Second-Order Sensitivity Analysis for Bilevel Optimization

TL;DR

This paper introduces a second-order sensitivity analysis method for bilevel optimization, enabling the use of faster second-order optimization techniques by deriving the IFT Hessian, which improves computational efficiency and broadens application scope.

Contribution

It extends first-order sensitivity analysis to include second-order derivatives (IFT Hessian), allowing more efficient optimization in bilevel problems.

Findings

IFT Hessian can be computed efficiently using existing computations.

Errors bounds for the IFT gradient apply to the IFT Hessian.

Using the IFT Hessian reduces overall computation in bilevel optimization.

Abstract

In this work we derive a second-order approach to bilevel optimization, a type of mathematical programming in which the solution to a parameterized optimization problem (the "lower" problem) is itself to be optimized (in the "upper" problem) as a function of the parameters. Many existing approaches to bilevel optimization employ first-order sensitivity analysis, based on the implicit function theorem (IFT), for the lower problem to derive a gradient of the lower problem solution with respect to its parameters; this IFT gradient is then used in a first-order optimization method for the upper problem. This paper extends this sensitivity analysis to provide second-order derivative information of the lower problem (which we call the IFT Hessian), enabling the usage of faster-converging second-order optimization methods at the upper level. Our analysis shows that (i) much of the computation already used to produce the IFT gradient can be reused for the IFT Hessian, (ii) errors bounds derived for the IFT gradient readily apply to the IFT Hessian, (iii) computing IFT Hessians can significantly reduce overall computation by extracting more information from each lower level solve. We corroborate our findings and demonstrate the broad range of applications of our method by applying it to problem instances of least squares hyperparameter auto-tuning, multi-class SVM auto-tuning, and inverse optimal control.