This study investigates the dynamics of alternating minimization applied to a bilinear regression task with normally distributed covariates, under the asymptotic system size limit where the number of parameters and observations diverge at the same rate. This is achieved by employing the replica method to a multi-temperature glassy system which unfolds the algorithm's time evolution. Our results show that the dynamics can be described effectively by a two-dimensional discrete stochastic process, where each step depends on all previous time steps, revealing the structure of the memory dependence in the evolution of alternating minimization. The theoretical framework developed in this work can be applied to the analysis of various iterative algorithms, extending beyond the scope of alternating minimization.
We analyze the performance of alternating minimization for loss functions optimized over two variables, where each variable may be restricted to lie in some potentially nonconvex constraint set. This type of setting arises naturally in high-dimensional statistics and signal processing, where the variables often reflect different structures or components within the signals being considered. Our analysis relies on the notion of local concavity coefficients, which has been proposed in Barber and Ha to measure and quantify the concavity of a general nonconvex set. Our results further reveal import…
{We consider alternating minimization procedures for convex optimization problems with variable divided in many block, each block being amenable for minimization with respect to its variable with freezed other variables blocks. In the case of two blocks, we prove a linear convergence rate for alternating minimization procedure under Polyak-Lojasiewicz condition, which can be seen as a relaxation of the strong convexity assumption. Under strong convexity assumption in many-blocks setting we provide an accelerated alternating minimization procedure with linear rate depending on the square root o…
Microsoft Research India（微软印度研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Alternating minimization represents a widely applicable and empirically successful approach for finding low-rank matrices that best fit the given data. For example, for the problem of low-rank matrix completion, this method is believed to be one of the most accurate and efficient, and formed a major component of the winning entry in the Netflix Challenge. In the alternating minimization approach, the low-rank target matrix is written in a bi-linear form, i.e. $X = UV^†$; the algorithm then alternates between finding the best $U$ and the best $V$. Typically, each alternating step in isolation i…
The majority of First Order methods for large-scale convex-concave saddle point problems and variational inequalities with monotone operators are proximal algorithms which at every iteration need to minimize over problem's domain X the sum of a linear form and a strongly convex function. To make such an algorithm practical, X should be proximal-friendly -- admit a strongly convex function with easy to minimize linear perturbations. As a byproduct, X admits a computationally cheap Linear Minimization Oracle (LMO) capable to minimize over X linear forms. There are, however, important situations …
We consider a multimarginal optimal transport, which includes as a particular case the Wasserstein barycenter problem. In this problem one has to find an optimal coupling between $m$ probability measures, which amounts to finding a tensor of the order $m$. We propose an accelerated method based on accelerated alternating minimization and estimate its complexity to find the approximate solution to the problem. We use entropic regularization with sufficiently small regularization parameter and apply accelerated alternating minimization to the dual problem. A novel primal-dual analysis is used to…
This technical report provides detailed theoretical analysis of the algorithm used in \textit{Alternating Minimization Based Trajectory Generation for Quadrotor Aggressive Flight}. An assumption is provided to ensure that settings for the objective function are meaningful. What's more, we explore the structure of the optimization problem and analyze the global/local convergence rate of the employed algorithm.
In this paper we make a first attempt at understanding how to build an optimal approximate normal factor analysis model. The criterion we have chosen to evaluate the distance between different models is the I-divergence between the corresponding normal laws. The algorithm that we propose for the construction of the best approximation is of an the alternating minimization kind.
School of Computer Science（计算机学院）；University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
Min-max formulations have attracted great attention in the ML community due to the rise of deep generative models and adversarial methods, while understanding the dynamics of gradient algorithms for solving such formulations has remained a grand challenge. As a first step, we restrict to bilinear zero-sum games and give a systematic analysis of popular gradient updates, for both simultaneous and alternating versions. We provide exact conditions for their convergence and find the optimal parameter setup and convergence rates. In particular, our results offer formal evidence that alternating upd…
Western Norway University of Applied Sciences（挪威西部应用科技大学）；University of Bergen（卑尔根大学）
The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating minimization of the objective functional whose first iteration is exactly the modified TV-Stokes model proposed earlier. The model is a generalization of the second order Total Generalized Variation model. A convergence analysis is given.
