Sampling from Gibbs distributions and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. While efficient algorithms are known for log-concave densities, the worst-case non-log-concave setting necessarily suffers from the curse of dimensionality. For many numerical problems, the curse of dimensionality can be alleviated when the target function is smooth, allowing the exponent in the rate to improve linearly with the number of available derivatives. Recently, it has been shown that similarly fast convergence rates can be achieved by efficient optimization algorithms. Since optimization can be seen as the low-temperature limit of sampling from Gibbs distributions, we pose the question of whether similarly fast convergence rates can be achieved for non-log-concave sampling. We first study the information-based complexity of the sampling and log-partition estimation problems and show that the optimal rates for sampling and log-partition computation are sometimes equal and sometimes faster than for optimization. We then analyze various polynomial-time sampling algorithms, including an extension of a recent promising optimization approach, and find that they sometimes exhibit interesting behavior but no near-optimal rates. Our results also give further insights into the relation between sampling, log-partition, and optimization problems.

翻译：从吉布斯分布中采样并计算其对数配分函数是统计学、机器学习与统计物理中的基础任务。对于对数凹密度函数，已有高效算法；但在非对数凹最坏情形下，必然面临维度灾难。针对许多数值问题，当目标函数光滑时，维度灾难可得到缓解，使得收敛速率中的指数随可用导数数量线性提升。近期研究表明，高效优化算法同样能实现如此快速的收敛速率。由于优化可视为吉布斯分布采样的低温极限，我们提出如下问题：非对数凹采样是否也能实现类似的快速收敛率？我们首先研究采样与对数配分估计问题的信息复杂度，并证明采样与对数配分计算的最优速率有时与优化速率相等，有时更快。随后分析多种多项式时间采样算法（包括对近期一种有前景优化方法的推广），发现它们虽有时呈现有趣行为，但未达到近最优速率。我们的结果还进一步揭示了采样、对数配分与优化问题之间的关联。