Adversarial Laws of Large Numbers and Optimal Regret in Online Classification
This provides foundational insights for online learning theory, connecting learnability to uniform convergence in an adversarial setting.
The paper tackles the problem of characterizing which classes admit a uniform law of large numbers in a sequential sampling model that interacts with the environment, showing these are exactly the online learnable classes, and resolves an open question by determining optimal regret bounds in online learning in terms of Littlestone's dimension.
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, Pál, and Shalev-Shwartz (2009), which was also posed by Rakhlin, Sridharan, and Tewari (2015).