Stochastic gradient descent (SGD), which dates back to the work by Robbins and Monro Robbins and Monro [1951a] is widely observed in training modern Deep Neural Networks (DNNs), which are widely used to achieve state-of-the-art results in multiple problem domains like image classification problems Krizhevsky et al. [2017, 2009], object detection Redmon and Farhadi [2017], and classification automatic machine translation Zhang et al. [2015]. The value of the step size (or learning rate) is crucial for the convergence rate of SGD. Selecting an appropriate step size value in each iteration ensures that SGD iterations converge to an optimal solution. If the step size value is too large, it may prevent SGD iterations from reaching the optimal point. Conversely, excessively small step size values can lead to slow convergence or mistakenly identify a local minimum as the optimal solution Mishra and Sarawadekar [2019]. To address these challenges, various schemes have been proposed. One popular approach is the Armijo line search method, initially introduced for SGD by Vaswani et al. Vaswani et al. [2019], which provides theoretical results for strong-convex, convex, and non-convex objective functions.

The sparse factorization of a large matrix is fundamental in modern statistical learning. In particular, the sparse singular value decomposition and its variants have been utilized in multivariate regression, factor analysis, biclustering, vector time series modeling, among others. The appeal of this factorization is owing to its power in discovering a highly-interpretable latent association network, either between samples and variables or between responses and predictors. However, many existing methods are either ad hoc without a general performance guarantee, or are computationally intensive, rendering them unsuitable for large-scale studies. We formulate the statistical problem as a sparse factor regression and tackle it with a divide-and-conquer approach. In the first stage of division, we consider both sequential and parallel approaches for simplifying the task into a set of co-sparse unit-rank estimation (CURE) problems, and establish the statistical underpinnings of these commonly-adopted and yet poorly understood deflation methods. In the second stage of division, we innovate a contended stagewise learning technique, consisting of a sequence of simple incremental updates, to efficiently trace out the whole solution paths of CURE. Our algorithm has a much lower computational complexity than alternating convex search, and the choice of the step size enables a flexible and principled tradeoff between statistical accuracy and computational efficiency. Our work is among the first to enable stagewise learning for non-convex problems, and the idea can be applicable in many multi-convex problems. Extensive simulation studies and an application in genetics demonstrate the effectiveness and scalability of our approach.

Distributed stochastic gradient descent~(DSGD) has been widely used for optimizing large-scale machine learning models, including both convex and non-convex models. With the rapid growth of model size, huge communication cost has been the bottleneck of traditional DSGD. Recently, many communication compression methods have been proposed. Memory-based distributed stochastic gradient descent~(M-DSGD) is one of the efficient methods since each worker communicates a sparse vector in each iteration so that the communication cost is small. Recent works propose the convergence rate of M-DSGD when it adopts vanilla SGD. However, there is still a lack of convergence theory for M-DSGD when it adopts momentum SGD. In this paper, we propose a universal convergence analysis for M-DSGD by introducing \emph{transformation equation}. The transformation equation describes the relation between traditional DSGD and M-DSGD so that we can transform M-DSGD to its corresponding DSGD. Hence we get the convergence rate of M-DSGD with momentum for both convex and non-convex problems. Furthermore, we combine M-DSGD and stagewise learning that the learning rate of M-DSGD in each stage is a constant and is decreased by stage, instead of iteration. Using the transformation equation, we propose the convergence rate of stagewise M-DSGD which bridges the gap between theory and practice.

Although stochastic gradient descent (SGD) method and its variants (e.g., stochastic momentum methods, AdaGrad) are the choice of algorithms for solving non-convex problems (especially deep learning), there still remain big gaps 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 {\bf non-smooth non-convex} (namely weakly 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 {\it increasing} as the stage number. Our theoretical results of stagewise AdaGrad exhibit its adaptive convergence, therefore shed insights on its faster convergence for problems with sparse stochastic gradients than stagewise SGD. 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.