A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks

In this paper we propose a distributed implementation of the relaxed Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization of a separable convex cost function, whose terms are stored by a set of interacting agents, one for each agent. Specifically the local cost stored by each node is in general a function of both the state of the node and the states of its neighbors, a framework that we refer to as `partition-based' optimization. This framework presents a great flexibility and can be adapted to a large number of different applications. We show that the partition-based R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford Splitting (R-PRS) operator which, historically, has been introduced in the literature to find the zeros of sum of functions. Interestingly, making use of non expansive operator theory, the proposed algorithm is shown to be provably robust against random packet losses that might occur in the communication between neighboring nodes. Finally, the effectiveness of the proposed algorithm is confirmed by a set of compelling numerical simulations run over random geometric graphs subject to i.i.d. random packet losses.Comment: Full version of the paper to be presented at Conference on Decision and Control (CDC) 201

Asynchronous Distributed Optimization over Lossy Networks via Relaxed ADMM: Stability and Linear Convergence

In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular, we are interested in the case in which communications between nodes are prone to failures and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed ADMM, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean to a neighborhood of the optimal solution, and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.Comment: To appear in IEEE Transactions on Automatic Contro

A Stochastic Operator Framework for Optimization and Learning with Sub-Weibull Errors

This paper proposes a framework to study the convergence of stochastic optimization and learning algorithms. The framework is modeled over the different challenges that these algorithms pose, such as (i) the presence of random additive errors (e.g. due to stochastic gradients), and (ii) random coordinate updates (e.g. due to asynchrony in distributed set-ups). The paper covers both convex and strongly convex problems, and it also analyzes online scenarios, involving changes in the data and costs. The paper relies on interpreting stochastic algorithms as the iterated application of stochastic operators, thus allowing us to use the powerful tools of operator theory. In particular, we consider operators characterized by additive errors with sub-Weibull distribution (which parameterize a broad class of errors by their tail probability), and random updates. In this framework we derive convergence results in mean and in high probability, by providing bounds to the distance of the current iteration from a solution of the optimization or learning problem. The contributions are discussed in light of federated learning applications

Online Distributed Learning with Quantized Finite-Time Coordination

In this paper we consider online distributed learning problems. Online distributed learning refers to the process of training learning models on distributed data sources. In our setting a set of agents need to cooperatively train a learning model from streaming data. Differently from federated learning, the proposed approach does not rely on a central server but only on peer-to-peer communications among the agents. This approach is often used in scenarios where data cannot be moved to a centralized location due to privacy, security, or cost reasons. In order to overcome the absence of a central server, we propose a distributed algorithm that relies on a quantized, finite-time coordination protocol to aggregate the locally trained models. Furthermore, our algorithm allows for the use of stochastic gradients during local training. Stochastic gradients are computed using a randomly sampled subset of the local training data, which makes the proposed algorithm more efficient and scalable than traditional gradient descent. In our paper, we analyze the performance of the proposed algorithm in terms of the mean distance from the online solution. Finally, we present numerical results for a logistic regression task.Comment: To be presented at IEEE CDC'2

ADMM-Tracking Gradient for Distributed Optimization over Asynchronous and Unreliable Networks

In this paper, we propose (i) a novel distributed algorithm for consensus optimization over networks and (ii) a robust extension tailored to deal with asynchronous agents and packet losses. The key idea is to achieve dynamic consensus on (i) the agents' average and (ii) the global descent direction by iteratively solving an online auxiliary optimization problem through a distributed implementation of the Alternating Direction Method of Multipliers (ADMM). Such a mechanism is suitably interlaced with a local proportional action steering each agent estimate to the solution of the original consensus optimization problem. First, in the case of ideal networks, by using tools from system theory, we prove the linear convergence of the scheme with strongly convex costs. Then, by exploiting the averaging theory, we extend such a first result to prove that the robust extension of our method preserves linear convergence in the case of asynchronous agents and packet losses. Further, by using the notion of Input-to-State Stability, we also guarantee the robustness of the schemes with respect to additional, generic errors affecting the agents' updates. Finally, some numerical simulations confirm our theoretical findings and show that the proposed methods outperform the existing state-of-the-art distributed methods for consensus optimization

Online Distributed Learning over Random Networks

The recent deployment of multi-agent systems in a wide range of scenarios has enabled the solution of learning problems in a distributed fashion. In this context, agents are tasked with collecting local data and then cooperatively train a model, without directly sharing the data. While distributed learning offers the advantage of preserving agents' privacy, it also poses several challenges in terms of designing and analyzing suitable algorithms. This work focuses specifically on the following challenges motivated by practical implementation: (i) online learning, where the local data change over time; (ii) asynchronous agent computations; (iii) unreliable and limited communications; and (iv) inexact local computations. To tackle these challenges, we introduce the Distributed Operator Theoretical (DOT) version of the Alternating Direction Method of Multipliers (ADMM), which we call the DOT-ADMM Algorithm. We prove that it converges with a linear rate for a large class of convex learning problems (e.g., linear and logistic regression problems) toward a bounded neighborhood of the optimal time-varying solution, and characterize how the neighborhood depends on~$\text{(i)--(iv)}$. We corroborate the theoretical analysis with numerical simulations comparing the DOT-ADMM Algorithm with other state-of-the-art algorithms, showing that only the proposed algorithm exhibits robustness to (i)--(iv)

