Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning
This work addresses sampling and optimization for non-convex losses with constraints, which is incremental as it extends Langevin algorithms to a more general setting.
The paper tackles the problem of constrained sampling and non-convex learning by analyzing the projected stochastic gradient Langevin algorithm (PSGLA), achieving a deviation of O(T^{-1/4}(log T)^{1/2}) from the target distribution in 1-Wasserstein distance and ε-suboptimal solutions in polynomial time.
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.