arXiv subjects

Sulaiman A. Alghunaim

Publications and source records attributed to Sulaiman A. Alghunaim.

At least 19 recordsLinked to original sources

arXiv2026

Achieving Linear Speedup with ProxSkip in Distributed Stochastic Optimization

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 decentralized ProxSkip and answer these questions affirmatively. We provide a unified convergence analysis for stochastic non-convex, convex, and strongly convex problems, revealing how gradient noise, local updates, network connectivity, and data heterogeneity jointly determine the convergence behavior. To the best of our knowledge, this is the first analysis showing that decentralized ProxSkip achieves linear speedup in the number of nodes under stochastic gradients. Moreover, our results demonstrate that local updates can effectively reduce communication frequency and improve communication efficiency.

cs.LG↗

arXiv2026

CED-EF: Compressed Exact Diffusion with Error Feedback for Multi-Agent Learning

We study decentralized stochastic optimization over a network of $N$ agents under compressed communication. We propose CED-EF, an exact diffusion-based method with error feedback that directly accommodates biased $δ$-contractive compressors while communicating one compressed model-sized vector per node per iteration. For smooth nonconvex objectives with unbiased stochastic gradients whose variance is bounded by $σ^2$, where $σ\geq0$, we establish a convergence rate whose leading stochastic term is $\mathcal O(σ/\sqrt{NK})$. For $σ>0$, the dominant dependence of the corresponding transient time on the number of agents, compression level, and spectral gap $Δ_λ$ is $\mathcal O(N^3/(δ^4Δ_λ^4))$, with fixed problem-dependent factors suppressed. Under the Polyak--Łojasiewicz condition, CED-EF attains a leading stochastic term $\widetilde{\mathcal O}(σ^2/(NK))$ with transient time on the order of $\widetilde{\mathcal O}(N/(δ^2Δ_λ^2))$. These dependencies improve the compression and/or network dependence of existing results. Numerical experiments on least-squares and logistic-regression problems illustrate the performance advantages of CED-EF.

math.OC↗

arXiv2026

Stochastic Gradient Tracking over Time-Varying Networks: One-Step Lyapunov Analysis

We study decentralized stochastic gradient tracking over a time-varying network of $N$ agents under a uniform window-mixing condition. Products of $τ$ consecutive doubly stochastic mixing matrices contract disagreement by a factor $λ<1$, although individual matrices need not contract disagreement strictly and individual communication graphs may be disconnected. We construct a time-varying quadratic norm that turns this window contraction into an exact one-step Lyapunov identity. This leads to coupled one-step recursions for the centroid and disagreement errors, without unrolling the dynamics over communication windows. For smooth strongly convex objectives, the leading stochastic term is $\widetilde{\mathcal O}(1/(NK))$; for smooth convex objectives, it is $\mathcal O(1/\sqrt{NK})$. Both match their centralized mini-batch counterparts and yield linear speedup after a network-dependent transient.

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arXiv2026

Accelerated Stochastic Min-Max Optimization Based on Bias-corrected Momentum

Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least $\mathcal{O}(\varepsilon^{-4})$ sample complexity to find an $\varepsilon$-stationary point. Some works indicate that this complexity can be improved to $\mathcal{O}(\varepsilon^{-3})$ when the stochastic loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions, remains open. 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 proposed method is validated through applications to robust logistic regression and robust adaptive cruise control.

cs.LG↗

arXiv2026

Communication-Efficient Decentralized Stochastic Minimax Optimization

In this work, we study decentralized stochastic nonconvex Polyak--Łojasiewicz minimax problems and propose a communication-efficient algorithm. Motivated by the efficiency of local SGD in federated learning, we investigate decentralized minimax algorithms that perform multiple local updates between gossip rounds to improve communication efficiency. Existing results show that the local decentralized gradient descent-ascent algorithm requires an excessive number of local updates, on the order of $\tilde{\mathcal{O}}(κ^2\varepsilon^{-2})$ per communication round, to achieve the communication complexity $\tilde{\mathcal{O}}(κ^3\varepsilon^{-2})$, where $\varepsilon$ denotes the target accuracy and $κ>1$ is the condition number. However, such a large number of local updates can be impractical: it can underexploit available communication resources and exacerbate local drift, as agents may move toward their own local optima. To address this limitation, we develop a local decentralized minimax method that integrates accelerated momentum with local updates. Our method reduces the number of local updates to $\tilde{\mathcal{O}}(κ\varepsilon^{-1})$ per communication round while achieving the best-known communication complexity $\mathcal{O}(κ^2\varepsilon^{-2})$. Compared with the local gradient descent-ascent method, the proposed method also achieves an enhanced sample complexity. Experiments on robust logistic regression with real-world datasets demonstrate that our method achieves superior communication efficiency over several existing baselines.

