This paper considers the problem of supervised learning with linear methods when both features and labels can be corrupted, either in the form of heavy tailed data and/or corrupted rows. We introduce a combination of coordinate gradient descent as a learning algorithm together with robust estimators of the partial derivatives. This leads to robust statistical learning methods that have a numerical complexity nearly identical to non-robust ones based on empirical risk minimization. The main idea is simple: while robust learning with gradient descent requires the computational cost of robustly estimating the whole gradient to update all parameters, a parameter can be updated immediately using a robust estimator of a single partial derivative in coordinate gradient descent. We prove upper bounds on the generalization error of the algorithms derived from this idea, that control both the optimization and statistical errors with and without a strong convexity assumption of the risk. Finally, we propose an efficient implementation of this approach in a new python library called linlearn, and demonstrate through extensive numerical experiments that our approach introduces a new interesting compromise between robustness, statistical performance and numerical efficiency for this problem.
Université Paris-Saclay（巴黎-萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能与替代能源委员会）；University of Genova（热那亚大学）
Acceleration of first order methods is mainly obtained via inertial techniques à la Nesterov, or via nonlinear extrapolation. The latter has known a recent surge of interest, with successful applications to gradient and proximal gradient techniques. On multiple Machine Learning problems, coordinate descent achieves performance significantly superior to full-gradient methods. Speeding up coordinate descent in practice is not easy: inertially accelerated versions of coordinate descent are theoretically accelerated, but might not always lead to practical speed-ups. We propose an accelerated versi…
We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known randomized online coordinate descent algorithms. Furthermore, the proposed algorithms for stochastic optimization exhibit as good convergence rates as the best known randomized coordinate descent algorithms. We also show simulation results to demonstrate performance of the proposed al…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Coordinate descent methods usually minimize a cost function by updating a random decision variable (corresponding to one coordinate) at a time. Ideally, we would update the decision variable that yields the largest decrease in the cost function. However, finding this coordinate would require checking all of them, which would effectively negate the improvement in computational tractability that coordinate descent is intended to afford. To address this, we propose a new adaptive method for selecting a coordinate. First, we find a lower bound on the amount the cost function decreases when a coord…
We demonstrate that distributed block coordinate descent can quickly solve kernel regression and classification problems with millions of data points. Armed with this capability, we conduct a thorough comparison between the full kernel, the Nyström method, and random features on three large classification tasks from various domains. Our results suggest that the Nyström method generally achieves better statistical accuracy than random features, but can require significantly more iterations of optimization. Lastly, we derive new rates for block coordinate descent which support our experimental f…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；ETH Zurich（苏黎世联邦理工学院）；EPFL（洛桑联邦理工学院）
Two popular examples of first-order optimization methods over linear spaces are coordinate descent and matching pursuit algorithms, with their randomized variants. While the former targets the optimization by moving along coordinates, the latter considers a generalized notion of directions. Exploiting the connection between the two algorithms, we present a unified analysis of both, providing affine invariant sublinear $\mathcal{O}(1/t)$ rates on smooth objectives and linear convergence on strongly convex objectives. As a byproduct of our affine invariant analysis of matching pursuit, our rates…
We study the problem of minimizing the sum of a smooth convex function and a convex block-separable regularizer and propose a new randomized coordinate descent method, which we call ALPHA. Our method at every iteration updates a random subset of coordinates, following an arbitrary distribution. No coordinate descent methods capable to handle an arbitrary sampling have been studied in the literature before for this problem. ALPHA is a remarkably flexible algorithm: in special cases, it reduces to deterministic and randomized methods such as gradient descent, coordinate descent, parallel coordin…
Recovering sparse signals from observed data is an important topic in signal/imaging processing, statistics and machine learning. Nonconvex penalized least squares have been attracted a lot of attentions since they enjoy nice statistical properties. Computationally, coordinate descent (CD) is a workhorse for minimizing the nonconvex penalized least squares criterion due to its simplicity and scalability. In this work, we prove the linear convergence rate to CD for solving MCP/SCAD penalized least squares problems.
