非对数凹采样与对数分区估计的收敛速率
Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation
AI总结：
研究非对数凹采样和对数分区估计的收敛速率，探讨其与优化问题的关系，发现采样和分区计算的最优速率有时等于或快于优化。
摘要：
从Gibbs分布采样和计算其对数分区函数是统计学、机器学习和统计物理中的基本任务。尽管已知log-concave密度有高效的算法，但非log-concave情况会面临维度灾难。对于许多数值问题，当目标函数光滑时，维度灾难可以缓解，使得收敛速率的指数随可用导数数量线性提升。最近研究表明，通过高效优化算法也能实现类似的快速收敛速率。由于优化可以视为Gibbs分布采样的低温极限，我们探讨非log-concave采样是否也能实现类似的快速收敛速率。我们首先研究采样和对数分区估计问题的信息复杂性，并显示采样和分区计算的最优速率有时等于或快于优化。然后分析各种多项式时间采样算法，包括最近一种有前途的优化方法的扩展，并发现它们有时表现出有趣的行为但没有接近最优的速率。我们的结果还进一步揭示了采样、对数分区和优化问题之间的关系。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