math.OC↗

arXiv2025

DAMA: A Unified Accelerated Approach for Decentralized Nonconvex Minimax Optimization-Part I: Algorithm Development and Results

In this work and its accompanying Part II [1], we develop an accelerated algorithmic framework, DAMA (Decentralized Accelerated Minimax Approach), for nonconvex Polyak-Lojasiewicz minimax optimization over decentralized multi-agent networks. Our approach integrates online and offline stochastic minimax algorithms with various decentralized learning strategies, yielding a versatile framework with broader flexibility than existing methods. Our unification is threefold: (i) we propose a unified decentralized learning strategy for minimax optimization that subsumes existing bias-correction techniques, such as gradient tracking, while introducing new variants that achieve tighter network-dependent bounds; (ii) we introduce a probabilistic gradient estimator, GRACE (Gradient Acceleration Estimator), which unifies momentum-based methods and loopless variance-reduction techniques for constructing accelerated gradients within DAMA, and is broadly applicable to general stochastic optimization problems; and (iii) we develop a unified analytical framework that establishes a general performance bound for DAMA, achieving state-of-the-art results with the best-known sample complexity. To the best of our knowledge, DAMA is the first framework to achieve a multi-level unification of decentralized learning strategies and accelerated gradient techniques. This work focuses on algorithm development and the main results, while Part II provides the theoretical analysis that substantiates these results and presents empirical validation across diverse network topologies using synthetic and real-world datasets.

math.OC↗

arXiv2025

DAMA: A Unified Accelerated Approach for Decentralized Nonconvex Minimax Optimization-Part II: Convergence and Performance Analyses

In Part I of this work [1], we developed an accelerated algorithmic framework, DAMA (Decentralized Accelerated Minimax Approach), for nonconvex Polyak-Lojasiewicz (PL) minimax optimization over decentralized multi-agent networks. To further enhance convergence in online and offline scenarios, Part I of this work [1] also proposed a novel accelerated gradient estimator, namely, GRACE (GRadient ACceleration Estimator), which unifies several momentum-based methods (e.g., STORM) and loopless variance-reduction techniques (e.g., PAGE, Loopless SARAH), thereby enabling accelerated gradient updates within DAMA. Part I reported a unified performance bound for DAMA and refined guarantees for specific algorithmic instances, demonstrating the superior performance of several new variants on sparsely connected networks. In this Part II, we focus on the convergence and performance bounds that substantiate the main results presented in Part I [1]. In particular, we establish a unified performance bound for DAMA using the transformed recursion derived in Part I and subsequently refine this bound for its various special cases.

math.OC↗

arXiv2024

Diffusion Stochastic Optimization for Min-Max Problems

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.

cs.LG↗

arXiv2023

Local Exact-Diffusion for Decentralized Optimization and Learning

Distributed optimization methods with local updates have recently attracted a lot of attention due to their potential to reduce the communication cost of distributed methods. In these algorithms, a collection of nodes performs several local updates based on their local data, and then they communicate with each other to exchange estimate information. While there have been many studies on distributed local methods with centralized network connections, there has been less work on decentralized networks. In this work, we propose and investigate a locally updated decentralized method called Local Exact-Diffusion (LED). We establish the convergence of LED in both convex and nonconvex settings for the stochastic online setting. Our convergence rate improves over the rate of existing decentralized methods. When we specialize the network to the centralized case, we recover the state-of-the-art bound for centralized methods. We also link LED to several other independently studied distributed methods, including Scaffnew, FedGate, and VRL-SGD. Additionally, we numerically investigate the benefits of local updates for decentralized networks and demonstrate the effectiveness of the proposed method.

math.OC↗

arXiv2023

An Enhanced Gradient-Tracking Bound for Distributed Online Stochastic Convex Optimization

Gradient-tracking (GT) based decentralized methods have emerged as an effective and viable alternative method to decentralized (stochastic) gradient descent (DSGD) when solving distributed online stochastic optimization problems. Initial studies of GT methods implied that GT methods have worse network dependent rate than DSGD, contradicting experimental results. This dilemma has recently been resolved, and tighter rates for GT methods have been established, which improves upon DSGD. In this work, we establish more enhanced rates for GT methods under the online stochastic convex settings. We present an alternative approach for analyzing GT methods for convex problems and over static graphs. When compared to previous analyses, this approach allows us to establish enhanced network dependent rates.

