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 $O(d \log(1/δ)/n)$ with probability $1 - δ$. 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 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 high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. One bound obtains a nearly optimal rate without requiring the loss to be Lipschitz continuous, and another requires Lipschitz continuity but obtains the optimal rate.