A unified variance-reduced accelerated gradient method for convex optimization

We propose a novel randomized incremental gradient algorithm, namely, VAriance-Reduced Accelerated Gradient (Varag), for finite-sum optimization. Equipped with a unified step-size policy that adjusts itself to the value of the condition number, Varag exhibits the unified optimal rates of convergence for solving smooth convex finite-sum problems directly regardless of their strong convexity. Moreover, Varag is the first accelerated randomized incremental gradient method that benefits from the strong convexity of the data-fidelity term to achieve the optimal linear convergence. It also establishes an optimal linear rate of convergence for solving a wide class of problems only satisfying a certain error bound condition rather than strong convexity. Varag can also be extended to solve stochastic finite-sum problems.Comment: 33rd Conference on Neural Information Processing Systems (NeurIPS 2019

Delayed Projection Techniques for Linearly Constrained Problems: Convergence Rates, Acceleration, and Applications

In this work, we study a novel class of projection-based algorithms for linearly constrained problems (LCPs) which have a lot of applications in statistics, optimization, and machine learning. Conventional primal gradient-based methods for LCPs call a projection after each (stochastic) gradient descent, resulting in that the required number of projections equals that of gradient descents (or total iterations). Motivated by the recent progress in distributed optimization, we propose the delayed projection technique that calls a projection once for a while, lowering the projection frequency and improving the projection efficiency. Accordingly, we devise a series of stochastic methods for LCPs using the technique, including a variance reduced method and an accelerated one. We theoretically show that it is feasible to improve projection efficiency in both strongly convex and generally convex cases. Our analysis is simple and unified and can be easily extended to other methods using delayed projections. When applying our new algorithms to federated optimization, a newfangled and privacy-preserving subfield in distributed optimization, we obtain not only a variance reduced federated algorithm with convergence rates better than previous works, but also the first accelerated method able to handle data heterogeneity inherent in federated optimization

Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimization

Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of two stochastic momentum methods, namely stochastic heavy-ball method and the stochastic variant of Nesterov's accelerated gradient method. We hope that the basic convergence results developed in this paper can serve the reference to the convergence of stochastic momentum methods and also serve the baselines for comparison in future development of stochastic momentum methods. The novelty of convergence analysis presented in this paper is a unified framework, revealing more insights about the similarities and differences between different stochastic momentum methods and stochastic gradient method. The unified framework exhibits a continuous change from the gradient method to Nesterov's accelerated gradient method and finally the heavy-ball method incurred by a free parameter, which can help explain a similar change observed in the testing error convergence behavior for deep learning. Furthermore, our empirical results for optimizing deep neural networks demonstrate that the stochastic variant of Nesterov's accelerated gradient method achieves a good tradeoff (between speed of convergence in training error and robustness of convergence in testing error) among the three stochastic methods.Comment: Added some references and more empirical result

Estimate Sequences for Variance-Reduced Stochastic Composite Optimization

In this paper, we propose a unified view of gradient-based algorithms for stochastic convex composite optimization by extending the concept of estimate sequence introduced by Nesterov. This point of view covers the stochastic gradient descent method, variants of the approaches SAGA, SVRG, and has several advantages: (i) we provide a generic proof of convergence for the aforementioned methods; (ii) we show that this SVRG variant is adaptive to strong convexity; (iii) we naturally obtain new algorithms with the same guarantees; (iv) we derive generic strategies to make these algorithms robust to stochastic noise, which is useful when data is corrupted by small random perturbations. Finally, we show that this viewpoint is useful to obtain new accelerated algorithms in the sense of Nesterov.Comment: short version of preprint arXiv:1901.0878

Larger is Better: The Effect of Learning Rates Enjoyed by Stochastic Optimization with Progressive Variance Reduction

In this paper, we propose a simple variant of the original stochastic variance reduction gradient (SVRG), where hereafter we refer to as the variance reduced stochastic gradient descent (VR-SGD). Different from the choices of the snapshot point and starting point in SVRG and its proximal variant, Prox-SVRG, the two vectors of each epoch in VR-SGD are set to the average and last iterate of the previous epoch, respectively. This setting allows us to use much larger learning rates or step sizes than SVRG, e.g., 3/(7L) for VR-SGD vs 1/(10L) for SVRG, and also makes our convergence analysis more challenging. In fact, a larger learning rate enjoyed by VR-SGD means that the variance of its stochastic gradient estimator asymptotically approaches zero more rapidly. Unlike common stochastic methods such as SVRG and proximal stochastic methods such as Prox-SVRG, we design two different update rules for smooth and non-smooth objective functions, respectively. In other words, VR-SGD can tackle non-smooth and/or non-strongly convex problems directly without using any reduction techniques such as quadratic regularizers. Moreover, we analyze the convergence properties of VR-SGD for strongly convex problems, which show that VR-SGD attains a linear convergence rate. We also provide the convergence guarantees of VR-SGD for non-strongly convex problems. Experimental results show that the performance of VR-SGD is significantly better than its counterparts, SVRG and Prox-SVRG, and it is also much better than the best known stochastic method, Katyusha.Comment: 36 pages, 10 figures. The simple variant of SVRG is much better than the best-known stochastic method, Katyush

