🤖 AI Summary
This paper addresses the optimal mistake bounds for randomized learners in online learning. **Problem:** Characterizing tight expected mistake bounds for randomized algorithms under both realizable and agnostic settings, and resolving the long-standing open problem of optimal randomized mistake bounds in expert prediction (Cesa-Bianchi et al., 1993). **Method:** The authors integrate average depth analysis of shattering trees, combinatorial tree complexity measures, and principled design of randomized online algorithms. **Contributions:** (1) They introduce and rigorously characterize the *randomized Littlestone dimension*, which precisely captures the optimal expected mistake bound for randomized learners in the realizable setting. (2) In the agnostic setting, for hypothesis classes with Littlestone dimension $d$, they establish a tight randomized bound of $k + Theta(sqrt{kd} + d)$ for achieving at most $k$ mistakes relative to the best function. (3) They fully resolve the expert prediction problem by constructing a universal optimal randomized algorithm whose expected mistake bound is half the deterministic lower bound, up to lower-order terms.
📝 Abstract
A classical result in online learning characterizes the optimal mistake bound achievable by deterministic learners using the Littlestone dimension (Littlestone '88). We prove an analogous result for randomized learners: we show that the optimal expected mistake bound in learning a class $mathcal{H}$ equals its randomized Littlestone dimension, which is the largest $d$ for which there exists a tree shattered by $mathcal{H}$ whose average depth is $2d$. We further study optimal mistake bounds in the agnostic case, as a function of the number of mistakes made by the best function in $mathcal{H}$, denoted by $k$. We show that the optimal randomized mistake bound for learning a class with Littlestone dimension $d$ is $k + Theta (sqrt{k d} + d )$. This also implies an optimal deterministic mistake bound of $2k + Theta(d) + O(sqrt{k d})$, thus resolving an open question which was studied by Auer and Long ['99]. As an application of our theory, we revisit the classical problem of prediction using expert advice: about 30 years ago Cesa-Bianchi, Freund, Haussler, Helmbold, Schapire and Warmuth studied prediction using expert advice, provided that the best among the $n$ experts makes at most $k$ mistakes, and asked what are the optimal mistake bounds. Cesa-Bianchi, Freund, Helmbold, and Warmuth ['93, '96] provided a nearly optimal bound for deterministic learners, and left the randomized case as an open problem. We resolve this question by providing an optimal learning rule in the randomized case, and showing that its expected mistake bound equals half of the deterministic bound of Cesa-Bianchi et al. ['93,'96], up to negligible additive terms. In contrast with previous works by Abernethy, Langford, and Warmuth ['06], and by Br^anzei and Peres ['19], our result applies to all pairs $n,k$.
