  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.
