In this paper, we analyze the sample and communication complexity of the federated linear stochastic approximation (FedLSA) algorithm. We explicitly quantify the effects of local training with agent heterogeneity. We show that the communication complexity of FedLSA scales polynomially with the inverse of the desired accuracy $\epsilon$. To overcome this, we propose SCAFFLSA a new variant of FedLSA that uses control variates to correct for client drift, and establish its sample and communication complexities. We show that for statistically heterogeneous agents, its communication complexity scales logarithmically with the desired accuracy, similar to Scaffnew. An important finding is that, compared to the existing results for Scaffnew, the sample complexity scales with the inverse of the number of agents, a property referred to as linear speed-up. Achieving this linear speed-up requires completely new theoretical arguments. We apply the proposed method to federated temporal difference learning with linear function approximation and analyze the corresponding complexity improvements.

翻译：本文分析了联邦线性随机逼近（FedLSA）算法的样本复杂度与通信复杂度。我们明确量化了智能体异构性对本地训练的影响。研究表明，FedLSA的通信复杂度随期望精度$\epsilon$的倒数呈多项式增长。为克服此问题，我们提出SCAFFLSA——一种利用控制变量校正客户端漂移的FedLSA新变体，并建立了其样本与通信复杂度。我们证明，对于统计异构的智能体，其通信复杂度随期望精度呈对数增长，与Scaffnew类似。一个重要发现是，相较于Scaffnew的现有结果，其样本复杂度随智能体数量的倒数变化，这一特性被称为线性加速。实现该线性加速需要全新的理论论证。我们将所提方法应用于线性函数逼近的联邦时序差分学习，并分析了相应的复杂度改进。