For some typical and widely used non-convex half-quadratic regularization models and the Ambrosio-Tortorelli approximate Mumford-Shah model, based on the Kurdyka-Łojasiewicz analysis and the recent nonconvex proximal algorithms, we developed an efficient preconditioned framework aiming at the linear subproblems that appeared in the nonlinear alternating minimization procedure. Solving large-scale linear subproblems is always important and challenging for lots of alternating minimization algorithms. By cooperating the efficient and classical preconditioned iterations into the nonlinear and nonc…
Microsoft Research（微软研究院）；University of California, Irvine（加州大学欧文分校）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）
We consider the problem of sparse coding, where each sample consists of a sparse linear combination of a set of dictionary atoms, and the task is to learn both the dictionary elements and the mixing coefficients. Alternating minimization is a popular heuristic for sparse coding, where the dictionary and the coefficients are estimated in alternate steps, keeping the other fixed. Typically, the coefficients are estimated via $\ell_1$ minimization, keeping the dictionary fixed, and the dictionary is estimated through least squares, keeping the coefficients fixed. In this paper, we establish local…
In this paper, we propose a novel symmetric alternating minimization algorithm to solve a broad class of total variation (TV) regularization problems. Unlike the usual $z^k\to x^k$ Gauss-Seidel cycle, the proposed algorithm performs the special $\overline{x}^{k}\to z^k\to x^k$ cycle. The main idea for our setting is the recent symmetric Gauss-Seidel (sGS) technique which is developed for solving the multi-block convex composite problem. This idea also enables us to build the equivalence between the proposed method and the well-known accelerated proximal gradient (APG) method. The faster conver…
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 propose NAMA (Newton-type Alternating Minimization Algorithm) for solving structured nonsmooth convex optimization problems where the sum of two functions is to be minimized, one being strongly convex and the other composed with a linear mapping. The proposed algorithm is a line-search method over a continuous, real-valued, exact penalty function for the corresponding dual problem, which is computed by evaluating the augmented Lagrangian at the primal points obtained by alternating minimizations. As a consequence, NAMA relies on exactly the same computations as the classical alternating min…
In this paper we study nonconvex and nonsmooth optimization problems with semi-algebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the Proximal Alternating Linearized Minimization (PALM) algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sp…
The Chinese University of Hong Kong（香港中文大学）；Shenzhen Research Institute of Big Data, the Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）深圳大数据研究院）；University of Minnesota（明尼苏达大学）；Peking University（北京大学）；Shanghai Jiao Tong University（上海交通大学）
The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to lp-norm minimization with two specific values of p, i.e., p=1/2 and p=2/3, we propose two…
In this paper, we consider the problem of minimizing the sum of two convex functions subject to linear linking constraints. The classical alternating direction type methods usually assume that the two convex functions have relatively easy proximal mappings. However, many problems arising from statistics, image processing and other fields have the structure that while one of the two functions has easy proximal mapping, the other function is smoothly convex but does not have an easy proximal mapping. Therefore, the classical alternating direction methods cannot be applied. To deal with the diffi…
In this paper we propose a proximal algorithm for minimizing an objective function of two block variables consisting of three terms: 1) a smooth function, 2) a nonsmooth function which is a composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function, and 3) a smooth function which couples the two block variables. We propose a variable metric composite proximal alternating linearized minimization (CPALM) to solve this class of problems. Building on the powerful Kurdyka-Łojasiewicz property, we derive the convergence analysis and establish that eac…
Block-coordinate descent algorithms and alternating minimization methods are fundamental optimization algorithms and an important primitive in large-scale optimization and machine learning. While various block-coordinate-descent-type methods have been studied extensively, only alternating minimization -- which applies to the setting of only two blocks -- is known to have convergence time that scales independently of the least smooth block. A natural question is then: is the setting of two blocks special? We show that the answer is "no" as long as the least smooth block can be optimized exactly…
This paper considers the problem of phase retrieval, where the goal is to recover a signal $z\in C^n$ from the observations $y_i=|a_i^* z|$, $i=1,2,\cdots,m$. While many algorithms have been proposed, the alternating minimization algorithm has been one of the most commonly used methods, and it is very simple to implement. Current work has proved that when the observation vectors $\{a_i\}_{i=1}^m$ are sampled from a complex Gaussian distribution $N(0, I)$, it recovers the underlying signal with a good initialization when $m=O(n)$, or with random initialization when $m=O(n^2)$, and it conjecture…
The alternating direction method of multipliers (ADMM) is a widely used method for solving many convex minimization models arising in signal and image processing. In this paper, we propose an inertial ADMM for solving a two-block separable convex minimization problem with linear equality constraints. This algorithm is obtained by making use of the inertial Douglas-Rachford splitting algorithm to the corresponding dual of the primal problem. We study the convergence analysis of the proposed algorithm in infinite-dimensional Hilbert spaces. Furthermore, we apply the proposed algorithm on the rob…
National University of Defense Technology（国防科技大学）；Hunan University（湖南大学）
