arXiv subjects

Peter Richtárik

Publications and source records attributed to Peter Richtárik.

At least 109 records · Page 6Linked to original sources

arXiv2024

Communication Compression for Byzantine Robust Learning: New Efficient Algorithms and Improved Rates

Byzantine robustness is an essential feature of algorithms for certain distributed optimization problems, typically encountered in collaborative/federated learning. These problems are usually huge-scale, implying that communication compression is also imperative for their resolution. These factors have spurred recent algorithmic and theoretical developments in the literature of Byzantine-robust learning with compression. In this paper, we contribute to this research area in two main directions. First, we propose a new Byzantine-robust method with compression - Byz-DASHA-PAGE - and prove that the new method has better convergence rate (for non-convex and Polyak-Lojasiewicz smooth optimization problems), smaller neighborhood size in the heterogeneous case, and tolerates more Byzantine workers under over-parametrization than the previous method with SOTA theoretical convergence guarantees (Byz-VR-MARINA). Secondly, we develop the first Byzantine-robust method with communication compression and error feedback - Byz-EF21 - along with its bidirectional compression version - Byz-EF21-BC - and derive the convergence rates for these methods for non-convex and Polyak-Lojasiewicz smooth case. We test the proposed methods and illustrate our theoretical findings in the numerical experiments.

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arXiv2024

Error Feedback Reloaded: From Quadratic to Arithmetic Mean of Smoothness Constants

Error Feedback (EF) is a highly popular and immensely effective mechanism for fixing convergence issues which arise in distributed training methods (such as distributed GD or SGD) when these are enhanced with greedy communication compression techniques such as TopK. While EF was proposed almost a decade ago (Seide et al., 2014), and despite concentrated effort by the community to advance the theoretical understanding of this mechanism, there is still a lot to explore. In this work we study a modern form of error feedback called EF21 (Richtarik et al., 2021) which offers the currently best-known theoretical guarantees, under the weakest assumptions, and also works well in practice. In particular, while the theoretical communication complexity of EF21 depends on the quadratic mean of certain smoothness parameters, we improve this dependence to their arithmetic mean, which is always smaller, and can be substantially smaller, especially in heterogeneous data regimes. We take the reader on a journey of our discovery process. Starting with the idea of applying EF21 to an equivalent reformulation of the underlying problem which (unfortunately) requires (often impractical) machine cloning, we continue to the discovery of a new weighted version of EF21 which can (fortunately) be executed without any cloning, and finally circle back to an improved analysis of the original EF21 method. While this development applies to the simplest form of EF21, our approach naturally extends to more elaborate variants involving stochastic gradients and partial participation. Further, our technique improves the best-known theory of EF21 in the rare features regime (Richtarik et al., 2023). Finally, we validate our theoretical findings with suitable experiments.

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arXiv2024

Consensus-Based Optimization with Truncated Noise

Consensus-based optimization (CBO) is a versatile multi-particle metaheuristic optimization method suitable for performing nonconvex and nonsmooth global optimizations in high dimensions. It has proven effective in various applications while at the same time being amenable to a theoretical convergence analysis. In this paper, we explore a variant of CBO, which incorporates truncated noise in order to enhance the well-behavedness of the statistics of the law of the dynamics. By introducing this additional truncation in the noise term of the CBO dynamics, we achieve that, in contrast to the original version, higher moments of the law of the particle system can be effectively bounded. As a result, our proposed variant exhibits enhanced convergence performance, allowing in particular for wider flexibility in choosing the noise parameter of the method as we confirm experimentally. By analyzing the time-evolution of the Wasserstein-$2$ distance between the empirical measure of the interacting particle system and the global minimizer of the objective function, we rigorously prove convergence in expectation of the proposed CBO variant requiring only minimal assumptions on the objective function and on the initialization. Numerical evidences demonstrate the benefit of truncating the noise in CBO.

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arXiv2024

On Biased Compression for Distributed Learning

In the last few years, various communication compression techniques have emerged as an indispensable tool helping to alleviate the communication bottleneck in distributed learning. However, despite the fact biased compressors often show superior performance in practice when compared to the much more studied and understood unbiased compressors, very little is known about them. In this work we study three classes of biased compression operators, two of which are new, and their performance when applied to (stochastic) gradient descent and distributed (stochastic) gradient descent. We show for the first time that biased compressors can lead to linear convergence rates both in the single node and distributed settings. We prove that distributed compressed SGD method, employed with error feedback mechanism, enjoys the ergodic rate $O\left( δL \exp \left[-\frac{μK}{δL}\right] + \frac{(C + δD)}{Kμ}\right)$, where $δ\ge 1$ is a compression parameter which grows when more compression is applied, $L$ and $μ$ are the smoothness and strong convexity constants, $C$ captures stochastic gradient noise ($C=0$ if full gradients are computed on each node) and $D$ captures the variance of the gradients at the optimum ($D=0$ for over-parameterized models). Further, via a theoretical study of several synthetic and empirical distributions of communicated gradients, we shed light on why and by how much biased compressors outperform their unbiased variants. Finally, we propose several new biased compressors with promising theoretical guarantees and practical performance.

