Beyond backpropagation: bilevel optimization through implicit differentiation and equilibrium propagation

TL;DR

This paper reviews gradient-based methods for bilevel optimization, focusing on implicit differentiation and equilibrium propagation, highlighting their mathematical foundations, algorithms, and comparative advantages.

Contribution

It provides a comprehensive overview of gradient-based bilevel optimization techniques, emphasizing implicit differentiation and equilibrium propagation methods.

Findings

Implicit differentiation enables efficient bilevel optimization.

Equilibrium propagation offers a biologically plausible alternative.

Comparison reveals trade-offs in efficiency and applicability.

Abstract

This paper reviews gradient-based techniques to solve bilevel optimization problems. Bilevel optimization is a general way to frame the learning of systems that are implicitly defined through a quantity that they minimize. This characterization can be applied to neural networks, optimizers, algorithmic solvers and even physical systems, and allows for greater modeling flexibility compared to an explicit definition of such systems. Here we focus on gradient-based approaches that solve such problems. We distinguish them in two categories: those rooted in implicit differentiation, and those that leverage the equilibrium propagation theorem. We present the mathematical foundations that are behind such methods, introduce the gradient-estimation algorithms in detail and compare the competitive advantages of the different approaches.