Fast and Provable ADMM for Learning with Generative Priors

TL;DR

This paper introduces a linearized ADMM algorithm tailored for convex optimization problems with nonconvex constraints derived from neural network generators, offering provable convergence and efficiency advantages over gradient descent.

Contribution

It develops a novel ADMM-based method for nonconvex constraints from neural networks, with theoretical convergence rates and practical efficiency improvements.

Findings

Algorithm converges under mild geometric conditions.

Handles non-smooth objectives efficiently.

Faster than gradient descent in practical scenarios.

Abstract

In this work, we propose a (linearized) Alternating Direction Method-of-Multipliers (ADMM) algorithm for minimizing a convex function subject to a nonconvex constraint. We focus on the special case where such constraint arises from the specification that a variable should lie in the range of a neural network. This is motivated by recent successful applications of Generative Adversarial Networks (GANs) in tasks like compressive sensing, denoising and robustness against adversarial examples. The derived rates for our algorithm are characterized in terms of certain geometric properties of the generator network, which we show hold for feedforward architectures, under mild assumptions. Unlike gradient descent (GD), it can efficiently handle non-smooth objectives as well as exploit efficient partial minimization procedures, thus being faster in many practical scenarios.