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、元学习和强化学习等领域具有广泛应用。然而，双层优化问题求解难度较大。近期可扩展双层算法的进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。本研究从惩罚方法的视角解决双层问题，证明了在特定条件下，惩罚重构能恢复原始双层问题的解。进一步，我们提出了基于惩罚的双层梯度下降（PBGD）算法，并建立了该算法在下层非强凸约束双层问题中的有限时间收敛性。实验结果表明了所提出的PBGD算法的高效性。