output will be an
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| element from a discrete set of labels. Classification is very common
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| for machine learning applications. In [[facial recognition system|facial recognition]], for
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| instance, a picture of a person's face would be the input, and the
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| output label would be that person's name. The input would be
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| represented by a large multidimensional vector, in which each
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| dimension represents the value of one of the pixels.
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| After learning a function based on the training set data, that
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| function is validated on a test set of data, data that did not appear
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| in the training set. Classification functions can use the percentage
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| of inputs that are correctly classified as a metric for how predictive the learned
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| function is, while regression functions must use some distance metric,
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| called a [[loss function]], for how accurate the predicted value is. A
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| familiar example of a loss function is the square of the difference
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| between the actual value and the predicted value; this is the loss
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| function used in ordinary least squares regression.
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| ==Formal Description==
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| Take <math>X</math> to be the vector space of all possible inputs, and <math>Y</math> to be
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| the vector space of all possible outputs. Statistical learning theory
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| takes the perspective that there is some unknown probability
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| distribution over the product space <math>Z = X \otimes Y</math>, i.e. there
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| exists some unknown <math>p(z) = p(\vec{x},y)</math>. The training
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| set is made up of <math>n</math> samples from this probability distribution, and is notated
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| :<math>S = \{(\vec{x}_1,y_1), \dots ,(\vec{x}_n,y_n)\} = \{\vec{z}_1, \dots ,\vec{z}_n\}</math>
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| Every <math>\vec{x}_i</math> is an input vector from the training data, and <math>y_i</math>
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| is the output that corresponds to it.
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| In this formalism, the inference problem consists of finding a
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| function <math>f: X \mapsto Y</math> such that <math>f(\vec{x}) \sim y</math>. Let
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| <math>\mathcal{H}</math> be a space of functions <math>f: X \mapsto Y</math> called the
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| hypothesis space. The hypothesis space is the space of functions the
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| algorithm will search through. Let <math>V(f(\vec{x}),y)</math> be the [[loss functional]], a metric for the difference between the predicted value
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| <math>f(\vec{x})</math> and the actual value <math>y</math>. The [[expected risk]] is defined to
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| be
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| :<math>I[f] = \displaystyle \int_{X \otimes Y} V(f(\vec{x}),y) p(\vec{x},y) d\vec{x} dy</math>
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| The target function, the best possible function <math>f</math> that can be
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| chosen, is given by the <math>f</math> that satisfies
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| :<math>\inf_{f \in \mathcal{H}} I[f]</math>
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| Because the probability distribution <math>p(\vec{x},y)</math> is unknown, a
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| proxy measure for the expected risk must be used. This measure is based on the
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| training set, a sample from this unknown probability distribution. It
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| is called the [[empirical risk]]
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| :<math>I_S[f] = \frac{1}{n} \displaystyle \sum_{i=1}^n V( f(\vec{x}_i),y_i)</math> | |
| A learning algorithm that chooses the function <math>f_S</math> that minimizes
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| the empirical risk is called [[empirical risk minimization]].
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| ==Loss Functions==
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| The choice of loss function is a determining factor on the function
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| <math>f_S</math> that will be chosen by the learning algorithm. The loss function
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| also affects the convergence rate for an algorithm. It is important
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| for the loss function to be convex.<ref>Rosasco, L., Vito, E.D., Caponnetto, A., Fiana, M., and Verri A. 2004. ''Neural computation'' Vol 16, pp 1063-1076</ref>
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| Different loss functions are used depending on whether the problem is
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| one of regression or one of classification.
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| ===Regression=== | |
| The most common loss function for regression is the square loss
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| function. This familiar loss function is used in ordinary least
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| squares regression. The form is:
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| :<math>V(f(\vec{x}),y) = (y - f(\vec{x}))^2</math> | |
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| The absolute value loss is also sometimes used:
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| :<math>V(f(\vec{x}),y) = |y - f(\vec{x})|</math>
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| ===Classification===
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| In some sense the 0-1 loss is the most natural loss function for
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| classification. It takes the value 0 if the predicted output is the
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| same as the actual output, and it takes the value 1 if the predicted output
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| is different from the actual output. For binary classification, this is:
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| :<math>V(f(\vec{x},y)) = \theta(- y f(\vec{x}))</math>
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| where <math>\Theta</math> is the Heaviside step function.
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| The 0-1 loss function, however, is not convex. The hinge loss is thus
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| often used:
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| :<math>V(f(\vec{x},y)) = (- y f(\vec{x}))_+</math>
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| ==Regularization==
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| [[File:Overfitting on Training Set Data.pdf|thumb|This image represents an example of overfitting in machine learning. The red dots represent training set data. The green line represents the true functional relationship, while the blue line shows the learned function, which has fallen victim to overfitting.]]
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| In machine learning problems, a major problem that arises is that of
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| overfitting. Because learning is a prediction problem, the goal is
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| not to find a function that most closely fits the data, but to find one
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| that will most accurately will predict output from future input.
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| Empirical risk minimization runs this risk of overfitting: finding a
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| function that matches the data exactly but does not predict future output well.
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| Overfitting is symptomatic of unstable solutions; a small perturbation
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| in the training set data would cause a large variation in the learned
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| function. It can be shown that if the stability for the solution can
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| be guaranteed, generalization and consistency are guaranteed as well.<ref>Vapnik, V.N. and Chervonenkis, A.Y. 1971. On the uniform convergence of relative frequencies of events to their probabilities. ''Theory of Probability and its Applications'' Vol 16, pp 264-280.</ref><ref>Mukherjee, S., Niyogi, P. Poggio, T., and Rifkin, R. 2006. Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization. ''Advances in Computational Mathematics''. Vol 25, pp 161-193.</ref> [[Regularization (mathematics)|Regularization]] can solve the overfitting problem and give
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| the problem stability.
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| Regularization can be accomplished by restricting the hypothesis space
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| <math>\mathcal{H}</math>. A common example would be restricting <math>\mathcal{H}</math> to
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| linear functions: this can be seen as a reduction to the standard problem of
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| [[linear regression]]. <math>\mathcal{H}</math> could also be restricted to
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| polynomial of degree <math>p</math>, exponentials, or bounded functions on
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| [[Lp space|L1]]. Restriction of the hypothesis space avoids overfitting because
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| the form of the potential functions are limited, and so does not allow
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| for the choice of a function that gives empirical risk arbitrarily
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| close to zero.
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| Regularization can also be accomplished through [[Tikhonov regularization]]. This
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| consists of minimizing
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| :<math>\frac{1}{n} \displaystyle \sum_{i=1}^n V(f(\vec{x}_i,y_i)) + \gamma
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| \|f\|_{\mathcal{H}}^2</math>
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| where <math>\gamma</math> is a fixed and positive parameter, the regularization
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| parameter. Tikhonov regularization ensures existence, uniqueness, and
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| stability of the solution.<ref>Tomaso Poggio, Lorenzo Rosasco, et al. ''Statistical Learning Theory and Applications'', 2012, Class 2 [http://www.mit.edu/~9.520/spring12/slides/class02/class02.pdf]</ref>
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| {{clear}}
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| ==See also==
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| * [[Reproducing kernel Hilbert spaces]] are a useful choice for <math>\mathcal{H}</math>.
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| * [[Proximal gradient methods for learning]]
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| ==References==
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| {{reflist}}
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| [[Category:Machine learning]]
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