Univ. Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）；Univ Montpellier（蒙彼利埃大学）；Institut Universitaire de France（法国大学研究院）
Machine learning models can leak information about the data used to train them. To mitigate this issue, Differentially Private (DP) variants of optimization algorithms like Stochastic Gradient Descent (DP-SGD) have been designed to trade-off utility for privacy in Empirical Risk Minimization (ERM) problems. In this paper, we propose Differentially Private proximal Coordinate Descent (DP-CD), a new method to solve composite DP-ERM problems. We derive utility guarantees through a novel theoretical analysis of inexact coordinate descent. Our results show that, thanks to larger step sizes, DP-CD c…
Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室）
By lifting the ReLU function into a higher dimensional space, we develop a smooth multi-convex formulation for training feed-forward deep neural networks (DNNs). This allows us to develop a block coordinate descent (BCD) training algorithm consisting of a sequence of numerically well-behaved convex optimizations. Using ideas from proximal point methods in convex analysis, we prove that this BCD algorithm will converge globally to a stationary point with R-linear convergence rate of order one. In experiments with the MNIST database, DNNs trained with this BCD algorithm consistently yielded bett…
We consider a generic convex-concave saddle point problem with separable structure, a form that covers a wide-ranged machine learning applications. Under this problem structure, we follow the framework of primal-dual updates for saddle point problems, and incorporate stochastic block coordinate descent with adaptive stepsize into this framework. We theoretically show that our proposal of adaptive stepsize potentially achieves a sharper linear convergence rate compared with the existing methods. Additionally, since we can select "mini-batch" of block coordinates to update, our method is also am…
Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other…
Nagoya Institute of Technology（名古屋工业大学）；Kanagawa University（神奈川大学）；Nagoya University（名古屋大学）
This paper provides a block coordinate descent algorithm to solve unconstrained optimization problems. In our algorithm, computation of function values or gradients is not required. Instead, pairwise comparison of function values is used. Our algorithm consists of two steps; one is the direction estimate step and the other is the search step. Both steps require only pairwise comparison of function values, which tells us only the order of function values over two points. In the direction estimate step, a Newton type search direction is estimated. A computation method like block coordinate desce…
The Hong Kong University of Science and Technology（香港科技大学）；Jiangxi Normal University（江西师范大学）；Tencent AI Lab（腾讯人工智能实验室）
Training deep neural networks (DNNs) efficiently is a challenge due to the associated highly nonconvex optimization. The backpropagation (backprop) algorithm has long been the most widely used algorithm for gradient computation of parameters of DNNs and is used along with gradient descent-type algorithms for this optimization task. Recent work have shown the efficiency of block coordinate descent (BCD) type methods empirically for training DNNs. In view of this, we propose a novel algorithm based on the BCD method for training DNNs and provide its global convergence results built upon the powe…
School of Mathematics, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom
We propose an efficient distributed randomized coordinate descent method for minimizing regularized non-strongly convex loss functions. The method attains the optimal $O(1/k^2)$ convergence rate, where $k$ is the iteration counter. The core of the work is the theoretical study of stepsize parameters. We have implemented the method on Archer - the largest supercomputer in the UK - and show that the method is capable of solving a (synthetic) LASSO optimization problem with 50 billion variables.
