# A Distributed Quasi-Newton Algorithm for Empirical Risk Minimization with Nonsmooth Regularization

## TL;DR

This paper introduces a distributed quasi-Newton algorithm that efficiently solves empirical risk minimization problems with nonsmooth regularization, reducing communication and computation costs while ensuring convergence.

## Contribution

It presents a novel distributed second-order optimization method that handles nonsmooth regularizers and converges globally for a wide class of ERM problems.

## Key findings

- Significantly reduces communication costs compared to existing methods
- Achieves global linear convergence for non-strongly convex problems
- Demonstrates improved running time in initial experiments

## Abstract

We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving ERM problems with a nonsmooth regularization term. Current second-order and quasi-Newton methods for this problem either do not work well in the distributed setting or work only for specific regularizers. Our algorithm uses successive quadratic approximations, and we describe how to maintain an approximation of the Hessian and solve subproblems efficiently in a distributed manner. The proposed method enjoys global linear convergence for a broad range of non-strongly convex problems that includes the most commonly used ERMs, thus requiring lower communication complexity. It also converges on non-convex problems, so has the potential to be used on applications such as deep learning. Initial computational results on convex problems demonstrate that our method significantly improves on communication cost and running time over the current state-of-the-art methods.