Variance-Reduced Proximal Stochastic Gradient Descent for Non-convex Composite optimization
This work addresses optimization challenges for machine learning practitioners dealing with non-convex problems, though it is incremental as it adapts existing variance reduction techniques to a new setting.
The paper tackles non-convex composite optimization by extending variance-reduced stochastic methods like prox-SVRG and prox-SAGA to this setting, proving they converge to a stationary point within O(1/ε) iterations, matching state-of-the-art rates and outperforming stochastic gradient descent.
Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend this problem into the non-convex setting using variance reduction techniques, such as prox-SVRG and prox-SAGA. We prove that, with a constant step size, both prox-SVRG and prox-SAGA are suitable for non-convex composite optimization, and help the problem converge to a stationary point within $O(1/ε)$ iterations. That is similar to the convergence rate seen with the state-of-the-art RSAG method and faster than stochastic gradient descent. Our analysis is also extended into the min-batch setting, which linearly accelerates the convergence. To the best of our knowledge, this is the first analysis of convergence rate of variance-reduced proximal stochastic gradient for non-convex composite optimization.