Some Notes on the Sample Complexity of Approximate Channel Simulation
<|reference_start|>Some Notes on the Sample Complexity of Approximate Channel Simulation: 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 $\exp_2\big((D_{KL}[Q \Vert P] + o(1)) \big/ \epsilon\big)$.<|reference_end|>
@article{flamich2024some,
    title={Some Notes on the Sample Complexity of Approximate Channel Simulation},
    author={Gergely Flamich and Lennie Wells},
    journal={arXiv preprint arXiv:2405.04363},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.04363},
    primaryClass={cs.IT cs.LG math.IT}
}