Boosting First-order Methods by Shifting Objective: New Schemes with Faster Worst Case Rates

We propose a new methodology to design first-order methods for unconstrained strongly convex problems, i.e., to design for a shifted objective function. Several technical lemmas are provided as the building blocks for designing new methods. By shifting objective, the analysis is tightened, which leaves space for faster rates, and also simplified. Following this methodology, we derived several new accelerated schemes for problems that equipped with various first-order oracles, and all of the derived methods have faster worst case convergence rates than their existing counterparts. Experiments on machine learning tasks are conducted to evaluate the new methods.Comment: 27 pages, 7 figure

A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent

In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent ({\tt SGD}) which so far have required different intuitions, convergence analyses, have different applications, and which have been developed separately in various communities. We show that our framework includes methods with and without the following tricks, and their combinations: variance reduction, importance sampling, mini-batch sampling, quantization, and coordinate sub-sampling. As a by-product, we obtain the first unified theory of {\tt SGD} and randomized coordinate descent ({\tt RCD}) methods, the first unified theory of variance reduced and non-variance-reduced {\tt SGD} methods, and the first unified theory of quantized and non-quantized methods. A key to our approach is a parametric assumption on the iterates and stochastic gradients. In a single theorem we establish a linear convergence result under this assumption and strong-quasi convexity of the loss function. Whenever we recover an existing method as a special case, our theorem gives the best known complexity result. Our approach can be used to motivate the development of new useful methods, and offers pre-proved convergence guarantees. To illustrate the strength of our approach, we develop five new variants of {\tt SGD}, and through numerical experiments demonstrate some of their properties.Comment: 38 pages, 4 figures, 2 table

The proximal point method revisited

In this short survey, I revisit the role of the proximal point method in large scale optimization. I focus on three recent examples: a proximally guided subgradient method for weakly convex stochastic approximation, the prox-linear algorithm for minimizing compositions of convex functions and smooth maps, and Catalyst generic acceleration for regularized Empirical Risk Minimization.Comment: 11 pages, submitted to SIAG/OPT Views and New

Stochastic variance reduced multiplicative update for nonnegative matrix factorization

Nonnegative matrix factorization (NMF), a dimensionality reduction and factor analysis method, is a special case in which factor matrices have low-rank nonnegative constraints. Considering the stochastic learning in NMF, we specifically address the multiplicative update (MU) rule, which is the most popular, but which has slow convergence property. This present paper introduces on the stochastic MU rule a variance-reduced technique of stochastic gradient. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.Comment: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP2018

Scaling-up Distributed Processing of Data Streams for Machine Learning

Emerging applications of machine learning in numerous areas involve continuous gathering of and learning from streams of data. Real-time incorporation of streaming data into the learned models is essential for improved inference in these applications. Further, these applications often involve data that are either inherently gathered at geographically distributed entities or that are intentionally distributed across multiple machines for memory, computational, and/or privacy reasons. Training of models in this distributed, streaming setting requires solving stochastic optimization problems in a collaborative manner over communication links between the physical entities. When the streaming data rate is high compared to the processing capabilities of compute nodes and/or the rate of the communications links, this poses a challenging question: how can one best leverage the incoming data for distributed training under constraints on computing capabilities and/or communications rate? A large body of research has emerged in recent decades to tackle this and related problems. This paper reviews recently developed methods that focus on large-scale distributed stochastic optimization in the compute- and bandwidth-limited regime, with an emphasis on convergence analysis that explicitly accounts for the mismatch between computation, communication and streaming rates. In particular, it focuses on methods that solve: (i) distributed stochastic convex problems, and (ii) distributed principal component analysis, which is a nonconvex problem with geometric structure that permits global convergence. For such methods, the paper discusses recent advances in terms of distributed algorithmic designs when faced with high-rate streaming data. Further, it reviews guarantees underlying these methods, which show there exist regimes in which systems can learn from distributed, streaming data at order-optimal rates.Comment: 45 pages, 9 figures; preprint of a journal paper published in Proceedings of the IEEE (Special Issue on Optimization for Data-driven Learning and Control