Core Concepts

The authors develop efficient distributed algorithms for fundamental convex optimization problems, including least squares regression, low-rank approximation, linear programming, and finite-sum minimization, with improved communication complexity in terms of the total number of bits exchanged.

Abstract

The paper addresses the communication complexity of several fundamental optimization problems in distributed settings, with a focus on improving the total number of bits communicated. The key contributions are:
Least Squares Regression, ℓp Regression, and Low-Rank Approximation:
For least squares regression in the coordinator model, the authors provide an algorithm with communication complexity of ̃O(sdL + d²L), improving upon the prior bound of ̃O(sd²L).
The authors' protocols extend to ℓp regression for 1 ≤ p < 2, achieving near-optimal communication.
For low-rank approximation in the coordinator model, the authors provide an algorithm with communication complexity of ̃O(kL · (dε⁻² + sε⁻¹)), improving upon prior work.
High-Accuracy Least Squares Regression:
The authors develop a protocol for high-accuracy least squares regression in the coordinator model, achieving communication complexity of ̃O(sd(L + log κ) log(ε⁻¹) + d²L), where κ is the condition number of the input matrix.
High-Accuracy Linear Programming:
The authors provide an algorithm for high-accuracy linear programming in the coordinator model, with communication complexity of ̃O(sd¹·⁵L + d²L), improving upon the prior bound of ̃O(sd³L + d⁴L).
Finite-Sum Minimization with Varying Supports:
In the blackboard model, the authors give an algorithm for minimizing a sum of convex, Lipschitz, and potentially non-smooth functions with varying supports, using ̃O(Σ₁ˢ d²ᵢ L) bits of communication, improving upon prior work.
Lower Bounds:
The authors establish lower bounds, showing that solving linear programs is exponentially harder than solving linear systems in the distributed setting, even when the number of constraints is polynomial in the problem size.
The authors develop novel techniques, including the use of block leverage scores, inverse maintenance, and cutting-plane methods, to achieve these improved communication complexity results.

Stats

For least squares regression in the coordinator model, the prior communication bound was ̃O(sd²L), which the authors improve to ̃O(sdL + d²L).
For high-accuracy least squares regression in the coordinator model, the authors achieve communication complexity of ̃O(sd(L + log κ) log(ε⁻¹) + d²L), where κ is the condition number of the input matrix.
For high-accuracy linear programming in the coordinator model, the authors achieve communication complexity of ̃O(sd¹·⁵L + d²L), improving upon the prior bound of ̃O(sd³L + d⁴L).
For finite-sum minimization with varying supports in the blackboard model, the authors achieve communication complexity of ̃O(Σ₁ˢ d²ᵢ L), improving upon the prior bound of ̃O(max₁ˢ dᵢ Σ₁ˢ dᵢL).

Quotes

"We strengthen known bounds for approximately solving linear regression, p-norm regression (for 1 ≤ p ≤ 2), linear programming, minimizing the sum of finitely many convex nonsmooth functions with varying supports, and low rank approximation; for a number of these fundamental problems our bounds are optimal, as proven by our lower bounds."
"Our lower bound can be used to show the first separation of linear programming and linear systems in the distributed model when the number of constraints is polynomial, addressing an open question in prior work."

Key Insights Distilled From

by Mehrdad Ghad... at **arxiv.org** 03-29-2024

Deeper Inquiries

The techniques developed in this work can be extended to other distributed optimization problems by leveraging the underlying principles and methodologies. For example, the use of block leverage scores and non-adaptive adaptive sketching can be applied to a wide range of optimization problems that involve distributed data and communication constraints. By adapting these techniques to suit the specific requirements of different optimization problems, researchers can potentially improve the communication complexity and efficiency of solving these problems in distributed settings. Additionally, the concept of subspace embeddings and preconditioning can be utilized in various optimization algorithms to enhance their performance in distributed environments.

The communication complexity bounds for the problems studied can potentially be further improved, especially in the high-precision setting or for ill-conditioned inputs, by exploring advanced algorithms and optimization techniques. For high-precision settings, refining the rounding procedures and leveraging more sophisticated iterative methods could lead to better communication efficiency. Additionally, for ill-conditioned inputs, developing specialized protocols that address the challenges posed by such matrices could help in reducing the communication complexity further. By continuously refining and optimizing the algorithms based on the insights gained from this work, researchers can strive to achieve even more efficient communication bounds for distributed optimization problems.

The theoretical results obtained in this work have significant practical implications for real-world distributed optimization systems. By reducing the communication complexity of fundamental optimization problems, such as least squares regression, low-rank approximation, and linear programming, the algorithms developed in this study can lead to faster and more efficient distributed optimization processes. These results can be leveraged in various applications, such as machine learning, federated learning, and distributed computing, where large-scale optimization tasks are common. Implementing these optimized algorithms in distributed systems can result in faster convergence, reduced resource consumption, and improved overall performance in handling complex optimization problems across distributed networks.

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