University of Michigan（密歇根大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Large-scale L1-regularized loss minimization problems arise in high-dimensional applications such as compressed sensing and high-dimensional supervised learning, including classification and regression problems. High-performance algorithms and implementations are critical to efficiently solving these problems. Building upon previous work on coordinate descent algorithms for L1-regularized problems, we introduce a novel family of algorithms called block-greedy coordinate descent that includes, as special cases, several existing algorithms such as SCD, Greedy CD, Shotgun, and Thread-Greedy. We g…
Coordinate descent methods employ random partial updates of decision variables in order to solve huge-scale convex optimization problems. In this work, we introduce new adaptive rules for the random selection of their updates. By adaptive, we mean that our selection rules are based on the dual residual or the primal-dual gap estimates and can change at each iteration. We theoretically characterize the performance of our selection rules and demonstrate improvements over the state-of-the-art, and extend our theory and algorithms to general convex objectives. Numerical evidence with hinge-loss su…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Google AI（谷歌人工智能）；Amazon Search（亚马逊搜索）
Geometric median (\textsc{Gm}) is a classical method in statistics for achieving a robust estimation of the uncorrupted data; under gross corruption, it achieves the optimal breakdown point of 0.5. However, its computational complexity makes it infeasible for robustifying stochastic gradient descent (SGD) for high-dimensional optimization problems. In this paper, we show that by applying \textsc{Gm} to only a judiciously chosen block of coordinates at a time and using a memory mechanism, one can retain the breakdown point of 0.5 for smooth non-convex problems, with non-asymptotic convergence r…
Université de Bourgogne（勃艮第大学）；Institut de Mathématiques de Bourgogne（勃艮第数学研究所）；Université Paris-Saclay（巴黎萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能委员会）；IMAG（格勒诺布尔阿尔卑斯大学综合理工学院（注：IMAG通常指Institut de Mathématiques et d'Informatique de Grenoble，中文常译为“格勒诺布尔数学与信息研究所”，但根据上下文与“Université de Montpellier”关联，此处更准确的常见译法为“格勒诺布尔阿尔卑斯大学数学与信息研究所”，但为简洁，按机构名直译或保留缩写的话，通常Institut de Mathématiques de Grenoble，此处IMAG为其缩写，可译为“格勒诺布尔数学与信息研究所”））；Université de Montpellier（蒙彼利埃大学）；CNRS（法国国家科学研究中心）；IMB（蒙彼利埃大学数学研究所（注：IMB通常指Institut de Mathématiques de Bordeaux，但此处根据上下文与“Université de Bourgogne”关联，应为Institut de Mathématiques de Bourgogne的缩写，即“勃艮第数学研究所”，但原文本中Samuel Vaiter的Affiliation同时有“CNRS and IMB, Université de Bourgogne”，需确认IMB是否为“IMB”，原文本中Samuel Vaiter的Affiliation为“CNRS and IMB, Université de Bourgogne, Dijon, France”，而前面Quentin Klopfenstein的Affiliation是“Université de Bourgogne, Institut de Mathématiques de Bourgogne, Dijon, France”，因此IMB应为“IMB”的独立机构，可能是Institut de Mathématiques de Bourgogne的缩写，即“勃艮第数学研究所”，此处按原文本保留“IMB”，中文译为“勃艮第数学研究所”））
For composite nonsmooth optimization problems, Forward-Backward algorithm achieves model identification (e.g. support identification for the Lasso) after a finite number of iterations, provided the objective function is regular enough. Results concerning coordinate descent are scarcer and model identification has only been shown for specific estimators, the support-vector machine for instance. In this work, we show that cyclic coordinate descent achieves model identification in finite time for a wide class of functions. In addition, we prove explicit local linear convergence rates for coordina…
Algorithms with fast convergence, small number of data access, and low per-iteration complexity are particularly favorable in the big data era, due to the demand for obtaining \emph{highly accurate solutions} to problems with \emph{a large number of samples} in \emph{ultra-high} dimensional space. Existing algorithms lack at least one of these qualities, and thus are inefficient in handling such big data challenge. In this paper, we propose a method enjoying all these merits with an accelerated convergence rate $O(\frac{1}{k^2})$. Empirical studies on large scale datasets with more than one mi…
Coordinate descent with random coordinate selection is the current state of the art for many large scale optimization problems. However, greedy selection of the steepest coordinate on smooth problems can yield convergence rates independent of the dimension $n$, and requiring upto $n$ times fewer iterations. In this paper, we consider greedy updates that are based on subgradients for a class of non-smooth composite problems, which includes $L1$-regularized problems, SVMs and related applications. For these problems we provide (i) the first linear rates of convergence independent of $n$, and sho…
Northwest University（西北大学）；Southwestern University of Finance and Economics（西南财经大学）；University of Birmingham（伯明翰大学）
Randomized coordinate descent (RCD) is a popular optimization algorithm with wide applications in solving various machine learning problems, which motivates a lot of theoretical analysis on its convergence behavior. As a comparison, there is no work studying how the models trained by RCD would generalize to test examples. In this paper, we initialize the generalization analysis of RCD by leveraging the powerful tool of algorithmic stability. We establish argument stability bounds of RCD for both convex and strongly convex objectives, from which we develop optimal generalization bounds by showi…
