Rejection sampling from shape-constrained distributions in sublinear time
This work addresses efficiency issues in sampling for machine learning and optimization, offering incremental improvements to existing methods like the Exp3 algorithm.
The paper tackles the problem of generating exact samples from discrete distributions using rejection sampling, and develops new algorithms with sublinear complexity in alphabet size, achieving a reduction in per-iteration complexity from O(K) to O(log^2 K) for adversarial bandits.
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.