Some Notes on the Sample Complexity of Approximate Channel Simulation

• URL: http://arxiv.org/abs/2405.04363v2
• Date: Tue, 14 May 2024 08:02:36 GMT
• Title: Some Notes on the Sample Complexity of Approximate Channel Simulation
• Authors: Gergely Flamich, Lennie Wells,
• Abstract summary: Channel simulation algorithms can efficiently encode random samples from a prescribed target distribution $Q$ and find applications in machine learning-based lossy data compression. This paper considers approximate schemes with a fixed runtime instead. We exploit global-bound, depth-limited A* coding to ensure $mathrmTV[Q Vert P] leq epsilon$ and maintain optimal coding performance with a sample complexity of only $expbig((D_KL[Q Vert P] + o(1)) big/ epsilonbig
• Score: 2.4554686192257424
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Channel simulation algorithms can efficiently encode random samples from a prescribed target distribution $Q$ and find applications in machine learning-based lossy data compression. However, algorithms that encode exact samples usually have random runtime, limiting their applicability when a consistent encoding time is desirable. Thus, this paper considers approximate schemes with a fixed runtime instead. First, we strengthen a result of Agustsson and Theis and show that there is a class of pairs of target distribution $Q$ and coding distribution $P$, for which the runtime of any approximate scheme scales at least super-polynomially in $D_\infty[Q \Vert P]$. We then show, by contrast, that if we have access to an unnormalised Radon-Nikodym derivative $r \propto dQ/dP$ and knowledge of $D_{KL}[Q \Vert P]$, we can exploit global-bound, depth-limited A* coding to ensure $\mathrm{TV}[Q \Vert P] \leq \epsilon$ and maintain optimal coding performance with a sample complexity of only $\exp_2\big((D_{KL}[Q \Vert P] + o(1)) \big/ \epsilon\big)$.

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