Distributed training enables large-scale deep learning, but suffers from high communication overhead, especially as models and datasets grow. Gradient compression, particularly quantization, is a promising approach to mitigate this bottleneck. However, existing quantization schemes are often incompatible with Allreduce, the dominant communication primitive in distributed deep learning, and many prior solutions rely on heuristics without theoretical guarantees. We introduce Global-QSGD, an Allreduce-compatible gradient quantization method that leverages global norm scaling to reduce communication overhead while preserving accuracy. Global-QSGD is backed by rigorous theoretical analysis, extending standard unbiased compressor frameworks to establish formal convergence guarantees. Additionally, we develop a performance model to evaluate its impact across different hardware configurations. Extensive experiments on NVLink, PCIe, and large-scale cloud environments show that Global-QSGD accelerates distributed training by up to 3.51× over baseline quantization methods, making it a practical and efficient solution for large-scale deep learning workloads.

In $D$istributed optimization and $L$earning, 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 $Lo$cal training, which reduces the communication frequency, and $Co$mpression, 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 heterogeneous regime with strongly convex functions. This is confirmed in practice, with LoCoDL outperforming existing algorithms.

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 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.

Non-convex Machine Learning problems typically do not adhere to the standard smoothness assumption. Based on empirical findings, Zhang et al. (2020b) proposed a more realistic generalized $(L_0,L_1)$-smoothness assumption, though it remains largely unexplored. Many existing algorithms designed for standard smooth problems need to be revised. However, in the context of Federated Learning, only a few works address this problem but rely on additional limiting assumptions. In this paper, we address this gap in the literature: we propose and analyze new methods with local steps, partial participation of clients, and Random Reshuffling without extra restrictive assumptions beyond generalized smoothness. The proposed methods are based on the proper interplay between clients' and server's stepsizes and gradient clipping. Furthermore, we perform the first analysis of these methods under the Polyak-Łojasiewicz condition. Our theory is consistent with the known results for standard smooth problems, and our experimental results support the theoretical insights.

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 on 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).

Asynchronous methods are fundamental for parallelizing computations in distributed machine learning. They aim to accelerate training by fully utilizing all available resources. However, their greedy approach can lead to inefficiencies using more computation than required, especially when computation times vary across devices. If the computation times were known in advance, training could be fast and resource-efficient by assigning more tasks to faster workers. The challenge lies in achieving this optimal allocation without prior knowledge of the computation time distributions. In this paper, we propose ATA (Adaptive Task Allocation), a method that adapts to heterogeneous and random distributions of worker computation times. Through rigorous theoretical analysis, we show that ATA identifies the optimal task allocation and performs comparably to methods with prior knowledge of computation times. Experimental results further demonstrate that ATA is resource-efficient, significantly reducing costs compared to the greedy approach, which can be arbitrarily expensive depending on the number of workers.

Asynchronous Stochastic Gradient Descent (Asynchronous SGD) is a cornerstone method for parallelizing learning in distributed machine learning. However, its performance suffers under arbitrarily heterogeneous computation times across workers, leading to suboptimal time complexity and inefficiency as the number of workers scales. While several Asynchronous SGD variants have been proposed, recent findings by Tyurin & Richtárik (NeurIPS 2023) reveal that none achieve optimal time complexity, leaving a significant gap in the literature. In this paper, we propose Ringmaster ASGD, a novel Asynchronous SGD method designed to address these limitations and tame the inherent challenges of Asynchronous SGD. We establish, through rigorous theoretical analysis, that Ringmaster ASGD achieves optimal time complexity under arbitrarily heterogeneous and dynamically fluctuating worker computation times. This makes it the first Asynchronous SGD method to meet the theoretical lower bounds for time complexity in such scenarios.

Vladimir Malinovskii, Andrei Panferov, Ivan Ilin, Han Guo, Peter Richtárik, Dan Alistarh. Proceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers). 2025.

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 $\mathcal{O}(1/\sqrt{K})$ 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.

We contribute to the growing body of knowledge on more powerful and adaptive stepsizes for convex optimization, empowered by local curvature information. We do not go the route of fully-fledged second-order methods, which require the expensive computation of the Hessian. Instead, our key observation is that, for some problems (e.g., when minimizing the sum of squares of absolutely convex functions), local curvature information is readily available, and can be used to obtain surprisingly powerful matrix-valued stepsizes, and meaningful theory. In particular, we develop three new methods — LCD1, LCD2, and LCD3 — where the abbreviation stands for local curvature descent. While LCD1 generalizes gradient descent with fixed stepsize, LCD2 generalizes gradient descent with Polyak stepsize. Our methods enhance these classical gradient descent baselines with local curvature information, and our theory recovers the known rates in the special case when no curvature information is used. Our last method, LCD3, is a variable-metric version of LCD2; this feature leads to a closed-form expression for the iterates. Our empirical results are encouraging and show that the local curvature descent improves upon gradient descent.

