arXiv subjects

Abdurakhmon Sadiev

Publications and source records attributed to Abdurakhmon Sadiev.

At least 19 recordsLinked to original sources

arXiv2026

A Nesterov-Accelerated Primal-Dual Splitting Algorithm for Convex Nonsmooth Optimization

We investigate the integration of Nesterov-type acceleration into primal-dual methods for structured convex optimization. While proximal splitting algorithms efficiently handle composite problems of the form $\min_x f(x) + g(x) + h(K x)$, accelerating their convergence with respect to the smooth term $f$ is notoriously challenging, due to the rotational dynamics in the primal-dual space. Our guiding principle is that Nesterov acceleration exploits the strong convexity of the dual problem. We first revisit accelerated proximal gradient descent and give a unified analysis of its convex and strongly convex regimes, including the weak convergence of its iterates. We then propose the Accelerated Proximal Alternating Predictor-Corrector algorithm (APAPC) for the setting where $g(x) = (μ_g/2)\|x\|^2$. APAPC is fully split: it only calls $\nabla f$, $K$, $K^*$ and the proximity operator of $h$, and never solves a linear system. From a single Lyapunov inequality, we derive optimal $O(1/t^2)$ convergence rates, as well as accelerated linear rates when $μ_g > 0$, in three regimes where the dual problem is strongly convex: when $h$ is smooth, when $K^*$ is bounded below, and for linearly constrained problems. In the last two regimes, the $O(1/t^2)$ rates without strong convexity of the primal problem are, to our knowledge, the first for fully split algorithms. Finally, leveraging recent results on accelerated gradient descent, we establish the weak convergence of the primal iterates and the strong convergence of the dual iterates to a saddle point.

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arXiv2026

A Unified Primal-Dual Recipe for Accelerating Three-Operator Splitting Methods

Composite optimization problems, formulated as the minimization of three functions, are ubiquitous in large-scale machine learning and signal processing. While state-of-the-art splitting methods such as Condat-Vũ (CV) [Condat, 2013, Vũ, 2013], Primal-Dual Davis-Yin (PDDY) [Salim et al., 2022b], and Primal-Dual Twice Reflected (PDTR) [Malitsky and Tam, 2026] are highly versatile, they inherently exhibit non-accelerated convergence rates. Existing accelerated primal-dual splitting results either focus on special structures like one of the functions being zero, or linearly constrained problems, or smooth regimes, or directly use Nesterov-type momentum. Our contribution is a unified Bregman primal-dual framework that yields four variants and a common Lyapunov analysis. By applying the Chambolle-Pock algorithm [Chambolle and Pock, 2011] to primal-dual reformulations, we systematically derive four novel accelerated algorithms: Accelerated Condat-Vũ (ACV-I and ACV-II) and Accelerated Primal-Dual Twice Reflected (APDTR-I and APDTR-II). Through a simplified Lyapunov-based analysis, we establish iteration complexities for both smooth and nonsmooth cases, successfully removing the restrictive assumptions restrictions required by prior works.

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arXiv2026

Ringmaster LMO: Asynchronous Linear Minimization Oracle Momentum Method

Muon has recently emerged as a strong alternative to AdamW for training neural networks, with encouraging large-scale pretraining results and growing evidence that matrix-structured updates can be faster in practice. Yet Muon, and more generally Linear Minimization Oracle (LMO) based methods, are typically used synchronously. This is problematic in heterogeneous distributed systems, where workers complete gradient computations at different speeds and synchronous training must repeatedly wait for slower workers. In this work, we introduce Ringmaster LMO, an asynchronous LMO-based momentum method for unconstrained stochastic nonconvex optimization. Our method builds on the delay-thresholding idea of Ringmaster ASGD. For SGD-type methods, Ringmaster ASGD achieves optimal time complexity by discarding overly stale gradients. Ringmaster LMO extends this mechanism to general LMO-based updates. We establish convergence guarantees under generalized $(L_0, L_1)$-smoothness and further develop a parameter-agnostic variant with decreasing stepsizes and adaptive delay thresholds. Finally, we translate our iteration guarantees into time complexity bounds under heterogeneous worker computation times. In the classical Euclidean smooth setting, these bounds recover the optimal time complexity of Ringmaster ASGD. Experiments on stochastic quadratic problems and NanoChat language-model pretraining show that the advantages of Ringmaster LMO grow with system heterogeneity and that the method outperforms strong synchronous and asynchronous baselines.

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arXiv2026

Tight Lower Bounds and Optimal Algorithms for Stochastic Nonconvex Optimization with Heavy-Tailed Noise

We study stochastic nonconvex optimization under heavy-tailed noise. In this setting, the stochastic gradients only have bounded $p$-th central moment ($p$-BCM) for some $p \in (1,2]$. Building on the foundational work of Arjevani et al. (2022) in stochastic optimization, we establish tight sample complexity lower bounds for all first-order methods under \emph{relaxed} mean-squared smoothness ($q$-WAS) and $δ$-similarity ($(q, δ)$-S) assumptions, allowing any exponent $q \in [1,2]$ instead of the standard $q = 2$. These results substantially broaden the scope of existing lower bounds. To complement them, we show that Normalized Stochastic Gradient Descent with Momentum Variance Reduction (NSGD-MVR), a known algorithm, matches these bounds in expectation. Beyond expectation guarantees, we introduce a new algorithm, Double-Clipped NSGD-MVR, which allows the derivation of high-probability convergence rates under weaker assumptions than in previous works. Finally, for second-order methods with stochastic Hessians satisfying bounded $q$-th central moment assumptions for some exponent $q \in [1, 2]$ (allowing $q \neq p$), we establish sharper lower bounds than previous works while improving over Sadiev et al. (2025) (where only $p = q$ is considered) and yielding stronger convergence exponents. Together, these results provide a nearly complete complexity characterization of stochastic nonconvex optimization in heavy-tailed regimes.

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arXiv2025

Better LMO-based Momentum Methods with Second-Order Information

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.

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arXiv2025

Improved Convergence in Parameter-Agnostic Error Feedback through Momentum

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.

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arXiv2025

Second-order Optimization under Heavy-Tailed Noise: Hessian Clipping and Sample Complexity Limits

Heavy-tailed noise is pervasive in modern machine learning applications, arising from data heterogeneity, outliers, and non-stationary stochastic environments. While second-order methods can significantly accelerate convergence in light-tailed or bounded-noise settings, such algorithms are often brittle and lack guarantees under heavy-tailed noise -- precisely the regimes where robustness is most critical. In this work, we take a first step toward a theoretical understanding of second-order optimization under heavy-tailed noise. We consider a setting where stochastic gradients and Hessians have only bounded $p$-th moments, for some $p\in (1,2]$, and establish tight lower bounds on the sample complexity of any second-order method. We then develop a variant of normalized stochastic gradient descent that leverages second-order information and provably matches these lower bounds. To address the instability caused by large deviations, we introduce a novel algorithm based on gradient and Hessian clipping, and prove high-probability upper bounds that nearly match the fundamental limits. Our results provide the first comprehensive sample complexity characterization for second-order optimization under heavy-tailed noise. This positions Hessian clipping as a robust and theoretically sound strategy for second-order algorithm design in heavy-tailed regimes.

