Adversarial Laws of Large Numbers and Optimal Regret in Online Classification

Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed subpopulation 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 [O. Ben-Eliezer and E. Yogev, The adversarial robustness of sampling, in Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS), 2020, pp. 49–62] and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are 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 Littlestone’s dimension, thus resolving the main open question from [S. Ben-David, D. Pál, and S. Shalev-Shwartz, Agnostic online learning, in Proceedings of the 22nd Conference on Learning Theory (COLT), 2009], which was also posed by [A. Rakhlin, K. Sridharan, and A. Tewari, J. Mach. Learn. Res., 16 (2015), pp. 155–186].