Fast rates with high probability in exp-concave statistical learning

We present an algorithm for the statistical learning setting with a bounded exp-concave loss in $d$ dimensions that obtains excess risk $O(d \log(1/\delta)/n)$ with probability at least $1 - \delta$. The core technique is to boost the confidence of recent in-expectation $O(d/n)$ excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability $1 - \delta$ the standard ERM method obtains excess risk $O(d (\log(n) + \log(1/\delta))/n)$. We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate.