Bilevel programming has recently received attention in the literature, due to a 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 our alternating algorithm to give the finite-time convergence analysis under different convexity assumptions. Owing to the generality of this complexity analysis, our result yields convergence rates for a wide variety of consensus problems including minimax and compositional optimization. Empirical results on both synthetic and real datasets demonstrate that the proposed method works well in practice.

翻译：双层规划因在强化学习和超参数优化等广泛应用中而近期受到文献关注。然而，现有研究普遍假设底层双层优化问题由单机求解，或由星形网络（即联邦学习场景）中的多台机器协作求解。后者在中心节点（如参数服务器）处存在通信成本高、隐私易泄露的缺陷。因此，开发通信高效的分布式双层优化方法具有重要研究价值。为此，本文提出了一种基于罚函数的分布式算法，并给出该类优化问题的理论保证。具体而言，本文开发了一种分布式交替梯度型算法，用于在分布式网络中求解共识双层规划问题。该算法的核心特征在于：通过分布式计算矩阵-向量积与少量向量通信来估计罚函数的超梯度，并将其集成到交替算法框架中，在不同凸性假设下给出了有限时间收敛性分析。得益于该复杂度分析的普适性，本文结果为包括极小极大优化与组合优化在内的广泛共识问题提供了收敛速率。在合成数据集与真实数据集上的实验结果表明，所提方法在实际应用中性能优异。