[Paper Review] Communication Efficient Distributed Optimization using an Approximate Newton-type Method

Ohad Shamir, Nathan Srebro+1 more|arXiv (Cornell University)|Dec 30, 2013

Stochastic Gradient Optimization Techniques18 references83 citations

TL;DR

This paper proposes DANE, a communication-efficient distributed optimization method based on an approximate Newton-type algorithm that leverages statistical similarity across machines. It achieves linear convergence with a rate that improves as data size increases, significantly outperforming one-shot averaging, gradient descent, and ADMM in both synthetic and real-world learning tasks with fewer communication rounds.

ABSTRACT

We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably \emph{improves} with the data size, requiring an essentially constant number of iterations under reasonable assumptions. We provide theoretical and empirical evidence of the advantages of our method compared to other approaches, such as one-shot parameter averaging and ADMM.

Motivation & Objective

• To address the high communication cost in distributed machine learning by designing a method that minimizes rounds of communication while maintaining fast convergence.
• To overcome the limitations of one-shot parameter averaging, which yields suboptimal solutions for stochastic optimization problems.
• To develop a distributed algorithm that scales favorably with data size, avoiding polynomial dependence on condition number common in gradient-based methods.
• To provide theoretical guarantees for convergence rate improvement with increasing data size per machine.
• To empirically validate the method on both synthetic quadratic and real-world non-quadratic learning problems.

Proposed method

• The method uses a distributed approximate Newton-type update that exploits the statistical similarity of local objectives across machines.
• At each iteration, each machine computes a local Newton-type update using its own data, followed by a global averaging of the update directions.
• The algorithm incorporates a regularization parameter μ and a step size η to balance convergence and stability, with η=1, μ=0 being optimal in favorable cases.
• It performs two communication rounds per iteration: one for averaging gradients and another for averaging update directions, enabling faster convergence than standard ADMM.
• The method is designed to converge linearly with a rate that improves as the number of samples per machine increases.
• Theoretical analysis shows that for quadratic objectives, the convergence rate improves with data size, requiring only a constant number of iterations under reasonable assumptions.

Experimental results

Research questions

• RQ1 Can a distributed optimization method achieve linear convergence with a rate that improves as data size increases?
• RQ2 How does the performance of DANE compare to one-shot parameter averaging, ADMM, and gradient descent in terms of communication efficiency and solution quality?
• RQ3 What is the impact of the regularization parameter μ and step size η on convergence and stability?
• RQ4 Does the method maintain fast convergence when data is split across many machines with small per-machine sample sizes?
• RQ5 Can the method be effectively applied to non-quadratic, real-world learning problems such as regularized loss minimization?

Key findings

• DANE achieves linear convergence with a rate that improves as the number of samples per machine increases, leading to a constant number of iterations under reasonable assumptions.
• In synthetic quadratic problems, DANE converges significantly faster than ADMM, and its convergence rate improves with larger data size, while ADMM's rate remains unchanged.
• On real-world datasets (COV1, ASTRO-PH, MNIST-47), DANE requires fewer iterations than ADMM to reach the same accuracy, especially when data is distributed across many machines.
• For small data per machine, DANE may fail to converge with μ=0, but convergence is restored by increasing μ to 3λ, albeit at a slight cost to average convergence speed.
• One-shot parameter averaging yields significantly suboptimal solutions on test loss, while DANE and ADMM converge to the optimal empirical risk minimizer.
• Empirical results show that DANE outperforms ADMM and one-shot averaging in terms of both convergence speed and final test accuracy across multiple datasets.