A Decentralized Primal-dual Method for Constrained Minimization of a Strongly Convex Function
Erfan Yazdandoost Hamedani, Necdet Serhat Aybat
备注：
A preliminary result of this paper was presented in arXiv:1706.07907 by Hamedani and Aybat. In this paper, we generalize our results to the setting where agent-specific constraints are defined by nonlinear functions rather than linear ones which greatly improves the modeling capability. This generalization requires a more complicated analysis which is studied in this separate arXiv submission
We propose decentralized primal-dual methods for cooperative multi-agent consensus optimization problems over both static and time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific convex functions over conic constraint sets defined by agent-specific nonlinear functions; hence, the optimal consensus decision should lie in the intersection of these private sets. Under the strong convexity assumption, we provide convergence rates for sub-optimality, infeasibility, and consensus violation in terms of the number of communications required; examine the effect of underlying network topology on the convergence rates.
In this paper, we consider the decentralized dynamic optimization problem defined over a multi-agent network. Each agent possesses a time-varying local objective function, and all agents aim to collaboratively track the drifting global optimal solution that minimizes the summation of all local objective functions. The decentralized dynamic optimization problem can be solved by primal or primal-dual methods, and when the problem degenerates to be static, it has been proved in literature that primal-dual methods are superior to primal ones. This motivates us to ask: are primal-dual methods neces…
In this paper, we develop a chance-constrained decentralized electric vehicle (EV) charging control framework to achieve "valley-filling" meanwhile meeting individual charging requirements and satisfying distribution network constraints. The control design is formulated as an optimization problem with a stochastic non-separable objective function and globally coupled chance constraints. We propose a novel chance-constrained shrunken-primal-dual subgradient (CC-SPDS) algorithm to support the chance-constrained decentralized control scheme and verify its efficacy and convergence with a represent…
We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data sc…
Decentralized optimization is a common paradigm used in distributed signal processing and sensing as well as privacy-preserving and large-scale machine learning. It is assumed that several computational entities locally hold objective functions and are connected by a network. The agents aim to commonly minimize the sum of the local objectives subject by making gradient updates and exchanging information with their immediate neighbors. Theory of decentralized optimization is pretty well-developed in the literature. In particular, it includes lower bounds and optimal algorithms. In this paper, w…
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…
In this work, we introduce ADAPD, $\textbf{A}$ $\textbf{D}$ecentr$\textbf{A}$lized $\textbf{P}$rimal-$\textbf{D}$ual algorithmic framework for solving non-convex and smooth consensus optimization problems over a network of distributed agents. The proposed framework relies on a novel problem formulation that elicits ADMM-type updates, where each agent first inexactly solves a local strongly convex subproblem with any method of its choice and then performs a neighbor communication to update a set of dual variables. We present two variants that allow for a single gradient step for the primal upda…
We consider the problem of minimizing a convex, separable, nonsmooth function subject to linear constraints. The numerical method we propose is a block-coordinate extension of the Chambolle-Pock primal-dual algorithm. We prove convergence of the method without resorting to assumptions like smoothness or strong convexity of the objective, full-rank condition on the matrix, strong duality or even consistency of the linear system. Freedom from imposing the latter assumption permits convergence guarantees for misspecified or noisy systems.
We present a primal-dual majorization-minimization method for solving large-scale linear programs. A smooth barrier augmented Lagrangian (SBAL) function with strict convexity for the dual linear program is derived. The majorization-minimization approach is naturally introduced to develop the smoothness and convexity of the SBAL function. Our method only depends on a factorization of the constant matrix independent of iterations and does not need any computation on step sizes, thus can be expected to be particularly appropriate for large-scale linear programs. The method shares some similar pro…
Many problems arising in image processing and signal recovery with multi-regularization can be formulated as minimization of a sum of three convex separable functions. Typically, the objective function involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function and a nonsmooth function. In this paper, we propose a primal-dual fixed-point (PDFP) scheme to solve the above class of problems. The proposed algorithm for three block problems is a fully splitting symmetric scheme, only involving explicit gradient and linear operators without inner iteration, w…
In this paper we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality constraints. A number of optimization problems in applications can be stated in this form, examples being the entropy-linear programming, the ridge regression, the elastic net, the regularized optimal transport, etc. We extend the Fast Gradient Method applied to the dual problem in order to make it primal-dual so that it allows not only to solve the dual problem, b…
We consider the decentralized convex optimization problem, where multiple agents must cooperatively minimize a cumulative objective function, with each local function expressible as an empirical average of data-dependent losses. State-of-the-art approaches for decentralized optimization rely on gradient tracking, where consensus is enforced via a doubly stochastic mixing matrix. Construction of such mixing matrices is not straightforward and requires coordination even prior to the start of the optimization algorithm. This paper puts forth a primal-dual framework for decentralized stochastic op…
