It is challenging to recover the large-scale internet traffic data purely from the link measurements. With the rapid growth of the problem scale, it will be extremely difficult to sustain the recovery accuracy and the computational cost. In this work, we propose a new Sparsity Low-Rank Recovery (SLRR) model and its Schur Complement Based semi-proximal Alternating Direction Method of Multipliers (SCB-spADMM) solver. Our approach distinguishes itself mainly for the following two aspects. First, we fully exploit the spatial low-rank property and the sparsity of traffic data, which are barely considered in the literature. Our model can be divided into a series of subproblems, which only relate to the traffics in a certain individual time interval. Thus, the model scale is significantly reduced. Second, we establish a globally convergent ADMM-type algorithm inspired by [Li et al., Math. Program., 155(2016)] to solve the SLRR model. In each iteration, all the intermediate variables' optimums can be calculated analytically, which makes the algorithm fast and accurate. Besides, due to the separability of the SLRR model, it is possible to design a parallel algorithm to further reduce computational time. According to the numerical results on the classic datasets Abilene and GEANT, our method achieves the best accuracy with a low computational cost. Moreover, in our newly released large-scale Huawei Origin-Destination (HOD) network traffics, our method perfectly reaches the seconds-level feedback, which meets the essential requirement for practical scenarios.
A new framework for analyzing Fejer convergent algorithms is presented. Using this framework we define a very general class of Fejer convergent algorithms and establish its convergence properties. We also introduce a new definition of approximations of resolvents which preserve some useful features of the exact resolvent, and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method. We show that methods based on families of approximate resolvents fall within the aforementioned class of Fej…
In this paper, we present a semi-proximal alternating direction method of multipliers (ADMM) for solving $3$-block separable convex minimization problems with the second block in the objective being a strongly convex function and one coupled linear equation constraint. By choosing the semi-proximal terms properly, we establish the global convergence of the proposed semi-proximal ADMM for the step-length $τ\in (0, (1+\sqrt{5})/2)$ and the penalty parameter $σ\in (0, +\infty)$. In particular, if $σ>0$ is smaller than a certain threshold and the first and third linear operators in the linear equa…
This paper aims to study a majorized alternating direction method of multipliers with indefinite proximal terms (iPADMM) for convex composite optimization problems. We show that the majorized iPADMM for 2-block convex optimization problems converges globally under weaker conditions than those used in the literature and exhibits a linear convergence rate under a local error bound condition. Based on these, we establish the linear rate convergence results for a symmetric Gaussian-Seidel based majorized iPADMM, which is designed for multi-block composite convex optimization problems. Moreover, we…
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of variational analysis, we establish implementable results on the global convergence of the proposed algorithm as well as its local convergence with superlinear and quadratic rates. For certain structured problems, the obtained local convergence conditions do not require the local Lipschitz continuity of the corresponding Hessian mappings that is a crucial assumptio…
We prove an abstract form of the strong convergence of the Halpern-type and Tikhonov-type proximal point algorithms in CAT(0) spaces. In addition, we derive uniform and computable rates of metastability (in the sense of Tao) for these iterations using proof mining techniques.
