A stochastic gradient descent algorithm for structural risk minimisation

Structural risk minimisation (SRM) is a general complexity regularization method which automatically selects the model complexity that approximately minimises the misclassification error probability of the empirical risk minimiser. It does so by adding a complexity penalty term ∊(m, k) to the empirical risk of the candidate hypotheses and then for any fixed sample size m it minimises the sum with respect to the model complexity variable k. When learning multicategory classification there are M subsamples m_i, corresponding to the M pattern classes with a priori probabilities p_i, 1 ≤ i ≤ M. Using the usual representation for a multi-category classifier as M individual boolean classifiers, the penalty becomes ∑^M_i=1 p_i∊(m_i, k_i). If the mi are given then the standard SRM trivially applies here by minimizing the penalised empirical risk with respect to k_i, 1,..., M. However, in situations where the total sample size ∑^M_i=1 mi needs to be minimal one needs to also minimize the penalised empirical risk with respect to the variables m_i, i = 1,..., M. The obvious problem is that the empirical risk can only be defined after the subsamples (and hence their sizes) are given (known). Utilising an on-line stochastic gradient descent approach, this paper overcomes this difficulty and introduces a sample-querying algorithm which extends the standard SRM principle. It minimises the penalised empirical risk not only with respect to the k_i, as the standard SRM does, but also with respect to the m_i, i = 1,..., M. The challenge here is in defining a stochastic empirical criterion which when minimised yields a sequence of subsample-size vectors which asymptotically achieve the Bayes’ optimal error convergence rate.