Fast Relative Entropy Coding with A* coding
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Relative entropy coding (REC) algorithms encode a sample from a target distribution Q using a proposal distribution P, such that the expected codelength is 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 Ω(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 ℝ, if the density ratio is unimodal, AS* has 𝒪(D_∞[Q || P]) expected runtime, where D_∞[Q || P] is the Rényi ∞-divergence. We provide experimental evidence that AD* also has 𝒪(D_∞[Q || P]) expected runtime. We prove that AS* and AD* achieve an expected codelength of 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.
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