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arXiv2024

Correlated Quantization for Faster Nonconvex Distributed Optimization

Quantization (Alistarh et al., 2017) is an important (stochastic) compression technique that reduces the volume of transmitted bits during each communication round in distributed model training. Suresh et al. (2022) introduce correlated quantizers and show their advantages over independent counterparts by analyzing distributed SGD communication complexity. We analyze the forefront distributed non-convex optimization algorithm MARINA (Gorbunov et al., 2022) utilizing the proposed correlated quantizers and show that it outperforms the original MARINA and distributed SGD of Suresh et al. (2022) with regard to the communication complexity. We significantly refine the original analysis of MARINA without any additional assumptions using the weighted Hessian variance (Tyurin et al., 2022), and then we expand the theoretical framework of MARINA to accommodate a substantially broader range of potentially correlated and biased compressors, thus dilating the applicability of the method beyond the conventional independent unbiased compressor setup. Extensive experimental results corroborate our theoretical findings.

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arXiv2024

A Computation and Communication Efficient Method for Distributed Nonconvex Problems in the Partial Participation Setting

We present a new method that includes three key components of distributed optimization and federated learning: variance reduction of stochastic gradients, partial participation, and compressed communication. We prove that the new method has optimal oracle complexity and state-of-the-art communication complexity in the partial participation setting. Regardless of the communication compression feature, our method successfully combines variance reduction and partial participation: we get the optimal oracle complexity, never need the participation of all nodes, and do not require the bounded gradients (dissimilarity) assumption.

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arXiv2023

Distributed Learning with Compressed Gradient Differences

Training large machine learning models requires a distributed computing approach, with communication of the model updates being the bottleneck. For this reason, several methods based on the compression (e.g., sparsification and/or quantization) of updates were recently proposed, including QSGD (Alistarh et al., 2017), TernGrad (Wen et al., 2017), SignSGD (Bernstein et al., 2018), and DQGD (Khirirat et al., 2018). However, none of these methods are able to learn the gradients, which renders them incapable of converging to the true optimum in the batch mode. In this work we propose a new distributed learning method -- DIANA -- which resolves this issue via compression of gradient differences. We perform a theoretical analysis in the strongly convex and nonconvex settings and show that our rates are superior to existing rates. We also provide theory to support non-smooth regularizers study the difference between quantization schemes. Our analysis of block-quantization and differences between $\ell_2$ and $\ell_{\infty}$ quantization closes the gaps in theory and practice. Finally, by applying our analysis technique to TernGrad, we establish the first convergence rate for this method.

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arXiv2023

Kimad: Adaptive Gradient Compression with Bandwidth Awareness

In distributed training, communication often emerges as a bottleneck. In response, we introduce Kimad, a solution that offers adaptive gradient compression. By consistently monitoring bandwidth, Kimad refines compression ratios to match specific neural network layer requirements. Our exhaustive tests and proofs confirm Kimad's outstanding performance, establishing it as a benchmark in adaptive compression for distributed deep learning.

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arXiv2023

2Direction: Theoretically Faster Distributed Training with Bidirectional Communication Compression

We consider distributed convex optimization problems in the regime when the communication between the server and the workers is expensive in both uplink and downlink directions. We develop a new and provably accelerated method, which we call 2Direction, based on fast bidirectional compressed communication and a new bespoke error-feedback mechanism which may be of independent interest. Indeed, we find that the EF and EF21-P mechanisms (Seide et al., 2014; Gruntkowska et al., 2023) that have considerable success in the design of efficient non-accelerated methods are not appropriate for accelerated methods. In particular, we prove that 2Direction improves the previous state-of-the-art communication complexity $\widetildeΘ\left(K \times \left(\frac{L}{αμ} + \frac{L_{\max} ω}{n μ} + ω\right)\right)$ (Gruntkowska et al., 2023) to $\widetildeΘ(K \times (\sqrt{\frac{L (ω+ 1)}{αμ}} + \sqrt{\frac{L_{\max} ω^2}{n μ}} + \frac{1}α + ω))$ in the $μ$-strongly-convex setting, where $L$ and $L_{\max}$ are smoothness constants, $n$ is # of workers, $ω$ and $α$ are compression errors of the Rand$K$ and Top$K$ sparsifiers (as examples), $K$ is # of coordinates/bits that the server and workers send to each other. Moreover, our method is the first that improves upon the communication complexity of the vanilla accelerated gradient descent (AGD) method (Nesterov, 2018). We obtain similar improvements in the general convex regime as well. Finally, our theoretical findings are corroborated by experimental evidence.