Difference-of-Convex (DC) minimization, referring to the problem of minimizing the difference of two convex functions, has been found rich applications in statistical learning and studied extensively for decades. However, existing methods are primarily based on multi-stage convex relaxation, only leading to weak optimality of critical points. This paper proposes a coordinate descent method for minimizing a class of DC functions based on sequential nonconvex approximation. Our approach iteratively solves a nonconvex one-dimensional subproblem globally, and it is guaranteed to converge to a coor…
National University of Defense Technology（国防科技大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
The method of block coordinate gradient descent (BCD) has been a powerful method for large-scale optimization. This paper considers the BCD method that successively updates a series of blocks selected according to a Markov chain. This kind of block selection is neither i.i.d. random nor cyclic. On the other hand, it is a natural choice for some applications in distributed optimization and Markov decision process, where i.i.d. random and cyclic selections are either infeasible or very expensive. By applying mixing-time properties of a Markov chain, we prove convergence of Markov chain BCD for m…
Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have high running times and are unsuitable for large-scale problems. Recent work have used convex optimization techniques to obtain very practical algorithms for minimizing functions that are sums of ``simple" functions. In this paper, we use random coordinate descent methods to obtain algorithms with faster linear convergence rates and cheaper iteration costs. C…
Nonconvex and nonsmooth problems have recently attracted considerable attention in machine learning. However, developing efficient methods for the nonconvex and nonsmooth optimization problems with certain performance guarantee remains a challenge. Proximal coordinate descent (PCD) has been widely used for solving optimization problems, but the knowledge of PCD methods in the nonconvex setting is very limited. On the other hand, the asynchronous proximal coordinate descent (APCD) recently have received much attention in order to solve large-scale problems. However, the accelerated variants of …
Shanghai University of Finance and Economics（上海财经大学）
Novel coordinate descent (CD) methods are proposed for minimizing nonconvex functions consisting of three terms: (i) a continuously differentiable term, (ii) a simple convex term, and (iii) a concave and continuous term. First, by extending randomized CD to nonsmooth nonconvex settings, we develop a coordinate subgradient method that randomly updates block-coordinate variables by using block composite subgradient mapping. This method converges asymptotically to critical points with proven sublinear convergence rate for certain optimality measures. Second, we develop a randomly permuted CD meth…
In this paper, we study two general classes of optimization algorithms for kernel methods with convex loss function and quadratic norm regularization, and analyze their convergence. The first approach, based on fixed-point iterations, is simple to implement and analyze, and can be easily parallelized. The second, based on coordinate descent, exploits the structure of additively separable loss functions to compute solutions of line searches in closed form. Instances of these general classes of algorithms are already incorporated into state of the art machine learning software for large scale pr…
EPFL（洛桑联邦理工学院）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎综合理工学院）
We introduce a randomly extrapolated primal-dual coordinate descent method that adapts to sparsity of the data matrix and the favorable structures of the objective function. Our method updates only a subset of primal and dual variables with sparse data, and it uses large step sizes with dense data, retaining the benefits of the specific methods designed for each case. In addition to adapting to sparsity, our method attains fast convergence guarantees in favorable cases \textit{without any modifications}. In particular, we prove linear convergence under metric subregularity, which applies to st…
North China Electric Power University（华北电力大学）；Shandong University of Finance and Economics（山东财经大学）；Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）
Distributed privacy-preserving regression schemes have been developed and extended in various fields, where multiparty collaboratively and privately run optimization algorithms, e.g., Gradient Descent, to learn a set of optimal parameters. However, traditional Gradient-Descent based methods fail to solve problems which contains objective functions with L1 regularization, such as Lasso regression. In this paper, we present Federated Coordinate Descent, a new distributed scheme called FCD, to address this issue securely under multiparty scenarios. Specifically, through secure aggregation and add…
This paper is about randomized iterative algorithms for solving a linear system of equations $X β= y$ in different settings. Recent interest in the topic was reignited when Strohmer and Vershynin (2009) proved the linear convergence rate of a Randomized Kaczmarz (RK) algorithm that works on the rows of $X$ (data points). Following that, Leventhal and Lewis (2010) proved the linear convergence of a Randomized Coordinate Descent (RCD) algorithm that works on the columns of $X$ (features). The aim of this paper is to simplify our understanding of these two algorithms, establish the direct relatio…
