Minimax Optimal Quantile and Semi-Adversarial Regret via Root-Logarithmic Regularizers

Quantile (and, more generally, KL) regret bounds, such as those achieved by NormalHedge (Chaudhuri, Freund, and Hsu 2009) and its variants, relax the goal of competing against the best individual expert to only competing against a majority of experts on adversarial data.