并行扩散采样的下界

中文摘要 AI 辅助

标准扩散采样器通过重复评估学习到的得分函数来生成样本。并行采样方法试图通过增加额外评估次数来减少顺序轮次，从而加速生成过程。这引发了一个问题：即使可以同时进行多次得分查询，顺序依赖在多大程度上是不可避免的。我们为具有近似得分的扩散采样建立了首个多项式并行轮次下界。具体而言，我们证明了（1）在$R^d$中对光滑、近各向同性高斯混合采样需要$\tilde{\Omega}(d^{1/3})$轮的下界，以及（2）从单位球内各向异性轴对齐盒子中均匀采样需要$\Omega(d)$轮的下界。这两个下界适用于任意随机算法，这些算法每轮进行多项式次查询，查询位置和噪声水平任意，得分误差为逆多项式，总变差精度为常数。线性下界对我们的盒子族是紧的。我们的构造使用固定的近似得分预言机，这些预言机强制对隐藏信息进行顺序访问，同时在每个噪声水平下满足精度保证。

英文摘要

Standard diffusion samplers generate samples through repeated evaluations of a learned score function. Parallel sampling methods seek to accelerate generation by trading additional evaluations for fewer sequential rounds. This raises the question of how much sequential dependence is unavoidable, even when many score queries can be made simultaneously. We establish the first polynomial parallel-round lower bounds for diffusion sampling with approximate scores. Specifically, we prove (1) a $\widetildeΩ(d^{1/3})$-round lower bound for sampling smooth, near-isotropic Gaussian mixtures in $R^d$, and (2) an $Ω(d)$-round lower bound for uniform sampling from anisotropic axis-aligned boxes contained in the unit ball. Both bounds hold for arbitrary randomized algorithms making polynomially many queries per round at arbitrary locations and noise levels, with inverse-polynomial score error and constant total variation accuracy. The linear bound is tight for our box family. Our constructions use fixed approximate score oracles that enforce sequential access to hidden information while satisfying the accuracy guarantee at every noise level.

发表机构

• UCLA（加州大学洛杉矶分校）

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