Sarit Khirirat

Xiangjian Hou, Sarit Khirirat, Mohammad Yaqub et al.

Federated learning (FL) is a distributed machine learning (ML) framework where multiple clients collaborate to train a model without exposing their private data. FL involves cycles of local computations and bi-directional communications between the clients and server. To bolster data security during this process, FL algorithms frequently employ a differential privacy (DP) mechanism that introduces noise into each client's model updates before sharing. However, while enhancing privacy, the DP mechanism often hampers convergence performance. In this paper, we posit that an optimal balance exists between the number of local steps and communication rounds, one that maximizes the convergence performance within a given privacy budget. Specifically, we present a proof for the optimal number of local steps and communication rounds that enhance the convergence bounds of the DP version of the ScaffNew algorithm. Our findings reveal a direct correlation between the optimal number of local steps, communication rounds, and a set of variables, e.g the DP privacy budget and other problem parameters, specifically in the context of strongly convex optimization. We furthermore provide empirical evidence to validate our theoretical findings.

Egor Shulgin, Grigory Malinovsky, Sarit Khirirat et al.

Federated Learning (FL) enables collaborative training on decentralized data. Differential privacy (DP) is crucial for FL, but current private methods often rely on unrealistic assumptions (e.g., bounded gradients or heterogeneity), hindering practical application. Existing works that relax these assumptions typically neglect practical FL features, including multiple local updates and partial client participation. We introduce Fed-$α$-NormEC, the first differentially private FL framework providing provable convergence and DP guarantees under standard assumptions while fully supporting these practical features. Fed-$α$-NormE integrates local updates (full and incremental gradient steps), separate server and client stepsizes, and, crucially, partial client participation, which is essential for real-world deployment and vital for privacy amplification. Our theoretical guarantees are corroborated by experiments on private deep learning tasks.

Sarit Khirirat, Abdurakhmon Sadiev, Yury Demidovich et al.

The use of momentum in stochastic optimization algorithms has shown empirical success across a range of machine learning tasks. Recently, a new class of stochastic momentum algorithms has emerged within the Linear Minimization Oracle (LMO) framework--leading to state-of-the-art methods, such as Muon, Scion, and Gluon, that effectively solve deep neural network training problems. However, traditional stochastic momentum methods offer convergence guarantees no better than the ${O}(1/K^{1/4})$ rate. While several approaches--such as Hessian-Corrected Momentum (HCM)--have aimed to improve this rate, their theoretical results are generally restricted to the Euclidean norm setting. This limitation hinders their applicability in problems, where arbitrary norms are often required. In this paper, we extend the LMO-based framework by integrating HCM, and provide convergence guarantees under relaxed smoothness and arbitrary norm settings. We establish improved convergence rates of ${O}(1/K^{1/3})$ for HCM, which can adapt to the geometry of the problem and achieve a faster rate than traditional momentum. Experimental results on training Multi-Layer Perceptrons (MLPs) and Long Short-Term Memory (LSTM) networks verify our theoretical observations.

Sarit Khirirat, Abdurakhmon Sadiev, Artem Riabinin et al.

We provide the first proof of convergence for normalized error feedback algorithms across a wide range of machine learning problems. Despite their popularity and efficiency in training deep neural networks, traditional analyses of error feedback algorithms rely on the smoothness assumption that does not capture the properties of objective functions in these problems. Rather, these problems have recently been shown to satisfy generalized smoothness assumptions, and the theoretical understanding of error feedback algorithms under these assumptions remains largely unexplored. Moreover, to the best of our knowledge, all existing analyses under generalized smoothness either i) focus on single-node settings or ii) make unrealistically strong assumptions for distributed settings, such as requiring data heterogeneity, and almost surely bounded stochastic gradient noise variance. In this paper, we propose distributed error feedback algorithms that utilize normalization to achieve the $O(1/\sqrt{K})$ convergence rate for nonconvex problems under generalized smoothness. Our analyses apply for distributed settings without data heterogeneity conditions, and enable stepsize tuning that is independent of problem parameters. Additionally, we provide strong convergence guarantees of normalized error feedback algorithms for stochastic settings. Finally, we show that due to their larger allowable stepsizes, our new normalized error feedback algorithms outperform their non-normalized counterparts on various tasks, including the minimization of polynomial functions, logistic regression, and ResNet-20 training.

