Convex and Non-convex Federated Learning with Stale Stochastic Gradients: Diminishing Step Size is All You Need

Decentralized stochastic optimization is a fundamental paradigm for large-scale learning over networks, where agents communicate only with their neighbors and no central coordinator is required. For strongly convex problems, communication efficiency is mainly determined by the condition number $κ=L/μ$ and the network spectral gap $1-β$. Although deterministic decentralized methods can simultaneously achieve accelerated $\sqrtκ$ and $1/\sqrt{1-β}$ dependences, no existing stochastic method attains both improvements at once. In this paper, we propose Multi-Gossip Accelerated DSGD (MG-ADSGD), a decentralized stochastic algorithm that combines Nesterov-type primal--dual extrapolation with multi-round fast gossip averaging. The key idea is to couple the gossip depth with the mini-batch size so that additional communication rounds simultaneously improve consensus accuracy and reduce gradient variance. We show that MG-ADSGD achieves the communication complexity $$\widetilde{\mathcal O}\!\left( \frac{σ^2}{μnε}\log\frac{1}ε + \sqrt{\fracκ{1-β}}\log\frac{1}ε \right),$$ where $ε$ denotes the target accuracy, $n$ is the number of nodes, and $σ^2$ is the gradient variance. To the best of our knowledge, this bound yields the best currently available communication complexity for decentralized stochastic strongly convex optimization, up to logarithmic factors that are independent of $ε$.

Federated learning (FL) is an emerging distributed machine learning paradigm that enables local devices to jointly train a global model while keeping data decentralized and private. We propose a variance-reduction based algorithm, VRA-FedSGD, for FL in the presence of heavy-tailed gradient noise and communication noise, where these noises are prevalent in large-scale machine learning over wireless networks and Internet of Things deployments. VRA-FedSGD employs a momentum variance reduction technique together with a nonlinear mapping to mitigate heavy-tailed gradient noise, and uses a variance-reduced aggregation mechanism to suppress heavy-tailed communication noise. In the mean sense, VRA-FedSGD achieves a convergence rate of {\small$\mathcal{O}\left(K^{-(p-1)/(2p-1)}\right)$} for nonconvex objective functions, where $p$ is the tail index of heavy-tailed noise. In the almost sure sense, VRA-FedSGD achieves a convergence rate of $\tilde{\mathcal{O}}\left(K^{-(1-1/(p-ε))}\right)$ for strongly convex objective functions, where $ε$ is an arbitrarily small constant. Simulated experiments on a logistic regression problem with real-world data verify the effectiveness of VRA-FedSGD.

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.