Near-Optimal Algorithms for Private Online Optimization in the Realizable Regime
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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( ε^-1log^1.5d) 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( ε^-1min{d, T^1/3log d}). We also develop an adaptive algorithm for the small-loss setting with regret O(L^⋆log d + ε^-1log^1.5d) 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 O (ε^-1 d^1.5), as well as an algorithm for the smooth case with regret O ( ε^-2/3 (dT)^1/3), both significantly improving over existing bounds in the non-realizable regime.
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