Federated learning (FL) is now recognized as a key framework for communicationefficient collaborative learning. Most theoretical and empirical studies, however, rely on the assumption that clients have access to pre-collected data sets, with limited investigation into scenarios where clients continuously collect data. In many real-world applications, particularly when data is generated by physical or biological processes, client data streams are often modeled by non-stationary Markov processes.

Local SGD, or Federated Averaging, is one of the most widely used algorithms for distributed optimization. Although it often outperforms alternatives such as mini-batch SGD, existing theory has not fully explained this advantage under realistic assumptions about data heterogeneity. Recent work has suggested that a second-order heterogeneity assumption may suffice to justify the empirical gains of local SGD. We confirm this conjecture by establishing new upper and lower bounds on the convergence of local SGD. These bounds demonstrate how a low secondorder heterogeneity, combined with third-order smoothness, enables local SGD to interpolate between heterogeneous and homogeneous regimes while maintaining communication efficiency. Our main technical contribution is a refined analysis of the consensus error, a central quantity in such results. We validate our theory with experiments on a distributed linear regression task.

Federated learning (FL) is now recognized as a key framework for communication-efficient collaborative learning. Most theoretical and empirical studies, however, rely on the assumption that clients have access to pre-collected data sets, with limited investigation into scenarios where clients continuously collect data. In many real-world applications, particularly when data is generated by physical or biological processes, client data streams are often modeled by non-stationary Markov processes.

Modern machine learning often requires training with large batch size, distributed data, and massively parallel compute hardware (like mobile and other edge devices or distributed data centers). Communication becomes a major bottleneck in such settings but methods like Local Stochastic Gradient Descent (Local SGD) show great promise to reduce the global communication need. Local SGD consists of three parts: a local optimization processes, an aggregation mechanism, and an outer optimizer that uses the aggregated updates from the nodes to produce a new model. While there exists an extensive literature on understanding the impact of hyperparameters in the local optimization process, the choice of outer optimizer and its hyperparameters is less clear. We study the role of the outer learning in Local SGD, and prove new convergence guarantees for the algorithm. In particular, we show that tuning the outer learning rate allows us to (a) trade off between optimization error and stochastic gradient noise variance, and (b) make up for ill-tuning of the inner learning rate. Our theory suggests that the outer learning rate should sometimes be set to values greater than $1$. We extend our results to apply to when we use momentum in the outer optimizer, and also introduce a novel data-dependent analysis of Local SGD that yields further insights on outer learning rate tuning. We conduct comprehensive experiments with standard language models and various outer optimizers to validate our theory.

Local SGD, or Federated Averaging, is one of the most widely used algorithms for distributed optimization. Although it often outperforms alternatives such as mini-batch SGD, existing theory has not fully explained this advantage under realistic assumptions about data heterogeneity. Recent work has suggested that a second-order heterogeneity assumption may suffice to justify the empirical gains of local SGD. We confirm this conjecture by establishing new upper and lower bounds on the convergence of local SGD. These bounds demonstrate how a low second-order heterogeneity, combined with third-order smoothness, enables local SGD to interpolate between heterogeneous and homogeneous regimes while maintaining communication efficiency. Our main technical contribution is a refined analysis of the consensus error, a central quantity in such results. We validate our theory with experiments on a distributed linear regression task.

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In this paper we consider linearly constrained stochastic approximation problems with federated learning asaspecial case.