Sampling can be faster than optimization

• Published in: Proceedings of the National Academy of Sciences - PNAS Vol. 116; no. 42; pp. 20881 - 20885
• Main Authors: Ma, Yi-An, Chen, Yuansi, Jin, Chi, Flammarion, Nicolas, Jordan, Michael I.
• Format: Journal Article
• Language: English
• Published: United States National Academy of Sciences 15.10.2019
• Subjects: Algorithms; Computer applications; Computer simulation; Learning algorithms; Machine learning; Optimization; Optimization algorithms; Physical Sciences; Sampling; Langevin Monte Carlo; nonconvex optimization; computational complexity
• ISSN: 0027-8424; 1091-6490; 1091-6490
• DOI: 10.1073/pnas.1820003116

Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these 2 kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multistable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.