This paper introduces AdaSDCA: an adaptive variant of stochastic dual coordinate ascent (SDCA) for solving the regularized empirical risk minimization problems. Our modification consists in allowing the method adaptively change the probability distribution over the dual variables throughout the iterative process. AdaSDCA achieves provably better complexity bound than SDCA with the best fixed probability distribution, known as importance sampling. However, it is of a theoretical character as it is expensive to implement. We also propose AdaSDCA+: a practical variant which in our experiments outperforms existing non-adaptive methods.
Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked good convergence analysis. This paper presents a new analysis of Stochastic Dual Coordinate Ascent (SDCA) showing that this class of methods enjoy strong theoretical guarantees that are comparable or better than SGD. This analysis justifies the effectiveness of SDCA for practical a…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
The stochastic dual coordinate-ascent (S-DCA) technique is a useful alternative to the traditional stochastic gradient-descent algorithm for solving large-scale optimization problems due to its scalability to large data sets and strong theoretical guarantees. However, the available S-DCA formulation is limited to finite sample sizes and relies on performing multiple passes over the same data. This formulation is not well-suited for online implementations where data keep streaming in. In this work, we develop an {\em online} dual coordinate-ascent (O-DCA) algorithm that is able to respond to st…
ETH Zurich（苏黎世联邦理工学院）；UC Berkeley（加州大学伯克利分校）；Lehigh University（理海大学）
Communication remains the most significant bottleneck in the performance of distributed optimization algorithms for large-scale machine learning. In this paper, we propose a communication-efficient framework, CoCoA, that uses local computation in a primal-dual setting to dramatically reduce the amount of necessary communication. We provide a strong convergence rate analysis for this class of algorithms, as well as experiments on real-world distributed datasets with implementations in Spark. In our experiments, we find that as compared to state-of-the-art mini-batch versions of SGD and SDCA alg…
We introduce a proximal version of dual coordinate ascent method. We demonstrate how the derived algorithmic framework can be used for numerous regularized loss minimization problems, including $\ell_1$ regularization and structured output SVM. The convergence rates we obtain match, and sometimes improve, state-of-the-art results.
We present a coordinate ascent method for a class of semidefinite programming problems that arise in non-convex quadratic integer optimization. These semidefinite programs are characterized by a small total number of active constraints and by low-rank constraint matrices. We exploit this special structure by solving the dual problem, using a barrier method in combination with a coordinate-wise exact line search. The main ingredient of our algorithm is the computationally cheap update at each iteration and an easy computation of the exact step size. Compared to interior point methods, our appro…
We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An extrapolation step on the primal variable is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an exten…
Dual first-order methods are essential techniques for large-scale constrained convex optimization. However, when recovering the primal solutions, we need $T(ε^{-2})$ iterations to achieve an $ε$-optimal primal solution when we apply an algorithm to the non-strongly convex dual problem with $T(ε^{-1})$ iterations to achieve an $ε$-optimal dual solution, where $T(x)$ can be $x$ or $\sqrt{x}$. In this paper, we prove that the iteration complexity of the primal solutions and dual solutions have the same $O\left(\frac{1}{\sqrtε}\right)$ order of magnitude for the accelerated randomized dual coordin…
The University of Edinburgh（爱丁堡大学）；Rutgers University（罗格斯大学）；Baidu Inc（百度公司）
We study the problem of minimizing the average of a large number of smooth convex functions penalized with a strongly convex regularizer. We propose and analyze a novel primal-dual method (Quartz) which at every iteration samples and updates a random subset of the dual variables, chosen according to an arbitrary distribution. In contrast to typical analysis, we directly bound the decrease of the primal-dual error (in expectation), without the need to first analyze the dual error. Depending on the choice of the sampling, we obtain efficient serial, parallel and distributed variants of the metho…
In prior works, stochastic dual coordinate ascent (SDCA) has been parallelized in a multi-core environment where the cores communicate through shared memory, or in a multi-processor distributed memory environment where the processors communicate through message passing. In this paper, we propose a hybrid SDCA framework for multi-core clusters, the most common high performance computing environment that consists of multiple nodes each having multiple cores and its own shared memory. We distribute data across nodes where each node solves a local problem in an asynchronous parallel fashion on its…
The distributed dual ascent is an established algorithm to solve strongly convex multi-agent optimization problems with separable cost functions, in the presence of coupling constraints. In this paper, we study its asynchronous counterpart. Specifically, we assume that each agent only relies on the outdated information received from some neighbors. Differently from the existing randomized and dual block-coordinate schemes, we show convergence under heterogeneous delays, communication and update frequencies. Consequently, our asynchronous dual ascent algorithm can be implemented without requiri…
Innopolis University（Innopolis 大学）；HSE University（俄罗斯国立高等经济大学）；Yandex（Yandex 公司）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）；Mila（米拉研究所）；DIRO, Université de Montréal（蒙特利尔大学 DIRO 学院）；AMS and MINDS, Johns Hopkins University（约翰斯·霍普金斯大学 AMS 与 MINDS 研究中心）