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arXiv2023

Optimal Time Complexities of Parallel Stochastic Optimization Methods Under a Fixed Computation Model

Parallelization is a popular strategy for improving the performance of iterative algorithms. Optimization methods are no exception: design of efficient parallel optimization methods and tight analysis of their theoretical properties are important research endeavors. While the minimax complexities are well known for sequential optimization methods, the theory of parallel optimization methods is less explored. In this paper, we propose a new protocol that generalizes the classical oracle framework approach. Using this protocol, we establish minimax complexities for parallel optimization methods that have access to an unbiased stochastic gradient oracle with bounded variance. We consider a fixed computation model characterized by each worker requiring a fixed but worker-dependent time to calculate stochastic gradient. We prove lower bounds and develop optimal algorithms that attain them. Our results have surprising consequences for the literature of asynchronous optimization methods.

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arXiv2023

Momentum Provably Improves Error Feedback!

Due to the high communication overhead when training machine learning models in a distributed environment, modern algorithms invariably rely on lossy communication compression. However, when untreated, the errors caused by compression propagate, and can lead to severely unstable behavior, including exponential divergence. Almost a decade ago, Seide et al [2014] proposed an error feedback (EF) mechanism, which we refer to as EF14, as an immensely effective heuristic for mitigating this issue. However, despite steady algorithmic and theoretical advances in the EF field in the last decade, our understanding is far from complete. In this work we address one of the most pressing issues. In particular, in the canonical nonconvex setting, all known variants of EF rely on very large batch sizes to converge, which can be prohibitive in practice. We propose a surprisingly simple fix which removes this issue both theoretically, and in practice: the application of Polyak's momentum to the latest incarnation of EF due to Richtárik et al. [2021] known as EF21. Our algorithm, for which we coin the name EF21-SGDM, improves the communication and sample complexities of previous error feedback algorithms under standard smoothness and bounded variance assumptions, and does not require any further strong assumptions such as bounded gradient dissimilarity. Moreover, we propose a double momentum version of our method that improves the complexities even further. Our proof seems to be novel even when compression is removed from the method, and as such, our proof technique is of independent interest in the study of nonconvex stochastic optimization enriched with Polyak's momentum.

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arXiv2023

Faster Rates for Compressed Federated Learning with Client-Variance Reduction

Due to the communication bottleneck in distributed and federated learning applications, algorithms using communication compression have attracted significant attention and are widely used in practice. Moreover, the huge number, high heterogeneity and limited availability of clients result in high client-variance. This paper addresses these two issues together by proposing compressed and client-variance reduced methods COFIG and FRECON. We prove an $O(\frac{(1+ω)^{3/2}\sqrt{N}}{Sε^2}+\frac{(1+ω)N^{2/3}}{Sε^2})$ bound on the number of communication rounds of COFIG in the nonconvex setting, where $N$ is the total number of clients, $S$ is the number of clients participating in each round, $ε$ is the convergence error, and $ω$ is the variance parameter associated with the compression operator. In case of FRECON, we prove an $O(\frac{(1+ω)\sqrt{N}}{Sε^2})$ bound on the number of communication rounds. In the convex setting, COFIG converges within $O(\frac{(1+ω)\sqrt{N}}{Sε})$ communication rounds, which, to the best of our knowledge, is also the first convergence result for compression schemes that do not communicate with all the clients in each round. We stress that neither COFIG nor FRECON needs to communicate with all the clients, and they enjoy the first or faster convergence results for convex and nonconvex federated learning in the regimes considered. Experimental results point to an empirical superiority of COFIG and FRECON over existing baselines.

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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

Understanding Progressive Training Through the Framework of Randomized Coordinate Descent

We propose a Randomized Progressive Training algorithm (RPT) -- a stochastic proxy for the well-known Progressive Training method (PT) (Karras et al., 2017). Originally designed to train GANs (Goodfellow et al., 2014), PT was proposed as a heuristic, with no convergence analysis even for the simplest objective functions. On the contrary, to the best of our knowledge, RPT is the first PT-type algorithm with rigorous and sound theoretical guarantees for general smooth objective functions. We cast our method into the established framework of Randomized Coordinate Descent (RCD) (Nesterov, 2012; Richtárik & Takáč, 2014), for which (as a by-product of our investigations) we also propose a novel, simple and general convergence analysis encapsulating strongly-convex, convex and nonconvex objectives. We then use this framework to establish a convergence theory for RPT. Finally, we validate the effectiveness of our method through extensive computational experiments.