Distributed Prediction-Correction ADMM for Time-Varying Convex Optimization

This paper introduces a dual-regularized ADMM approach to distributed, time-varying optimization. The proposed algorithm is designed in a prediction-correction framework, in which the computing nodes predict the future local costs based on past observations, and exploit this information to solve the time-varying problem more effectively. In order to guarantee linear convergence of the algorithm, a regularization is applied to the dual, yielding a dual-regularized ADMM. We analyze the convergence properties of the time-varying algorithm, as well as the regularization error of the dual-regularized ADMM. Numerical results show that in time-varying settings, despite the regularization error, the performance of the dual-regularized ADMM can outperform inexact gradient-based methods, as well as exact dual decomposition techniques, in terms of asymptotical error and consensus constraint violation.Comment: Presented at Asilomar Conference on Signals, Systems, and Computers 202

Teoria degli operatori per ottimizzazione e learning

Optimization is an important tool in many science and engineering applications, ranging from machine learning to control, from signal processing to power systems management, to name a few. The relevance of optimization in this wide range of applications requires the design of algorithms that can overcome different challenges. Indeed, depending on the application, optimization algorithms may have access to limited computational power, or have hard constraints on the time available for computations. To address these challenges, optimization research in recent years has increasingly relied on operator theory, with its powerful tools and results. Importantly, this is allowed by the fact that optimization algorithms can be interpreted as the recursive application of an operator, thus translating a minimization problem into a fixed point problem. The central theme of this thesis is therefore the intersection of optimization and operator theory, and how we can leverage their interplay to solve different challenges and design novel algorithms. We focus in particular on two broad areas of research: stochastic optimization and online optimization. Stochastic optimization groups all problems in which an algorithms is constrained by the presence of randomness during its execution, e.g. because of additive noise perturbing the computation. In this context, the thesis proposes two contribution. The first is the study of the alternating direction method of multipliers (ADMM) applied to distributed optimization problems when the network is asynchronous and peer-to-peer communications may randomly fail. The second contribution is a framework for stochastic operator theory that allows us to analyze the convergence of a large number of stochastic optimization algorithms in a unified way, with applications in e.g. machine learning. In this framework we derive both mean and high probability convergence guarantees, the latter leveraging the powerful formalism of sub-Weibull random variables. In online optimization we are faced with the challenge of solving a problem whose cost and constraints change over time, and thus the goal is to track a sequence of optimizers. The first contribution we propose is a general prediction-correction method, which abstracts many online algorithms and can be used to study their convergence. The prediction-correction method is characterized by the fact that past information is used to warm-start the solution of future problems, and we propose a novel polynomial extrapolation-based strategy to do so. Secondly, in the area of learning to optimize we propose a novel approach to accelerate online algorithms for weakly convex problems. The method is based on the concept of operator regression, which learns a faster algorithm from samples of the original one. Finally, the thesis reports numerical results that showcase the practical applications of the theoretical contributions, and discusses the tvopt Python module implemented for the purpose of prototyping and benchmarking optimization algorithms.Optimization is an important tool in many science and engineering applications, ranging from machine learning to control, from signal processing to power systems management, to name a few. The relevance of optimization in this wide range of applications requires the design of algorithms that can overcome different challenges. Indeed, depending on the application, optimization algorithms may have access to limited computational power, or have hard constraints on the time available for computations. To address these challenges, optimization research in recent years has increasingly relied on operator theory, with its powerful tools and results. Importantly, this is allowed by the fact that optimization algorithms can be interpreted as the recursive application of an operator, thus translating a minimization problem into a fixed point problem. The central theme of this thesis is therefore the intersection of optimization and operator theory, and how we can leverage their interplay to solve different challenges and design novel algorithms. We focus in particular on two broad areas of research: stochastic optimization and online optimization. Stochastic optimization groups all problems in which an algorithms is constrained by the presence of randomness during its execution, e.g. because of additive noise perturbing the computation. In this context, the thesis proposes two contribution. The first is the study of the alternating direction method of multipliers (ADMM) applied to distributed optimization problems when the network is asynchronous and peer-to-peer communications may randomly fail. The second contribution is a framework for stochastic operator theory that allows us to analyze the convergence of a large number of stochastic optimization algorithms in a unified way, with applications in e.g. machine learning. In this framework we derive both mean and high probability convergence guarantees, the latter leveraging the powerful formalism of sub-Weibull random variables. In online optimization we are faced with the challenge of solving a problem whose cost and constraints change over time, and thus the goal is to track a sequence of optimizers. The first contribution we propose is a general prediction-correction method, which abstracts many online algorithms and can be used to study their convergence. The prediction-correction method is characterized by the fact that past information is used to warm-start the solution of future problems, and we propose a novel polynomial extrapolation-based strategy to do so. Secondly, in the area of learning to optimize we propose a novel approach to accelerate online algorithms for weakly convex problems. The method is based on the concept of operator regression, which learns a faster algorithm from samples of the original one. Finally, the thesis reports numerical results that showcase the practical applications of the theoretical contributions, and discusses the tvopt Python module implemented for the purpose of prototyping and benchmarking optimization algorithms

tvopt: A Python Framework for Time-Varying Optimization

This paper introduces tvopt, a Python framework for prototyping and benchmarking time-varying (or online) optimization algorithms. The paper first describes the theoretical approach that informed the development of tvopt. Then it discusses the different components of the framework and their use for modeling and solving time-varying optimization problems. In particular, tvopt provides functionalities for defining both centralized and distributed online problems, and a collection of built-in algorithms to solve them, for example gradient-based methods, ADMM and other splitting methods. Moreover, the framework implements prediction strategies to improve the accuracy of the online solvers. The paper then proposes some numerical results on a benchmark problem and discusses their implementation using tvopt. The code for tvopt is available at https://github.com/nicola-bastianello/tvopt.Comment: Code available here: https://github.com/nicola-bastianello/tvop