Universal Online Learning with Bounded Loss: Reduction to Binary Classification

TL;DR

This paper demonstrates that for bounded loss functions, the class of processes allowing universal online learning is identical for binary and multiclass classification, enabling reduction to binary classification with practical algorithms like nearest neighbor.

Contribution

It proves the equivalence of universal online learnability between binary and multiclass settings under bounded loss, resolving an open problem and providing a constructive reduction.

Findings

Universal online learnability is the same for binary and multiclass classification.

Any bounded loss setting can be reduced to binary classification.

Nearest neighbor algorithm effectively learns on processes admitting universal learning.

Abstract

We study universal consistency of non-i.i.d. processes in the context of online learning. A stochastic process is said to admit universal consistency if there exists a learner that achieves vanishing average loss for any measurable response function on this process. When the loss function is unbounded, Blanchard et al. showed that the only processes admitting strong universal consistency are those taking a finite number of values almost surely. However, when the loss function is bounded, the class of processes admitting strong universal consistency is much richer and its characterization could be dependent on the response setting (Hanneke). In this paper, we show that this class of processes is independent from the response setting thereby closing an open question (Hanneke, Open Problem 3). Specifically, we show that the class of processes that admit universal online learning is the same for binary classification as for multiclass classification with countable number of classes. Consequently, any output setting with bounded loss can be reduced to binary classification. Our reduction is constructive and practical. Indeed, we show that the nearest neighbor algorithm is transported by our construction. For binary classification on a process admitting strong universal learning, we prove that nearest neighbor successfully learns at least all finite unions of intervals.