Bilevel programming has recently received attention in the literature, due to its wide range of applications, including reinforcement learning and hyper-parameter optimization. However, it is widely assumed that the underlying bilevel optimization problem is solved either by a single machine or in the case of multiple machines connected in a star-shaped network, i.e., federated learning setting. The latter approach suffers from a high communication cost on the central node (e.g., parameter server) and exhibits privacy vulnerabilities. Hence, it is of interest to develop methods that solve bilevel optimization problems in a communication-efficient decentralized manner. To that end, this paper introduces a penalty function based decentralized algorithm with theoretical guarantees for this class of optimization problems. Specifically, a distributed alternating gradient-type algorithm for solving consensus bilevel programming over a decentralized network is developed. A key feature of the proposed algorithm is to estimate the hyper-gradient of the penalty function via decentralized computation of matrix-vector products and few vector communications, which is then integrated within an alternating algorithm to obtain finite-time convergence analysis under different convexity assumptions. Our theoretical result highlights improvements in the iteration complexity of decentralized bilevel optimization, all while making efficient use of vector communication. Empirical results on both synthetic and real datasets demonstrate that the proposed method performs well in real-world settings.

翻译：双层规划因其在强化学习和超参数优化等领域的广泛应用，近期在文献中受到关注。然而，现有研究普遍假设底层双层优化问题由单机解决，或由星形网络（即联邦学习设置）中的多台机器完成。后者在中心节点（如参数服务器）上存在高通信成本，并暴露出隐私漏洞。因此，开发一种通信高效且去中心化的双层优化方法具有重要意义。为此，本文提出一种基于惩罚函数的去中心化算法，并为该类优化问题提供理论保障。具体而言，我们设计了一种分布式交替梯度型算法，用于在去中心化网络中解决共识双层规划问题。该算法的一个关键特征是通过去中心化计算矩阵-向量积和少量向量通信来估计惩罚函数的超梯度，随后将其集成到交替算法中，从而在不同凸性假设下获得有限时间收敛分析。我们的理论结果突出了去中心化双层优化迭代复杂度的改进，同时高效利用向量通信。在合成数据集和真实数据集上的实验结果表明，所提出的方法在实际场景中表现优异。