Faster Relative Entropy Coding with Greedy Rejection Coding

Relative entropy coding (REC) algorithms encode a sample from a target distribution $Q$ using a proposal distribution $P$ using as few bits as possible. Unlike entropy coding, REC does not assume discrete distributions or require quantisation. As such, it can be naturally integrated into communication pipelines such as learnt compression and differentially private federated learning. Unfortunately, despite their practical benefits, REC algorithms have not seen widespread application, due to their prohibitively slow runtimes or restrictive assumptions. In this paper, we make progress towards addressing these issues. We introduce Greedy Rejection Coding (GRC), which generalises the rejection based-algorithm of Harsha et al. (2007) to arbitrary probability spaces and partitioning schemes. We first show that GRC terminates almost surely and returns unbiased samples from $Q$, after which we focus on two of its variants: GRCS and GRCD. We show that for continuous $Q$ and $P$ over $\mathbb{R}$ with unimodal density ratio $dQ/dP$, the expected runtime of GRCS is upper bounded by $\beta D_{KL}[Q || P] + O(1)$ where $\beta \approx 4.82$, and its expected codelength is optimal. This makes GRCS the first REC algorithm with guaranteed optimal runtime for this class of distributions, up to the multiplicative constant $\beta$. This significantly improves upon the previous state-of-the-art method, A* coding (Flamich et al., 2022). Under the same assumptions, we experimentally observe and conjecture that the expected runtime and codelength of GRCD are upper bounded by $D_{KL}[Q || P] + O(1)$. Finally, we evaluate GRC in a variational autoencoder-based compression pipeline on MNIST, and show that a modified ELBO and an index-compression method can further improve compression efficiency.