# Implicit Bilevel Optimization: Differentiating through Bilevel Optimization Programming

## TL;DR

This paper introduces BiGrad, a novel method for differentiating through bilevel optimization problems, enabling end-to-end learning in models that incorporate bilevel programming, applicable to both continuous and combinatorial cases.

## Contribution

It extends single-level optimization approaches to bilevel programming, providing a general, efficient framework for differentiating through complex bilevel problems in machine learning.

## Key findings

- BiGrad effectively extends single-level methods to bilevel programming.
- The approach reduces computational complexity for combinatorial problems.
- Experiments demonstrate successful integration of bilevel optimization in learning models.

## Abstract

Bilevel Optimization Programming is used to model complex and conflicting interactions between agents, for example in Robust AI or Privacy-preserving AI. Integrating bilevel mathematical programming within deep learning is thus an essential objective for the Machine Learning community. Previously proposed approaches only consider single-level programming. In this paper, we extend existing single-level optimization programming approaches and thus propose Differentiating through Bilevel Optimization Programming (BiGrad) for end-to-end learning of models that use Bilevel Programming as a layer. BiGrad has wide applicability and can be used in modern machine learning frameworks. BiGrad is applicable to both continuous and combinatorial Bilevel optimization problems. We describe a class of gradient estimators for the combinatorial case which reduces the requirements in terms of computation complexity; for the case of the continuous variable, the gradient computation takes advantage of the push-back approach (i.e. vector-jacobian product) for an efficient implementation. Experiments show that the BiGrad successfully extends existing single-level approaches to Bilevel Programming.