arXiv subjects

Peter Richtárik

Publications and source records attributed to Peter Richtárik.

At least 73 records · Page 4Linked to original sources

arXiv2025

LoCoDL: Communication-Efficient Distributed Learning with Local Training and Compression

In Distributed optimization and Learning, and even more in the modern framework of federated learning, communication, which is slow and costly, is critical. We introduce LoCoDL, a communication-efficient algorithm that leverages the two popular and effective techniques of Local training, which reduces the communication frequency, and Compression, in which short bitstreams are sent instead of full-dimensional vectors of floats. LoCoDL works with a large class of unbiased compressors that includes widely-used sparsification and quantization methods. LoCoDL provably benefits from local training and compression and enjoys a doubly-accelerated communication complexity, with respect to the condition number of the functions and the model dimension, in the general heterogenous regime with strongly convex functions. This is confirmed in practice, with LoCoDL outperforming existing algorithms.

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arXiv2025

Sparse-ProxSkip: Accelerated Sparse-to-Sparse Training in Federated Learning

In Federated Learning (FL), both client resource constraints and communication costs pose major problems for training large models. In the centralized setting, sparse training addresses resource constraints, while in the distributed setting, local training addresses communication costs. Recent work has shown that local training provably improves communication complexity through acceleration. In this work we show that in FL, naive integration of sparse training and acceleration fails, and we provide theoretical and empirical explanations of this phenomenon. We introduce Sparse-ProxSkip, addressing the issue and implementing the efficient technique of Straight-Through Estimator pruning into sparse training. We demonstrate the performance of Sparse-ProxSkip in extensive experiments.

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arXiv2025

Smoothed Normalization for Efficient Distributed Private Optimization

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.

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arXiv2025

A Novel Unified Parametric Assumption for Nonconvex Optimization

Nonconvex optimization is central to modern machine learning, but the general framework of nonconvex optimization yields weak convergence guarantees that are too pessimistic compared to practice. On the other hand, while convexity enables efficient optimization, it is of limited applicability to many practical problems. To bridge this gap and better understand the practical success of optimization algorithms in nonconvex settings, we introduce a novel unified parametric assumption. Our assumption is general enough to encompass a broad class of nonconvex functions while also being specific enough to enable the derivation of a unified convergence theorem for gradient-based methods. Notably, by tuning the parameters of our assumption, we demonstrate its versatility in recovering several existing function classes as special cases and in identifying functions amenable to efficient optimization. We derive our convergence theorem for both deterministic and stochastic optimization, and conduct experiments to verify that our assumption can hold practically over optimization trajectories.

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arXiv2025

Symmetric Pruning of Large Language Models

Popular post-training pruning methods such as Wanda and RIA are known for their simple, yet effective, designs that have shown exceptional empirical performance. Wanda optimizes performance through calibrated activations during pruning, while RIA emphasizes the relative, rather than absolute, importance of weight elements. Despite their practical success, a thorough theoretical foundation explaining these outcomes has been lacking. This paper introduces new theoretical insights that redefine the standard minimization objective for pruning, offering a deeper understanding of the factors contributing to their success. Our study extends beyond these insights by proposing complementary strategies that consider both input activations and weight significance. We validate these approaches through rigorous experiments, demonstrating substantial enhancements over existing methods. Furthermore, we introduce a novel training-free fine-tuning approach $R^2$-DSnoT that incorporates relative weight importance and a regularized decision boundary within a dynamic pruning-and-growing framework, significantly outperforming strong baselines and establishing a new state of the art.

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arXiv2024

On the Convergence of DP-SGD with Adaptive Clipping

Stochastic Gradient Descent (SGD) with gradient clipping is a powerful technique for enabling differentially private optimization. Although prior works extensively investigated clipping with a constant threshold, private training remains highly sensitive to threshold selection, which can be expensive or even infeasible to tune. This sensitivity motivates the development of adaptive approaches, such as quantile clipping, which have demonstrated empirical success but lack a solid theoretical understanding. This paper provides the first comprehensive convergence analysis of SGD with quantile clipping (QC-SGD). We demonstrate that QC-SGD suffers from a bias problem similar to constant-threshold clipped SGD but show how this can be mitigated through a carefully designed quantile and step size schedule. Our analysis reveals crucial relationships between quantile selection, step size, and convergence behavior, providing practical guidelines for parameter selection. We extend these results to differentially private optimization, establishing the first theoretical guarantees for DP-QC-SGD. Our findings provide theoretical foundations for widely used adaptive clipping heuristic and highlight open avenues for future research.