This paper is devoted to the design of an efficient and convergent {semi-proximal} alternating direction method of multipliers (ADMM) for finding a solution of low to medium accuracy to convex quadratic conic programming and related problems. For this class of problems, the convergent two block semi-proximal ADMM can be employed to solve their primal form in a straightforward way. However, it is known that it is more efficient to apply the directly extended multi-block semi-proximal ADMM, though its convergence is not guaranteed, to the dual form of these problems. Naturally, one may ask the f…
Nowadays, analysing data from different classes or over a temporal grid has attracted a great deal of interest. As a result, various multiple graphical models for learning a collection of graphical models simultaneously have been derived by introducing sparsity in graphs and similarity across multiple graphs. This paper focuses on the fused multiple graphical Lasso model which encourages not only shared pattern of sparsity, but also shared values of edges across different graphs. For solving this model, we develop an efficient regularized proximal point algorithm, where the subproblem in each …
We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic algorithms, making the resulting framework more scalable facing the ever-increasing problem dimension. Each subproblem is solved inexactly with an explicit and self-adaptive stopping criterion, without requiring to set an a priori target accuracy. When the primal and dual domain are bounded, our method achieves $O(1/\sqrtε)$ and $O(1/ε)$ complexity bound in…
The objective of this paper is to design an efficient and convergent alternating direction method of multipliers (ADMM) for finding a solution of medium accuracy to conic programming problems whose constraints consist of linear equalities, linear inequalities, a non-polyhedral cone and a polyhedral cone. For this class of problems, one may apply the directly extended ADMM to their dual, which can be written in the form of convex programming with four separable blocks in the objective function and a coupling linear equation constraint. Indeed, the directly extended ADMM, though may diverge in t…
Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based proximal decentralized methods are known to converge to the optimal solution with sublinear rates, and it remains unclear whether this family of methods can achieve global linear convergence. To tackle this problem, this work assumes the non-smooth regularization term is common across all networked agents, which is the case for many machine learning problem…
This paper considers an $N$-player stochastic Nash game in which the $i$th player minimizes a composite objective $f_i(x) + r_i(x_i)$, where $f_i$ is expectation-valued and $r_i$ has an efficient prox-evaluation. In this context, we make the following contributions. (i) Under a strong monotonicity assumption on the concatenated gradient map, we derive ({\bf optimal}) rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme; (ii) We overlay (VS-PGR) with a consensus phase with a view towards developing distributed p…
Plug-and-Play (PnP) methods solve ill-posed inverse problems through iterative proximal algorithms by replacing a proximal operator by a denoising operation. When applied with deep neural network denoisers, these methods have shown state-of-the-art visual performance for image restoration problems. However, their theoretical convergence analysis is still incomplete. Most of the existing convergence results consider nonexpansive denoisers, which is non-realistic, or limit their analysis to strongly convex data-fidelity terms in the inverse problem to solve. Recently, it was proposed to train th…
We investigate the proximal point algorithm (PPA) and its inexact extensions under an error bound condition, which guarantees a global linear convergence if the proximal regularization parameter is larger than the error bound condition parameter. We propose an adaptive generalized proximal point algorithm (AGPPA), which adaptively updates the proximal regularization parameters based on some implementable criteria. We show that AGPPA achieves linear convergence without any knowledge of the error bound condition parameter, and the rate only differs from the optimal one by a logarithm term. We ap…
Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Each of its iterations requires the sequential solution of two proximal subproblems. The aim of this work is to present a fully inexact version of Douglas-Rachford method wherein both proximal subproblems are solved approximately within a relative error tolerance. We also present a semi-inexact variant in which the first subproblem is solved exactly and the second one inexactly. We prove that both methods generate sequences weakly convergent to the solution of the underlying inclus…
We introduce a unified algorithmic framework, called proximal-like incremental aggregated gradient (PLIAG) method, for minimizing the sum of a convex function that consists of additive relatively smooth convex components and a proper lower semi-continuous convex regularization function, over an abstract feasible set whose geometry can be captured by using the domain of a Legendre function. The PLIAG method includes many existing algorithms in the literature as special cases such as the proximal gradient method, the Bregman proximal gradient method (also called NoLips algorithm), the incrementa…
University of Utah（犹他大学）；Lawrence Livermore National Lab（劳伦斯利弗莫尔国家实验室）
Many important machine learning applications involve regularized nonconvex bi-level optimization. However, the existing gradient-based bi-level optimization algorithms cannot handle nonconvex or nonsmooth regularizers, and they suffer from a high computation complexity in nonconvex bi-level optimization. In this work, we study a proximal gradient-type algorithm that adopts the approximate implicit differentiation (AID) scheme for nonconvex bi-level optimization with possibly nonconvex and nonsmooth regularizers. In particular, the algorithm applies the Nesterov's momentum to accelerate the com…
