[Paper Review] Iterative Pre-Conditioning for Expediting the Gradient-Descent Method: The Distributed Linear Least-Squares Problem

Kushal Chakrabarti, Nirupam Gupta+1 more|arXiv (Cornell University)|Aug 6, 2020

Sparse and Compressive Sensing Techniques18 references4 citations

TL;DR

This paper proposes an iterative pre-conditioning technique for accelerating the gradient-descent method in distributed linear least-squares problems across a server-agent network. By adaptively updating a pre-conditioning matrix, the method achieves superlinear convergence when the solution is unique and significantly improves convergence rates over standard and accelerated gradient methods, especially in ill-conditioned settings, as validated on real-world datasets under both ideal and noisy conditions.

ABSTRACT

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.

Motivation & Objective

• Address the slow convergence of gradient-descent methods in distributed linear least-squares problems due to ill-conditioned data.
• Enable privacy-preserving collaborative learning where agents do not share raw local data with the server.
• Develop a robust, iterative pre-conditioning mechanism that adapts to data conditioning without requiring prior knowledge of the condition number.
• Improve convergence speed over existing methods—especially in high-condition-number scenarios—while maintaining distributed, communication-efficient operation.
• Demonstrate performance gains in both noise-free and noisy computational environments using real-world datasets.

Proposed method

• Introduces a server-agent network architecture where agents compute local gradients and send them to a central server for aggregation.
• Proposes an iterative pre-conditioning matrix $ K(t) $ that is updated at each iteration to improve the conditioning of the Hessian approximation.
• The pre-conditioned gradient update is defined as $ x(t+1) = x(t) - K(t)^{-1} \sum_{i=1}^m g^i(t) $, where $ g^i(t) $ is the local gradient from agent $ i $.
• The pre-conditioning matrix $ K(t) $ is updated using a recursive formula based on gradient differences, enabling adaptive conditioning without explicit Hessian computation.
• The method operates in a synchronous, distributed manner with minimal communication overhead—only gradients and pre-conditioning updates are exchanged.
• Theoretical analysis shows that under unique solution conditions, the method achieves superlinear convergence, while in general cases, it exhibits improved linear convergence rates compared to GD, NAG, HBM, and APC.

Experimental results

Research questions

• RQ1 Can iterative pre-conditioning mitigate the negative impact of high data condition numbers on gradient-descent convergence in distributed least-squares problems?
• RQ2 Does the proposed method achieve faster convergence than traditional and accelerated gradient methods in both ideal and noisy environments?
• RQ3 Can the pre-conditioning mechanism be updated iteratively in a distributed setting without requiring global Hessian information?
• RQ4 What is the convergence behavior of the proposed algorithm when the least-squares problem has a unique solution versus when it does not?
• RQ5 How does the algorithm perform in the presence of system noise, particularly in comparison to BFGS and other quasi-Newton methods?

Key findings

• The proposed algorithm achieves superlinear convergence when the least-squares problem has a unique solution, significantly outperforming standard gradient-descent and accelerated methods.
• In general cases, the algorithm demonstrates a faster linear convergence rate than gradient descent (GD), Nesterov’s accelerated gradient (NAG), heavy-ball method (HBM), and accelerated projection-consensus (APC).
• On real-world datasets like "ash608" and "gr_30_30", the asymptotic estimation error of the proposed method is zero, while GD, NAG, HBM, and APC exhibit non-zero errors.
• In noisy environments, the algorithm maintains bounded error and outperforms BFGS, which exhibits unbounded error growth after 360 iterations due to singularity issues in the Hessian approximation.
• The algorithm’s convergence rate is robust to system noise, with asymptotic error norms significantly lower than those of GD, NAG, HBM, APC, and BFGS across tested datasets.
• Experimental results confirm that the proposed method converges faster and more reliably than state-of-the-art methods, particularly in ill-conditioned and noisy settings.