Some Notes on the Sample Complexity of Approximate Channel Simulation

Gergely Flamich, Lennie Wells

Keywords: channel simulation, relative entropy coding, approximate sampling, sample complexity

TL;DR: We study, under various computational settings, how many proposal samples channel simulation algorithms need to examine before they can output a good approximate sample from the target distribution.

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 $TV[Q \Vert P] \leq \epsilon$ and maintain optimal coding performance with a sample complexity of only $\exp_2\big((D_{KL}[Q \Vert P] + o(1)) \big/ \epsilon\big)$.

Submission Number: 12