Suboptimality of penalized empirical risk minimization in classification

Let F 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 F. 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.