Bilevel optimization enjoys a wide range of applications in emerging machine learning and signal processing problems such as hyper-parameter optimization, image reconstruction, meta-learning, adversarial training, and reinforcement learning. However, bilevel optimization problems are traditionally known to be difficult to solve. Recent progress on bilevel algorithms mainly focuses on bilevel optimization problems through the lens of the implicit-gradient method, where the lower-level objective is either strongly convex or unconstrained. In this work, we tackle a challenging class of bilevel problems through the lens of the penalty method. We show that under certain conditions, the penalty reformulation recovers the (local) 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 with lower-level constraints yet without lower-level strong convexity. Experiments on synthetic and real datasets showcase the efficiency of the proposed PBGD algorithm.

翻译：双层优化在机器学习与信号处理的新兴问题中具有广泛应用，如超参数优化、图像重建、元学习、对抗训练和强化学习等。然而，传统上双层优化问题因其求解困难而备受挑战。近年来双层算法的研究进展主要集中于通过隐式梯度方法处理下层目标函数强凸或无约束的双层优化问题。本文通过惩罚方法处理一类具有挑战性的双层优化问题。我们证明，在特定条件下，惩罚重构能够恢复原双层问题的（局部）解。进一步，我们提出基于惩罚的双层梯度下降（PBGD）算法，并针对具有下层约束但下层目标非强凸的约束双层问题，建立了该算法的有限时间收敛性。在合成数据集与真实数据集上的实验验证了所提PBGD算法的有效性。