Bilevel optimization enjoys a wide range of applications in hyper-parameter optimization, meta-learning and reinforcement learning. However, bilevel optimization problems are difficult to solve. Recent progress on scalable bilevel algorithms mainly focuses on bilevel optimization problems where the lower-level objective is either strongly convex or unconstrained. In this work, we tackle the bilevel problem through the lens of the penalty method. We show that under certain conditions, the penalty reformulation recovers the solutions of the original bilevel problem. Further, we propose the penalty-based bilevel gradient descent (PBGD) algorithm and establish its finite-time convergence for the constrained bilevel problem without lower-level strong convexity. Experiments showcase the efficiency of the proposed PBGD algorithm.

翻译：双层优化在超参数优化、元学习和强化学习中有着广泛应用。然而，双层优化问题难以求解。近期可扩展双层算法的发展主要聚焦于下层目标函数为强凸或无约束的双层优化问题。本研究通过惩罚方法的视角处理双层问题，证明了在特定条件下惩罚重构能够恢复原双层问题的解。进一步地，我们提出了基于惩罚的双层梯度下降（PBGD）算法，并建立了该算法在无下层强凸性约束条件下的有限时间收敛性。实验结果表明，所提出的PBGD算法具有高效性。