In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1/t) convergence rate, wheret denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of m…
In this paper, we first study nonsmooth steepest descent method for nonsmooth functions defined on Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitzian continuity. As one application, the established algorithm is used to search the minimizer of lower semicontinuous and convex functions on finite-dimensional space. The convergent theorem, as one extension and improvement of the existing converging result for twice continuously differentiable convex f…
In this paper we propose a corrected semi-proximal ADMM (alternating direction method of multipliers) for the general $p$-block $(p\!\ge 3)$ convex optimization problems with linear constraints, aiming to resolve the dilemma that almost all the existing modified versions of the directly extended ADMM, although with convergent guarantee, often perform substantially worse than the directly extended ADMM itself with no convergent guarantee. Specifically, in each iteration, we use the multi-block semi-proximal ADMM with step-size at least $1$ as the prediction step to generate a good prediction po…
In this paper, we adapt proximal incremental aggregated gradient methods to saddle point problems, which is motivated by decoupling linear transformations in regularized empirical risk minimization models. First, the Primal-Dual Proximal Incremental Aggregated (PD-PIAG) methods with extrapolations were proposed. We proved that the primal-dual gap of the averaged iteration sequence sublinearly converges to 0, and the iteration sequence converges to some saddle point. Under the strong convexity of $f$ and $h^\ast$, we proved that the iteration sequence linearly converges to the saddle point. The…
Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their generalizations like proximal Newton and quasi-Newton methods. In this paper, we propose a regularized proximal quasi-Newton method whose main features are: (a) the method is globally convergent to stationary points, (b) the globalization is controlled by a regularization parameter, no line search is required, (c) the method can be implemented very efficientl…
We consider the estimation of the state transition matrix in vector autoregressive models, when time sequence data is limited but nonsequence steady-state data is abundant. To leverage both sources of data, we formulate the least squares minimization problem regularized by a Lyapunov penalty. We impose cardinality or rank constraints to reduce the complexity of the autoregressive model. We solve the resulting nonconvex, nonsmooth problem by using the proximal alternating linearization method (PALM). We show that PALM is globally convergent to a critical point and that the estimation error mono…
We consider the problem of recovering a signal from nonlinear transformations, under convex constraints modeling a priori information. Standard feasibility and optimization methods are ill-suited to tackle this problem due to the nonlinearities. We show that, in many common applications, the transformation model can be associated with fixed point equations involving firmly nonexpansive operators. In turn, the recovery problem is reduced to a tractable common fixed point formulation, which is solved efficiently by a provably convergent, block-iterative algorithm. Applications to signal and imag…
We consider the co-design problem of sparse output feedback and row/column-sparse output matrix. A row-sparse (resp. column-sparse) output matrix implies a small number of outputs (resp. sensor measurements). We impose row/column-cardinality constraint on the output matrix and the cardinality constraint on the output feedback gain. The resulting nonconvex, nonsmooth optimal control problem is solved by using the proximal alternating linearization method (PALM). One advantage of PALM is that the proximal operators for sparsity constraints admit closed-form expressions and are easy to implement.…
In this paper we introduce and study a class of structured set-valued operators which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situation…
We introduce and analyze BPALM and A-BPALM, two multi-block proximal alternating linearized minimization algorithms using Bregman distances for solving structured nonconvex problems. The objective function is the sum of a multi-block relatively smooth function (i.e., relatively smooth by fixing all the blocks except one) and block separable (nonsmooth) nonconvex functions. It turns out that the sequences generated by our algorithms are subsequentially convergent to critical points of the objective function, while they are globally convergent under KL inequality assumption. Further, the rate of…
Motivated by the increasing availability of high-performance parallel computing, we design a distributed parallel algorithm for linearly-coupled block-structured nonconvex constrained optimization problems. Our algorithm performs Jacobi-type proximal updates of the augmented Lagrangian function, requiring only local solutions of separable block nonlinear programming (NLP) problems. We provide a cheap and explicitly computable Lyapunov function that allows us to establish global and local sublinear convergence of our algorithm, its iteration complexity, as well as simple, practical and theoreti…
Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms
Stochastic variance reduced optimization methods are known to be globally convergent while they suffer from slow local convergence, especially when moderate or high accuracy is needed. To alleviate this problem, we propose an optimization algorithm -- which we refer to as a hybrid acceleration scheme -- for a class of proximal variance reduced stochastic optimization algorithms. The proposed optimization scheme combines a fast locally convergent algorithm, such as a quasi--Newton method, with a globally convergent variance reduced stochastic algorithm, for instance SAGA or L--SVRG. Our global …
We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.