In this paper, we consider a class of nonconvex problems with linear constraints appearing frequently in the area of image processing. We solve this problem by the penalty method and propose the iteratively reweighted alternating minimization algorithm. To speed up the algorithm, we also apply the continuation strategy to the penalty parameter. A convergence result is proved for the algorithm. Compared with the nonconvex ADMM, the proposed algorithm enjoys both theoretical and computational advantages like weaker convergence requirements and faster speed. Numerical results demonstrate the effi…
In this paper, we consider a class of nonsmooth nonconvex optimization problems whose objective is the sum of a block relative smooth function and a proper and lower semicontinuous block separable function. Although the analysis of block proximal gradient (BPG) methods for the class of block $L$-smooth functions have been successfully extended to Bregman BPG methods that deal with the class of block relative smooth functions, accelerated Bregman BPG methods are scarce and challenging to design. Taking our inspiration from Nesterov-type acceleration and the majorization-minimization scheme, we …
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 introduce a new weighted averaging scheme using "Fenchel-type" operators to recover primal solutions in the alternating minimization-type algorithm (AMA) for prototype constrained convex optimization. Our approach combines the classical AMA idea in \cite{Tseng1991} and Nesterov's prox-function smoothing technique without requiring the strong convexity of the objective function. We develop a new non-accelerated primal-dual AMA method and estimate its primal convergence rate both on the objective residual and on the feasibility gap. Then, we incorporate Nesterov's accelerated step into this a…
The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem. This algorithm appears to perform extremely well and is extremely fast even when the given covariance matrix has a very large dimension. The effectiveness of the algorithm is assessed through simulation studies and by applications to three real datasets that are considered as benchmark for the problem. A local convergence analysis of the algorithm is also pres…
The alternating direction method of multipliers (ADMM) is a popular method for solving convex separable minimization problems with linear equality constraints. The generalization of the two-block ADMM to the three-block ADMM is not trivial since the three-block ADMM is not convergence in general. Many variants of three-block ADMM have been developed with guarantee convergence. Besides the ADMM, the alternating minimization algorithm (AMA) is also an important algorithm for solving the convex separable minimization problem with linear equality constraints. The AMA is first proposed by Tseng, an…
We propose an algorithm for the Wireless Sensor Network localization problem, which is based on the well-known algorithmic framework of Alternating Minimization. We start with a non-smooth and non-convex minimization, and transform it into an equivalent smooth and non-convex problem, which stands at the heart of our study. This paves the way to a new method which is globally convergent: not only does the sequence of objective function values converge, but the sequence of the location estimates also converges to a unique location that is a critical point of the corresponding (original) objectiv…
The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector $\tilde{x}\in\mathbb{C}^d$ from a set of $N$ measurements $b_n=|f^*_n\tilde{x}|^2,\ n=1,\cdots, N$, where $\{f_n\}_{n=1}^N$ forms a frame of $\mathbb{C}^d$. %It is generally a non-convex minimization problem, which is NP-hard. Existing algorithms usually use a least squares fitting to the measurements, yielding a quartic polynomial minimization. In this paper, we employ a new strategy by splitting the variables, and we solve a bi-variate optimization pro…
National University of Defense Technology（国防科技大学）；Beihang University（北京航空航天大学）；Imperial College London（帝国理工学院）；University of Modena and Reggio Emilia (UNIMORE)（摩德纳-雷焦艾米利亚大学（UNIMORE））
In this paper, we propose a novel solution for non-convex problems of multiple variables, especially for those typically solved by an alternating minimization (AM) strategy that splits the original optimization problem into a set of sub-problems corresponding to each variable, and then iteratively optimize each sub-problem using a fixed updating rule. However, due to the intrinsic non-convexity of the original optimization problem, the optimization can usually be trapped into spurious local minimum even when each sub-problem can be optimally solved at each iteration. Meanwhile, learning-based …
The alternating minimization (AM) method is a fundamental method for minimizing convex functions whose variable consists of two blocks. How to efficiently solve each subproblems when applying the AM method is the most concerned task. In this paper, we investigate this task and design two new variants of the AM method by borrowing proximal linearized techniques. The first variant is very suitable for the case where half of the subproblems are hard to be solved and the other half can be directly computed. The second variant is designed for parallel computation. Both of them are featured by simpl…
Mean-Field is an efficient way to approximate a posterior distribution in complex graphical models and constitutes the most popular class of Bayesian variational approximation methods. In most applications, the mean field distribution parameters are computed using an alternate coordinate minimization. However, the convergence properties of this algorithm remain unclear. In this paper, we show how, by adding an appropriate penalization term, we can guarantee convergence to a critical point, while keeping a closed form update at each step. A convergence rate estimate can also be derived based on…
In the first part of the paper we consider accelerated first order optimization method for convex functions with $L$-Lipschitz-continuous gradient, that is able to automatically adapts to problems which satisfies Polyak-Łojasiewicz condition or which is strongly convex with the value of parameter equal to $μ$. In these cases method poses linear convergence with factor $\fracμ{L}$, if $μ$ is unknown. If the problem is strongly convex and $μ$ is known, than the method possesses linear convergence with the factor $\sqrt{\fracμ{L}}$. If that are not the cases, the method converges with a rate $O(1…
