Rejection sampling from shape-constrained distributions in sublinear time

TL;DR

This paper investigates the fundamental limits of rejection sampling for discrete distributions, introducing new algorithms with sublinear complexity and applying these insights to improve adversarial bandit algorithms.

Contribution

It provides the first minimax analysis of rejection sampling complexity and develops algorithms with sublinear query complexity, enhancing efficiency in sampling and bandit problems.

Findings

Rejection sampling complexity can be reduced to sublinear in alphabet size.

Modified Exp3 algorithm achieves logarithmic per-iteration complexity.

New algorithms outperform classical methods in terms of query efficiency.

Abstract

We consider the task of generating exact samples from a target distribution, known up to normalization, over a finite alphabet. The classical algorithm for this task is rejection sampling, and although it has been used in practice for decades, there is surprisingly little study of its fundamental limitations. In this work, we study the query complexity of rejection sampling in a minimax framework for various classes of discrete distributions. Our results provide new algorithms for sampling whose complexity scales sublinearly with the alphabet size. When applied to adversarial bandits, we show that a slight modification of the Exp3 algorithm reduces the per-iteration complexity from $\mathcal O(K)$ to $\mathcal O(\log^2 K)$, where $K$ is the number of arms.