Notes on the runtime of A* sampling

TL;DR

This paper analyzes the runtime of A* sampling for simulating from a target distribution and shows that under certain unimodality assumptions, the runtime can be significantly reduced from exponential to linear in the Renyi divergence.

Contribution

The paper demonstrates that imposing unimodality conditions on the Radon-Nikodym derivative of distributions on the real line leads to much faster A* sampling runtimes.

Findings

Runtime of A* sampling is exponentially bounded by Renyi divergence.

Under unimodality assumptions, runtime improves to linear in divergence.

Faster sampling is achievable with additional distribution restrictions.

Abstract

The challenge of simulating random variables is a central problem in Statistics and Machine Learning. Given a tractable proposal distribution $P$, from which we can draw exact samples, and a target distribution $Q$ which is absolutely continuous with respect to $P$, the A* sampling algorithm allows simulating exact samples from $Q$, provided we can evaluate the Radon-Nikodym derivative of $Q$ with respect to $P$. Maddison et al. originally showed that for a target distribution $Q$ and proposal distribution $P$, the runtime of A* sampling is upper bounded by $\mathcal{O}(\exp(D_{\infty}[Q||P]))$ where $D_{\infty}[Q||P]$ is the Renyi divergence from $Q$ to $P$. This runtime can be prohibitively large for many cases of practical interest. Here, we show that with additional restrictive assumptions on $Q$ and $P$, we can achieve much faster runtimes. Specifically, we show that if $Q$ and $P$ are distributions on $\mathbb{R}$ and their Radon-Nikodym derivative is unimodal, the runtime of A* sampling is $\mathcal{O}(D_{\infty}[Q||P])$, which is exponentially faster than A* sampling without assumptions.