In this paper, the convergence of alternating minimization is established for non-smooth convex optimization in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth and a non-smooth part is considered with the latter being block-separable, e.g., corresponding to convex constraints or regularization. For the smooth part, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. Linear convergence is established for the first two cases, generalizin…
Alternating minimization methods have recently been proposed as alternatives to the gradient descent for deep neural network optimization. Alternating minimization methods can typically decompose a deep neural network into layerwise subproblems, which can then be optimized in parallel. Despite the significant parallelism, alternating minimization methods are rarely explored in training deep neural networks because of the severe accuracy degradation. In this paper, we analyze the reason and propose to achieve a compelling trade-off between parallelism and accuracy by a reformulation called Tuna…
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 …
With much research has been conducted into trajectory planning for quadrotors, planning with spatial and temporal optimal trajectories in real-time is still challenging. In this paper, we propose a framework for generating large-scale piecewise polynomial trajectories for aggressive autonomous flights, with highlights on its superior computational efficiency and simultaneous spatial-temporal optimality. Exploiting the implicitly decoupled structure of the planning problem, we conduct alternating minimization between boundary conditions and time durations of trajectory pieces. In each minimizat…
In this paper, we propose the inexact alternating minimization algorithm (inexact AMA), which allows inexact iterations in the algorithm, and its accelerated variant, called the inexact fast alternating minimization algorithm (inexact FAMA). We show that inexact AMA and inexact FAMA are equivalent to the inexact proximal-gradient method and its accelerated variant applied to the dual problem. Based on this equivalence, we derive complexity upper-bounds on the number of iterations for the inexact algorithms. We apply inexact AMA and inexact FAMA to distributed optimization problems, with an emp…
The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables. The iterative nature and simplicity of the algorithm has led to its application to many areas such as signal processing, information theory, control, and finance. A general set of sufficient conditions for the convergence and correctness of the algorithm is quite well-known when the underlying problem parameters are fixed. In many practical situations, however, the underlying problem parameters are changing over time, and the use of an adaptiv…
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机研究所）
Alternating minimization (AM) procedures are practically efficient in many applications for solving convex and non-convex optimization problems. On the other hand, Nesterov's accelerated gradient is theoretically optimal first-order method for convex optimization. In this paper we combine AM and Nesterov's acceleration to propose an accelerated alternating minimization algorithm. We prove $1/k^2$ convergence rate in terms of the objective for convex problems and $1/k$ in terms of the squared gradient norm for non-convex problems, where $k$ is the iteration counter. Our method does not require …
This paper examines the problem of state estimation in power distribution systems under low-observability conditions. The recently proposed constrained matrix completion method which combines the standard matrix completion method and power flow constraints has been shown to be effective in estimating voltage phasors under low-observability conditions using single-snapshot information. However, the method requires solving a semidefinite programming (SDP) problem, which becomes computationally infeasible for large systems and if multiple-snapshot (time-series) information is used. This paper pro…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Google Research（谷歌研究院）；University of Washington（华盛顿大学）
Meta-learning synthesizes and leverages the knowledge from a given set of tasks to rapidly learn new tasks using very little data. Meta-learning of linear regression tasks, where the regressors lie in a low-dimensional subspace, is an extensively-studied fundamental problem in this domain. However, existing results either guarantee highly suboptimal estimation errors, or require $Ω(d)$ samples per task (where $d$ is the data dimensionality) thus providing little gain over separately learning each task. In this work, we study a simple alternating minimization method (MLLAM), which alternately l…
Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation
Simons Institute for the Theory of Computing, UC Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；University of Illinois, Chicago（伊利诺伊大学芝加哥分校）；Rice University（莱斯大学）；Massachusetts Institute of Technology（麻省理工学院）
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…
In this paper, we present a network manipulation algorithm based on an alternating minimization scheme from (Nesterov 2020). In our context, the latter mimics the natural behavior of agents and organizations operating on a network. By selecting starting distributions, the organizations determine the short-term dynamics of the network. While choosing an organization in accordance with their manipulation goals, agents are prone to errors. This rational inattentive behavior leads to discrete choice probabilities. We extend the analysis of our algorithm to the inexact case, where the corresponding…
Bridging Identification and Second-Order Acceleration: A Fast Alternating Minimization Framework for Composite Optimization
School of Mathematics, Nanjing University（南京大学数学系）；School of Mathematics, National Key Laboratory for Novel Software Technology, Nanjing University（南京大学数学学院，软件新技术国家重点实验室）
