Suboptimality of Penalized Empirical Risk Minimization in Classification

Details Suboptimality of Penalized Empirical Risk Minimization in Classification Suboptimality of Penalized Empirical Risk Minimization in Classification

2007-03-27

arxiv.org/abs/math/0703811v1

Mathematics

Statistics Theory

15 pages

Scientific paper

Let $\cF$ be a set of $M$ classification procedures with values in $[-1,1]$. Given a loss function, we want to construct a procedure which mimics at the best possible rate the best procedure in $\cF$. This fastest rate is called optimal rate of aggregation. Considering a continuous scale of loss functions with various types of convexity, we prove that optimal rates of aggregation can be either $((\log M)/n)^{1/2}$ or $(\log M)/n$. We prove that, if all the $M$ classifiers are binary, the (penalized) Empirical Risk Minimization procedures are suboptimal (even under the margin/low noise condition) when the loss function is somewhat more than convex, whereas, in that case, aggregation procedures with exponential weights achieve the optimal rate of aggregation.

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