In this paper, we propose a new algorithm to speed-up the convergence of accelerated proximal gradient (APG) methods. In order to minimize a convex function $f(\mathbf{x})$, our algorithm introduces a simple line search step after each proximal gradient step in APG so that a biconvex function $f(θ\mathbf{x})$ is minimized over scalar variable $θ>0$ while fixing variable $\mathbf{x}$. We propose two new ways of constructing the auxiliary variables in APG based on the intermediate solutions of the proximal gradient and the line search steps. We prove that at arbitrary iteration step $t (t\geq1)$, our algorithm can achieve a smaller upper-bound for the gap between the current and optimal objective values than those in the traditional APG methods such as FISTA, making it converge faster in practice. In fact, our algorithm can be potentially applied to many important convex optimization problems, such as sparse linear regression and kernel SVMs. Our experimental results clearly demonstrate that our algorithm converges faster than APG in all of the applications above, even comparable to some sophisticated solvers.
Zhejiang University City College（浙大城市学院）；Southwest Jiaotong University（西南交通大学）；Zhejiang University（浙江大学）
In this paper, we introduce a proximal-proximal majorization-minimization (PPMM) algorithm for nonconvex tuning-free robust regression problems. The basic idea is to apply the proximal majorization-minimization algorithm to solve the nonconvex problem with the inner subproblems solved by a sparse semismooth Newton (SSN) method based proximal point algorithm (PPA). We must emphasize that the main difficulty in the design of the algorithm lies in how to overcome the singular difficulty of the inner subproblem. Furthermore, we also prove that the PPMM algorithm converges to a d-stationary point. …
We propose a new first-order optimisation algorithm to solve high-dimensional non-smooth composite minimisation problems. Typical examples of such problems have an objective that decomposes into a non-smooth empirical risk part and a non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal Mirror-Prox, leverages the Fenchel-type representation of one part of the objective while handling the other part of the objective via linear minimization over the domain. The algorithm stands in contrast with more classical proximal gradient algorithms with smoothing, which require th…
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 …
In recent years, the Deep Learning Alternating Minimization (DLAM), which is actually the alternating minimization applied to the penalty form of the deep neutral networks training, has been developed as an alternative algorithm to overcome several drawbacks of Stochastic Gradient Descent (SGD) algorithms. This work develops an improved DLAM by the well-known inertial technique, namely iPDLAM, which predicts a point by linearization of current and last iterates. To obtain further training speed, we apply a warm-up technique to the penalty parameter, that is, starting with a small initial one a…
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.
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）
For first-order smooth optimization, the research on the acceleration phenomenon has a long-time history. Until recently, the mechanism leading to acceleration was not successfully uncovered by the gradient correction term and its equivalent implicit-velocity form. Furthermore, based on the high-resolution differential equation framework with the corresponding emerging techniques, phase-space representation and Lyapunov function, the squared gradient norm of Nesterov's accelerated gradient descent (\texttt{NAG}) method at an inverse cubic rate is discovered. However, this result cannot be dire…
In this paper we consider the composite self-concordant (CSC) minimization problem, which minimizes the sum of a self-concordant function $f$ and a (possibly nonsmooth) proper closed convex function $g$. The CSC minimization is the cornerstone of the path-following interior point methods for solving a broad class of convex optimization problems. It has also found numerous applications in machine learning. The proximal damped Newton (PDN) methods have been well studied in the literature for solving this problem that enjoy a nice iteration complexity. Given that at each iteration these methods t…
University of Utah（犹他大学）；The Ohio State University（俄亥俄州立大学）；Duke University（杜克大学）
Various types of parameter restart schemes have been proposed for accelerated gradient algorithms to facilitate their practical convergence in convex optimization. However, the convergence properties of accelerated gradient algorithms under parameter restart remain obscure in nonconvex optimization. In this paper, we propose a novel accelerated proximal gradient algorithm with parameter restart (named APG-restart) for solving nonconvex and nonsmooth problems. Our APG-restart is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global …
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aims to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By leveraging the momentum acceleration technique, we pr…
University of Pittsburgh（匹兹堡大学）；University of Texas at Arlington（德克萨斯大学阿灵顿分校）
In machine learning research, the proximal gradient methods are popular for solving various optimization problems with non-smooth regularization. Inexact proximal gradient methods are extremely important when exactly solving the proximal operator is time-consuming, or the proximal operator does not have an analytic solution. However, existing inexact proximal gradient methods only consider convex problems. The knowledge of inexact proximal gradient methods in the non-convex setting is very limited. % Moreover, for some machine learning models, there is still no proposed solver for exactly solv…