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arXiv2023

Improving Accelerated Federated Learning with Compression and Importance Sampling

Federated Learning is a collaborative training framework that leverages heterogeneous data distributed across a vast number of clients. Since it is practically infeasible to request and process all clients during the aggregation step, partial participation must be supported. In this setting, the communication between the server and clients poses a major bottleneck. To reduce communication loads, there are two main approaches: compression and local steps. Recent work by Mishchenko et al. [2022] introduced the new ProxSkip method, which achieves an accelerated rate using the local steps technique. Follow-up works successfully combined local steps acceleration with partial participation [Grudzień et al., 2023, Condat et al. 2023] and gradient compression [Condat et al. [2022]. In this paper, we finally present a complete method for Federated Learning that incorporates all necessary ingredients: Local Training, Compression, and Partial Participation. We obtain state-of-the-art convergence guarantees in the considered setting. Moreover, we analyze the general sampling framework for partial participation and derive an importance sampling scheme, which leads to even better performance. We experimentally demonstrate the advantages of the proposed method in practice.

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arXiv2023

Clip21: Error Feedback for Gradient Clipping

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.

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arXiv2023

A Guide Through the Zoo of Biased SGD

Stochastic Gradient Descent (SGD) is arguably the most important single algorithm in modern machine learning. Although SGD with unbiased gradient estimators has been studied extensively over at least half a century, SGD variants relying on biased estimators are rare. Nevertheless, there has been an increased interest in this topic in recent years. However, existing literature on SGD with biased estimators (BiasedSGD) lacks coherence since each new paper relies on a different set of assumptions, without any clear understanding of how they are connected, which may lead to confusion. We address this gap by establishing connections among the existing assumptions, and presenting a comprehensive map of the underlying relationships. Additionally, we introduce a new set of assumptions that is provably weaker than all previous assumptions, and use it to present a thorough analysis of BiasedSGD in both convex and non-convex settings, offering advantages over previous results. We also provide examples where biased estimators outperform their unbiased counterparts or where unbiased versions are simply not available. Finally, we demonstrate the effectiveness of our framework through experimental results that validate our theoretical findings.

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arXiv2023

Error Feedback Shines when Features are Rare

We provide the first proof that gradient descent $\left({\color{green}\sf GD}\right)$ with greedy sparsification $\left({\color{green}\sf TopK}\right)$ and error feedback $\left({\color{green}\sf EF}\right)$ can obtain better communication complexity than vanilla ${\color{green}\sf GD}$ when solving the distributed optimization problem $\min_{x\in \mathbb{R}^d} {f(x)=\frac{1}{n}\sum_{i=1}^n f_i(x)}$, where $n$ = # of clients, $d$ = # of features, and $f_1,\dots,f_n$ are smooth nonconvex functions. Despite intensive research since 2014 when ${\color{green}\sf EF}$ was first proposed by Seide et al., this problem remained open until now. We show that ${\color{green}\sf EF}$ shines in the regime when features are rare, i.e., when each feature is present in the data owned by a small number of clients only. To illustrate our main result, we show that in order to find a random vector $\hat{x}$ such that $\lVert {\nabla f(\hat{x})} \rVert^2 \leq \varepsilon$ in expectation, ${\color{green}\sf GD}$ with the ${\color{green}\sf Top1}$ sparsifier and ${\color{green}\sf EF}$ requires ${\cal O} \left(\left( L+{\color{blue}r} \sqrt{ \frac{{\color{red}c}}{n} \min \left( \frac{{\color{red}c}}{n} \max_i L_i^2, \frac{1}{n}\sum_{i=1}^n L_i^2 \right) }\right) \frac{1}{\varepsilon} \right)$ bits to be communicated by each worker to the server only, where $L$ is the smoothness constant of $f$, $L_i$ is the smoothness constant of $f_i$, ${\color{red}c}$ is the maximal number of clients owning any feature ($1\leq {\color{red}c} \leq n$), and ${\color{blue}r}$ is the maximal number of features owned by any client ($1\leq {\color{blue}r} \leq d$). Clearly, the communication complexity improves as ${\color{red}c}$ decreases (i.e., as features become more rare), and can be much better than the ${\cal O}({\color{blue}r} L \frac{1}{\varepsilon})$ communication complexity of ${\color{green}\sf GD}$ in the same regime.

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