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arXiv2022

On the Performance of Gradient Tracking with Local Updates

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).

math.OC↗

arXiv2022

A Unified and Refined Convergence Analysis for Non-Convex Decentralized Learning

We study the consensus decentralized optimization problem where the objective function is the average of $n$ agents private non-convex cost functions; moreover, the agents can only communicate to their neighbors on a given network topology. The stochastic learning setting is considered in this paper where each agent can only access a noisy estimate of its gradient. Many decentralized methods can solve such problem including EXTRA, Exact-Diffusion/D$^2$, and gradient-tracking. Unlike the famed DSGD algorithm, these methods have been shown to be robust to the heterogeneity across the local cost functions. However, the established convergence rates for these methods indicate that their sensitivity to the network topology is worse than DSGD. Such theoretical results imply that these methods can perform much worse than DSGD over sparse networks, which, however, contradicts empirical experiments where DSGD is observed to be more sensitive to the network topology. In this work, we study a general stochastic unified decentralized algorithm (SUDA) that includes the above methods as special cases. We establish the convergence of SUDA under both non-convex and the Polyak-Lojasiewicz condition settings. Our results provide improved network topology dependent bounds for these methods (such as Exact-Diffusion/D$^2$ and gradient-tracking) compared with existing literature. Moreover, our results show that these methods are often less sensitive to the network topology compared to DSGD, which agrees with numerical experiments.

cs.DC↗

arXiv2022

Removing Data Heterogeneity Influence Enhances Network Topology Dependence of Decentralized SGD

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.

math.OC↗

arXiv2020

Decentralized Proximal Gradient Algorithms with Linear Convergence Rates

This work studies a class of non-smooth decentralized multi-agent optimization problems where the agents aim at minimizing a sum of local strongly-convex smooth components plus a common non-smooth term. We propose a general primal-dual algorithmic framework that unifies many existing state-of-the-art algorithms. We establish linear convergence of the proposed method to the exact solution in the presence of the non-smooth term. Moreover, for the more general class of problems with agent specific non-smooth terms, we show that linear convergence cannot be achieved (in the worst case) for the class of algorithms that uses the gradients and the proximal mappings of the smooth and non-smooth parts, respectively. We further provide a numerical counterexample that shows how some state-of-the-art algorithms fail to converge linearly for strongly-convex objectives and different local non-smooth terms.

math.OC↗

arXiv2020

A Multi-Agent Primal-Dual Strategy for Composite Optimization over Distributed Features

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.

math.OC↗

arXiv2020

Linear Convergence of Primal-Dual Gradient Methods and their Performance in Distributed Optimization

In this work, we revisit a classical incremental implementation of the primal-descent dual-ascent gradient method used for the solution of equality constrained optimization problems. We provide a short proof that establishes the linear (exponential) convergence of the algorithm for smooth strongly-convex cost functions and study its relation to the non-incremental implementation. We also study the effect of the augmented Lagrangian penalty term on the performance of distributed optimization algorithms for the minimization of aggregate cost functions over multi-agent networks.

math.OC↗

arXiv2019

A Proximal Diffusion Strategy for Multi-Agent Optimization with Sparse Affine Constraints

This work develops a proximal primal-dual decentralized strategy for multi-agent optimization problems that involve multiple coupled affine constraints, where each constraint may involve only a subset of the agents. The constraints are generally sparse, meaning that only a small subset of the agents are involved in them. This scenario arises in many applications including decentralized control formulations, resource allocation problems, and smart grids. Traditional decentralized solutions tend to ignore the structure of the constraints and lead to degraded performance. We instead develop a decentralized solution that exploits the sparsity structure. Under constant step-size learning, the asymptotic convergence of the proposed algorithm is established in the presence of non-smooth terms, and it occurs at a linear rate in the smooth case. We also examine how the performance of the algorithm is influenced by the sparsity of the constraints. Simulations illustrate the superior performance of the proposed strategy.

math.OC↗

arXiv2019

A Linearly Convergent Proximal Gradient Algorithm for Decentralized Optimization

Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based proximal decentralized methods are known to converge to the optimal solution with sublinear rates, and it remains unclear whether this family of methods can achieve global linear convergence. To tackle this problem, this work assumes the non-smooth regularization term is common across all networked agents, which is the case for many machine learning problems. Under this condition, we design a proximal gradient decentralized algorithm whose fixed point coincides with the desired minimizer. We then provide a concise proof that establishes its linear convergence. In the absence of the non-smooth term, our analysis technique covers the well known EXTRA algorithm and provides useful bounds on the convergence rate and step-size.

math.OC↗