Strong Convexity of Feasible Sets in Off-line and Online Optimization

It is known that the curvature of the feasible set in convex optimization allows for algorithms with better convergence rates, and there is renewed interest in this topic for both off-line and online problems. In this paper, leveraging results on geometry and convex analysis, we further our understanding of the role of curvature in optimization: We first show the equivalence of two notions of curvature, namely, strong convexity and gauge bodies, proving a conjecture of Abernethy et al. As a consequence, this shows that the Frank–Wolfe–type method of Wang and Abernethy has accelerated convergence rate O ( 1 t 2 ) over strongly convex feasible sets without additional assumptions on the (convex) objective function. In online linear optimization, we identify two main properties that help explaining why/when follow the leader (FTL) has only logarithmic regret over strongly convex sets. This allows one to directly recover and slightly extend a recent result of Huang et al., and to show that FTL has logarithmic regret over strongly convex sets whenever the gain vectors are nonnegative. We provide an efficient procedure for approximating convex bodies by strongly convex ones while smoothly trading off approximation error and curvature. This allows one to extend the improved algorithms over strongly convex sets to general convex sets. As a concrete application, we extend results on online linear optimization with hints to general convex sets.