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:我们考虑的是利用将经验风险降到最低程度来将变形损失与变形损失相优化的问题。在回答前几次工程中提出的问题时,我们提供了美元(d/ n +\log(1 /\ delta) / n) 的超额风险,它适用于广泛的捆绑的变形损失,其中美元是变形参照的维度,美元是样本大小,美元是置信度。我们的结果基于对损失梯度和当地规范概念的统一几何假设。