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：双级优化在超参数优化、元学习和强化学习方面有着广泛的应用,但双级优化问题难以解决。在可缩放双级算法方面最近的进展主要侧重于双级优化问题,低级目标要么是强烈的弯曲,要么是不受约束。在这项工作中,我们通过惩罚方法的透镜来解决双级问题。我们表明,在某些条件下,重订刑罚恢复了最初的双级问题的解决办法。此外,我们建议采用基于惩罚的双级梯度下调算法(PBGD),并在没有较低水平强的弯曲的情况下,为受限制的双级问题建立固定时间趋同点。实验展示了拟议的PBGD算法的效率。</s>