Fast Relative Entropy Coding with A* coding

TL;DR

This paper introduces A* based relative entropy coding algorithms that achieve near-optimal compression with efficient runtime, especially for continuous distributions, and demonstrates their effectiveness on image datasets.

Contribution

It presents A* sampling-based REC algorithms with provable runtime and codelength guarantees, and introduces the IKVAE model for improved compression.

Findings

A* REC algorithms have expected runtime proportional to Rènyi divergence.

The algorithms achieve expected codelength close to the KL divergence.

Experimental results show near-optimal lossless image compression.

Abstract

Relative entropy coding (REC) algorithms encode a sample from a target distribution $Q$ using a proposal distribution $P$, such that the expected codelength is $\mathcal{O}(D_{KL}[Q \,||\, P])$. REC can be seamlessly integrated with existing learned compression models since, unlike entropy coding, it does not assume discrete $Q$ or $P$, and does not require quantisation. However, general REC algorithms require an intractable $\Omega(e^{D_{KL}[Q \,||\, P]})$ runtime. We introduce AS* and AD* coding, two REC algorithms based on A* sampling. We prove that, for continuous distributions over $\mathbb{R}$, if the density ratio is unimodal, AS* has $\mathcal{O}(D_{\infty}[Q \,||\, P])$ expected runtime, where $D_{\infty}[Q \,||\, P]$ is the R\'enyi $\infty$-divergence. We provide experimental evidence that AD* also has $\mathcal{O}(D_{\infty}[Q \,||\, P])$ expected runtime. We prove that AS* and AD* achieve an expected codelength of $\mathcal{O}(D_{KL}[Q \,||\, P])$. Further, we introduce DAD*, an approximate algorithm based on AD* which retains its favourable runtime and has bias similar to that of alternative methods. Focusing on VAEs, we propose the IsoKL VAE (IKVAE), which can be used with DAD* to further improve compression efficiency. We evaluate A* coding with (IK)VAEs on MNIST, showing that it can losslessly compress images near the theoretically optimal limit.