Stochastic Gradient Descent-Ascent (SGDA) is one of the most prominent algorithms for solving min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. The success of the method led to several advanced extensions of the classical SGDA, including variants with arbitrary sampling, variance reduction, coordinate randomization, and distributed variants with compression, which were extensively studied in the literature, especially during the last few years. In this paper, we propose a unified convergence analysis that covers a large variety of sto…
We propose and analyze a new parallel coordinate descent method---`NSync---in which at each iteration a random subset of coordinates is updated, in parallel, allowing for the subsets to be chosen non-uniformly. We derive convergence rates under a strong convexity assumption, and comment on how to assign probabilities to the sets to optimize the bound. The complexity and practical performance of the method can outperform its uniform variant by an order of magnitude. Surprisingly, the strategy of updating a single randomly selected coordinate per iteration---with optimal probabilities---may requ…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；University of Pittsburgh（匹兹堡大学）
In the paper, we propose a class of faster adaptive Gradient Descent Ascent (GDA) methods for solving the nonconvex-strongly-concave minimax problems by using the unified adaptive matrices, which include almost all existing coordinate-wise and global adaptive learning rates. In particular, we provide an effective convergence analysis framework for our adaptive GDA methods. Specifically, we propose a fast Adaptive Gradient Descent Ascent (AdaGDA) method based on the basic momentum technique, which reaches a lower gradient complexity of $\tilde{O}(κ^4ε^{-4})$ for finding an $ε$-stationary point …
Lehigh University（理海大学）；University of Edinburgh（爱丁堡大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
We present an improved analysis of mini-batched stochastic dual coordinate ascent for regularized empirical loss minimization (i.e. SVM and SVM-type objectives). Our analysis allows for flexible sampling schemes, including where data is distribute across machines, and combines a dependence on the smoothness of the loss and/or the data spread (measured through the spectral norm).
A Stochastic Riemannian Alternating Descent Ascent Method for Nonsmooth Composite Expectation Optimization on Riemannian Manifolds
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Nanjing Normal University（南京师范大学）；Beijing University of Posts and Telecommunications（北京邮电大学）；The Chinese University of Hong Kong（香港中文大学）
University of Edinburgh（爱丁堡大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent (SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss.
We consider the problem of choosing design parameters to minimize the probability of an undesired rare event that is described through the average of $n$ iid random variables. Since the probability of interest for near optimal design parameters is very small, one needs to develop suitable accelerated Monte-Carlo methods for estimating the objective function of interest. One of the challenges in the study is that simulating from exponential twists of the laws of the summands may be computationally demanding since these transformed laws may be non-standard and intractable. We consider a setting …
MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
We consider stochastic gradient methods under the interpolation regime where a perfect fit can be obtained (minimum loss at each observation). While previous work highlighted the implicit regularization of such algorithms, we consider an explicit regularization framework as a minimum Bregman divergence convex feasibility problem. Using convex duality, we propose randomized Dykstra-style algorithms based on randomized dual coordinate ascent. For non-accelerated coordinate descent, we obtain an algorithm which bears strong similarities with (non-averaged) stochastic mirror descent on specific fu…
In this paper we develop an adaptive dual free Stochastic Dual Coordinate Ascent (adfSDCA) algorithm for regularized empirical risk minimization problems. This is motivated by the recent work on dual free SDCA of Shalev-Shwartz (2016). The novelty of our approach is that the coordinates to update at each iteration are selected non-uniformly from an adaptive probability distribution, and this extends the previously mentioned work which only allowed for a uniform selection of "dual" coordinates from a fixed probability distribution. We describe an efficient iterative procedure for generating the…
We develop a new randomized iterative algorithm---stochastic dual ascent (SDA)---for finding the projection of a given vector onto the solution space of a linear system. The method is dual in nature: with the dual being a non-strongly concave quadratic maximization problem without constraints. In each iteration of SDA, a dual variable is updated by a carefully chosen point in a subspace spanned by the columns of a random matrix drawn independently from a fixed distribution. The distribution plays the role of a parameter of the method. Our complexity results hold for a wide family of distributi…
Lehigh University（理海大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
Distributed optimization algorithms are essential for training machine learning models on very large-scale datasets. However, they often suffer from communication bottlenecks. Confronting this issue, a communication-efficient primal-dual coordinate ascent framework (CoCoA) and its improved variant CoCoA+ have been proposed, achieving a convergence rate of $\mathcal{O}(1/t)$ for solving empirical risk minimization problems with Lipschitz continuous losses. In this paper, an accelerated variant of CoCoA+ is proposed and shown to possess a convergence rate of $\mathcal{O}(1/t^2)$ in terms of redu…
We study the problem of minimizing the sum of potentially non-differentiable convex cost functions with partially overlapping dependences in an asynchronous manner, where communication in the network is not coordinated. We study the behavior of an asynchronous algorithm based on dual decomposition and block coordinate subgradient methods under assumptions weaker than those used in the literature. At the same time, we allow different agents to use local stepsizes with no global coordination. Sufficient conditions are provided for almost sure convergence to the solution of the optimization probl…
