Sampling from Gibbs distributions $p(x) \propto \exp(-V(x)/\varepsilon)$ and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. However, while efficient algorithms are known for convex potentials $V$, the situation is much more difficult in the non-convex case, where algorithms necessarily suffer from the curse of dimensionality in the worst case. For optimization, which can be seen as a low-temperature limit of sampling, it is known that smooth functions $V$ allow faster convergence rates. Specifically, for $m$-times differentiable functions in $d$ dimensions, the optimal rate for algorithms with $n$ function evaluations is known to be $O(n^{-m/d})$, where the constant can potentially depend on $m, d$ and the function to be optimized. Hence, the curse of dimensionality can be alleviated for smooth functions at least in terms of the convergence rate. Recently, it has been shown that similarly fast rates can also be achieved with polynomial runtime $O(n^{3.5})$, where the exponent $3.5$ is independent of $m$ or $d$. Hence, it is natural to ask whether similar rates for sampling and log-partition computation are possible, and whether they can be realized in polynomial time with an exponent independent of $m$ and $d$. We 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 on the relation between sampling, log-partition, and optimization problems.

翻译：从吉布斯分布 $p(x) \propto \exp(-V(x)/\varepsilon)$ 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学中的基础任务。然而，尽管对于凸势函数 $V$ 存在高效算法，但非凸情形下的求解难度显著增加——最坏情况下算法必然面临维度灾难。对于可视为采样低温极限的优化问题，已知光滑函数 $V$ 能实现更快的收敛速率。具体而言，对于 $d$ 维空间中 $m$ 次可微函数，使用 $n$ 次函数评估的算法最优速率可达 $O(n^{-m/d})$，其中常数可能依赖于 $m$、$d$ 及待优化函数。因此，光滑函数至少在收敛速率层面可缓解维度灾难。近期研究表明，此类快速速率可通过多项式时间算法 $O(n^{3.5})$ 实现，其中指数 $3.5$ 与 $m$ 或 $d$ 无关。这自然引发疑问：采样与对数配分计算是否也能达到类似速率，且能否通过指数与 $m$、$d$ 无关的多项式时间算法实现？我们证明：采样与对数配分计算的最优速率有时与优化相同，有时甚至更快。进一步分析多种多项式时间采样算法（包括近期有前景的优化方法的扩展）后发现：这些算法虽偶有有趣特性，但均未达到近最优速率。我们的研究还为采样、对数配分及优化问题之间的关联提供了更深入见解。