Structural risk minimization: Difference between revisions

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'''Structural risk minimization (SRM)''' is an inductive principle of use in [[machine learning]]. Commonly in machine learning, a generalized model must be selected from a finite data set, with the consequent problem of [[overfitting]] – the model becoming too strongly tailored to the particularities of the training set and generalizing poorly to new data. The SRM principle addresses this problem by balancing the model's complexity against its success at fitting the training data. This principle was first set out in a 1974 paper by [[Vladimir Vapnik]] and [[Alexey Chervonenkis]] and uses the [[VC dimension]].

In practical terms, Structural Risk Minimization is implemented by minimizing <math>E_{train} + \beta H(W)</math>, where <math>E_{train}</math> is the train error, the function <math>H(W)</math> is called a regularization function, and <math>\beta</math> is a constant. <math>H(W)</math> is chosen such that it takes large values on parameters <math>W</math> that belong to high-capacity subsets of the parameter space. Minimizing <math>H(W)</math> in effect limits the capacity of the accessible subsets of the parameter space, thereby controlling the trade-off between minimizing the training error and minimizing the expected gap between the training error and test error.<ref>{{Cite web|url=http://vision.stanford.edu/cs598_spring07/papers/Lecun98.pdf|title=Gradient-Based Learning Applied to Document Recognition|last=LeCun|first=Yann|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref>

The SRM problem can be formulated in terms of data. Given n data points consisting of data x and labels y, the objective <math>J(\theta)</math> is often expressed in the following manner: