We consider the problem of stochastic convex optimization with exp-concave losses using Empirical Risk Minimization in a convex class. Answering a question raised in several prior works, we provide a $O( d / n + \log( 1 / \delta) / n )$ excess risk bound valid for a wide class of bounded exp-concave losses, where $d$ is the dimension of the convex reference set, $n$ is the sample size, and $\delta$ is the confidence level. Our result is based on a unified geometric assumption on the gradient of losses and the notion of local norms.

翻译：我们考虑在凸类别中使用经验风险最小化处理具有指数凹损失的随机凸优化问题。针对先前若干研究中提出的问题，我们提供了一个 $O( d / n + \log( 1 / \delta) / n )$ 的超额风险界，该结果适用于一大类有界指数凹损失，其中 $d$ 是凸参考集的维度，$n$ 是样本量，$\delta$ 是置信水平。我们的结果基于损失函数梯度的统一几何假设以及局部范数的概念。