Abstract
Langevin algorithms are gradient descent methods with additive noise. They have been used for decades in Markov Chain Monte Carlo (MCMC) sampling, optimization, and learning. Their convergence properties for unconstrained non-convex optimization and learning problems have been studied widely in the last few years. Other work has examined projected Langevin algorithms for sampling from log-concave distributions restricted to convex compact sets. For learning and optimization, log-concave distributions correspond to convex losses. In this paper, we analyze the case of non-convex losses with compact convex constraint sets and IID external data variables. We term the resulting method the projected stochastic gradient Langevin algorithm (PSGLA). We show the algorithm achieves a deviation of O(T-1/4(log T)1/2) from its target distribution in 1-Wasserstein distance. For optimization and learning, we show that the algorithm achieves ε-suboptimal solutions, on average, provided that it is run for a time that is polynomial in ε-1 and slightly super-exponential in the problem dimension.
Original language | English (US) |
---|---|
Pages (from-to) | 2891-2937 |
Number of pages | 47 |
Journal | Proceedings of Machine Learning Research |
Volume | 134 |
State | Published - 2021 |
Event | 34th Conference on Learning Theory, COLT 2021 - Boulder, United States Duration: Aug 15 2021 → Aug 19 2021 |
Bibliographical note
Publisher Copyright:© 2021 A. Lamperski.
Keywords
- Langevin Methods
- Markov Chain Monte Carlo Sampling
- Non-Asymptotic Analysis
- Non-Convex Learning
- Stochastic Gradient Algorithms