Non-convex Stochastic Composite Optimization with Polyak Momentum
This addresses convergence issues in non-convex optimization for machine learning practitioners using stochastic methods with small batches.
The paper tackles the problem of stochastic proximal gradient methods failing to converge in non-convex settings with significant stochastic noise, and shows that adding Polyak momentum achieves optimal convergence rates for non-convex composite optimization regardless of batch size.
The stochastic proximal gradient method is a powerful generalization of the widely used stochastic gradient descent (SGD) method and has found numerous applications in Machine Learning. However, it is notoriously known that this method fails to converge in non-convex settings where the stochastic noise is significant (i.e. when only small or bounded batch sizes are used). In this paper, we focus on the stochastic proximal gradient method with Polyak momentum. We prove this method attains an optimal convergence rate for non-convex composite optimization problems, regardless of batch size. Additionally, we rigorously analyze the variance reduction effect of the Polyak momentum in the composite optimization setting and we show the method also converges when the proximal step can only be solved inexactly. Finally, we provide numerical experiments to validate our theoretical results.