Near-Optimal Algorithms for Private Online Optimization in the Realizable Regime

We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret O big( varepsilon-1 log1.5d big) where d is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are O big( varepsilon-1 minbigd, T1/3log dbig big). We also develop an adaptive algorithm for the small-loss setting with regret O(L^starlog d + varepsilon-1 log1.5d) where L^star is the total loss of the best expert. Additionally, we consider DP online convex optimization in the realizable setting and propose an algorithm with near-optimal regret O big(varepsilon-1 d1.5 big), as well as an algorithm for the smooth case with regret O big( varepsilon-2/3 (dT)1/3 big), both significantly improving over existing bounds in the non-realizable regime.