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'''Empirical risk minimization''' (ERM) is a principle in [[statistical learning theory]] which defines a family of learning algorithms and is used to give theoretical bounds on the performance of [[machine learning|learning algorithms]].
 
== Background ==
Consider the following situation, which is a general setting of many [[supervised learning]] problems. We have two spaces of objects <math>X</math> and <math>Y</math> and would like to learn a function <math>\! h: X \to Y</math> (often called ''hypothesis'') which outputs an object <math>y \in Y</math>, given <math>x \in X</math>. To do so, we have at our disposal a ''training set'' of a few examples <math>\! (x_1, y_1), \ldots, (x_m, y_m)</math> where <math>x_i \in X</math> is an input and <math>y_i \in Y</math> is the corresponding response that we wish to get from <math>\! h(x_i)</math>.
 
To put it more formally, we assume that there is a [[joint probability distribution]] <math>P(x, y)</math> over <math>X</math> and <math>Y</math>, and that the training set consists of <math>m</math> instances <math>\! (x_1, y_1), \ldots, (x_m, y_m)</math> drawn [[i.i.d.]] from <math>P(x, y)</math>. Note that the assumption of a joint probability distribution allows us to model uncertainty in predictions (e.g. from noise in data) because <math>y</math> is not a deterministic function of <math>x</math>, but rather a [[random variable]] with [[conditional distribution]] <math>P(y | x)</math> for a fixed <math>x</math>.
 
We also assume that we are given a non-negative real-valued [[loss function]] <math>L(\hat{y}, y)</math> which measures how different the prediction <math>\hat{y}</math> of a hypothesis is from the true outcome <math>y</math>. The [[Risk (statistics)|risk]] associated with hypothesis <math>h(x)</math> is then defined as the [[Expected value|expectation]] of the loss function:
: <math>R(h) = \mathbf{E}[L(h(x), y)] = \int L(h(x), y)\,dP(x, y).</math>
 
A loss function commonly used in theory is the [[0-1 loss function]]: <math>L(\hat{y}, y) = I(\hat{y} \ne y)</math>, where <math>I(...)</math> is the [[indicator notation]].
 
The ultimate goal of a learning algorithm is to find a hypothesis <math> h^*</math> among a fixed class of functions <math>\mathcal{H}</math> for which the risk <math>R(h)</math> is minimal:
: <math>h^* = \arg \min_{h \in \mathcal{H}} R(h).</math>
 
== Empirical risk minimization ==
In general, the risk <math>R(h)</math> cannot be computed because the distribution <math>P(x, y)</math> is unknown to the learning algorithm (this situation is referred to as [[agnostic learning]]). However, we can compute an approximation, called ''empirical risk'', by averaging the loss function on the training set:
: <math>\! R_\mbox{emp}(h) = \frac{1}{m} \sum_{i=1}^m L(h(x_i), y_i).</math>
 
''Empirical risk minimization'' principle states that the learning algorithm should choose a hypothesis <math>\hat{h}</math> which minimizes the empirical risk:
: <math>\hat{h} = \arg \min_{h \in \mathcal{H}} R_{\mbox{emp}}(h).</math>
Thus the learning algorithm defined by the ERM principle consists in solving the above [[Mathematical optimization|optimization]] problem.
 
== Properties ==
{{Expand section|date=February 2010}}
 
=== Computational complexity ===
Empirical risk minimization for a classification problem with [[0-1 loss function]] is known to be an [[NP-hard]] problem even for such relatively simple class of functions as [[linear classifier]]s.<ref>V. Feldman, V. Guruswami, P. Raghavendra and Yi Wu (2009). [http://www.almaden.ibm.com/cs/people/vitaly/papers/FGRW09_MonoHalf_FOCS.pdf ''Agnostic Learning of Monomials by Halfspaces is Hard.''] (See the paper and references therein)</ref> Though, it can be solved efficiently when minimal empirical risk is zero, i.e. data is [[linearly separable]].
 
In practice, machine learning algorithms cope with that either by employing a convex approximation to 0-1 loss function (like [[hinge loss]] for [[Support vector machine|SVM]]), which is easier to optimize, or by posing assumptions on the distribution <math>P(x, y)</math> (and thus stop being agnostic learning algorithms to which the above result applies,)
 
== References ==
{{Reflist}}
 
== Literature ==
* {{cite book
    | last=Vapnik
    | first=V.
    | authorlink = Vladimir Vapnik
    | title=The Nature of Statistical Learning Theory
    | publisher = [[Springer-Verlag]]
    | series=Information Science and Statistics
    | year = 2000
    | isbn=978-0-387-98780-4}}
 
[[Category:Machine learning]]

Latest revision as of 23:59, 28 November 2014

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