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Xun Qian

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8 papers

2 author rows

TMLR Journal 2023 Journal Article

Distributed Newton-Type Methods with Communication Compression and Bernoulli Aggregation

Rustem Islamov
Xun Qian
Slavomir Hanzely
Mher Safaryan
Peter Richtárik

Despite their high computation and communication costs, Newton-type methods remain an appealing option for distributed training due to their robustness against ill-conditioned convex problems. In this work, we study communication compression and aggregation mechanisms for curvature information in order to reduce these costs while preserving theoretically superior local convergence guarantees. We prove that the recently developed class of three point compressors (3PC) of [Richtarik et al., 2022] for gradient communication can be generalized to Hessian communication as well. This result opens up a wide variety of communication strategies, such as contractive compression and lazy aggregation, available to our disposal to compress prohibitively costly curvature information. Moreover, we discovered several new 3PC mechanisms, such as adaptive thresholding and Bernoulli aggregation, which require reduced communication and occasional Hessian computations. Furthermore, we extend and analyze our approach to bidirectional communication compression and partial device participation setups to cater to the practical considerations of applications in federated learning. For all our methods, we derive fast condition-number-independent local linear and/or superlinear convergence rates. Finally, with extensive numerical evaluations on convex optimization problems, we illustrate that our designed schemes achieve state-of-the-art communication complexity compared to several key baselines using second-order information.

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ICML Conference 2022 Conference Paper

FedNL: Making Newton-Type Methods Applicable to Federated Learning

Mher Safaryan
Rustem Islamov
Xun Qian
Peter Richtárik

Inspired by recent work of Islamov et al (2021), we propose a family of Federated Newton Learn (\algname{FedNL}) methods, which we believe is a marked step in the direction of making second-order methods applicable to FL. In contrast to the aforementioned work, \algname{FedNL} employs a different Hessian learning technique which i) enhances privacy as it does not rely on the training data to be revealed to the coordinating server, ii) makes it applicable beyond generalized linear models, and iii) provably works with general contractive compression operators for compressing the local Hessians, such as Top-$K$ or Rank-$R$, which are vastly superior in practice. Notably, we do not need to rely on error feedback for our methods to work with contractive compressors. Moreover, we develop \algname{FedNL-PP}, \algname{FedNL-CR} and \algname{FedNL-LS}, which are variants of \algname{FedNL} that support partial participation, and globalization via cubic regularization and line search, respectively, and \algname{FedNL-BC}, which is a variant that can further benefit from bidirectional compression of gradients and models, i. e. , smart uplink gradient and smart downlink model compression. We prove local convergence rates that are independent of the condition number, the number of training data points, and compression variance. Our communication efficient Hessian learning technique provably learns the Hessian at the optimum. Finally, we perform a variety of numerical experiments that show that our \algname{FedNL} methods have state-of-the-art communication complexity when compared to key baselines.

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ICML Conference 2021 Conference Paper

Distributed Second Order Methods with Fast Rates and Compressed Communication

Rustem Islamov
Xun Qian
Peter Richtárik

We develop several new communication-efficient second-order methods for distributed optimization. Our first method, NEWTON-STAR, is a variant of Newton’s method from which it inherits its fast local quadratic rate. However, unlike Newton’s method, NEWTON-STAR enjoys the same per iteration communication cost as gradient descent. While this method is impractical as it relies on the use of certain unknown parameters characterizing the Hessian of the objective function at the optimum, it serves as the starting point which enables us to design practical variants thereof with strong theoretical guarantees. In particular, we design a stochastic sparsification strategy for learning the unknown parameters in an iterative fashion in a communication efficient manner. Applying this strategy to NEWTON-STAR leads to our next method, NEWTON-LEARN, for which we prove local linear and superlinear rates independent of the condition number. When applicable, this method can have dramatically superior convergence behavior when compared to state-of-the-art methods. Finally, we develop a globalization strategy using cubic regularization which leads to our next method, CUBIC-NEWTON-LEARN, for which we prove global sublinear and linear convergence rates, and a fast superlinear rate. Our results are supported with experimental results on real datasets, and show several orders of magnitude improvement on baseline and state-of-the-art methods in terms of communication complexity.

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NeurIPS Conference 2021 Conference Paper

Error Compensated Distributed SGD Can Be Accelerated

Xun Qian
Peter Richtarik
Tong Zhang

Gradient compression is a recent and increasingly popular technique for reducing the communication cost in distributed training of large-scale machine learning models. In this work we focus on developing efficient distributed methods that can work for any compressor satisfying a certain contraction property, which includes both unbiased (after appropriate scaling) and biased compressors such as RandK and TopK. Applied naively, gradient compression introduces errors that either slow down convergence or lead to divergence. A popular technique designed to tackle this issue is error compensation/error feedback. Due to the difficulties associated with analyzing biased compressors, it is not known whether gradient compression with error compensation can be combined with acceleration. In this work, we show for the first time that error compensated gradient compression methods can be accelerated. In particular, we propose and study the error compensated loopless Katyusha method, and establish an accelerated linear convergence rate under standard assumptions. We show through numerical experiments that the proposed method converges with substantially fewer communication rounds than previous error compensated algorithms.