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arXiv2025

Bernoulli-LoRA: A Theoretical Framework for Randomized Low-Rank Adaptation

Parameter-efficient fine-tuning (PEFT) has emerged as a crucial approach for adapting large foundational models to specific tasks, particularly as model sizes continue to grow exponentially. Among PEFT methods, Low-Rank Adaptation (LoRA) (arXiv:2106.09685) stands out for its effectiveness and simplicity, expressing adaptations as a product of two low-rank matrices. While extensive empirical studies demonstrate LoRA's practical utility, theoretical understanding of such methods remains limited. Recent work on RAC-LoRA (arXiv:2410.08305) took initial steps toward rigorous analysis. In this work, we introduce Bernoulli-LoRA, a novel theoretical framework that unifies and extends existing LoRA approaches. Our method introduces a probabilistic Bernoulli mechanism for selecting which matrix to update. This approach encompasses and generalizes various existing update strategies while maintaining theoretical tractability. Under standard assumptions from non-convex optimization literature, we analyze several variants of our framework: Bernoulli-LoRA-GD, Bernoulli-LoRA-SGD, Bernoulli-LoRA-PAGE, Bernoulli-LoRA-MVR, Bernoulli-LoRA-QGD, Bernoulli-LoRA-MARINA, and Bernoulli-LoRA-EF21, establishing convergence guarantees for each variant. Additionally, we extend our analysis to convex non-smooth functions, providing convergence rates for both constant and adaptive (Polyak-type) stepsizes. Through extensive experiments on various tasks, we validate our theoretical findings and demonstrate the practical efficacy of our approach. This work is a step toward developing theoretically grounded yet practically effective PEFT methods.

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arXiv2024

Differentially Private Random Block Coordinate Descent

Coordinate Descent (CD) methods have gained significant attention in machine learning due to their effectiveness in solving high-dimensional problems and their ability to decompose complex optimization tasks. However, classical CD methods were neither designed nor analyzed with data privacy in mind, a critical concern when handling sensitive information. This has led to the development of differentially private CD methods, such as DP-CD (Differentially Private Coordinate Descent) proposed by Mangold et al. (ICML 2022), yet a disparity remains between non-private CD and DP-CD methods. In our work, we propose a differentially private random block coordinate descent method that selects multiple coordinates with varying probabilities in each iteration using sketch matrices. Our algorithm generalizes both DP-CD and the classical DP-SGD (Differentially Private Stochastic Gradient Descent), while preserving the same utility guarantees. Furthermore, we demonstrate that better utility can be achieved through importance sampling, as our method takes advantage of the heterogeneity in coordinate-wise smoothness constants, leading to improved convergence rates.

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arXiv2024

Error Feedback under $(L_0,L_1)$-Smoothness: Normalization and Momentum

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.

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arXiv2024

High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise

High-probability analysis of stochastic first-order optimization methods under mild assumptions on the noise has been gaining a lot of attention in recent years. Typically, gradient clipping is one of the key algorithmic ingredients to derive good high-probability guarantees when the noise is heavy-tailed. However, if implemented naïvely, clipping can spoil the convergence of the popular methods for composite and distributed optimization (Prox-SGD/Parallel SGD) even in the absence of any noise. Due to this reason, many works on high-probability analysis consider only unconstrained non-distributed problems, and the existing results for composite/distributed problems do not include some important special cases (like strongly convex problems) and are not optimal. To address this issue, we propose new stochastic methods for composite and distributed optimization based on the clipping of stochastic gradient differences and prove tight high-probability convergence results (including nearly optimal ones) for the new methods. Using similar ideas, we also develop new methods for composite and distributed variational inequalities and analyze the high-probability convergence of these methods.

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arXiv2024

SPAM: Stochastic Proximal Point Method with Momentum Variance Reduction for Non-convex Cross-Device Federated Learning

Cross-device training is a crucial subfield of federated learning, where the number of clients can reach into the billions. Standard approaches and local methods are prone to issues such as client drift and insensitivity to data similarities. We propose a novel algorithm (SPAM) for cross-device federated learning with non-convex losses, which solves both issues. We provide sharp analysis under second-order (Hessian) similarity, a condition satisfied by a variety of machine learning problems in practice. Additionally, we extend our results to the partial participation setting, where a cohort of selected clients communicate with the server at each communication round. Our method is the first in its kind, that does not require the smoothness of the objective and provably benefits from clients having similar data.

