Convergence of First-Order Methods for Constrained Nonconvex Optimization with Dependent Data

TL;DR

This paper analyzes stochastic projected gradient methods for constrained nonconvex optimization with dependent data, achieving improved convergence rates under mild mixing conditions and extending results to various algorithms and applications.

Contribution

It provides the first convergence analysis for constrained nonconvex optimization with dependent data under mild mixing conditions, improving complexity bounds and extending to multiple algorithms.

Findings

Achieves $ ilde{O}(t^{-1/4})$ convergence rate.

Reduces complexity from $ ilde{O}( ext{} ext{varepsilon}^{-8})$ to $ ilde{O}( ext{} ext{varepsilon}^{-4})$.

Extends results to stochastic proximal gradient, AdaGrad, and heavy ball momentum.

Abstract

We focus on analyzing the classical stochastic projected gradient methods under a general dependent data sampling scheme for constrained smooth nonconvex optimization. We show the worst-case rate of convergence $\tilde{O}(t^{-1/4})$ and complexity $\tilde{O}(\varepsilon^{-4})$ for achieving an $\varepsilon$-near stationary point in terms of the norm of the gradient of Moreau envelope and gradient mapping. While classical convergence guarantee requires i.i.d. data sampling from the target distribution, we only require a mild mixing condition of the conditional distribution, which holds for a wide class of Markov chain sampling algorithms. This improves the existing complexity for the constrained smooth nonconvex optimization with dependent data from $\tilde{O}(\varepsilon^{-8})$ to $\tilde{O}(\varepsilon^{-4})$ with a significantly simpler analysis. We illustrate the generality of our approach by deriving convergence results with dependent data for stochastic proximal gradient methods, adaptive stochastic gradient algorithm AdaGrad and stochastic gradient algorithm with heavy ball momentum. As an application, we obtain first online nonnegative matrix factorization algorithms for dependent data based on stochastic projected gradient methods with adaptive step sizes and optimal rate of convergence.