This paper presents a family of algorithms for decentralized convex composite problems. We consider the setting of a network of agents that cooperatively minimize a global objective function composed of a sum of local functions plus a regularizer. Through the use of intermediate consensus variables, we remove the need for inner communication loops between agents when computing curvature-guided updates. A general scheme is presented which unifies the analysis for a plethora of computing choices, including gradient descent, Newton updates, and BFGS updates. Our analysis establishes sublinear con…
We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrate how practical and fully decoupled schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in di…
We analyze a solution method for minimization problems over a space of $\mathbb{R}^d$-valued functions of bounded variation on an interval $I$. The presented method relies on piecewise constant iterates. In each iteration the algorithm alternates between proposing a new point at which the iterate is allowed to be discontinuous and optimizing the magnitude of its jumps as well as the offset. A sublinear $\mathcal{O}(1/k)$ convergence rate for the objective function values is obtained in general settings. Under additional structural assumptions on the dual variable this can be improved to a loca…
We introduce the primal-dual quasi-Newton (PD-QN) method as an approximated second order method for solving decentralized optimization problems. The PD-QN method performs quasi-Newton updates on both the primal and dual variables of the consensus optimization problem to find the optimal point of the augmented Lagrangian. By optimizing the augmented Lagrangian, the PD-QN method is able to find the exact solution to the consensus problem with a linear rate of convergence. We derive fully decentralized quasi-Newton updates that approximate second order information to reduce the computational burd…
We propose a new first-order primal-dual optimization framework for a convex optimization template with broad applications. Our optimization algorithms feature optimal convergence guarantees under a variety of common structure assumptions on the problem template. Our analysis relies on a novel combination of three classic ideas applied to the primal-dual gap function: smoothing, acceleration, and homotopy. The algorithms due to the new approach achieve the best known convergence rate results, in particular when the template consists of only non-smooth functions. We also outline a restart strat…
We consider minimizing the composite function that consists of a strongly convex function and a convex function. The fast dual proximal gradient (FDPG) method decreases the dual function with a rate $O(1/k^2)$, leading to a rate $O(1/k)$ for decreasing the primal function. We propose a generalized FDPG method that guarantees an $O(1/k^{1.5})$ rate for the dual proximal gradient norm decrease. By relating this to the primal function decrease, the proposed approach decreases the primal function with the improved $O(1/k^{1.5})$ rate.
In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures from computational and statistical sides. The statistical focus is estimating the sample size of measures necessary to calculate an approximation for Fréchet mean (barycenter) of a probability distribution with a given precision. For empirical risk minimization approaches, the question of the regularization is also studied together with proposing a new regularization which contributes to the better complexity bounds in comparison with quadratic regularization. The computational focus is developing a…
We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An extrapolation step on the primal variable is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an exten…
We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov's excessive gap technique in a structured fashion and unifies it with smoothing and primal-dual methods. For instance, through the choices of a dual smoothing strategy and a center point, our framework subsumes decomposition algorithms, augmented Lagrangian as well as the alternating direction met…
In this work, we show that for linearly constrained optimization problems the primal-dual hybrid gradient algorithm, analyzed by Chambolle and Pock [3], can be written as an entirely primal algorithm. This allows us to prove convergence of the iterates even in the degenerate cases when the linear system is inconsistent or when the strong duality does not hold. We also obtain new convergence rates which seem to improve existing ones in the literature. For a decentralized distributed optimization we show that the new scheme is much more efficient than the original one.
The Chinese University of Hong Kong（香港中文大学）；Tencent AI Lab（腾讯人工智能实验室）；University of California, Davis（加州大学戴维斯分校）；University of Rochester（罗切斯特大学）
Regularized empirical risk minimization problem with linear predictor appears frequently in machine learning. In this paper, we propose a new stochastic primal-dual method to solve this class of problems. Different from existing methods, our proposed methods only require O(1) operations in each iteration. We also develop a variance-reduction variant of the algorithm that converges linearly. Numerical experiments suggest that our methods are faster than existing ones such as proximal SGD, SVRG and SAGA on high-dimensional problems.
In this paper, we propose a model-free feedback solution method to solve generic constrained optimization problems, without knowing the specific formulations of the objective and constraint functions. This solution method is termed projected primal-dual zeroth-order dynamics (P-PDZD) and is developed based on projected primal-dual gradient dynamics and extremum seeking control. In particular, the P-PDZD method can be interpreted as a model-free controller that autonomously drives an unknown system to the solution of the optimization problem using only output feedback. The P-PDZD can properly h…
We consider strongly convex optimization problems with affine-type restrictions. We build dual problem and solve dual problem by Fast Gradient Method. We use primal-dual structure of this method to construct the solution of the primal problem. The paper contain a lot of different tricks that allows to generalize mentioned above results for almost all methods we would like to choose to solve the dual problem.
