Decentralized optimization often relies on achieving consensus among disparate agents. This article addresses the consensus problem in decentralized networks, focusing on the challenges posed by a nonconvex compact submanifold constraint. We identify conditions on network topology that facilitate local linear convergence to global consensus, where the achieved linear rate matches that of the Euclidean setting. Central to our analysis are the convex-like properties, specifically proximal smoothness and the restricted secant inequality, which form the foundation of our theoretical framework. These results will be useful for the design and analysis of decentralized manifold optimization algorithms. Numerical experiments are conducted to validate our theoretical findings.

This paper explores constrained non-convex personalized federated learning (PFL), in which a group of workers train local models and a global model, under the coordination of a server. To address the challenges of efficient information exchange and robustness against the so-called Byzantine workers, we propose a projected stochastic gradient descent algorithm for PFL that simultaneously ensures Byzantine-robustness and communication efficiency. We implement personalized learning at the workers aided by the global model, and employ a Huber function-based robust aggregation with an adaptive threshold-selecting strategy at the server to reduce the effects of Byzantine attacks. To improve communication efficiency, we incorporate random communication that allows multiple local updates per communication round. We establish the convergence of our algorithm, showing the effects of Byzantine attacks, random communication, and stochastic gradients on the learning error. Numerical experiments demonstrate the superiority of our algorithm in neural network training compared to existing ones.

Decentralized training of deep neural networks has attracted significant attention for its theoretically superior scalability compared to synchronous data-parallel methods like All-Reduce. However, realizing this potential in multi-node training is challenging due to the complex design space that involves communication topologies, computation patterns, and optimization algorithms. This paper identifies three key factors that can lead to speedups over All-Reduce training and constructs a runtime model to determine when and how decentralization can shorten the per-iteration runtimes. To support the decentralized training of transformer-based models, we introduce a decentralized Adam algorithm that overlaps communications with computations, prove its convergence, and propose an accumulation technique to mitigate the high variance caused by small local batch sizes. We deploy our solution in clusters with up to 64 GPUs, demonstrating its practical advantages in both runtime and generalization performance under a fixed iteration budget.

We introduce a locally differentially private (LDP) algorithm for online federated learning that employs temporally correlated noise to improve utility while preserving privacy. To address challenges posed by the correlated noise and local updates with streaming non-IID data, we develop a perturbed iterate analysis that controls the impact of the noise on the utility. Moreover, we demonstrate how the drift errors from local updates can be effectively managed for several classes of nonconvex loss functions. Subject to an (ε, δ)-LDP budget, we establish a dynamic regret bound that quantifies the impact of key parameters and the intensity of changes in the dynamic environment on the learning performance. Numerical experiments confirm the efficacy of the proposed algorithm.

We propose a novel algorithm, termed soft quasi-Newton (soft QN), for optimization in the presence of bounded noise. Traditional quasi-Newton algorithms are vulnerable to such noise-induced perturbations. To develop a more robust quasi-Newton method, we replace the secant condition in the matrix optimization problem for the Hessian update with a penalty term in its objective and derive a closed-form update formula. A key feature of our approach is its ability to maintain positive definiteness of the Hessian inverse approximation throughout the iterations. Furthermore, we establish the following properties of soft QN: it recovers the BFGS method under specific limits, it treats positive and negative curvature equally, and it is scale invariant. Collectively, these features enhance the efficacy of soft QN in noisy environments. For strongly convex objective functions and Hessian approximations obtained using soft QN, we develop an algorithm that exhibits linear convergence toward a neighborhood of the optimal solution even when gradient and function evaluations are subject to bounded perturbations. Through numerical experiments, we demonstrate that soft QN consistently outperforms state-of-the-art methods across a range of scenarios.

We propose a novel algorithm for solving the composite Federated Learning (FL) problem. This algorithm manages non-smooth regularization by strategically decoupling the proximal operator and communication, and addresses client drift without any assumptions about data similarity. Moreover, each worker uses local updates to reduce the communication frequency with the server and transmits only a d-dimensional vector per communication round. We prove that our algorithm converges linearly to a neighborhood of the optimal solution and demonstrate the superiority of our algorithm over state-of-the-art methods in numerical experiments.

We introduce a novel differentially private algorithm for online federated learning that employs temporally correlated noise to enhance utility while ensuring privacy of continuously released models. To address challenges posed by DP noise and local updates with streaming non-iid data, we develop a perturbed iterate analysis to control the impact of the DP noise on the utility. Moreover, we demonstrate how the drift errors from local updates can be effectively managed under a quasi-strong convexity condition. Subject to an (, δ) DP budget, we establish a dynamic regret bound over the entire time horizon, quantifying the impact of key parameters and the intensity of changes in dynamic environments. Numerical experiments confirm the efficacy of the proposed algorithm.

This paper proposes a locally differentially private federated learning algorithm for strongly convex but possibly nonsmooth problems that protects the gradients of each worker against an honest but curious server. The proposed algorithm adds artificial noise to the shared information to ensure privacy and dynamically allocates the time-varying noise variance to minimize an upper bound of the optimization error subject to a predefined privacy budget constraint. This allows for an arbitrarily large but finite number of iterations to achieve both privacy protection and utility up to a neighborhood of the optimal solution, removing the need for tuning the number of iterations. Numerical results show the superiority of the proposed algorithm over state-of-the-art methods.

Many machine learning tasks, such as principal component analysis and low-rank matrix completion, give rise to manifold optimization problems. Although there is a large body of work studying the design and analysis of algorithms for manifold optimization in the centralized setting, there are currently very few works addressing the federated setting. In this paper, we consider nonconvex federated learning over a compact smooth submanifold in the setting of heterogeneous client data. We propose an algorithm that leverages stochastic Riemannian gradients and a manifold projection operator to improve computational efficiency, uses local updates to improve communication efficiency, and avoids client drift. Theoretically, we show that our proposed algorithm converges sub-linearly to a neighborhood of a first-order optimal solution by using a novel analysis that jointly exploits the manifold structure and properties of the loss functions. Numerical experiments demonstrate that our algorithm has significantly smaller computational and communication overhead than existing methods.

In this article, we consider a strongly convex finite-sum minimization problem over a decentralized network and pro-pose a communication-efficient decentralized Newton's method for solving it. The main challenges in designing such an algorithm come from three aspects: (i) mismatch between local gradients/Hessians and the global ones; (ii) cost of sharing second-order information; (iii) tradeoff among computation and communication. To handle these challenges, we first apply dynamic average consensus (DAC) so that each node is able to use a local gradient approximation and a local Hessian approximation to track the global gradient and Hessian, respectively. Second, since exchanging Hessian approxi-mations is far from communication-efficient, we require the nodes to exchange the compressed ones instead and then apply an error compensation mechanism to correct for the compression noise. Third, we introduce multi-step consensus for exchanging local variables and local gradient approximations to balance between computation and communication. With novel analysis, we establish the globally linear (resp., asymptotically super-linear) convergence rate of the proposed algorithm when m is constant (resp., tends to infinity), where m = 1 is the number of consensus inner steps. To the best of our knowledge, this is the first super-linear conver-gence result for a communication-efficient decentralized Newton's method. Moreover, the rate we establish is provably faster than those of first-order methods. Our numerical results on various applications corroborate the theoretical findings.