In the setting of CAT(k) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky-Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.
Proximal algorithms have gained popularity in recent years in large-scale and distributed optimization problems. One such problem is the phase retrieval problem, for which proximal operators have been proposed recently. The phase retrieval problem commonly refers to the task of recovering a target signal based on the magnitude of linear projections of that signal onto known vectors, usually under the presence of noise. A more general problem is the multispectral phase retrieval problem, where sums of these magnitudes are observed instead. In this paper we study the proximal operator for this p…
A Semismooth Newton-Based Proximal Augmented Lagrangian Method for Joint Estimation of Multiple Gaussian Graphical Models with Clustered Structure
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；Center for Applied Mathematics of Fujian Province, School of Mathematics and Statistics, Fuzhou University（福建省应用数学中心，福州大学数学与统计学院）
We propose an accelerated forward-backward method with fast convergence rate for finding a minimizer of a decomposable nonsmooth convex function over a closed convex set, and name it smoothing accelerated proximal gradient (SAPG) algorithm. The proposed algorithm combines the smoothing method with the proximal gradient algorithm with extrapolation $\frac{k-1}{k+α-1}$ and $α>3$. The updating rule of smoothing parameter $μ_k$ is a smart scheme and guarantees the global convergence rate of $o(\ln^σk/k)$ with $σ\in(\frac{1}{2},1]$ on the objective function values. Moreover, we prove that the seque…
We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: $L(x,y)=f(x)+Q(x,y)+g(y)$, where $f:\R^n\rightarrow\R\cup{+\infty}$ and $g:\R^m\rightarrow\R\cup{+\infty}$ are proper lower semicontinuous functions, and $Q:\R^n\times\R^m\rightarrow \R$ is a smooth $C^1$ function which couples the variables $x$ and $y$. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize $L$. We work in a nonconvex setting, just assuming that the function $L$ satisfies the Kurdyka-Łojasiewicz in…
Plug-and-Play methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser. Although Plug-and-Play methods can lead to tremendous visual performance for various image problems, the few existing convergence guarantees are based on unrealistic (or suboptimal) hypotheses on the denoiser, or limited to strongly convex data terms. In this work, we propose a new type of Plug-and-Play methods, based on half-quadratic splitting, for which the denoiser is realized as a gradient descent step on a functional parameterized by a deep…
Shenzhen Research Institute of Big Data（深圳大数据研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Science and Technology of China（中国科学技术大学）；Hong Kong University of Science and Technology（香港科技大学）
This paper considers the decentralized composite optimization problem. We propose a novel decentralized variance-reduction proximal-gradient algorithmic framework, called PMGT-VR, which is based on a combination of several techniques including multi-consensus, gradient tracking, and variance reduction. The proposed framework relies on an imitation of centralized algorithms and we demonstrate that algorithms under this framework achieve convergence rates similar to that of their centralized counterparts. We also describe and analyze two representative algorithms, PMGT-SAGA and PMGT-LSVRG, and c…
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…
The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting alg…
To do dimensionality reduction on the datasets with outliers, the $\ell_1$-norm principal component analysis (L1-PCA) as a typical robust alternative of the conventional PCA has enjoyed great popularity over the past years. In this work, we consider a rotationally invariant L1-PCA, which is hardly studied in the literature. To tackle it, we propose a proximal alternating linearized minimization method with a nonlinear extrapolation for solving its two-block reformulation. Moreover, we show that the proposed method converges at least linearly to a limiting critical point of the reformulated pro…