In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with asymptotically vanishing damping term. The system is formulated in terms of the augmented Lagrangian associated to the minimization problem. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. In case the objective function has Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically weakly converges to a primal-du…
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.
This paper considers decentralized dynamic optimization problems where nodes of a network try to minimize a sequence of time-varying objective functions in a real-time scheme. At each time slot, nodes have access to different summands of an instantaneous global objective function and they are allowed to exchange information only with their neighbors. This paper develops the application of the Exact Second-Order Method (ESOM) to solve the dynamic optimization problem in a decentralized manner. The proposed dynamic ESOM algorithm operates by primal descending and dual ascending on a quadratic ap…
In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with an asymptotically vanishing damping term. The system under consideration is a time rescaled version of another system previously found in the literature. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. These convergence rates now depend on the rescaling parameter, and thus can be improved by choosing said parameter appropriately. When the …
We consider stochastic convex optimization problems with affine constraints and develop several methods using either primal or dual approach to solve it. In the primal case, we use a special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm to solve the same problems in the non-smooth case. We prove the convergence guarantees in the smoo…
The charging processes of a large number of electric vehicles (EVs) require coordination and control for the alleviation of their impacts on the distribution network and for the provision of various grid services. However, the scalability of existing EV charging control paradigms are limited by either the number of EVs or the distribution network dimension, largely impairing EVs' aggregate service capability and applicability. To overcome the scalability barrier, this paper, motivated by the optimal scheduling problem for the valley-filling service, (1) proposes a novel dimension reduction met…
Consider the problem of minimizing the sum of a smooth convex function and a separable nonsmooth convex function subject to linear coupling constraints. Problems of this form arise in many contemporary applications including signal processing, wireless networking and smart grid provisioning. Motivated by the huge size of these applications, we propose a new class of first order primal-dual algorithms called the block successive upper-bound minimization method of multipliers (BSUM-M) to solve this family of problems. The BSUM-M updates the primal variable blocks successively by minimizing local…
A class of convex constrained minimization problems over polyhedral cones for geometry-dependent quadratic objective functions is considered in a functional analysis framework. Shape differentiability of the primal minimization problem needs a bijective property for mapping of the primal cone. This restrictive assumption is relaxed to bijection of the dual cone within the Lagrangian formulation as a primal-dual minimax problem. In this paper, we give results on primal-dual shape sensitivity analysis that extends the class of shape-differentiable problems supported by explicit formula of the sh…
This work proposes an accelerated primal-dual dynamical system for affine constrained convex optimization and presents a class of primal-dual methods with nonergodic convergence rates. In continuous level, exponential decay of a novel Lyapunov function is established and in discrete level, implicit, semi-implicit and explicit numerical discretizations for the continuous model are considered sequentially and lead to new accelerated primal-dual methods for solving linearly constrained optimization problems. Special structures of the subproblems in those schemes are utilized to develop efficient …
Consider the minimization of a nonconvex differentiable function over a polyhedron. A popular primal-dual first-order method for this problem is to perform a gradient projection iteration for the augmented Lagrangian function and then update the dual multiplier vector using the constraint residual. However, numerical examples show that this approach can exhibit "oscillation" and may not converge. In this paper, we propose a proximal alternating direction method of multipliers for the multi-block version of this problem. A distinctive feature of this method is the introduction of a "smoothed" (…
This paper suggests two novel ideas to develop new proximal variable-metric methods for solving a class of composite convex optimization problems. The first idea is a new parameterization of the optimality condition which allows us to develop a class of homotopy proximal variable-metric methods. We show that under appropriate assumptions such as strong convexity-type and smoothness, or self-concordance, our new schemes can achieve finite global iteration-complexity bounds. Our second idea is a primal-dual-primal framework for proximal-Newton methods which can lead to some useful computational …
SPARKLE: A Unified Single-Loop Primal-Dual Framework for Decentralized Bilevel Optimization
Peking University（北京大学）；IBM Research（IBM研究院）；University of Pennsylvania（宾夕法尼亚大学）；National Engineering Labratory for Big Data Analytics and Applications（国家大数据分析与应用工程实验室）；AI for Science Institute（科学智能研究所）
Kyoto University（京都大学）；RIKEN AIP（理化学研究所人工智能研究中心）；NTT Communication Science Laboratories（NTT通信科学实验室）
In recent years, decentralized learning has emerged as a powerful tool not only for large-scale machine learning, but also for preserving privacy. One of the key challenges in decentralized learning is that the data distribution held by each node is statistically heterogeneous. To address this challenge, the primal-dual algorithm called the Edge-Consensus Learning (ECL) was proposed and was experimentally shown to be robust to the heterogeneity of data distributions. However, the convergence rate of the ECL is provided only when the objective function is convex, and has not been shown in a sta…
