# On Penalty-based Bilevel Gradient Descent Method

**Authors:** Han Shen, Quan Xiao, Tianyi Chen  
**arXiv:** [2302.05185](https://arxiv.org/abs/2302.05185)  
**Date:** 2023-02-10  
**Summary page:** https://www.summarizepaper.com/en/paper/2302.05185v4/

## Key points

- Bilevel optimization is valuable in hyper-parameter optimization, meta-learning, and reinforcement learning
- Existing scalable algorithms focus on strongly convex or unconstrained lower-level objectives
- The authors propose a penalty-based approach to solve bilevel problems
- They introduce the penalty-based bilevel gradient descent (PBGD) algorithm
- PBGD has finite-time convergence for constrained bilevel problems without lower-level strong convexity
- Experiments validate the effectiveness of their approach in solving bilevel optimization problems with constraints and non-strongly convex lower-level objectives
- This research contributes to advancing scalable algorithms for difficult bilevel optimization problems with potential practical applications in various domains.

## Summary

Bilevel optimization is a valuable tool in various fields such as hyper-parameter optimization, meta-learning, and reinforcement learning. Solving bilevel optimization problems is challenging; existing scalable algorithms mainly focus on cases where the lower-level objective is strongly convex or unconstrained. In this study, the authors approach the bilevel problem using the penalty method. They demonstrate that under certain conditions, the penalty reformulation can recover solutions of the original bilevel problem and propose a new algorithm called penalty-based bilevel gradient descent (PBGD). They establish its finite-time convergence for constrained bilevel problems without lower-level strong convexity. Experiments validate their approach and indicate that their method effectively solves bilevel optimization problems even when faced with constraints and non-strongly convex lower-level objectives. This research contributes to advancing scalable algorithms for solving difficult bilevel optimization problems by offering a promising approach to tackle these challenges with potential for practical applications in various domains.