Wepresentanovelcolumngenerationbasedboostingmethod for multi-class classification. Our multi-class boosting is formulated in a single optimization problem as in Shen and Hao (2011). Different from most existing multi-class boosting methods, which use the same set of weak learners for all the classes, we train class specified weak learners (i.e., each class has a different set of weak learners). We show that using separate weak learner sets for each class leads to fast convergence, without introducing additional computational overhead in the training procedure. To further make the training more…
Asynchronous parallel implementations for stochastic optimization have received huge successes in theory and practice recently. Asynchronous implementations with lock-free are more efficient than the one with writing or reading lock. In this paper, we focus on a composite objective function consisting of a smooth convex function $f$ and a block separable convex function, which widely exists in machine learning and computer vision. We propose an asynchronous stochastic block coordinate descent algorithm with the accelerated technology of variance reduction (AsySBCDVR), which are with lock-free …
Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being …
We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex non-smooth function, while the denominator part is a convex or concave function. This problem is difficult to solve since it is non-convex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. The proposed methods iteratively solve a one-dimensional subproblem \textit{globally}, and they are guaranteed to converge to coordinate-wise stationary points. In the ca…
In this paper we tackle the challenge of making the stochastic coordinate descent algorithm differentially private. Compared to the classical gradient descent algorithm where updates operate on a single model vector and controlled noise addition to this vector suffices to hide critical information about individuals, stochastic coordinate descent crucially relies on keeping auxiliary information in memory during training. This auxiliary information provides an additional privacy leak and poses the major challenge addressed in this work. Driven by the insight that under independent noise additio…
Institut für Neuroinformatik（神经信息学研究所）；Ruhr-Universität Bochum（波鸿鲁尔大学）；Institut für Mathematik（数学研究所）；Universität Potsdam（波茨坦大学）
Coordinate descent (CD) algorithms have become the method of choice for solving a number of optimization problems in machine learning. They are particularly popular for training linear models, including linear support vector machine classification, LASSO regression, and logistic regression. We consider general CD with non-uniform selection of coordinates. Instead of fixing selection frequencies beforehand we propose an online adaptation mechanism for this important parameter, called the adaptive coordinate frequencies (ACF) method. This mechanism removes the need to estimate optimal coordinate…
Worcester Polytechnic Institute（伍斯特理工学院）；University of California, Davis（加利福尼亚大学戴维斯分校）
In this paper, we study the randomized distributed coordinate descent algorithm with quantized updates. In the literature, the iteration complexity of the randomized distributed coordinate descent algorithm has been characterized under the assumption that machines can exchange updates with an infinite precision. We consider a practical scenario in which the messages exchange occurs over channels with finite capacity, and hence the updates have to be quantized. We derive sufficient conditions on the quantization error such that the algorithm with quantized update still converge. We further veri…
This monograph presents a class of algorithms called coordinate descent algorithms for mathematicians, statisticians, and engineers outside the field of optimization. This particular class of algorithms has recently gained popularity due to their effectiveness in solving large-scale optimization problems in machine learning, compressed sensing, image processing, and computational statistics. Coordinate descent algorithms solve optimization problems by successively minimizing along each coordinate or coordinate hyperplane, which is ideal for parallelized and distributed computing. Avoiding deta…
We propose a new randomized coordinate descent method for a convex optimization template with broad applications. Our analysis relies on a novel combination of four ideas applied to the primal-dual gap function: smoothing, acceleration, homotopy, and coordinate descent with non-uniform sampling. As a result, our method features the first convergence rate guarantees among the coordinate descent methods, that are the best-known under a variety of common structure assumptions on the template. We provide numerical evidence to support the theoretical results with a comparison to state-of-the-art al…
We design a randomised parallel version of Adaboost based on previous studies on parallel coordinate descent. The algorithm uses the fact that the logarithm of the exponential loss is a function with coordinate-wise Lipschitz continuous gradient, in order to define the step lengths. We provide the proof of convergence for this randomised Adaboost algorithm and a theoretical parallelisation speedup factor. We finally provide numerical examples on learning problems of various sizes that show that the algorithm is competitive with concurrent approaches, especially for large scale problems.
