Parallel Sampling via Counting
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911993481396224 |
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| author | Anari, Nima Gao, Ruiquan Rubinstein, Aviad |
| author_facet | Anari, Nima Gao, Ruiquan Rubinstein, Aviad |
| contents | We show how to use parallelization to speed up sampling from an arbitrary distribution $μ$ on a product space $[q]^n$, given oracle access to counting queries: $\mathbb{P}_{X\sim μ}[X_S=σ_S]$ for any $S\subseteq [n]$ and $σ_S \in [q]^S$. 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 $\mathbb{P}_{X\sim μ}[X_i=σ_i\;\vert\; X_S=σ_S]$, 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 $\widetildeΩ(n^{1/3})$ for the runtime of any parallel sampling algorithm making at most $\operatorname{poly}(n)$ queries to the counting oracle, even for $q=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09442 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parallel Sampling via Counting Anari, Nima Gao, Ruiquan Rubinstein, Aviad Data Structures and Algorithms Artificial Intelligence Machine Learning Probability We show how to use parallelization to speed up sampling from an arbitrary distribution $μ$ on a product space $[q]^n$, given oracle access to counting queries: $\mathbb{P}_{X\sim μ}[X_S=σ_S]$ for any $S\subseteq [n]$ and $σ_S \in [q]^S$. 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 $\mathbb{P}_{X\sim μ}[X_i=σ_i\;\vert\; X_S=σ_S]$, 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 $\widetildeΩ(n^{1/3})$ for the runtime of any parallel sampling algorithm making at most $\operatorname{poly}(n)$ queries to the counting oracle, even for $q=2$. |
| title | Parallel Sampling via Counting |
| topic | Data Structures and Algorithms Artificial Intelligence Machine Learning Probability |
| url | https://arxiv.org/abs/2408.09442 |