PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting

2026-05-25

AI Summary

This work investigates the sample complexity of multiclass PAC learning in the realizable setting when only binary bandit feedback is available. To address this problem, the authors introduce a novel combinatorial dimension—termed the bandit DS dimension—based on a pseudo-box structure that captures the complexity of a concept class by aggregating the number of neighboring labels across coordinates, and establish its connection to list learning. Building upon this dimension, they design a general-purpose algorithm, ListCascade, which achieves a tight upper bound on sample complexity that matches the information-theoretic lower bound up to logarithmic factors. This result yields the first universally applicable learning framework for this setting with nearly matching upper and lower bounds.

Abstract

We study the problem of multiclass PAC learning with bandit feedback in the realizable setting. In this framework, there is an unknown data distribution over an instance space $\mathcal{X}$ and a label space $\mathcal{Y}$, as in classical multiclass PAC learning, but the learner does not observe the labels of the i.i.d. training examples. Instead, in each round, it receives an unlabeled instance, predicts its label, and receives bandit feedback indicating only whether the prediction is correct. Despite this restriction, the goal remains the same as in classical PAC learning. We provide a general characterization of the optimal sample complexity of this problem, sharp for every concept class up to logarithmic factors. Our characterization is based on a new combinatorial dimension, termed the bandit $\mathrm{DS}$ dimension, defined via generalized combinatorial structures we call pseudo-boxes. These extend the pseudo-cubes underlying the $\mathrm{DS}$ dimension by allowing a different number of neighbors in each coordinate. In contrast to the $\mathrm{DS}$ dimension, which governs the full-information setting by counting the number of coordinates in the pseudo-cube, the bandit $\mathrm{DS}$ dimension aggregates the number of neighbors across coordinates, leading to a characterization in which the sample complexity scales with the total number of neighbors. We also propose a general learning algorithm achieving the upper bound, based on an algorithmic principle called ListCascade, which connects bandit learning to list learning and may be of independent interest.

Problem

• PAC learning
• bandit feedback
• multiclass classification
• sample complexity
• realizable setting

Innovation

• bandit feedback
• PAC learning
• sample complexity
• combinatorial dimension
• ListCascade