arXiv subjects

Peter Richtárik

Publications and source records attributed to Peter Richtárik.

At least 37 records · Page 2Linked to original sources

arXiv2026

Rennala MVR: Improved Time Complexity for Parallel Stochastic Optimization via Momentum-Based Variance Reduction

Large-scale machine learning models are trained on clusters of machines that exhibit heterogeneous performance due to hardware variability, network delays, and system-level instabilities. In such environments, time complexity rather than iteration complexity becomes the relevant performance metric for optimization algorithms. Recent work by Tyurin and Richtárik (2023) established the first time complexity analysis for parallel first-order stochastic optimization, proposing Rennala SGD as a time-optimal method for smooth nonconvex optimization. However, Rennala SGD is fundamentally a modification of SGD, and variance reduction techniques are known to improve the iteration complexity of SGD. In this work, we investigate whether variance reduction can also improve time complexity in heterogeneous systems. We show that, under a mean-squared smoothness assumption, variance reduction can improve time complexity in relevant parameter regimes. To this end, we propose Rennala MVR, a variance-reduced extension of Rennala SGD based on momentum-based variance reduction, and analyze its oracle and time complexity. We establish lower bounds for time complexity under these assumptions. On a stochastic quadratic benchmark, experiments with the exact method support the theory, while neural-network experiments with a practical inexact variant show similar empirical gains over Rennala SGD.

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arXiv2026

Tighter Performance Theory of FedExProx

We revisit FedExProx - a recently proposed distributed optimization method designed to enhance convergence properties of parallel proximal algorithms via extrapolation. In the process, we uncover a surprising flaw: its known theoretical guarantees on quadratic optimization tasks are no better than those offered by the vanilla Gradient Descent (GD) method. Motivated by this observation, we develop a novel analysis framework, establishing a tighter linear convergence rate for non-strongly convex quadratic problems. By incorporating both computation and communication costs, we demonstrate that FedExProx can indeed provably outperform GD, in stark contrast to the original analysis. Furthermore, we consider partial participation scenarios and analyze two adaptive extrapolation strategies - based on gradient diversity and Polyak stepsizes - again significantly outperforming previous results. Moving beyond quadratics, we extend the applicability of our analysis to general functions satisfying the Polyak-Lojasiewicz condition, outperforming the previous strongly convex analysis while operating under weaker assumptions. Backed by empirical results, our findings point to a new and stronger potential of FedExProx, paving the way for further exploration of the benefits of extrapolation in federated learning.

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arXiv2026

Broximal Alignment for Global Non-Convex Optimization

Most non-convex optimization theory is built around gradient dynamics, leaving global convergence largely unexplored. The dominant paradigm focuses on stationarity, certifying only that the gradient norm vanishes, which is often a weak proxy for actual optimization success. In practice, gradient norms can stagnate or even increase during training, and stationary points may be far from global solutions. In this work, we propose a new framework for global non-convex optimization that avoids gradient-based reasoning altogether. We revisit the Ball Proximal Point Method (BPM), a trust-region-style algorithm introduced by Gruntkowska et al. (2025), and propose a novel structural condition - Broximal Alignment - under which BPM provably converges to a global minimizer. Our condition requires no convexity, smoothness, or Lipschitz assumptions, and it permits multiple and disconnected global minima as well as non-optimal local minima. We show that this class generalizes standard non-convex frameworks such as quasiconvexity, star convexity, quasar convexity, and the aiming condition. Our results provide a new conceptual foundation for global non-convex optimization beyond stationarity.

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arXiv2026

Communication-Efficient Gluon in Federated Learning

Recent developments have shown that Muon-type optimizers based on linear minimization oracles (LMOs) over non-Euclidean norm balls have the potential to get superior practical performance than Adam-type methods in the training of large language models. Since large-scale neural networks are trained across massive machines, communication cost becomes the bottleneck. To address this bottleneck, we investigate Gluon, which is an extension of Muon under the more general layer-wise $(L^0, L^1)$-smooth setting, with both unbiased and contraction compressors. In order to reduce the compression error, we employ the variance reduced technique in SARAH in our compressed methods. The convergence rates and improved communication cost are achieved under certain conditions. As a byproduct, a new variance reduced algorithm with faster convergence rate than Gluon is obtained. We also incorporate momentum variance reduction (MVR) to these compressed algorithms and comparable communication cost is derived under weaker conditions when $L_i^1 \neq 0$. Finally, several numerical experiments are conducted to verify the superior performance of our compressed algorithms in terms of communication cost.