We propose a Projected Proximal Point Algorithm (ProPPA) for solving a class of optimization problems. The algorithm iteratively computes the proximal point of the last estimated solution projected into an affine space which itself is parallel and approaching to the feasible set. We provide convergence analysis theoretically supporting the general algorithm, and then apply it for solving $\ell_1$-minimization problems and the matrix completion problem. These problems arise in many applications including machine learning, image and signal processing. We compare our algorithm with the existing s…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Random Reshuffling (RR), also known as Stochastic Gradient Descent (SGD) without replacement, is a popular and theoretically grounded method for finite-sum minimization. We propose two new algorithms: Proximal and Federated Random Reshuffing (ProxRR and FedRR). The first algorithm, ProxRR, solves composite convex finite-sum minimization problems in which the objective is the sum of a (potentially non-smooth) convex regularizer and an average of $n$ smooth objectives. We obtain the second algorithm, FedRR, as a special case of ProxRR applied to a reformulation of distributed problems with eithe…
Hong Kong University of Science and Technology（香港科技大学）
Nonconvex optimization problems arise in different research fields and arouse lots of attention in signal processing, statistics and machine learning. In this work, we explore the accelerated proximal gradient method and some of its variants which have been shown to converge under nonconvex context recently. We show that a novel variant proposed here, which exploits adaptive momentum and block coordinate update with specific update rules, further improves the performance of a broad class of nonconvex problems. In applications to sparse linear regression with regularizations like Lasso, grouped…
Peking University（北京大学）；Massachusetts Institute of Technology（麻省理工学院）
We contribute to advancing the understanding of Riemannian accelerated gradient methods. In particular, we revisit Accelerated Hybrid Proximal Extragradient(A-HPE), a powerful framework for obtaining Euclidean accelerated methods \citep{monteiro2013accelerated}. Building on A-HPE, we then propose and analyze Riemannian A-HPE. The core of our analysis consists of two key components: (i) a set of new insights into Euclidean A-HPE itself; and (ii) a careful control of metric distortion caused by Riemannian geometry. We illustrate our framework by obtaining a few existing and new Riemannian accele…
We propose an algorithmic framework for convex minimization problems of a composite function with two terms: a self-concordant function and a possibly nonsmooth regularization term. Our method is a new proximal Newton algorithm that features a local quadratic convergence rate. As a specific instance of our framework, we consider the sparse inverse covariance matrix estimation in graph learning problems. Via a careful dual formulation and a novel analytic step-size selection procedure, our approach for graph learning avoids Cholesky decompositions and matrix inversions in its iteration making i…
Department of Mathematics, University of California, Los Angeles, CA 90095（加州大学洛杉矶分校数学系）；Department of Mathematics, University of South Carolina, Columbia, SC 29208（南卡罗来纳大学数学系）
Stanford University（斯坦福大学）；Microsoft Research, New England（微软新英格兰研究院）；MIT（麻省理工学院）
We develop a family of accelerated stochastic algorithms that minimize sums of convex functions. Our algorithms improve upon the fastest running time for empirical risk minimization (ERM), and in particular linear least-squares regression, across a wide range of problem settings. To achieve this, we establish a framework based on the classical proximal point algorithm. Namely, we provide several algorithms that reduce the minimization of a strongly convex function to approximate minimizations of regularizations of the function. Using these results, we accelerate recent fast stochastic algorith…
Many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constraint set affords a self-concordant barrier. We provide a new joint treatment of proximal and self-concordant barrier concepts and illustrate that such problems can be efficiently solved, without the need of lifting the problem dimensions, as in disciplined convex optimization approach. We propose an in…
In this paper, we study the proximal gradient algorithm with extrapolation for minimizing the sum of a Lipschitz differentiable function and a proper closed convex function. Under the error bound condition used in [19] for analyzing the convergence of the proximal gradient algorithm, we show that there exists a threshold such that if the extrapolation coefficients are chosen below this threshold, then the sequence generated converges $R$-linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also $R$-linearly convergent. In addition, the thre…
Zhejiang University City College（浙大城市学院）；Southwest Jiaotong University（西南交通大学）；The Hong Kong Polytechnic University（香港理工大学）；National University of Singapore（新加坡国立大学）
In this paper, we consider high-dimensional nonconvex square-root-loss regression problems and introduce a proximal majorization-minimization (PMM) algorithm for these problems. Our key idea for making the proposed PMM to be efficient is to develop a sparse semismooth Newton method to solve the corresponding subproblems. By using the Kurdyka-Łojasiewicz property exhibited in the underlining problems, we prove that the PMM algorithm converges to a d-stationary point. We also analyze the oracle property of the initial subproblem used in our algorithm. Extensive numerical experiments are presente…
WeBank AI Group（微众银行AI集团）；Tencent Inc.（腾讯公司）；University of California, Davis（加州大学戴维斯分校）；The Hong Kong University of Science and Technology（香港科技大学）
Dual averaging-type methods are widely used in industrial machine learning applications due to their ability to promoting solution structure (e.g., sparsity) efficiently. In this paper, we propose a novel accelerated dual-averaging primal-dual algorithm for minimizing a composite convex function. We also derive a stochastic version of the proposed method which solves empirical risk minimization, and its advantages on handling sparse data are demonstrated both theoretically and empirically.
