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

TL;DR

This paper introduces root-logarithmic regularizers in FTRL algorithms to achieve minimax optimal quantile and semi-adversarial regret bounds, extending theoretical guarantees to broader settings and improving adaptivity.

Contribution

It develops novel regularizers for FTRL that attain minimax optimal regret in quantile and semi-adversarial paradigms, extending bounds to uncountable expert classes and providing tight lower bounds.

Findings

Achieves minimax optimal regret bounds in both paradigms.

Extends KL regret bounds to uncountable expert classes with arbitrary priors.

Provides tight lower bounds for quantile regret on finite expert classes.

Abstract

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. More recently, the semi-adversarial paradigm (Bilodeau, Negrea, and Roy 2020) provides an alternative relaxation of adversarial online learning by considering data that may be neither fully adversarial nor stochastic (i.i.d.). We achieve the minimax optimal regret in both paradigms using FTRL with separate, novel, root-logarithmic regularizers, both of which can be interpreted as yielding variants of NormalHedge. We extend existing KL regret upper bounds, which hold uniformly over target distributions, to possibly uncountable expert classes with arbitrary priors; provide the first full-information lower bounds for quantile regret on finite expert classes (which are tight); and provide an adaptively minimax optimal algorithm for the semi-adversarial paradigm that adapts to the true, unknown constraint faster, leading to uniformly improved regret bounds over existing methods.