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 Õ(ε^-1 log^1.5 d) where d is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are Õ(ε^-1 min {d,T^1/3 log d ). We also develop an adaptive algorithm for the small-loss setting with regret O(L^★ log d + ε^-1 log^1.5 d) where L^★ 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 Õ⁽ε^-1d^1.5), as well as an algorithm for the smooth case with regret Õ⁽ε^-2/3(dT)^1/3), both significantly improving over existing bounds in the non-realizable regime.