Computational Separations between Sampling and Optimization
This work addresses foundational computational complexity issues in machine learning, providing theoretical insights into when sampling or optimization is harder, which is incremental but clarifies prior results.
The paper tackles the computational relationship between sampling and optimization in Bayesian learning, showing that they are provably incomparable with families of functions where one is easy while the other is NP-hard, and it identifies sharp phase transitions in sampling complexity with temperature changes.
Two commonly arising computational tasks in Bayesian learning are Optimization (Maximum A Posteriori estimation) and Sampling (from the posterior distribution). In the convex case these two problems are efficiently reducible to each other. Recent work (Ma et al. 2019) shows that in the non-convex case, sampling can sometimes be provably faster. We present a simpler and stronger separation. We then compare sampling and optimization in more detail and show that they are provably incomparable: there are families of continuous functions for which optimization is easy but sampling is NP-hard, and vice versa. Further, we show function families that exhibit a sharp phase transition in the computational complexity of sampling, as one varies the natural temperature parameter. Our results draw on a connection to analogous separations in the discrete setting which are well-studied.