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JMLR Journal 2021 Journal Article

L-SVRG and L-Katyusha with Arbitrary Sampling

Xun Qian
Zheng Qu
Peter Richtárik

We develop and analyze a new family of nonaccelerated and accelerated loopless variance-reduced methods for finite-sum optimization problems. Our convergence analysis relies on a novel expected smoothness condition which upper bounds the variance of the stochastic gradient estimation by a constant times a distance-like function. This allows us to handle with ease arbitrary sampling schemes as well as the nonconvex case. We perform an in-depth estimation of these expected smoothness parameters and propose new importance samplings which allow linear speedup when the expected minibatch size is in a certain range. Furthermore, a connection between these expected smoothness parameters and expected separable overapproximation (ESO) is established, which allows us to exploit data sparsity as well. Our general methods and results recover as special cases the loopless SVRG and loopless Katyusha methods. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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ICML Conference 2020 Conference Paper

Acceleration for Compressed Gradient Descent in Distributed and Federated Optimization

Zhize Li 0001
Dmitry Kovalev
Xun Qian
Peter Richtárik

Due to the high communication cost in distributed and federated learning problems, methods relying on compression of communicated messages are becoming increasingly popular. While in other contexts the best performing gradient-type methods invariably rely on some form of acceleration/momentum to reduce the number of iterations, there are no methods which combine the benefits of both gradient compression and acceleration. In this paper, we remedy this situation and propose the first \emph{accelerated compressed gradient descent (ACGD)} methods. In the single machine regime, we prove that ACGD enjoys the rate $O\Big((1+\omega)\sqrt{\frac{L}{\mu}}\log \frac{1}{\epsilon}\Big)$ for $\mu$-strongly convex problems and $O\Big((1+\omega)\sqrt{\frac{L}{\epsilon}}\Big)$ for convex problems, respectively, where $\omega$ is the compression parameter. Our results improve upon the existing non-accelerated rates $O\Big((1+\omega)\frac{L}{\mu}\log \frac{1}{\epsilon}\Big)$ and $O\Big((1+\omega)\frac{L}{\epsilon}\Big)$, respectively, and recover the optimal rates of accelerated gradient descent as a special case when no compression ($\omega=0$) is applied. We further propose a distributed variant of ACGD (called ADIANA) and prove the convergence rate $\widetilde{O}\Big(\omega+\sqrt{\frac{L}{\mu}}+\sqrt{\big(\frac{\omega}{n}+\sqrt{\frac{\omega}{n}}\big)\frac{\omega L}{\mu}}\Big)$, where $n$ is the number of devices/workers and $\widetilde{O}$ hides the logarithmic factor $\log \frac{1}{\epsilon}$. This improves upon the previous best result $\widetilde{O}\Big(\omega + \frac{L}{\mu}+\frac{\omega L}{n\mu} \Big)$ achieved by the DIANA method of Mishchenko et al. (2019). Finally, we conduct several experiments on real-world datasets which corroborate our theoretical results and confirm the practical superiority of our accelerated methods.

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ICML Conference 2019 Conference Paper

SAGA with Arbitrary Sampling

Xun Qian
Zheng Qu 0001
Peter Richtárik

We study the problem of minimizing the average of a very large number of smooth functions, which is of key importance in training supervised learning models. One of the most celebrated methods in this context is the SAGA algorithm of Defazio et al. (2014). Despite years of research on the topic, a general-purpose version of SAGA—one that would include arbitrary importance sampling and minibatching schemes—does not exist. We remedy this situation and propose a general and flexible variant of SAGA following the arbitrary sampling paradigm. We perform an iteration complexity analysis of the method, largely possible due to the construction of new stochastic Lyapunov functions. We establish linear convergence rates in the smooth and strongly convex regime, and under certain error bound conditions also in a regime without strong convexity. Our rates match those of the primal-dual method Quartz (Qu et al. , 2015) for which an arbitrary sampling analysis is available, which makes a significant step towards closing the gap in our understanding of complexity of primal and dual methods for finite sum problems. Finally, we show through experiments that specific variants of our general SAGA method can perform better in practice than other competing methods.

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ICML Conference 2019 Conference Paper

SGD with Arbitrary Sampling: General Analysis and Improved Rates

Xun Qian
Peter Richtárik
Robert M. Gower
Alibek Sailanbayev
Nicolas Loizou
Egor Shulgin

We propose a general yet simple theorem describing the convergence of SGD under the arbitrary sampling paradigm. Our theorem describes the convergence of an infinite array of variants of SGD, each of which is associated with a specific probability law governing the data selection rule used to form minibatches. This is the first time such an analysis is performed, and most of our variants of SGD were never explicitly considered in the literature before. Our analysis relies on the recently introduced notion of expected smoothness and does not rely on a uniform bound on the variance of the stochastic gradients. By specializing our theorem to different mini-batching strategies, such as sampling with replacement and independent sampling, we derive exact expressions for the stepsize as a function of the mini-batch size. With this we can also determine the mini-batch size that optimizes the total complexity, and show explicitly that as the variance of the stochastic gradient evaluated at the minimum grows, so does the optimal mini-batch size. For zero variance, the optimal mini-batch size is one. Moreover, we prove insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime.

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