Proximal Multitask Learning over Networks with Sparsity-inducing Coregularization

In this work, we consider multitask learning problems where clusters of nodes are interested in estimating their own parameter vector. Cooperation among clusters is beneficial when the optimal models of adjacent clusters have a good number of similar entries. We propose a fully distributed algorithm for solving this problem. The approach relies on minimizing a global mean-square error criterion regularized by non-differentiable terms to promote cooperation among neighboring clusters. A general diffusion forward-backward splitting strategy is introduced. Then, it is specialized to the case of sparsity promoting regularizers. A closed-form expression for the proximal operator of a weighted sum of $\ell_1$-norms is derived to achieve higher efficiency. We also provide conditions on the step-sizes that ensure convergence of the algorithm in the mean and mean-square error sense. Simulations are conducted to illustrate the effectiveness of the strategy

Distributed Learning for Stochastic Generalized Nash Equilibrium Problems

This work examines a stochastic formulation of the generalized Nash equilibrium problem (GNEP) where agents are subject to randomness in the environment of unknown statistical distribution. We focus on fully-distributed online learning by agents and employ penalized individual cost functions to deal with coupled constraints. Three stochastic gradient strategies are developed with constant step-sizes. We allow the agents to use heterogeneous step-sizes and show that the penalty solution is able to approach the Nash equilibrium in a stable manner within $O(\mu_\text{max})$, for small step-size value $\mu_\text{max}$ and sufficiently large penalty parameters. The operation of the algorithm is illustrated by considering the network Cournot competition problem

Distributed Coupled Multi-Agent Stochastic Optimization

This work develops effective distributed strategies for the solution of constrained multi-agent stochastic optimization problems with coupled parameters across the agents. In this formulation, each agent is influenced by only a subset of the entries of a global parameter vector or model, and is subject to convex constraints that are only known locally. Problems of this type arise in several applications, most notably in disease propagation models, minimum-cost flow problems, distributed control formulations, and distributed power system monitoring. This work focuses on stochastic settings, where a stochastic risk function is associated with each agent and the objective is to seek the minimizer of the aggregate sum of all risks subject to a set of constraints. Agents are not aware of the statistical distribution of the data and, therefore, can only rely on stochastic approximations in their learning strategies. We derive an effective distributed learning strategy that is able to track drifts in the underlying parameter model. A detailed performance and stability analysis is carried out showing that the resulting coupled diffusion strategy converges at a linear rate to an $O(\mu)-$neighborhood of the true penalized optimizer

On the Performance and Linear Convergence of Decentralized Primal-Dual Methods

This dissertation studies the performance and linear convergence properties of primal-dual methods for the solution of decentralized multi-agent optimization problems. Decentralized multi-agent optimization is a powerful paradigm that finds applications in diverse fields in learning and engineering design. In these setups, a network of agents is connected through some topology and agents are allowed to share information only locally. Their overall goal is to seek the minimizer of a global optimization problem through localized interactions. In decentralized consensus problems, the agents are coupled through a common consensus variable that they need to agree upon. While in decentralized resource allocation problems, the agents are coupled through global affine constraints. Various decentralized consensus optimization algorithms already exist in the literature. Some methods are derived from a primal-dual perspective, while other methods are derived as gradient tracking mechanisms meant to track the average of local gradients. Among the gradient tracking methods are the adapt-then-combine implementations motivated by diffusion strategies, which have been observed to perform better than other implementations. In this dissertation, we develop a novel adapt-then-combine primal-dual algorithmic framework that captures most state-of-the-art gradient based methods as special cases including all the variations of the gradient-tracking methods. We also develop a concise and novel analysis technique that establishes the linear convergence of this general framework under strongly-convex objectives. Due to our unified framework, the analysis reveals important characteristics for these methods such as their convergence rates and step-size stability ranges. Moreover, the analysis reveals how the augmented Lagrangian penalty term, which is utilized in most of these methods, affects the performance of decentralized algorithms. Another important question that we answer is whether decentralized proximal gradient methods can achieve global linear convergence for non-smooth composite optimization. For centralized algorithms, linear convergence has been established in the presence of a non-smooth composite term. In this dissertation, we close the gap between centralized and decentralized proximal gradient algorithms and show that decentralized proximal algorithms can also achieve linear convergence in the presence of a non-smooth term. Furthermore, we show that when each agent possesses a different local non-smooth term then global linear convergence cannot be established in the worst case. Most works that study decentralized optimization problems assume that all agents are involved in computing all variables. However, in many applications the coupling across agents is sparse in the sense that only a few agents are involved in computing certain variables. We show how to design decentralized algorithms in sparsely coupled consensus and resource allocation problems. More importantly, we establish analytically the importance of exploiting the sparsity structure in coupled large-scale networks

Networked signal and information processing

The article reviews significant advances in networked signal and information processing, which have enabled in the last 25 years extending decision making and inference, optimization, control, and learning to the increasingly ubiquitous environments of distributed agents. As these interacting agents cooperate, new collective behaviors emerge from local decisions and actions. Moreover, and significantly, theory and applications show that networked agents, through cooperation and sharing, are able to match the performance of cloud or federated solutions, while offering the potential for improved privacy, increasing resilience, and saving resources

Data-Reserved Periodic Diffusion LMS With Low Communication Cost Over Networks

In this paper, we analyze diffusion strategies in which all nodes attempt to estimate a common vector parameter for achieving distributed estimation in adaptive networks. Under diffusion strategies, each node essentially needs to share processed data with predefined neighbors. Although the use of internode communication has contributed significantly to improving convergence performance based on diffusion, such communications consume a huge quantity of power in data transmission. In developing low-power consumption diffusion strategies, it is very important to reduce the communication cost without significant degradation of convergence performance. For that purpose, we propose a data-reserved periodic diffusion least-mean-squares (LMS) algorithm in which each node updates and transmits an estimate periodically while reserving its measurement data even during non-update time. By applying these reserved data in an adaptation step at update time, the proposed algorithm mitigates the decline in convergence speed incurred by most conventional periodic schemes. For a period p, the total cost of communication is reduced to a factor of 1/p relative to the conventional adapt-then-combine (ATC) diffusion LMS algorithm. The loss of combination steps in this process leads naturally to a slight increase in the steady-state error as the period p increases, as is theoretically confirmed through mathematical analysis. We also prove an interesting property of the proposed algorithm, namely, that it suffers less degradation of the steady-state error than the conventional diffusion in a noisy communication environment. Experimental results show that the proposed algorithm outperforms related conventional algorithms and, in particular, outperforms ATC diffusion LMS over a network with noisy links.11Ysciescopu

Networked Signal and Information Processing

The article reviews significant advances in networked signal and information processing, which have enabled in the last 25 years extending decision making and inference, optimization, control, and learning to the increasingly ubiquitous environments of distributed agents. As these interacting agents cooperate, new collective behaviors emerge from local decisions and actions. Moreover, and significantly, theory and applications show that networked agents, through cooperation and sharing, are able to match the performance of cloud or federated solutions, while offering the potential for improved privacy, increasing resilience, and saving resources