Ali Beikmohammadi, Sarit Khirirat, Sindri Magnússon

Reinforcement learning has recently gained unprecedented popularity, yet it still grapples with sample inefficiency. Addressing this challenge, federated reinforcement learning (FedRL) has emerged, wherein agents collaboratively learn a single policy by aggregating local estimations. However, this aggregation step incurs significant communication costs. In this paper, we propose CompFedRL, a communication-efficient FedRL approach incorporating both \textit{periodic aggregation} and (direct/error-feedback) compression mechanisms. Specifically, we consider compressed federated $Q$-learning with a generative model setup, where a central server learns an optimal $Q$-function by periodically aggregating compressed $Q$-estimates from local agents. For the first time, we characterize the impact of these two mechanisms (which have remained elusive) by providing a finite-time analysis of our algorithm, demonstrating strong convergence behaviors when utilizing either direct or error-feedback compression. Our bounds indicate improved solution accuracy concerning the number of agents and other federated hyperparameters while simultaneously reducing communication costs. To corroborate our theory, we also conduct in-depth numerical experiments to verify our findings, considering Top-$K$ and Sparsified-$K$ sparsification operators.

Ali Beikmohammadi, Sarit Khirirat, Sindri Magnússon

Data similarity assumptions have traditionally been relied upon to understand the convergence behaviors of federated learning methods. Unfortunately, this approach often demands fine-tuning step sizes based on the level of data similarity. When data similarity is low, these small step sizes result in an unacceptably slow convergence speed for federated methods. In this paper, we present a novel and unified framework for analyzing the convergence of federated learning algorithms without the need for data similarity conditions. Our analysis centers on an inequality that captures the influence of step sizes on algorithmic convergence performance. By applying our theorems to well-known federated algorithms, we derive precise expressions for three widely used step size schedules: fixed, diminishing, and step-decay step sizes, which are independent of data similarity conditions. Finally, we conduct comprehensive evaluations of the performance of these federated learning algorithms, employing the proposed step size strategies to train deep neural network models on benchmark datasets under varying data similarity conditions. Our findings demonstrate significant improvements in convergence speed and overall performance, marking a substantial advancement in federated learning research.

Ali Beikmohammadi, Sarit Khirirat, Sindri Magnússon

Parallel stochastic gradient methods are gaining prominence in solving large-scale machine learning problems that involve data distributed across multiple nodes. However, obtaining unbiased stochastic gradients, which have been the focus of most theoretical research, is challenging in many distributed machine learning applications. The gradient estimations easily become biased, for example, when gradients are compressed or clipped, when data is shuffled, and in meta-learning and reinforcement learning. In this work, we establish worst-case bounds on parallel momentum methods under biased gradient estimation on both general non-convex and $μ$-PL problems. Our analysis covers general distributed optimization problems, and we work out the implications for special cases where gradient estimates are biased, i.e. in meta-learning and when the gradients are compressed or clipped. Our numerical experiments verify our theoretical findings and show faster convergence performance of momentum methods than traditional biased gradient descent.

Abdurakhmon Sadiev, Yury Demidovich, Igor Sokolov et al.

Communication compression is essential for scalable distributed training of modern machine learning models, but it often degrades convergence due to the noise it introduces. Error Feedback (EF) mechanisms are widely adopted to mitigate this issue of distributed compression algorithms. Despite their popularity and training efficiency, existing distributed EF algorithms often require prior knowledge of problem parameters (e.g., smoothness constants) to fine-tune stepsizes. This limits their practical applicability especially in large-scale neural network training. In this paper, we study normalized error feedback algorithms that combine EF with normalized updates, various momentum variants, and parameter-agnostic, time-varying stepsizes, thus eliminating the need for problem-dependent tuning. We analyze the convergence of these algorithms for minimizing smooth functions, and establish parameter-agnostic complexity bounds that are close to the best-known bounds with carefully-tuned problem-dependent stepsizes. Specifically, we show that normalized EF21 achieve the convergence rate of near ${O}(1/T^{1/4})$ for Polyak's heavy-ball momentum, ${O}(1/T^{2/7})$ for Iterative Gradient Transport (IGT), and ${O}(1/T^{1/3})$ for STORM and Hessian-corrected momentum. Our results hold with decreasing stepsizes and small mini-batches. Finally, our empirical experiments confirm our theoretical insights.

Ali Beikmohammadi, Sarit Khirirat, Peter Richtárik et al.

Federated reinforcement learning (FedRL) enables collaborative learning while preserving data privacy by preventing direct data exchange between agents. However, many existing FedRL algorithms assume that all agents operate in identical environments, which is often unrealistic. In real-world applications, such as multi-robot teams, crowdsourced systems, and large-scale sensor networks, each agent may experience slightly different transition dynamics, leading to inherent model mismatches. In this paper, we first establish linear convergence guarantees for single-agent temporal difference learning (TD(0)) in policy evaluation and demonstrate that under a perturbed environment, the agent suffers a systematic bias that prevents accurate estimation of the true value function. This result holds under both i.i.d. and Markovian sampling regimes. We then extend our analysis to the federated TD(0) (FedTD(0)) setting, where multiple agents, each interacting with its own perturbed environment, periodically share value estimates to collaboratively approximate the true value function of a common underlying model. Our theoretical results indicate the impact of model mismatch, network connectivity, and mixing behavior on the convergence of FedTD(0). Empirical experiments corroborate our theoretical gains, highlighting that even moderate levels of information sharing significantly mitigate environment-specific errors.

