Nonconvex Generalization of ADMM for Nonlinear Equality Constrained Problems

The ever-increasing demand for efficient and distributed optimization algorithms for large-scale data has led to the growing popularity of the Alternating Direction Method of Multipliers (ADMM). However, although the use of ADMM to solve linear equality constrained problems is well understood, we lacks a generic framework for solving problems with nonlinear equality constraints, which are common in practical applications (e.g., spherical constraints). To address this problem, we are proposing a new generic ADMM framework for handling nonlinear equality constraints, neADMM. After introducing the generalized problem formulation and the neADMM algorithm, the convergence properties of neADMM are discussed, along with its sublinear convergence rate $o(1/k)$, where $k$ is the number of iterations. Next, two important applications of neADMM are considered and the paper concludes by describing extensive experiments on several synthetic and real-world datasets to demonstrate the convergence and effectiveness of neADMM compared to existing state-of-the-art methods

Some New Results in Distributed Tracking and Optimization

The current age of Big Data is built on the foundation of distributed systems, and efficient distributed algorithms to run on these systems.With the rapid increase in the volume of the data being fed into these systems, storing and processing all this data at a central location becomes infeasible. Such a central \textit{server} requires a gigantic amount of computational and storage resources. Even when it is possible to have central servers, it is not always desirable, due to privacy concerns. Also, sending huge amounts of data to such servers incur often infeasible bandwidth requirements. In this dissertation, we consider two kinds of distributed architectures: 1) star-shaped topology, where multiple worker nodes are connected to, and communicate with a server, but the workers do not communicate with each other; and 2) mesh topology or network of interconnected workers, where each worker can communicate with a small number of neighboring workers. In the first half of this dissertation (Chapters 2 and 3), we consider distributed systems with mesh topology.We study two different problems in this context. First, we study the problem of simultaneous localization and multi-target tracking. Multiple mobile agents localize themselves cooperatively, while also tracking multiple, unknown number of mobile targets, in the presence of measurement-origin uncertainty. In situations with limited GPS signal availability, agents (like self-driving cars in urban canyons, or autonomous vehicles in hazardous environments) need to rely on inter-agent measurements for localization. The agents perform the additional task of tracking multiple targets (pedestrians and road-signs for self-driving cars). We propose a decentralized algorithm for this problem. To be effective in real-time applications, we propose efficient Gaussian and Gaussian-mixture based filters, rather than the computationally expensive particle-based methods in the existing literature. Our novel factor-graph based approach gives better performance, in terms of both agent localization errors, and target-location and cardinality errors. Next, we study an online convex optimization problem, where a network of agents cooperate to minimize a global time-varying objective function. Only the local functions are revealed to individual agents. The agents also need to satisfy their individual constraints. We propose a primal-dual update based decentralized algorithm for this problem. Under standard assumptions, we prove that the proposed algorithm achieves sublinear regret and constraint violation across the network. In other words, over a long enough time horizon, the decisions taken by the agents are, on average, as good as if all the information was revealed ahead of time. In addition, the individual constraint violations of the agents, averaged over time, are zero. In the next part of the dissertation (Chapters 4), we study distributed systems with a star-shaped topology. The problem we study is distributed nonconvex optimization. With the recent success of deep learning, coupled with the use of distributed systems to solve large-scale problems, this problem has gained prominence over the past decade. The recently proposed paradigm of Federated Learning (which has already been deployed by Google/Apple in Android/iOS phones) has further catalyzed research in this direction. The problem we consider is minimizing the average of local smooth, nonconvex functions. Each node has access only to its own loss function, but can communicate with the server, which aggregates updates from all the nodes, before distributing them to all the nodes. With the advent of more and more complex neural network architectures, these updates can be high dimensional. To save resources, the problem needs to be solved via communication-efficient approaches. We propose a novel algorithm, which combines the idea of variance-reduction, with the paradigm of carrying out multiple local updates at each node before averaging. We prove the convergence of the approach to a first-order stationary point. Our algorithm is optimal in terms of computation, and state-of-the-art in terms of the communication requirements. Lastly in Chapter 5, we consider the situation when the nodes do not have access to function gradients, and need to minimize the loss function using only function values. This problem lies in the domain of zeroth-order optimization. For simplicity of analysis, we study this problem only in the single-node case. This problem finds application in simulation-based optimization, and adversarial example generation for attacking deep neural networks. We propose a novel function value based gradient estimator, which has better variance, and better query-efficiency compared to existing estimators. The proposed estimator covers the most commonly used existing estimators as special cases. We conduct a comprehensive convergence analysis under different conditions. We also demonstrate its effectiveness through a real-world application to generating adversarial examples from a black-box deep neural network

Twenty-Five Years of Advances in Beamforming: From Convex and Nonconvex Optimization to Learning Techniques

Beamforming is a signal processing technique to steer, shape, and focus an electromagnetic wave using an array of sensors toward a desired direction. It has been used in several engineering applications such as radar, sonar, acoustics, astronomy, seismology, medical imaging, and communications. With the advances in multi-antenna technologies largely for radar and communications, there has been a great interest on beamformer design mostly relying on convex/nonconvex optimization. Recently, machine learning is being leveraged for obtaining attractive solutions to more complex beamforming problems. This article captures the evolution of beamforming in the last twenty-five years from convex-to-nonconvex optimization and optimization-to-learning approaches. It provides a glimpse of this important signal processing technique into a variety of transmit-receive architectures, propagation zones, paths, and conventional/emerging applications