标题： 分类中惩罚经验风险最小化的次优性 显示英文标题

标题： Suboptimality of Penalized Empirical Risk Minimization in Classification

摘要： 设 $\cF$ 是一个取值于 $[-1,1]$ 的 $M$ 种分类程序的集合。 给定一个损失函数，我们希望构造一个程序，以尽可能快的速率模仿 $\cF$ 中的最佳程序。 这个最快的速率被称为最优聚合速率。 考虑具有各种凸性类型的连续尺度的损失函数，我们证明了最优聚合速率可以是 $((\log M)/n)^{1/2}$ 或 $(\log M)/n$。 我们证明了，如果所有 $M$ 分类器均为二元分类器，在损失函数稍微比凸函数更复杂的情况下，（惩罚）经验风险最小化程序是次优的（即使在边界/低噪声条件下），而在这种情况下，带指数权重的聚合程序能够达到最优聚合速率。 显示英文摘要

摘要： 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.