Parallel Sampling via Counting

• URL: http://arxiv.org/abs/2408.09442v1
• Date: Sun, 18 Aug 2024 11:42:54 GMT
• Title: Parallel Sampling via Counting
• Authors: Nima Anari, Ruiquan Gao, Aviad Rubinstein,
• Abstract summary: We show how to use parallelization to speed up sampling from an arbitrary distribution. Our algorithm takes $O(n2/3cdot operatornamepolylog(n,q))$ parallel time. Our results have implications for sampling in autoregressive models.
• Score: 3.6854430278041264
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show how to use parallelization to speed up sampling from an arbitrary distribution $\mu$ on a product space $[q]^n$, given oracle access to counting queries: $\mathbb{P}_{X\sim \mu}[X_S=\sigma_S]$ for any $S\subseteq [n]$ and $\sigma_S \in [q]^S$. Our algorithm takes $O({n^{2/3}\cdot \operatorname{polylog}(n,q)})$ parallel time, to the best of our knowledge, the first sublinear in $n$ runtime for arbitrary distributions. Our results have implications for sampling in autoregressive models. Our algorithm directly works with an equivalent oracle that answers conditional marginal queries $\mathbb{P}_{X\sim \mu}[X_i=\sigma_i\;\vert\; X_S=\sigma_S]$, whose role is played by a trained neural network in autoregressive models. This suggests a roughly $n^{1/3}$-factor speedup is possible for sampling in any-order autoregressive models. We complement our positive result by showing a lower bound of $\widetilde{\Omega}(n^{1/3})$ for the runtime of any parallel sampling algorithm making at most $\operatorname{poly}(n)$ queries to the counting oracle, even for $q=2$.

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