Optimal Multiclass Online Regret with Arbitrary Label Spaces

We consider binary prediction with i.i.d. contexts from an unknown distribution and adaptively chosen losses. We show that a simple Follow-the-Perturbed-Leader algorithm using a Gaussian perturbation for each observed context achieves $\widetilde O(\sqrt{T\log N})$ regret for a class of $N$ experts, while requiring one optimization-oracle call per round and no explicit enumeration of the class. For an infinite hypothesis class $\mathcal H$, the same algorithm achieves $\widetilde O(\sqrt{T\operatorname{VC}(\mathcal H)})$ regret. This resolves an open problem posed by Lazaric and Munos (2012), showing that hybrid classification is computationally as easy as statistical learning. As an application, we reduce the problem of contextual bandits with $K$ actions to classification through uniform exploration, achieving $\widetilde O(K^{2/3}T^{2/3}(\log N)^{1/3})$ regret. This matches the best known dependence on the horizon while removing the context-distribution access required by prior oracle-efficient methods.

We prove that the optimal expected mistake bound of online learning a function class $\mathcal{H}$ by a randomized proper learning algorithm is $O(\mathtt{L}(\mathcal{H}) \log T)$, where $\mathtt{L}(\mathcal{H})$ is the Littlestone dimension of $\mathcal{H}$ and $T$ is the time horizon. Our result improves upon the previously best known bound of $O(\mathtt{L}(\mathcal{H}) \log^6 T)$ given by Daskalakis and Golowich (STOC 2022), and is optimal up to a universal constant for worst-case classes.

We consider the problem of universal transductive online classification with a possibly unbounded label space. This setting considers online learning, with the sequence of instances (without labels) known to the learner in advance. We say a concept class $\mathcal{H}$ is learnable if there is a learning algorithm $\mathcal{A}$, such that for every realizable sequence, the number of mistakes made by $\mathcal{A}$ grows at most sublinearly with the number of predictions. We characterize the learnability of this setting and show that there are only two possible optimal rates for the learnable classes: either bounded or increasing logarithmically. We introduce a new combinatorial structure, called ``Level-Constrained-Littlestone-Littlestone (LCLL) tree'', which, along with the indifference property, characterizes the learnability. We also extend the learnability result to the agnostic case and the case where only the stochastic process that generates the instance sequence is known.