2021-07-21-lamperski21a.md

title	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning	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 $\epsilon$-suboptimal solutions, on average, provided that it is run for a time that is polynomial in $\epsilon$ and slightly super-exponential in the problem dimension.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	lamperski21a	0	Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning	2891	2937	2891-2937	2891	false	Lamperski, Andrew	given family Andrew Lamperski	2021-07-21		Proceedings of Thirty Fourth Conference on Learning Theory	134	inproceedings	date-parts 2021 7 21	http://proceedings.mlr.press/v134/lamperski21a/lamperski21a.pdf