Convergence results of a nested decentralized gradient method for non-strongly convex problems
Woocheol Choi, Doheon Kim, Seok-Bae Yun
备注：
28 pages, Typos were fixed and new convergence results were added for the case when the communication number is constant, to appear in J. Optim. Theory Appl
We are concerned with the convergence of NEAR-DGD$^+$ (Nested Exact Alternating Recursion Distributed Gradient Descent) method introduced to solve the distributed optimization problems. Under the assumption of the strong convexity of local objective functions and the Lipschitz continuity of their gradients, the linear convergence is established in \cite{BBKW - Near DGD}. In this paper, we investigate the convergence property of NEAR-DGD$^+$ in the absence of strong convexity. More precisely, we establish the convergence results in the following two cases: (1) When only the convexity is assumed on the objective function. (2) When the objective function is represented as a composite function of a strongly convex function and a rank deficient matrix, which falls into the class of convex and quasi-strongly convex functions. Numerical results are provided to support the convergence results.
In this paper, we investigate a decentralized control problem with nested subsystems, which is a general model for one-directional communication amongst many subsystems. The noises in our dynamics are modelled as uncertain variables which take values in finite sets. The objective is to minimize a worst-case shared cost. We demonstrate how the prescription approach can simplify the information structure and derive a structural form for optimal control strategies. The structural form allows us to restrict attention to control strategies whose domains do not grow in size with time, and thus, this…
In this paper, we investigate a decentralized stochastic control problem with two agents, where a part of the memory of the second agent is also available to the first agent at each instance of time. We derive a structural form for optimal control strategies which allows us to restrict their domain to a set which does not grow in size with time. We also present a dynamic programming (DP) decomposition which can utilize our results to derive optimal strategies for arbitrarily long time horizons. Since obtaining optimal control strategies by solving this DP decomposition is computationally inten…
In this paper, we consider minimizing a sum of local convex objective functions in a distributed setting, where the cost of communication and/or computation can be expensive. We extend and generalize the analysis for a class of nested gradient-based distributed algorithms (NEAR-DGD; Berahas, Bollapragada, Keskar and Wei, 2018) to account for multiple gradient steps at every iteration. We show the effect of performing multiple gradient steps on the rate of convergence and on the size of the neighborhood of convergence, and prove R-Linear convergence to the exact solution with a fixed number of …
University of Notre Dame（圣母大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We study the problem of control policy design for decentralized state-feedback linear quadratic control with a partially nested information structure, when the system model is unknown. We propose a model-based learning solution, which consists of two steps. First, we estimate the unknown system model from a single system trajectory of finite length, using least squares estimation. Next, based on the estimated system model, we design a control policy that satisfies the desired information structure. We show that the suboptimality gap between our control policy and the optimal decentralized cont…
In the last few years, the theory of decentralized distributed convex optimization has made significant progress. The lower bounds on communications rounds and oracle calls have appeared, as well as methods that reach both of these bounds. In this paper, we focus on how these results can be explained based on optimal algorithms for the non-distributed setup. In particular, we provide our recent results that have not been published yet and that could be found in details only in arXiv preprints.
Decentralized optimization is a promising parallel computation paradigm for large-scale data analytics and machine learning problems defined over a network of nodes. This paper is concerned with decentralized non-convex composite problems with population or empirical risk. In particular, the networked nodes are tasked to find an approximate stationary point of the average of local, smooth, possibly non-convex risk functions plus a possibly non-differentiable extended valued convex regularizer. Under this general formulation, we propose the first provably efficient, stochastic proximal gradient…
The problem of minimizing a multilinear function of binary variables is a well-studied NP-hard problem. The set of solutions of the standard linearization of this problem is called the multilinear set. We study a cardinality constrained version of it with upper and lower bounds on the number of nonzero variables. We call the set of solutions of the standard linearization of this problem a multilinear set with cardinality constraints. We characterize a set of conditions on these multilinear terms (called properness) and observe that under these conditions the convex hull description of the set …
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…
We describe a convex programming approach to the calculation of lower bounds on the minimum cost of constrained decentralized control problems with nonclassical information structures. The class of problems we consider entail the decentralized output feedback control of a linear time-varying system over a finite horizon, subject to polyhedral constraints on the state and input trajectories, and sparsity constraints on the controller's information structure. As the determination of optimal control policies for such systems is known to be computationally intractable in general, considerable effo…
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…
Large-scale saddle-point problems arise in such machine learning tasks as GANs and linear models with affine constraints. In this paper, we study distributed saddle-point problems (SPP) with strongly-convex-strongly-concave smooth objectives that have different strong convexity and strong concavity parameters of composite terms, which correspond to min and max variables, and bilinear saddle-point part. We consider two types of first-order oracles: deterministic (returns gradient) and stochastic (returns unbiased stochastic gradient). Our method works in both cases and takes several consensus s…
We study decentralized asynchronous multiagent optimization over networks, modeled as static (possibly directed) graphs. The optimization problem consists of minimizing a (possibly nonconvex) smooth function--the sum of the agents' local costs--plus a convex (possibly nonsmooth) regularizer, subject to convex constraints. Agents can perform their local computations as well as communicate with their immediate neighbors at any time, without any form of coordination or centralized scheduling; furthermore, when solving their local subproblems, they can use outdated information from their neighbors…
