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Hoffman's classical result gives a bound on the distance of a point from aconvex and compact polytope in terms of the magnitude of violation of theconstraints. Recently, several results showed that Hoffman's bound can be usedto derive strongly-convex-like rates for first-order methods for\textit{offline} convex optimization of curved, though not strongly convex,functions, over polyhedral sets. In this work, we use this classical result forthe first time to obtain faster rates for \textit{online convex optimization}over polyhedral sets with curved convex, though not strongly convex, lossfunctions. We show that under several reasonable assumptions on the data, thestandard \textit{Online Gradient Descent} algorithm guarantees logarithmicregret. To the best of our knowledge, the only previous algorithm to achievelogarithmic regret in the considered settings is the \textit{Online NewtonStep} algorithm which requires quadratic (in the dimension) memory and at leastquadratic runtime per iteration, which greatly limits its applicability tolarge-scale problems. In particular, our results hold for\textit{semi-adversarial} settings in which the data is a combination of anarbitrary (adversarial) sequence and a stochastic sequence, which might providereasonable approximation for many real-world sequences, or under a naturalassumption that the data is low-rank. We demonstrate via experiments that theregret of OGD is indeed comparable to that of ONS (and even far better) oncurved though not strongly-convex losses.

Logarithmic Regret for Online Gradient Descent Beyond Strong Convexity