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arXiv2024

A Unified Theory of Stochastic Proximal Point Methods without Smoothness

This paper presents a comprehensive analysis of a broad range of variations of the stochastic proximal point method (SPPM). Proximal point methods have attracted considerable interest owing to their numerical stability and robustness against imperfect tuning, a trait not shared by the dominant stochastic gradient descent (SGD) algorithm. A framework of assumptions that we introduce encompasses methods employing techniques such as variance reduction and arbitrary sampling. A cornerstone of our general theoretical approach is a parametric assumption on the iterates, correction and control vectors. We establish a single theorem that ensures linear convergence under this assumption and the $μ$-strong convexity of the loss function, and without the need to invoke smoothness. This integral theorem reinstates best known complexity and convergence guarantees for several existing methods which demonstrates the robustness of our approach. We expand our study by developing three new variants of SPPM, and through numerical experiments we elucidate various properties inherent to them.

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arXiv2024

Stochastic Proximal Point Methods for Monotone Inclusions under Expected Similarity

Monotone inclusions have a wide range of applications, including minimization, saddle-point, and equilibria problems. We introduce new stochastic algorithms, with or without variance reduction, to estimate a root of the expectation of possibly set-valued monotone operators, using at every iteration one call to the resolvent of a randomly sampled operator. We also introduce a notion of similarity between the operators, which holds even for discontinuous operators. We leverage it to derive linear convergence results in the strongly monotone setting.

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arXiv2024

Decentralized Personalized Federated Learning for Min-Max Problems

Personalized Federated Learning (PFL) has witnessed remarkable advancements, enabling the development of innovative machine learning applications that preserve the privacy of training data. However, existing theoretical research in this field has primarily focused on distributed optimization for minimization problems. This paper is the first to study PFL for saddle point problems encompassing a broader range of optimization problems, that require more than just solving minimization problems. In this work, we consider a recently proposed PFL setting with the mixing objective function, an approach combining the learning of a global model together with locally distributed learners. Unlike most previous work, which considered only the centralized setting, we work in a more general and decentralized setup that allows us to design and analyze more practical and federated ways to connect devices to the network. We proposed new algorithms to address this problem and provide a theoretical analysis of the smooth (strongly) convex-(strongly) concave saddle point problems in stochastic and deterministic cases. Numerical experiments for bilinear problems and neural networks with adversarial noise demonstrate the effectiveness of the proposed methods.

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arXiv2024

Stochastic Gradient Methods with Preconditioned Updates

This work considers the non-convex finite sum minimization problem. There are several algorithms for such problems, but existing methods often work poorly when the problem is badly scaled and/or ill-conditioned, and a primary goal of this work is to introduce methods that alleviate this issue. Thus, here we include a preconditioner based on Hutchinson's approach to approximating the diagonal of the Hessian, and couple it with several gradient-based methods to give new scaled algorithms: Scaled SARAH and Scaled L-SVRG. Theoretical complexity guarantees under smoothness assumptions are presented. We prove linear convergence when both smoothness and the PL condition are assumed. Our adaptively scaled methods use approximate partial second-order curvature information and, therefore, can better mitigate the impact of badly scaled problems. This improved practical performance is demonstrated in the numerical experiments also presented in this work.

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arXiv2023

High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance

During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central $α$-th moment for $α\in (1,2]$ in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization.

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arXiv2023

Optimal Algorithms for Decentralized Stochastic Variational Inequalities

Variational inequalities are a formalism that includes games, minimization, saddle point, and equilibrium problems as special cases. Methods for variational inequalities are therefore universal approaches for many applied tasks, including machine learning problems. This work concentrates on the decentralized setting, which is increasingly important but not well understood. In particular, we consider decentralized stochastic (sum-type) variational inequalities over fixed and time-varying networks. We present lower complexity bounds for both communication and local iterations and construct optimal algorithms that match these lower bounds. Our algorithms are the best among the available literature not only in the decentralized stochastic case, but also in the decentralized deterministic and non-distributed stochastic cases. Experimental results confirm the effectiveness of the presented algorithms.

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