Distributed Optimization is an increasingly important subject area with the rise of multi-agent control and optimization. We consider a decentralized stochastic optimization problem where the agents on a graph aim to asynchronously optimize a collective (additive) objective function consisting of agents' individual (possibly non-convex) local objective functions. Each agent only has access to a noisy estimate of the gradient of its own function (one component of the sum of objective functions). We proposed an asynchronous distributed algorithm for such a class of problems. The algorithm combin…
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 study the problem of networked online convex optimization, where each agent individually decides on an action at every time step and agents cooperatively seek to minimize the total global cost over a finite horizon. The global cost is made up of three types of local costs: convex node costs, temporal interaction costs, and spatial interaction costs. In deciding their individual action at each time, an agent has access to predictions of local cost functions for the next $k$ time steps in an $r$-hop neighborhood. Our work proposes a novel online algorithm, Localized Predictive Control (LPC), …
This paper is concerned with the properties of the sets of strategic measures induced by admissible team policies in decentralized stochastic control and the convexity properties in dynamic team problems. To facilitate a convex analytical approach, strategic measures for team problems are introduced. Properties such as convexity, and compactness and Borel measurability under weak convergence topology are studied, and sufficient conditions for each of these properties are presented. These lead to existence of and structural results for optimal policies. It will be shown that the set of strategi…
The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be non-smooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-conti…
In this paper, a class of Decentralized Approximate Newton (DEAN) methods for addressing convex optimization on a networked system are developed, where nodes in the networked system seek for a consensus that minimizes the sum of their individual objective functions through local interactions only. The proposed DEAN algorithms allow each node to repeatedly perform a local approximate Newton update, which leverages tracking the global Newton direction and dissipating the discrepancies among the nodes. Under less restrictive problem assumptions in comparison with most existing second-order method…
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 c…
In this letter, we first propose a \underline{Z}eroth-\underline{O}rder c\underline{O}ordinate \underline{M}ethod~(ZOOM) to solve the stochastic optimization problem over a decentralized network with only zeroth-order~(ZO) oracle feedback available. Moreover, we equip a simple mechanism "powerball" to ZOOM and propose ZOOM-PB to accelerate the convergence of ZOOM. Compared with the existing methods, we verify the proposed algorithms through two benchmark examples in the literature, namely the black-box binary classification and the generating adversarial examples from black-box DNNs in order t…
We consider multi-agent, convex optimization programs subject to separable constraints, where the constraint function of each agent involves only its local decision vector, while the decision vectors of all agents are coupled via a common objective function. We focus on a regularized variant of the so called Jacobi algorithm for decentralized computation in such problems. We first consider the case where the objective function is quadratic, and provide a fixed-point theoretic analysis showing that the algorithm converges to a minimizer of the centralized problem. Moreover, we quantify the pote…
We study a convex resource allocation problem in which lower and upper bounds are imposed on partial sums of allocations. This model is linked to a large range of applications, including production planning, speed optimization, stratified sampling, support vector machines, portfolio management, and telecommunications. We propose an efficient gradient-free divide-and-conquer algorithm, which uses monotonicity arguments to generate valid bounds from the recursive calls, and eliminate linking constraints based on the information from sub-problems. This algorithm does not need strict convexity or …
Decentralized Bilevel Optimization: A Perspective from Transient Iteration Complexity
Center for Data Science, Peking University（北京大学数据科学中心）；Center for Machine Learning Research, Peking University AI for Science Institute（北京大学人工智能科学研究院机器学习研究中心）
We consider a resource allocation problem over an undirected network of agents, where edges of the network define communication links. The goal is to minimize the sum of agent-specific convex objective functions, while the agents' decisions are coupled via a convex conic constraint. We derive two methods by applying the alternating direction method of multipliers (ADMM) for decentralized consensus optimization to the dual of our resource allocation problem. Both methods are fully parallelizable and decentralized in the sense that each agent exchanges information only with its neighbors in the …
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…
This paper presents a decentralized algorithm for non-convex optimization over tree-structured networks. We assume that each node of this network can solve small-scale optimization problems and communicate approximate value functions with its neighbors based on a novel multi-sweep communication protocol. In contrast to existing parallelizable optimization algorithms for non-convex optimization the nodes of the network are neither synchronized nor assign any central entity. None of the nodes needs to know the whole topology of the network, but all nodes know that the network is tree-structured.…
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…
DAMO Academy, Alibaba Group（阿里巴巴集团达摩院）；Peking University（北京大学）；University of Pennsylvania（宾夕法尼亚大学）；MetaCarbon
Decentralized optimization is effective to save communication in large-scale machine learning. Although numerous algorithms have been proposed with theoretical guarantees and empirical successes, the performance limits in decentralized optimization, especially the influence of network topology and its associated weight matrix on the optimal convergence rate, have not been fully understood. While (Lu and Sa, 2021) have recently provided an optimal rate for non-convex stochastic decentralized optimization with weight matrices defined over linear graphs, the optimal rate with general weight matri…
In this paper, we consider the convex, finite-sum minimization problem with explicit convex constraints over strongly connected directed graphs. The constraint is an intersection of several convex sets each being known to only one node. To solve this problem, we propose a novel decentralized projected gradient scheme based on local averaging and prove its convergence using only local functions' smoothness.