Xidian University（西安电子科技大学）；The Chinese University of Hong Kong（香港中文大学）
This paper proposes an accelerated proximal stochastic variance reduced gradient (ASVRG) method, in which we design a simple and effective momentum acceleration trick. Unlike most existing accelerated stochastic variance reduction methods such as Katyusha, ASVRG has only one additional variable and one momentum parameter. Thus, ASVRG is much simpler than those methods, and has much lower per-iteration complexity. We prove that ASVRG achieves the best known oracle complexities for both strongly convex and non-strongly convex objectives. In addition, we extend ASVRG to mini-batch and non-smooth …
The Chinese University of Hong Kong（香港中文大学）；Tencent AI Lab（腾讯AI实验室）
In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate $(1-1/\sqrtκ)$ and thus achieves the optimal rate among first-order methods, where $κ$ is the condition number of the problem. Numerical results on linear regression and logistic regression with elastic net regularization show that GeoPG compares favorably with Nesterov's accelerated proximal gradient method, especially whe…
Istituto Italiano di Tecnologia（意大利技术研究院）；MIT（麻省理工学院）；Università di Genova（热那亚大学）
We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual $\ell_1$ and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we de…
The classical non-greedy algorithm (NGA) and the recently proposed proximal alternating minimization method with extrapolation (PAMe) for $L_1$-norm PCA are revisited and their finite-step convergence are studied. It is first shown that NGA can be interpreted as a conditional subgradient or an alternating maximization method. By recognizing it as a conditional subgradient, we prove that the iterative points generated by the algorithm will be constant in finitely many steps under a certain full-rank assumption; such an assumption can be removed when the projection dimension is one. By treating …
Optimization is at the heart of machine learning, statistics and many applied scientific disciplines. It also has a long history in physics, ranging from the minimal action principle to finding ground states of disordered systems such as spin glasses. Proximal algorithms form a class of methods that are broadly applicable and are particularly well-suited to nonsmooth, constrained, large-scale, and distributed optimization problems. There are essentially five proximal algorithms currently known: Forward-backward splitting, Tseng splitting, Douglas-Rachford, alternating direction method of multi…
We propose a new stochastic coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel and PROXimal; this is the first time such a method is proposed. In the special case when the number of processors is equal to the number of coordinates, the method converges at the rate $2\barω\bar{L} R^2/(k+1)^2 $, where $k$ is the iteration counter, $\barω$ is an average degree of separability of the loss function, $\bar{L}$ is the average of Lipschitz constants associated with…
In this paper, we investigate the attractive properties of the proximal gradient algorithm with inertia. Notably, we show that using alternated inertia yields monotonically decreasing functional values, which contrasts with usual accelerated proximal gradient methods. We also provide convergence rates for the algorithm with alternated inertia based on local geometric properties of the objective function. The results are put into perspective by discussions on several extensions and illustrations on common regularized problems.
Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling
University of Edinburgh（爱丁堡大学）；Heriot-Watt University（赫瑞-瓦特大学）；Maxwell Institute for Mathematical Sciences（麦克斯韦数学科学研究所）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
