Convergence of First-Order Methods for Constrained Nonconvex Optimization with Dependent Data
This work addresses the challenge of optimization with dependent data for researchers in machine learning and optimization, providing improved theoretical guarantees for practical algorithms like AdaGrad and momentum methods, though it is incremental as it builds on existing methods with new analysis.
The paper tackles the problem of analyzing stochastic projected gradient methods for constrained nonconvex optimization with dependent data, showing a worst-case convergence rate of $ ilde{O}(t^{-1/4})$ and complexity of $ ilde{O}(\varepsilon^{-4})$ for achieving an $\varepsilon$-near stationary point, improving from $ ilde{O}(\varepsilon^{-8})$ with a simpler analysis.
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.