Fast Rates for Nonparametric Online Learning: From Realizability to Learning in Games
This work addresses the challenge of improving regret bounds in online learning for researchers and practitioners, offering incremental advances by generalizing prior results from finite to nonparametric settings.
The paper tackles the problem of achieving fast convergence rates in nonparametric online learning, showing that in the realizable setting with absolute loss, a proper learning algorithm achieves near-optimal cumulative loss scaling with sequential fat-shattering dimension, reducing to d·poly log T for online classification, improving from previous bounds of ~O(√(dT)). It also extends this to general-sum binary games, where each player achieves regret ~O(d^{3/4}·T^{1/4}), accelerating from O(√T) to O(T^{1/4}) in game settings.
We study fast rates of convergence in the setting of nonparametric online regression, namely where regret is defined with respect to an arbitrary function class which has bounded complexity. Our contributions are two-fold: - In the realizable setting of nonparametric online regression with the absolute loss, we propose a randomized proper learning algorithm which gets a near-optimal cumulative loss in terms of the sequential fat-shattering dimension of the hypothesis class. In the setting of online classification with a class of Littlestone dimension $d$, our bound reduces to $d \cdot {\rm poly} \log T$. This result answers a question as to whether proper learners could achieve near-optimal cumulative loss; previously, even for online classification, the best known cumulative loss was $\tilde O( \sqrt{dT})$. Further, for the real-valued (regression) setting, a cumulative loss bound with near-optimal scaling on sequential fat-shattering dimension was not even known for improper learners, prior to this work. - Using the above result, we exhibit an independent learning algorithm for general-sum binary games of Littlestone dimension $d$, for which each player achieves regret $\tilde O(d^{3/4} \cdot T^{1/4})$. This result generalizes analogous results of Syrgkanis et al. (2015) who showed that in finite games the optimal regret can be accelerated from $O(\sqrt{T})$ in the adversarial setting to $O(T^{1/4})$ in the game setting. To establish the above results, we introduce several new techniques, including: a hierarchical aggregation rule to achieve the optimal cumulative loss for real-valued classes, a multi-scale extension of the proper online realizable learner of Hanneke et al. (2021), an approach to show that the output of such nonparametric learning algorithms is stable, and a proof that the minimax theorem holds in all online learnable games.