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arXiv2024

Methods for Convex $(L_0,L_1)$-Smooth Optimization: Clipping, Acceleration, and Adaptivity

Due to the non-smoothness of optimization problems in Machine Learning, generalized smoothness assumptions have been gaining a lot of attention in recent years. One of the most popular assumptions of this type is $(L_0,L_1)$-smoothness (Zhang et al., 2020). In this paper, we focus on the class of (strongly) convex $(L_0,L_1)$-smooth functions and derive new convergence guarantees for several existing methods. In particular, we derive improved convergence rates for Gradient Descent with (Smoothed) Gradient Clipping and for Gradient Descent with Polyak Stepsizes. In contrast to the existing results, our rates do not rely on the standard smoothness assumption and do not suffer from the exponential dependency from the initial distance to the solution. We also extend these results to the stochastic case under the over-parameterization assumption, propose a new accelerated method for convex $(L_0,L_1)$-smooth optimization, and derive new convergence rates for Adaptive Gradient Descent (Malitsky and Mishchenko, 2020).

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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

MARINA-P: Superior Performance in Non-smooth Federated Optimization with Adaptive Stepsizes

Non-smooth communication-efficient federated optimization is crucial for many machine learning applications, yet remains largely unexplored theoretically. Recent advancements have primarily focused on smooth convex and non-convex regimes, leaving a significant gap in understanding the non-smooth convex setting. Additionally, existing literature often overlooks efficient server-to-worker communication (downlink), focusing primarily on worker-to-server communication (uplink). We consider a setup where uplink costs are negligible and focus on optimizing downlink communication by improving state-of-the-art schemes like EF21-P (arXiv:2209.15218) and MARINA-P (arXiv:2402.06412) in the non-smooth convex setting. We extend the non-smooth convex theory of EF21-P [Anonymous, 2024], originally developed for single-node scenarios, to the distributed setting, and extend MARINA-P to the non-smooth convex setting. For both algorithms, we prove an optimal $O(1/\sqrt{T})$ convergence rate and establish communication complexity bounds matching classical subgradient methods. We provide theoretical guarantees under constant, decreasing, and adaptive (Polyak-type) stepsizes. Our experiments demonstrate that MARINA-P with correlated compressors outperforms other methods in both smooth non-convex and non-smooth convex settings. This work presents the first theoretical results for distributed non-smooth optimization with server-to-worker compression, along with comprehensive analysis for various stepsize schemes.

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arXiv2024

Speeding up Stochastic Proximal Optimization in the High Hessian Dissimilarity Setting

Stochastic proximal point methods have recently garnered renewed attention within the optimization community, primarily due to their desirable theoretical properties. Notably, these methods exhibit a convergence rate that is independent of the Lipschitz smoothness constants of the loss function, a feature often missing in the loss functions of modern ML applications. In this paper, we revisit the analysis of the Loopless Stochastic Variance Reduced Proximal Point Method (L-SVRP). Building on existing work, we establish a theoretical improvement in the convergence rate in scenarios characterized by high Hessian dissimilarity among the functions. Our concise analysis, which does not require smoothness assumptions, demonstrates a significant improvement in communication complexity compared to standard stochastic gradient descent.

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arXiv2024

Unlocking FedNL: Self-Contained Compute-Optimized Implementation

Federated Learning (FL) is an emerging paradigm that enables intelligent agents to collaboratively train Machine Learning (ML) models in a distributed manner, eliminating the need for sharing their local data. The recent work (arXiv:2106.02969) introduces a family of Federated Newton Learn (FedNL) algorithms, marking a significant step towards applying second-order methods to FL and large-scale optimization. However, the reference FedNL prototype exhibits three serious practical drawbacks: (i) It requires 4.8 hours to launch a single experiment in a sever-grade workstation; (ii) The prototype only simulates multi-node setting; (iii) Prototype integration into resource-constrained applications is challenging. To bridge the gap between theory and practice, we present a self-contained implementation of FedNL, FedNL-LS, FedNL-PP for single-node and multi-node settings. Our work resolves the aforementioned issues and reduces the wall clock time by x1000. With this FedNL outperforms alternatives for training logistic regression in a single-node -- CVXPY (arXiv:1603.00943), and in a multi-node -- Apache Spark (arXiv:1505.06807), Ray/Scikit-Learn (arXiv:1712.05889). Finally, we propose two practical-orientated compressors for FedNL - adaptive TopLEK and cache-aware RandSeqK, which fulfill the theory of FedNL.

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arXiv2024

Pushing the Limits of Large Language Model Quantization via the Linearity Theorem

Quantizing large language models has become a standard way to reduce their memory and computational costs. Typically, existing methods focus on breaking down the problem into individual layer-wise sub-problems, and minimizing per-layer error, measured via various metrics. Yet, this approach currently lacks theoretical justification and the metrics employed may be sub-optimal. In this paper, we present a "linearity theorem" establishing a direct relationship between the layer-wise $\ell_2$ reconstruction error and the model perplexity increase due to quantization. This insight enables two novel applications: (1) a simple data-free LLM quantization method using Hadamard rotations and MSE-optimal grids, dubbed HIGGS, which outperforms all prior data-free approaches such as the extremely popular NF4 quantized format, and (2) an optimal solution to the problem of finding non-uniform per-layer quantization levels which match a given compression constraint in the medium-bitwidth regime, obtained by reduction to dynamic programming. On the practical side, we demonstrate improved accuracy-compression trade-offs on Llama-3.1 and 3.2-family models, as well as on Qwen-family models. Further, we show that our method can be efficiently supported in terms of GPU kernels at various batch sizes, advancing both data-free and non-uniform quantization for LLMs.

