Iterative Pre-Conditioning for Expediting the Gradient-Descent Method: The Distributed Linear Least-Squares Problem
This addresses the slow convergence of gradient-descent in distributed settings with ill-conditioned data, offering a method to expedite computations while preserving data privacy, though it is incremental as it builds on existing gradient-descent frameworks.
The paper tackles the distributed linear least-squares problem in a server-agent network, where agents aim to compute an optimal linear model without sharing local data, and proposes an iterative pre-conditioning technique that achieves superlinear convergence for unique solutions and improved linear convergence rates compared to traditional and accelerated gradient-descent methods, with experimental validation on real-world problems.
This paper considers the multi-agent linear least-squares problem in a server-agent network. In this problem, the system comprises multiple agents, each having a set of local data points, that are connected to a server. The goal for the agents is to compute a linear mathematical model that optimally fits the collective data points held by all the agents, without sharing their individual local data points. This goal can be achieved, in principle, using the server-agent variant of the traditional iterative gradient-descent method. The gradient-descent method converges linearly to a solution, and its rate of convergence is lower bounded by the conditioning of the agents' collective data points. If the data points are ill-conditioned, the gradient-descent method may require a large number of iterations to converge. We propose an iterative pre-conditioning technique that mitigates the deleterious effect of the conditioning of data points on the rate of convergence of the gradient-descent method. We rigorously show that the resulting pre-conditioned gradient-descent method, with the proposed iterative pre-conditioning, achieves superlinear convergence when the least-squares problem has a unique solution. In general, the convergence is linear with improved rate of convergence in comparison to the traditional gradient-descent method and the state-of-the-art accelerated gradient-descent methods. We further illustrate the improved rate of convergence of our proposed algorithm through experiments on different real-world least-squares problems in both noise-free and noisy computation environment.