Optimal Prediction Using Expert Advice and Randomized Littlestone Dimension

TL;DR

This work extends the classical Littlestone framework to randomized online learning by introducing the randomized Littlestone dimension, showing it exactly characterizes the optimal expected mistake bound in the realizable setting. It further develops the k-realizable and agnostic cases, establishing tight bounds M*(H) = RL(H) and M*(H,k) = k + Θ(√(kL(H)) + L(H)), linking performance to the Littlestone dimension. A key application is prediction with expert advice, where the optimal randomized bound is shown to be essentially half of the deterministic bound across all n,k, with precise bounds in the n=2 case and generalization via the universal experts class. The paper also introduces quasi-balanced trees and a weighted SOA framework to handle bounded horizons and adaptive strategies, yielding near-optimal procedures without prior knowledge of the horizon or k. Overall, the results unify online learning performance with combinatorial dimensions, delivering exact characterizations, concrete learning rules, and broad applicability to expert-advice settings.

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 $k + Θ(\sqrt{k d} + d )$. This also implies an optimal deterministic mistake bound of $2k + Θ(d) + O(\sqrt{k d})$, 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 Brânzei and Peres ['19], our result applies to all pairs $n,k$.

Figures (5)

• Figure 1: A quasi-balanced tree. The edges are labeled with the unique weights. The sum of weights in each branch is $\tfrac78$, which is half the expected branch length $\tfrac74$.
• Figure 2: The randomized $\mathsf{SOA}$ is a variation of $\mathsf{SOA}$ that finds an optimal randomized prediction in every round. $\mathsf{SOA}$ is the name of the original deterministic algorithm by Littlestone littlestone1988learning, and it stands for "Standard Optimal Algorithm".
• Figure 3: $\mathsf{BoundedRandSOA}$ is a bounded variant of $\mathsf{RandSOA}$.
• Figure 4: The weighted version of $\mathsf{SOA}$.
• Figure 5: The weighted version of $\mathsf{RandSOA}$.

Theorems & Definitions (71)

• Theorem 2.1: Deterministic Mistake Bound littlestone1988learning
• Definition 2.2: Randomized Littlestone Dimension
• Theorem 2.3: Main Result (i): Randomized Mistake Bound
• Theorem 2.4: Main Result (ii): $k$-Littlestone Dimension
• Theorem 2.5: Main Result (iii): Optimal Regret Bounds for Littlestone Classes
• Theorem 2.6: Main Result (iv): Bounds for Randomized Predictors
• Corollary 2.7
• Theorem 2.8: Main Result (v): Bounded vs Unbounded Horizon
• Theorem 2.9: Optimal Mistake Bounds: Bounded Horizon
• Theorem 2.10