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_∞[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 DKL[Q Vert P], we can exploit global-bound, depth-limited A* coding to ensure TV[Q Vert P] ≤ ε and maintain optimal coding performance with a sample complexity of only exp2big((DKL[Q Vert P] + o(1)) big/ εbig).