Local MixVR: Breaking the Communication-Sample Dependence in Distributed Learning

Communication is a major bottleneck in distributed learning, especially in large-scale settings and in federated learning environments with slow links. Three standard ways to reduce this cost are communication compression, local training, and communication-computation overlap. Methods that combine these ingredients are used in practice and have been found to be effective for large-scale training, but there is little theory for methods that combine all three. We study a heterogeneous-compute setting in which different workers may take different numbers of local steps, and we propose LOSCAR-SGD, a Local SGD method that communicates only a sparse subset of model coordinates and continues optimizing while communication is in flight. A key ingredient is a delay-corrected merge rule that incorporates delayed synchronized information without discarding the progress made during the overlap phase. We give convergence guarantees for smooth non-convex objectives and show how sparsity, overlap, and worker heterogeneity affect the rate. To the best of our knowledge, this is the first theory for this combination of ingredients. Experiments further show that communication-computation overlap reduces training time and that the delay-corrected merge outperforms naive overwriting.

Local SGD is a popular optimization method in distributed learning, often outperforming other algorithms in practice, including mini-batch SGD. Despite this success, theoretically proving the dominance of local SGD in settings with reasonable data heterogeneity has been difficult, creating a significant gap between theory and practice. In this paper, we provide new lower bounds for local SGD under existing first-order data heterogeneity assumptions, showing that these assumptions are insufficient to prove the effectiveness of local update steps. Furthermore, under these same assumptions, we demonstrate the min-max optimality of accelerated mini-batch SGD, which fully resolves our understanding of distributed optimization for several problem classes. Our results emphasize the need for better models of data heterogeneity to understand the effectiveness of local SGD in practice. Towards this end, we consider higher-order smoothness and heterogeneity assumptions, providing new upper bounds that imply the dominance of local SGD over mini-batch SGD when data heterogeneity is low.

Machine learning and optimization have advanced together, with practical demands motivating new theory and theoretical breakthroughs enabling new applications. Modern large-scale training relies on classical optimization principles, but the constraints of distributed systems require these foundations to be reconsidered. This thesis addresses seven challenges at the intersection of theory and practice, focusing on key bottlenecks in federated learning and distributed optimization. First, we introduce ProxSkip and prove that local gradient steps can accelerate communication, providing a theoretical foundation for this widely used heuristic. Second, we develop Variance Reduced ProxSkip, which eliminates the neighborhood error of stochastic local updates while balancing communication and local computation. Third, we show that local steps retain their communication acceleration under partial client participation. Fourth, we prove that server-side stepsizes and sampling without replacement improve convergence in heterogeneous settings. Fifth, for Random Reshuffling, we demonstrate that compressing gradient differences rather than gradients yields better theoretical and practical performance. Sixth, we establish that Byzantine robustness and partial participation can be achieved simultaneously using gradient-difference clipping. Finally, we develop the first theoretical framework for low-rank adaptation based on randomized asymmetric chains, providing new insights into fine-tuning large models. Across these contributions, we introduce novel algorithmic frameworks, establish sharp guarantees under realistic assumptions, and support the theory with numerical experiments.