A Penalty-Based Method for Communication-Efficient Decentralized Bilevel Programming

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., in a federated learning setting. The latter approach suffers from a high communication cost on the central node (e.g., parameter server). Hence, there is an interest in developing 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 the estimation of the hyper-gradient of the penalty function through decentralized computation of matrix-vector products and a few vector communications. The estimation is integrated into an alternating algorithm for solving the penalized reformulation of the bilevel optimization problem. Under appropriate step sizes and penalty parameters, our theoretical framework ensures non-asymptotic convergence to the optimal solution of the original problem under various convexity conditions. Our theoretical result highlights improvements in the iteration complexity of decentralized bilevel optimization, all while making efficient use of vector communication. Empirical results demonstrate that the proposed method performs well in real-world settings.