Data-driven nonsmooth optimization

TL;DR

This paper introduces a unified framework for large-scale nonsmooth optimization, combining classical algorithms with neural network-based parameter learning to improve convergence and performance in applications like image reconstruction.

Contribution

It develops a generic iterative scheme encompassing modern primal-dual solvers, proves convergence for a new method, and proposes a neural network approach to learn optimal parameters within this class.

Findings

Convergence established for a new optimization method.

Neural network training improves parameter selection for specific problems.

Effective in applications like tomographic reconstruction and image deconvolution.

Abstract

In this work, we consider methods for solving large-scale optimization problems with a possibly nonsmooth objective function. The key idea is to first specify a class of optimization algorithms using a generic iterative scheme involving only linear operations and applications of proximal operators. This scheme contains many modern primal-dual first-order solvers like the Douglas-Rachford and hybrid gradient methods as special cases. Moreover, we show convergence to an optimal point for a new method which also belongs to this class. Next, we interpret the generic scheme as a neural network and use unsupervised training to learn the best set of parameters for a specific class of objective functions while imposing a fixed number of iterations. In contrast to other approaches of "learning to optimize", we present an approach which learns parameters only in the set of convergent schemes. As use cases, we consider optimization problems arising in tomographic reconstruction and image deconvolution, and in particular a family of total variation regularization problems.