Stagewise training strategy is widely used for learning neural networks, which runs a stochastic algorithm (e.g., SGD) starting with a relatively large step size (aka learning rate) and geometrically decreasing the step size after a number of iterations. It has been observed that the stagewise SGD has much faster convergence than the vanilla SGD with a polynomially decaying step size in terms of both training error and testing error. But how to explain this phenomenon has been largely ignored by existing studies. This paper provides some theoretical evidence for explaining this faster convergence. In particular, we consider a stagewise training strategy for minimizing empirical risk that satisfies the Polyak-Łojasiewicz (PL) condition, which has been observed/proved for neural networks and also holds for a broad family of convex functions. For convex loss functions and two classes of "nice-behaved" non-convex objectives that are close to a convex function, we establish faster convergence of stagewise training than the vanilla SGD under the PL condition on both training error and testing error. Experiments on stagewise learning of deep neural networks exhibits that it satisfies one type of non-convexity assumption and therefore can be explained by our theory.

In this paper, we develop a novel homotopy smoothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving such non-smooth optimization problems is O(1/ɛ) without any assumption on the strong convexity.