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:文献最近注意到,由于应用范围广泛,包括强化学习和超参数优化,文献中最近对双层编程进行了广泛关注,因为应用范围广泛,包括强化学习和超参数优化;然而,人们广泛认为,潜在的双层优化问题要么通过单一机器解决,要么通过星形网络连接的多台机器解决,即联合学习设置;后者在中央节点(例如参数服务器)和隐私脆弱性方面有很高的通信成本;因此,我们有兴趣制定方法,以通信效率低的分散方式解决双层优化问题;为此,本文件采用基于惩罚的分散算法,为这类优化问题提供理论保障;具体地说,为在一个分散的网络上解决协商一致双层编程的交替的梯度型算法;拟议的算法的一个主要特点是,通过分散计算矩阵-变量产品和几乎没有矢量通信,估计惩罚功能的高度高度高度高度的等级,然后将之纳入我们的交替算法,以便在不同的交替算法假设下进行有限的时间趋同式分析;为此,本文件引入了基于理论保证这一类优化问题的分级算法的分级算法;具体地,我们关于为在广泛共识化方法上的结果趋同率的合并率,以最接近法和最接近法的模型的模型显示,包括最接近率制成。