Title: Universal  Stagewise Learning for Non-Convex Problems with  Convergence on  Averaged Solutions. Abstract: Although stochastic gradient descent (SGD) method and its variants (e.g., stochastic momentum methods, AdaGrad) are algorithms of choice for solving non-convex problems (especially deep learning),  big gaps still remain between the theory and the practice with many questions unresolved. For example, there is still a lack of theories of convergence for SGD and its variants that use stagewise step size and return an averaged solution in practice. In addition, theoretical insights of why adaptive step size of AdaGrad could improve non-adaptive step size of SGD is still missing for non-convex optimization.   This paper aims to address these questions and fill the gap between theory and practice. We propose a universal stagewise optimization framework for a broad family of non-smooth non-convex problems with the following key features: (i) at each stage any suitable stochastic convex optimization algorithms (e.g., SGD  or AdaGrad)  that return an averaged solution can be employed for minimizing a regularized convex problem; (ii) the step size is decreased in  a stagewise manner; (iii)  an averaged solution  is returned as the final solution. % that is selected from all stagewise averaged solutions with sampling probabilities  increasing as the stage number. 
Our theoretical results of stagewise {\ada}  exhibit its adaptive convergence, therefore shed insights on its faster convergence than stagewise SGD  for problems with slowly growing cumulative stochastic gradients. To the best of our knowledge, these new results are the first of their kind for addressing the unresolved issues of existing theories  mentioned earlier. Besides theoretical contributions, our empirical studies show that our stagewise variants of SGD, AdaGrad  improve the generalization performance of existing variants/implementations of SGD and AdaGrad. 