We consider optimal control of decentralized LQG problems for plants controlled by two players having asymmetric information sharing patterns between them. In one scenario, players are assumed to have a bidirectional error-free, unlimited rate communication channel with no delay in one direction and a unit delay in the other. In another scenario, the communication channel is assumed to be unidirectional with a unit delay. Delayed information sharing patterns in general do not admit linear optimal control laws and are thus difficult to control optimally. However, in these scenarios, we show tha…
This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized algorithm for solving convex composite optimization problems over undirected and connected networks. The local loss function in these problems contains both smooth and nonsmooth terms. The proposed algorithm uses uncoordinated network-independent constant stepsizes and only needs to approximately solve a sequence of proximal mappings, which is advantageous for solving decentralized composite optimization problems where the proximal mappings of the nonsmooth loss functions may not have analytical solutions. For the general…
Modern power systems are now in continuous process of massive changes. Increased penetration of distributed generation, usage of energy storage and controllable demand require introduction of a new control paradigm that does not rely on massive information exchange required by centralized approaches. Distributed algorithms can rely only on limited information from neighbours to obtain an optimal solution for various optimization problems, such as optimal power flow, unit commitment etc. As a generalization of these problems we consider the problem of decentralized minimization of the smooth an…
We consider the decentralized control of radial distribution systems with controllable photovoltaic inverters and energy storage resources. For such systems, we investigate the problem of designing fully decentralized controllers that minimize the expected cost of balancing demand, while guaranteeing the satisfaction of individual resource and distribution system voltage constraints. Employing a linear approximation of the branch flow model, we formulate this problem as the design of a decentralized disturbance-feedback controller that minimizes the expected value of a convex quadratic cost fu…
Recently, convex nested stochastic composite optimization (NSCO) has received considerable attention for its applications in reinforcement learning and risk-averse optimization. The current NSCO algorithms have worse stochastic oracle complexities, by orders of magnitude, than those for simpler stochastic composite optimization problems (e.g., sum of smooth and nonsmooth functions) without the nested structure. Moreover, they require all outer-layer functions to be smooth, which is not satisfied by some important applications. These discrepancies prompt us to ask: ``does the nested composition…
We consider the task of decentralized minimization of the sum of smooth strongly convex functions stored across the nodes of a network. For this problem, lower bounds on the number of gradient computations and the number of communication rounds required to achieve $\varepsilon$ accuracy have recently been proven. We propose two new algorithms for this decentralized optimization problem and equip them with complexity guarantees. We show that our first method is optimal both in terms of the number of communication rounds and in terms of the number of gradient computations. Unlike existing optima…
In this work, we introduce an asynchronous decentralized accelerated stochastic gradient descent type of method for decentralized stochastic optimization, considering communication and synchronization are the major bottlenecks. We establish $\mathcal{O}(1/ε)$ (resp., $\mathcal{O}(1/\sqrtε)$) communication complexity and $\mathcal{O}(1/ε^2)$ (resp., $\mathcal{O}(1/ε)$) sampling complexity for solving general convex (resp., strongly convex) problems.
Gradient-tracking (GT) based decentralized methods have emerged as an effective and viable alternative method to decentralized (stochastic) gradient descent (DSGD) when solving distributed online stochastic optimization problems. Initial studies of GT methods implied that GT methods have worse network dependent rate than DSGD, contradicting experimental results. This dilemma has recently been resolved, and tighter rates for GT methods have been established, which improves upon DSGD. In this work, we establish more enhanced rates for GT methods under the online stochastic convex settings. We pr…
University of Pennsylvania（宾夕法尼亚大学）；Peking University（北京大学）
Decentralized optimization with time-varying networks is an emerging paradigm in machine learning. It saves remarkable communication overhead in large-scale deep training and is more robust in wireless scenarios especially when nodes are moving. Federated learning can also be regarded as decentralized optimization with time-varying communication patterns alternating between global averaging and local updates. While numerous studies exist to clarify its theoretical limits and develop efficient algorithms, it remains unclear what the optimal complexity is for non-convex decentralized stochastic …