We present and analyze a computational hybrid architecture for performing multi-agent optimization. The optimization problems under consideration have convex objective and constraint functions with mild smoothness conditions imposed on them. For such problems, we provide a primal-dual algorithm implemented in the hybrid architecture, which consists of a decentralized network of agents into which centralized information is occasionally injected, and we establish its convergence properties. To accomplish this, a central cloud computer aggregates global information, carries out computations of th…
We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that communication is a major bottleneck in decentralized optimization, our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter-node communications. We first propose a decentralized primal-dual method which can find an $ε$-solution both in terms of functional optimality gap and feasibility residual in $O(1/ε)$ inter-node communication rounds when the objective functions are convex…
We propose a new self-adaptive, double-loop smoothing algorithm to solve composite, nonsmooth, and constrained convex optimization problems. Our algorithm is based on Nesterov's smoothing technique via general Bregman distance functions. It self-adaptively selects the number of iterations in the inner loop to achieve a desired complexity bound without requiring the accuracy a priori as in variants of Augmented Lagrangian methods (ALM). We prove $\BigO{\frac{1}{k}}$-convergence rate on the last iterate of the outer sequence for both unconstrained and constrained settings in contrast to ergodic …
The problem of minimizing the difference of two convex functions is called polyhedral d.c. optimization problem if at least one of the two component functions is polyhedral. We characterize the existence of global optimal solutions of polyhedral d.c. optimization problems. This result is used to show that, whenever the existence of an optimal solution can be certified, polyhedral d.c. optimization problems can be solved by certain concave minimization algorithms. No further assumptions are necessary in case of the first component being polyhedral and just some mild assumptions to the first com…
In this paper we propose a primal-dual homotopy method for $\ell_1$-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem and a respective piecewise constant path for the dual problem. We show that by solving a small linear program, one can jump to the next primal break point and then, solving another small linear program, a new optimal dual solution is calculate…
We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method, thereby providing a systematic way for deriving several well-known decentralized algorithms including EXTRA arXiv:1404.6264 and SSDA arXiv:1702.08704. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower …
In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objec…
This paper presents a canonical dual approach for solving nonconvex quadratic minimization problem. By using the canonical duality theory, nonconvex primal minimization problems over n-dimensional Lorentz cone can be transformed into certain canonical dual problems with only one dual variable, which can be solved by using standard convex minimization methods. Extremality conditions of these solutions are classified by the triality theory. Applications are illustrated.
We develop a novel primal-dual algorithm to solve a class of nonsmooth and nonlinear compositional convex minimization problems, which covers many existing and brand-new models as special cases. Our approach relies on a combination of a new nonconvex potential function, Nesterov's accelerated scheme, and an adaptive parameter updating strategy. Our algorithm is single-loop and has low per-iteration complexity. Under only general convexity and mild assumptions, our algorithm achieves $\mathcal{O}(1/k)$ convergence rates through three different criteria: primal objective residual, dual objective…
In this work, we revisit a classical incremental implementation of the primal-descent dual-ascent gradient method used for the solution of equality constrained optimization problems. We provide a short proof that establishes the linear (exponential) convergence of the algorithm for smooth strongly-convex cost functions and study its relation to the non-incremental implementation. We also study the effect of the augmented Lagrangian penalty term on the performance of distributed optimization algorithms for the minimization of aggregate cost functions over multi-agent networks.
We introduce a novel primal-dual flow for affine constrained convex optimization problems. As a modification of the standard saddle-point system, our primal-dual flow is proved to possess the exponential decay property, in terms of a tailored Lyapunov function. Then two primal-dual methods are obtained from numerical discretizations of the continuous model, and global nonergodic linear convergence rate is established via a discrete Lyapunov function. Instead of solving the subproblem of the primal variable, we apply the semi-smooth Newton iteration to the subproblem with respect to the multipl…
In this paper, we introduce a novel iterative algorithm for the problem of phase-retrieval where the measurements consist of only the magnitude of linear function of the unknown signal, and the noise in the measurements follow Poisson distribution. The proposed algorithm is based on the principle of majorization-minimization (MM); however, the application of MM here is very novel and distinct from the way MM has been usually used to solve optimization problems in the literature. More precisely, we reformulate the original minimization problem into a saddle point problem by invoking Fenchel dua…
