Adversarial Laws of Large Numbers and Optimal Regret in Online Classification

TL;DR

This paper explores the conditions under which laws of large numbers hold in adaptive online sampling processes, linking these conditions to online learnability and deriving optimal regret bounds based on Littlestone's dimension.

Contribution

It characterizes classes that admit uniform laws of large numbers in adaptive sampling, establishing an online analogue to classical learnability and uniform convergence results.

Findings

Characterization of classes with uniform laws of large numbers in adaptive sampling.

Tight sample-complexity bounds for various regimes.

Resolution of the open problem on optimal regret bounds in online learning.

Abstract

Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed sub-population is well-estimated by its frequency in the sample. We study laws of large numbers in sampling processes that can affect the environment they are acting upon and interact with it. Specifically, we consider the sequential sampling model proposed by Ben-Eliezer and Yogev (2020), and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are \emph{online learnable}. Our characterization may be interpreted as an online analogue to the equivalence between learnability and uniform convergence in statistical (PAC) learning. The sample-complexity bounds we obtain are tight for many parameter regimes, and as an application, we determine the optimal regret bounds in online learning, stated in terms of \emph{Littlestone's dimension}, thus resolving the main open question from Ben-David, P\'al, and Shalev-Shwartz (2009), which was also posed by Rakhlin, Sridharan, and Tewari (2015).