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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 at least 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 with probability 1 − δ the standard ERM method obtains excess risk O(d(log(n) + log(1/δ))/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.

We consider two variants of private stochastic online learning. The first variant is differentially private stochastic bandits. Previously, (Sajed & Sheffet, 2019) devised the DP Successive Elimination (DP-SE) algorithm that achieves the optimal O 1≤j≤K:∆j >0 log T ∆j + K log T ǫ problem-dependent regret bound, where K is the number of arms, ∆ j is the mean reward gap of arm j, T is the time horizon, and ǫ is the required privacy parameter. However, like other elimination style algorithms, it is not an anytime algorithm. Until now, it was not known whether UCB-based algorithms could achieve this optimal regret bound. We present an anytime, UCB-based algorithm that achieves optimality. Our experiments show that the UCB-based algorithm is competitive with DP-SE. The second variant is the full information version of private stochastic online learning. Specifically, for the problems of decision-theoretic online learning with stochastic rewards, we present the first algorithm that achieves an O log K ∆min + log K ǫ regret bound, where ∆ min is the minimum mean reward gap. The key idea behind our good theoretical guarantees in both settings is the forgetfulness, i.e., decisions are made based on a certain amount of newly obtained observations instead of all the observations obtained from the very beginning.

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