Sulaiman A. Alghunaim

Edward Duc Hien Nguyen, Sulaiman A. Alghunaim, Kun Yuan et al.

We study the decentralized optimization problem where a network of $n$ agents seeks to minimize the average of a set of heterogeneous non-convex cost functions distributedly. State-of-the-art decentralized algorithms like Exact Diffusion~(ED) and Gradient Tracking~(GT) involve communicating every iteration. However, communication is expensive, resource intensive, and slow. In this work, we analyze a locally updated GT method (LU-GT), where agents perform local recursions before interacting with their neighbors. While local updates have been shown to reduce communication overhead in practice, their theoretical influence has not been fully characterized. We show LU-GT has the same communication complexity as the Federated Learning setting but allows arbitrary network topologies. In addition, we prove that the number of local updates does not degrade the quality of the solution achieved by LU-GT. Numerical examples reveal that local updates can lower communication costs in certain regimes (e.g., well-connected graphs).

Luyao Guo, Sulaiman A. Alghunaim, Kun Yuan et al.

The ProxSkip algorithm for distributed optimization is gaining increasing attention due to its effectiveness in reducing communication. However, existing analyses of ProxSkip are limited to the strongly convex setting and fail to achieve linear speedup with respect to the number of nodes. Key questions regarding its behavior in the non-convex setting and the achievability of linear speedup remain open. In this paper, we revisit ProxSkip and address both questions. We provide a comprehensive analysis for stochastic non-convex, convex, and strongly convex problems, revealing the effects of gradient noise, local updates, network connectivity, and data heterogeneity on its convergence. We prove that ProxSkip achieves linear speedup across all three settings, and can further achieve linear speedup with network-independent stepsizes in the strongly convex setting. Moreover, we show that properly increasing local updates effectively reduces communication complexity.

Haoyuan Cai, Sulaiman A. Alghunaim, Ali H. Sayed

Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least $\mathcal{O}(\varepsilon^{-4})$ oracle complexity to find an $\varepsilon$-stationary point. Some works indicate that this complexity can be improved to $\mathcal{O}(\varepsilon^{-3})$ when the loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions, remains unresolved. In this work, we address this question for optimization problems that are nonconvex in the minimization variable and strongly concave or Polyak-Lojasiewicz (PL) in the maximization variable. We introduce novel bias-corrected momentum algorithms utilizing efficient Hessian-vector products. We establish convergence conditions and demonstrate a lower iteration complexity of $\mathcal{O}(\varepsilon^{-3})$ for the proposed algorithms. The effectiveness of the method is validated through applications to robust logistic regression using real-world datasets.

Haoyuan Cai, Sulaiman A. Alghunaim, Ali H. Sayed

The optimistic gradient method is useful in addressing minimax optimization problems. Motivated by the observation that the conventional stochastic version suffers from the need for a large batch size on the order of $\mathcal{O}(\varepsilon^{-2})$ to achieve an $\varepsilon$-stationary solution, we introduce and analyze a new formulation termed Diffusion Stochastic Same-Sample Optimistic Gradient (DSS-OG). We prove its convergence and resolve the large batch issue by establishing a tighter upper bound, under the more general setting of nonconvex Polyak-Lojasiewicz (PL) risk functions. We also extend the applicability of the proposed method to the distributed scenario, where agents communicate with their neighbors via a left-stochastic protocol. To implement DSS-OG, we can query the stochastic gradient oracles in parallel with some extra memory overhead, resulting in a complexity comparable to its conventional counterpart. To demonstrate the efficacy of the proposed algorithm, we conduct tests by training generative adversarial networks.

Kun Yuan, Sulaiman A. Alghunaim, Xinmeng Huang

We consider the decentralized stochastic optimization problems, where a network of $n$ nodes, each owning a local cost function, cooperate to find a minimizer of the globally-averaged cost. A widely studied decentralized algorithm for this problem is decentralized SGD (D-SGD), in which each node averages only with its neighbors. D-SGD is efficient in single-iteration communication, but it is very sensitive to the network topology. For smooth objective functions, the transient stage (which measures the number of iterations the algorithm has to experience before achieving the linear speedup stage) of D-SGD is on the order of $Ω(n/(1-β)^2)$ and $Ω(n^3/(1-β)^4)$ for strongly and generally convex cost functions, respectively, where $1-β\in (0,1)$ is a topology-dependent quantity that approaches $0$ for a large and sparse network. Hence, D-SGD suffers from slow convergence for large and sparse networks. In this work, we study the non-asymptotic convergence property of the D$^2$/Exact-diffusion algorithm. By eliminating the influence of data heterogeneity between nodes, D$^2$/Exact-diffusion is shown to have an enhanced transient stage that is on the order of $\tildeΩ(n/(1-β))$ and $Ω(n^3/(1-β)^2)$ for strongly and generally convex cost functions, respectively. Moreover, when D$^2$/Exact-diffusion is implemented with gradient accumulation and multi-round gossip communications, its transient stage can be further improved to $\tildeΩ(1/(1-β)^{\frac{1}{2}})$ and $\tildeΩ(n/(1-β))$ for strongly and generally convex cost functions, respectively. These established results for D$^2$/Exact-Diffusion have the best (i.e., weakest) dependence on network topology to our knowledge compared to existing decentralized algorithms. We also conduct numerical simulations to validate our theories.

Sulaiman A. Alghunaim, Ming Yan, Ali H. Sayed

This work studies multi-agent sharing optimization problems with the objective function being the sum of smooth local functions plus a convex (possibly non-smooth) function coupling all agents. This scenario arises in many machine learning and engineering applications, such as regression over distributed features and resource allocation. We reformulate this problem into an equivalent saddle-point problem, which is amenable to decentralized solutions. We then propose a proximal primal-dual algorithm and establish its linear convergence to the optimal solution when the local functions are strongly-convex. To our knowledge, this is the first linearly convergent decentralized algorithm for multi-agent sharing problems with a general convex (possibly non-smooth) coupling function.

Kun Yuan, Sulaiman A. Alghunaim, Bicheng Ying et al.

Various bias-correction methods such as EXTRA, gradient tracking methods, and exact diffusion have been proposed recently to solve distributed {\em deterministic} optimization problems. These methods employ constant step-sizes and converge linearly to the {\em exact} solution under proper conditions. However, their performance under stochastic and adaptive settings is less explored. It is still unknown {\em whether}, {\em when} and {\em why} these bias-correction methods can outperform their traditional counterparts (such as consensus and diffusion) with noisy gradient and constant step-sizes. This work studies the performance of exact diffusion under the stochastic and adaptive setting, and provides conditions under which exact diffusion has superior steady-state mean-square deviation (MSD) performance than traditional algorithms without bias-correction. In particular, it is proven that this superiority is more evident over sparsely-connected network topologies such as lines, cycles, or grids. Conditions are also provided under which exact diffusion method match or may even degrade the performance of traditional methods. Simulations are provided to validate the theoretical findings.