In large-data applications, such as the inference process of diffusion models, it is desirable to design sampling algorithms with a high degree of parallelization. In this work, we study the adaptive complexity of sampling, which is the minimal number of sequential rounds required to achieve sampling given polynomially many queries executed in parallel at each round. For unconstrained sampling, we examine distributions that are log-smooth or log-Lipschitz and log strongly or non-strongly concave. We show that an almost linear iteration algorithm cannot return a sample with a specific exponentially small accuracy under total variation distance. For box-constrained sampling, we show that an almost linear iteration algorithm cannot return a sample with sup-polynomially small accuracy under total variation distance for log-concave distributions. Our proof relies upon novel analysis with the characterization of the output for the hardness potentials based on the chain-like structure with random partition and classical smoothing techniques.

翻译：在大规模数据应用中，例如扩散模型的推断过程，设计具有高度并行化的采样算法是至关重要的。本文研究采样的自适应复杂度，即在每轮并行执行多项式数量查询的条件下，实现采样所需的最小顺序轮数。对于无约束采样，我们考察对数平滑或对数Lipschitz且对数强凹或非强凹的分布。我们证明，在总变差距离下，几乎线性迭代算法无法返回具有特定指数级小精度的样本。对于盒约束采样，我们证明对于对数凹分布，在总变差距离下，几乎线性迭代算法无法返回具有超多项式小精度的样本。我们的证明依赖于基于随机划分链式结构的硬度势函数输出表征分析，并结合经典平滑技术的新颖分析方法。