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arXiv2024

Shadowheart SGD: Distributed Asynchronous SGD with Optimal Time Complexity Under Arbitrary Computation and Communication Heterogeneity

We consider nonconvex stochastic optimization problems in the asynchronous centralized distributed setup where the communication times from workers to a server can not be ignored, and the computation and communication times are potentially different for all workers. Using an unbiassed compression technique, we develop a new method-Shadowheart SGD-that provably improves the time complexities of all previous centralized methods. Moreover, we show that the time complexity of Shadowheart SGD is optimal in the family of centralized methods with compressed communication. We also consider the bidirectional setup, where broadcasting from the server to the workers is non-negligible, and develop a corresponding method.

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arXiv2024

Improving the Worst-Case Bidirectional Communication Complexity for Nonconvex Distributed Optimization under Function Similarity

Effective communication between the server and workers plays a key role in distributed optimization. In this paper, we focus on optimizing the server-to-worker communication, uncovering inefficiencies in prevalent downlink compression approaches. Considering first the pure setup where the uplink communication costs are negligible, we introduce MARINA-P, a novel method for downlink compression, employing a collection of correlated compressors. Theoretical analyses demonstrates that MARINA-P with permutation compressors can achieve a server-to-worker communication complexity improving with the number of workers, thus being provably superior to existing algorithms. We further show that MARINA-P can serve as a starting point for extensions such as methods supporting bidirectional compression. We introduce M3, a method combining MARINA-P with uplink compression and a momentum step, achieving bidirectional compression with provable improvements in total communication complexity as the number of workers increases. Theoretical findings align closely with empirical experiments, underscoring the efficiency of the proposed algorithms.

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arXiv2024

Freya PAGE: First Optimal Time Complexity for Large-Scale Nonconvex Finite-Sum Optimization with Heterogeneous Asynchronous Computations

In practical distributed systems, workers are typically not homogeneous, and due to differences in hardware configurations and network conditions, can have highly varying processing times. We consider smooth nonconvex finite-sum (empirical risk minimization) problems in this setup and introduce a new parallel method, Freya PAGE, designed to handle arbitrarily heterogeneous and asynchronous computations. By being robust to "stragglers" and adaptively ignoring slow computations, Freya PAGE offers significantly improved time complexity guarantees compared to all previous methods, including Asynchronous SGD, Rennala SGD, SPIDER, and PAGE, while requiring weaker assumptions. The algorithm relies on novel generic stochastic gradient collection strategies with theoretical guarantees that can be of interest on their own, and may be used in the design of future optimization methods. Furthermore, we establish a lower bound for smooth nonconvex finite-sum problems in the asynchronous setup, providing a fundamental time complexity limit. This lower bound is tight and demonstrates the optimality of Freya PAGE in the large-scale regime, i.e., when $\sqrt{m} \geq n$, where $n$ is # of workers, and $m$ is # of data samples.

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arXiv2024

On the Optimal Time Complexities in Decentralized Stochastic Asynchronous Optimization

We consider the decentralized stochastic asynchronous optimization setup, where many workers asynchronously calculate stochastic gradients and asynchronously communicate with each other using edges in a multigraph. For both homogeneous and heterogeneous setups, we prove new time complexity lower bounds under the assumption that computation and communication speeds are bounded. We develop a new nearly optimal method, Fragile SGD, and a new optimal method, Amelie SGD, that converge under arbitrary heterogeneous computation and communication speeds and match our lower bounds (up to a logarithmic factor in the homogeneous setting). Our time complexities are new, nearly optimal, and provably improve all previous asynchronous/synchronous stochastic methods in the decentralized setup.

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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

Variance Reduced Distributed Non-Convex Optimization Using Matrix Stepsizes

Matrix-stepsized gradient descent algorithms have been shown to have superior performance in non-convex optimization problems compared to their scalar counterparts. The det-CGD algorithm, as introduced by Li et al. (2023), leverages matrix stepsizes to perform compressed gradient descent for non-convex objectives and matrix-smooth problems in a federated manner. The authors establish the algorithm's convergence to a neighborhood of a weighted stationarity point under a convex condition for the symmetric and positive-definite matrix stepsize. In this paper, we propose two variance-reduced versions of the det-CGD algorithm, incorporating MARINA and DASHA methods. Notably, we establish theoretically and empirically, that det-MARINA and det-DASHA outperform MARINA, DASHA and the distributed det-CGD algorithms in terms of iteration and communication complexities.

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