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arXiv2026

Byzantine-Robust and Differentially Private Federated Optimization under Weaker Assumptions

Federated Learning (FL) enables heterogeneous clients to collaboratively train a shared model without centralizing their raw data, offering an inherent level of privacy. However, gradients and model updates can still leak sensitive information, while malicious servers may mount adversarial attacks such as Byzantine manipulation. These vulnerabilities highlight the need to address differential privacy (DP) and Byzantine robustness within a unified framework. Existing approaches, however, often rely on unrealistic assumptions such as bounded gradients, require auxiliary server-side datasets, or fail to provide convergence guarantees. We address these limitations by proposing Byz-Clip21-SGD2M, a new algorithm that integrates robust aggregation with double momentum and carefully designed clipping. We prove high-probability convergence guarantees under standard $L$-smoothness and $σ$-sub-Gaussian gradient noise assumptions, thereby relaxing conditions that dominate prior work. Our analysis recovers state-of-the-art convergence rates in the absence of adversaries and improves utility guarantees under Byzantine and DP settings. Empirical evaluations on CNN and MLP models trained on MNIST further validate the effectiveness of our approach.

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arXiv2026

Stabilized Proximal Point Method via Trust Region Control

The Proximal Point Method (PPM) (Rockafellar, 1976) is a fundamental tool for nonsmooth convex optimization. However, its convergence is not linear under general convexity in the absence of strong convexity or other structural assumptions. To address this limitation, we study a trust-region stabilized proximal point scheme in which each proximal update is computed over a localized feasible region. We show that this simple stabilization enforces non-vanishing steps and yields a linear decrease in objective values outside any prescribed neighborhood, without assuming smoothness or strong convexity. Our analysis identifies a displacement condition as the key driver of linear descent and provides two complementary parameter regimes to guarantee it: fixing the trust-region radius and choosing the regularization properly, or fixing the regularization and selecting radii via a uniform displacement lower bound. We further give explicit characterization of the linear regime conditions respectively, and prove that the trust-region is redundant under strong convexity, Finally, we establish an exact equivalence with the Broximal Point Method (BPM) (Gruntkowska et al., 2025) in the active constraint regime.

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arXiv2026

CompressedScaffnew: The First Theoretical Double Acceleration of Communication from Local Training and Compression in Distributed Optimization

In distributed optimization, a large number of machines alternate between local computations and communication with a coordinating server. Communication, which can be slow and costly, is the main bottleneck in this setting. To reduce this burden and therefore accelerate distributed gradient descent, two strategies are popular: 1) communicate less frequently; that is, perform several iterations of local computations between the communication rounds; and 2) communicate compressed information instead of full-dimensional vectors. We propose CompressedScaffnew, the first algorithm for distributed optimization that jointly harnesses these two strategies and converges linearly to an exact solution in the strongly convex setting, with a doubly accelerated rate: it benefits from the two acceleration mechanisms provided by local training and compression, namely a better dependency on the condition number of the functions and on the dimension of the model, respectively.

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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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arXiv2026

Revisiting Stochastic Proximal Point Methods: Generalized Smoothness and Similarity

The growing prevalence of nonsmooth optimization problems in machine learning has spurred significant interest in generalized smoothness assumptions. Among these, the (L0, L1)-smoothness assumption has emerged as one of the most prominent. While proximal methods are well-suited and effective for nonsmooth problems in deterministic settings, their stochastic counterparts remain underexplored. This work focuses on the stochastic proximal point method (SPPM), valued for its stability and minimal hyperparameter tuning-advantages often missing in stochastic gradient descent (SGD). We propose a novel phi-smoothness framework and provide a comprehensive analysis of SPPM without relying on traditional smoothness assumptions. Our results are highly general, encompassing existing findings as special cases. Furthermore, we examine SPPM under the widely adopted expected similarity assumption, thereby extending its applicability to a broader range of scenarios. Our theoretical contributions are illustrated and validated by practical experiments.