Egor Shulgin, Sarit Khirirat, Peter Richtárik

Federated learning enables training machine learning models while preserving the privacy of participants. Surprisingly, there is no differentially private distributed method for smooth, non-convex optimization problems. The reason is that standard privacy techniques require bounding the participants' contributions, usually enforced via $\textit{clipping}$ of the updates. Existing literature typically ignores the effect of clipping by assuming the boundedness of gradient norms or analyzes distributed algorithms with clipping but ignores DP constraints. In this work, we study an alternative approach via $\textit{smoothed normalization}$ of the updates motivated by its favorable performance in the single-node setting. By integrating smoothed normalization with an error-feedback mechanism, we design a new distributed algorithm $α$-$\sf NormEC$. We prove that our method achieves a superior convergence rate over prior works. By extending $α$-$\sf NormEC$ to the DP setting, we obtain the first differentially private distributed optimization algorithm with provable convergence guarantees. Finally, our empirical results from neural network training indicate robust convergence of $α$-$\sf NormEC$ across different parameter settings.

Sarit Khirirat, Eduard Gorbunov, Samuel Horváth et al.

Motivated by the increasing popularity and importance of large-scale training under differential privacy (DP) constraints, we study distributed gradient methods with gradient clipping, i.e., clipping applied to the gradients computed from local information at the nodes. While gradient clipping is an essential tool for injecting formal DP guarantees into gradient-based methods [1], it also induces bias which causes serious convergence issues specific to the distributed setting. Inspired by recent progress in the error-feedback literature which is focused on taming the bias/error introduced by communication compression operators such as Top-$k$ [2], and mathematical similarities between the clipping operator and contractive compression operators, we design Clip21 -- the first provably effective and practically useful error feedback mechanism for distributed methods with gradient clipping. We prove that our method converges at the same $\mathcal{O}\left(\frac{1}{K}\right)$ rate as distributed gradient descent in the smooth nonconvex regime, which improves the previous best $\mathcal{O}\left(\frac{1}{\sqrt{K}}\right)$ rate which was obtained under significantly stronger assumptions. Our method converges significantly faster in practice than competing methods.

Sarit Khirirat, Sindri Magnússon, Arda Aytekin et al.

With the increasing scale of machine learning tasks, it has become essential to reduce the communication between computing nodes. Early work on gradient compression focused on the bottleneck between CPUs and GPUs, but communication-efficiency is now needed in a variety of different system architectures, from high-performance clusters to energy-constrained IoT devices. In the current practice, compression levels are typically chosen before training and settings that work well for one task may be vastly suboptimal for another dataset on another architecture. In this paper, we propose a flexible framework which adapts the compression level to the true gradient at each iteration, maximizing the improvement in the objective function that is achieved per communicated bit. Our framework is easy to adapt from one technology to the next by modeling how the communication cost depends on the compression level for the specific technology. Theoretical results and practical experiments indicate that the automatic tuning strategies significantly increase communication efficiency on several state-of-the-art compression schemes.

Dan Alistarh, Torsten Hoefler, Mikael Johansson et al.

Distributed training of massive machine learning models, in particular deep neural networks, via Stochastic Gradient Descent (SGD) is becoming commonplace. Several families of communication-reduction methods, such as quantization, large-batch methods, and gradient sparsification, have been proposed. To date, gradient sparsification methods - where each node sorts gradients by magnitude, and only communicates a subset of the components, accumulating the rest locally - are known to yield some of the largest practical gains. Such methods can reduce the amount of communication per step by up to three orders of magnitude, while preserving model accuracy. Yet, this family of methods currently has no theoretical justification. This is the question we address in this paper. We prove that, under analytic assumptions, sparsifying gradients by magnitude with local error correction provides convergence guarantees, for both convex and non-convex smooth objectives, for data-parallel SGD. The main insight is that sparsification methods implicitly maintain bounds on the maximum impact of stale updates, thanks to selection by magnitude. Our analysis and empirical validation also reveal that these methods do require analytical conditions to converge well, justifying existing heuristics.

Sarit Khirirat, Hamid Reza Feyzmahdavian, Mikael Johansson

Asynchronous computation and gradient compression have emerged as two key techniques for achieving scalability in distributed optimization for large-scale machine learning. This paper presents a unified analysis framework for distributed gradient methods operating with staled and compressed gradients. Non-asymptotic bounds on convergence rates and information exchange are derived for several optimization algorithms. These bounds give explicit expressions for step-sizes and characterize how the amount of asynchrony and the compression accuracy affect iteration and communication complexity guarantees. Numerical results highlight convergence properties of different gradient compression algorithms and confirm that fast convergence under limited information exchange is indeed possible.