Efficient First Order Methods for Linear Composite Regularizers

TL;DR

This paper introduces a general, efficient approach for computing proximity operators of linear composite regularizers in machine learning, improving computational efficiency and convergence rates over existing methods.

Contribution

It presents a novel fixed point iteration method for proximity operators of composite regularizers, applicable to various regularization problems, outperforming current first order optimization techniques.

Findings

Outperforms state-of-the-art O(1/T) methods for overlapping Group Lasso

Matches optimal O(1/T^2) convergence for Fused Lasso

More general and computationally efficient than existing approaches

Abstract

A wide class of regularization problems in machine learning and statistics employ a regularization term which is obtained by composing a simple convex function \omega with a linear transformation. This setting includes Group Lasso methods, the Fused Lasso and other total variation methods, multi-task learning methods and many more. In this paper, we present a general approach for computing the proximity operator of this class of regularizers, under the assumption that the proximity operator of the function \omega is known in advance. Our approach builds on a recent line of research on optimal first order optimization methods and uses fixed point iterations for numerically computing the proximity operator. It is more general than current approaches and, as we show with numerical simulations, computationally more efficient than available first order methods which do not achieve the optimal rate. In particular, our method outperforms state of the art O(1/T) methods for overlapping Group Lasso and matches optimal O(1/T^2) methods for the Fused Lasso and tree structured Group Lasso.