No-Regret Learnability for Piecewise Linear Losses

In the convex optimization approach to online regret minimization, many methodshave been developed to guarantee a O(√T ) bound on regret for subdifferentiableconvex loss functions with bounded subgradients, by using a reduction to linearloss functions. This suggests that linear loss functions tend to be the hardest onesto learn against, regardless of the underlying decision spaces. We investigate thisquestion in a systematic fashion looking at the interplay between the set of pos-sible moves for both the decision maker and the adversarial environment. Thisallows us to highlight sharp distinctive behaviors about the learnability of piece-wise linear loss functions. On the one hand, when the decision set of the decisionmaker is a polyhedron, we establish Ω(√T ) lower bounds on regret for a largeclass of piecewise linear loss functions with important applications in online lin-ear optimization, repeated zero-sum Stackelberg games, online prediction withside information, and online two-stage optimization. On the other hand, we ex-hibit O(log T ) learning rates, achieved by the Follow-The-Leader algorithm, inonline linear optimization when the boundary of the decision maker’s decision setis curved and when 0 does not lie in the convex hull of the environment’s decisionset. These results hold in a completely adversarial setting.