An Asymptotically Optimal Coordinate Descent Algorithm for Learning Bayesian Networks from Gaussian Models
Department of Industrial Engineering and Management Sciences Northwestern University（工业工程与管理科学系西北大学）；Department of Biostatistics University of Washington（生物统计学系华盛顿大学）；Department of Statistics University of Washington（统计学系华盛顿大学）
Pacific Northwest National Laboratory（太平洋西北国家实验室）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We present a generic framework for parallel coordinate descent (CD) algorithms that includes, as special cases, the original sequential algorithms Cyclic CD and Stochastic CD, as well as the recent parallel Shotgun algorithm. We introduce two novel parallel algorithms that are also special cases---Thread-Greedy CD and Coloring-Based CD---and give performance measurements for an OpenMP implementation of these.
Rakuten Institute of Technology（乐天技术研究所）；University of California, Merced（加州大学默塞德分校）；University of Utah（犹他大学）
We propose and analyze a block coordinate descent proximal algorithm (BCD-prox) for simultaneous filtering and parameter estimation of ODE models. As we show on ODE systems with up to d=40 dimensions, as compared to state-of-the-art methods, BCD-prox exhibits increased robustness (to noise, parameter initialization, and hyperparameters), decreased training times, and improved accuracy of both filtered states and estimated parameters. We show how BCD-prox can be used with multistep numerical discretizations, and we establish convergence of BCD-prox under hypotheses that include real systems of …
We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. Unlike existing coordinate descent type algorithms, the SNCD updates each regression coefficient and its corresponding subgradient simultaneously in each iteration. It combines the strengths of the coordinate descent and the semismooth Newton algorithm, and effectively solves the computational challenges posed by dimensionality and nonsmoothness. We establish the convergence properties of the algorithm. In addition, we p…
In this paper we develop and analyze Hydra: HYbriD cooRdinAte descent method for solving loss minimization problems with big data. We initially partition the coordinates (features) and assign each partition to a different node of a cluster. At every iteration, each node picks a random subset of the coordinates from those it owns, independently from the other computers, and in parallel computes and applies updates to the selected coordinates based on a simple closed-form formula. We give bounds on the number of iterations sufficient to approximately solve the problem with high probability, and …
Université de Mons（蒙斯大学）；Université de Rennes（雷恩大学）；INRIA（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；IRISA（雷恩第一大学计算机科学研究所）
This paper is concerned with improving the empirical convergence speed of block-coordinate descent algorithms for approximate nonnegative tensor factorization (NTF). We propose an extrapolation strategy in-between block updates, referred to as heuristic extrapolation with restarts (HER). HER significantly accelerates the empirical convergence speed of most existing block-coordinate algorithms for dense NTF, in particular for challenging computational scenarios, while requiring a negligible additional computational budget.
Nonnegative matrix factorization (NMF) has attracted much attention in the last decade as a dimension reduction method in many applications. Due to the explosion in the size of data, naturally the samples are collected and stored distributively in local computational nodes. Thus, there is a growing need to develop algorithms in a distributed memory architecture. We propose a novel distributed algorithm, called \textit{distributed incremental block coordinate descent} (DID), to solve the problem. By adapting the block coordinate descent framework, closed-form update rules are obtained in DID. M…
We consider convex-concave saddle point problems with a separable structure and non-strongly convex functions. We propose an efficient stochastic block coordinate descent method using adaptive primal-dual updates, which enables flexible parallel optimization for large-scale problems. Our method shares the efficiency and flexibility of block coordinate descent methods with the simplicity of primal-dual methods and utilizing the structure of the separable convex-concave saddle point problem. It is capable of solving a wide range of machine learning applications, including robust principal compon…
Solving logistic regression with L1-regularization in distributed settings is an important problem. This problem arises when training dataset is very large and cannot fit the memory of a single machine. We present d-GLMNET, a new algorithm solving logistic regression with L1-regularization in the distributed settings. We empirically show that it is superior over distributed online learning via truncated gradient.
In this paper we propose a novel parallel stochastic coordinate descent (SCD) algorithm with convergence guarantees that exhibits strong scalability. We start by studying a state-of-the-art parallel implementation of SCD and identify scalability as well as system-level performance bottlenecks of the respective implementation. We then take a principled approach to develop a new SCD variant which is designed to avoid the identified system bottlenecks, such as limited scaling due to coherence traffic of model sharing across threads, and inefficient CPU cache accesses. Our proposed system-aware pa…
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…