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arXiv2026

Ringleader ASGD: The First Asynchronous SGD with Optimal Time Complexity under Data Heterogeneity

Asynchronous stochastic gradient methods are central to scalable distributed optimization, particularly when devices differ in computational capabilities. Such settings arise naturally in federated learning, where training takes place on smartphones and other heterogeneous edge devices. In addition to varying computation speeds, these devices often hold data from different distributions. However, existing asynchronous SGD methods struggle in such heterogeneous settings and face two key limitations. First, many rely on unrealistic assumptions of similarity across workers' data distributions. Second, methods that relax this assumption still fail to achieve theoretically optimal performance under heterogeneous computation times. We introduce Ringleader ASGD, the first asynchronous SGD algorithm that attains the theoretical lower bounds for parallel first-order stochastic methods in the smooth nonconvex regime, thereby achieving optimal time complexity under data heterogeneity and without restrictive similarity assumptions. Our analysis further establishes that Ringleader ASGD remains optimal under arbitrary and even time-varying worker computation speeds, closing a fundamental gap in the theory of asynchronous optimization.

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arXiv2026

BiCoLoR: Communication-Efficient Optimization with Bidirectional Compression and Local Training

Slow and costly communication is often the main bottleneck in distributed optimization, especially in federated learning where it occurs over wireless networks. We introduce BiCoLoR, a communication-efficient optimization algorithm that combines two widely used and effective strategies: local training, which increases computation between communication rounds, and compression, which encodes high-dimensional vectors into short bitstreams. While these mechanisms have been combined before, compression has typically been applied only to uplink (client-to-server) communication, leaving the downlink (server-to-client) side unaddressed. In practice, however, both directions are costly. We propose BiCoLoR, the first algorithm to combine local training with bidirectional compression using arbitrary unbiased compressors. This joint design achieves accelerated complexity guarantees in both convex and strongly convex heterogeneous settings. Empirically, BiCoLoR outperforms existing algorithms and establishes a new standard in communication efficiency.

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arXiv2026

MAST: Model-Agnostic Sparsified Training

We introduce a novel optimization problem formulation that departs from the conventional way of minimizing machine learning model loss as a black-box function. Unlike traditional formulations, the proposed approach explicitly incorporates an initially pre-trained model and random sketch operators, allowing for sparsification of both the model and gradient during training. We establish the insightful properties of the proposed objective function and highlight its connections to the standard formulation. Furthermore, we present several variants of the Stochastic Gradient Descent (SGD) method adapted to the new problem formulation, including SGD with general sampling, a distributed version, and SGD with variance reduction techniques. We achieve tighter convergence rates and relax assumptions, bridging the gap between theoretical principles and practical applications, covering several important techniques such as Dropout and Sparse training. This work presents promising opportunities to enhance the theoretical understanding of model training through a sparsification-aware optimization approach.

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arXiv2025

First Provable Guarantees for Practical Private FL: Beyond Restrictive Assumptions

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.

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arXiv2025

Muon is Provably Faster with Momentum Variance Reduction

Recent empirical research has demonstrated that deep learning optimizers based on the linear minimization oracle (LMO) over specifically chosen Non-Euclidean norm balls, such as Muon and Scion, outperform Adam-type methods in the training of large language models. In this work, we show that such optimizers can be provably improved by replacing their vanilla momentum by momentum variance reduction (MVR). Instead of proposing and analyzing MVR variants of Muon and Scion separately, we incorporate MVR into the recently proposed Gluon framework, which captures Muon, Scion and other specific Non-Euclidean LMO-based methods as special cases, and at the same time works with a more general smoothness assumption which better captures the layer-wise structure of neural networks. In the non-convex case, we incorporate MVR into Gluon in three different ways. All of them improve the convergence rate from ${\cal O} (\frac{1}{K^{1/4}})$ to ${\cal O} (\frac{1}{K^{1/3}})$. Additionally, we provide improved rates in the star-convex case. Finally, we conduct several numerical experiments that verify the superior performance of our proposed algorithms in terms of iteration complexity.

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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

Beyond the Ideal: Analyzing the Inexact Muon Update

The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon's efficiency relies on fast, approximate orthogonalization, yet all prior theoretical work analyzes an idealized, computationally intractable version assuming exact SVD-based updates. This work moves beyond the ideal by providing the first analysis of the inexact orthogonalized update at Muon's core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure (e.g., the number of Newton-Schulz steps) from an implementation detail to a critical parameter that must be co-tuned with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.

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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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