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.

Federated sampling algorithms have recently gained great popularity in the community of machine learning and statistics. This paper proposes a new federated sampling algorithm called Error Feedback Langevin algorithms (ELF). In particular, we analyze the combinations of EF21 and EF21-P with the federated Langevin Monte-Carlo. We propose three algorithms, P-ELF, D-ELF, and B-ELF, that use primal, dual, and bidirectional compressors. We analyze the proposed methods under Log-Sobolev inequality and provide non-asymptotic convergence guarantees. Simple experimental results support our theoretical findings.

We investigate the problem of minimizing the expectation of smooth nonconvex functions in a distributed setting with multiple parallel workers that are able to compute stochastic gradients. A significant challenge in this context is the presence of arbitrarily heterogeneous and stochastic compute times among workers, which can severely degrade the performance of existing parallel stochastic gradient descent (SGD) methods. While some parallel SGD algorithms achieve optimal performance under deterministic but heterogeneous delays, their effectiveness diminishes when compute times are random-a scenario not explicitly addressed in their design. To bridge this gap, we introduce MindFlayer SGD, a novel parallel SGD method specifically designed to handle stochastic and heterogeneous compute times. Through theoretical analysis and empirical evaluation, we demonstrate that MindFlayer SGD consistently outperforms existing baselines, particularly in environments with heavy-tailed noise. Our results highlight its robustness and scalability, making it a compelling choice for large-scale distributed learning tasks.

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 fore- front 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 re- fine 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.

First proposed by Seide (2014) as a heuristic, error feedback (EF) is a very popular mechanism for enforcing convergence of distributed gradient-based optimization methods enhanced with communication compression strategies based on the application of contractive compression operators. However, existing theory of EF relies on very strong assumptions (e.g., bounded gradients), and provides pessimistic convergence rates (e.g., while the best known rate for EF in the smooth nonconvex regime, and when full gradients are compressed, is $O(1/T^{2/3})$, the rate of gradient descent in the same regime is $O(1/T)$). Recently, Richtàrik et al. (2021) proposed a new error feedback mechanism, EF21, based on the construction of a Markov compressor induced by a contractive compressor. EF21 removes the aforementioned theoretical deficiencies of EF and at the same time works better in practice. In this work we propose six practical extensions of EF21, all supported by strong convergence theory: partial participation, stochastic approximation, variance reduction, proximal setting, momentum, and bidirectional compression. To the best of our knowledge, several of these techniques have not been previously analyzed in combination with EF, and in cases where prior analysis exists---such as for bidirectional compression---our theoretical convergence guarantees significantly improve upon existing results.

We present a new zero-order optimization method called Minibatch Stochastic Three Points (MiSTP), specifically designed to solve stochastic unconstrained minimization problems when only an approximate evaluation of the objective function is possible. MiSTP is an extension of the Stochastic Three Point Method (STP). The key innovation of MiSTP is that it selects the next point solely based on the objective function approximation, without relying on its exact evaluation. At each iteration, MiSTP generates a random search direction and compares the approximations of the objective function at the current point, the randomly generated direction and its opposite. The best of these three points is chosen as the next iterate. We analyze the worst-case complexity of MiSTP in the convex and non-convex cases and demonstrate that it matches the most accurate complexity bounds known in the literature for zero-order optimization methods. We perform extensive numerical evaluations to assess the computational efficiency of MiSTP and compare its performance to other state-of-the-art methods by testing it on several machine learning tasks. The results show that MiSTP outperforms or has comparable performance against state-of-the-art methods indicating its potential for a wide range of practical applications.

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 bi-directional 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.

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; Richtarik & Takac, 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.

This paper introduces a new method for minimizing matrix-smooth non-convex objectives through the use of novel Compressed Gradient Descent (CGD) algorithms enhanced with a matrix-valued stepsize. The proposed algorithms are theoretically analyzed first in the single-node and subsequently in the distributed settings. Our theoretical results reveal that the matrix stepsize in CGD can capture the objective’s structure and lead to faster convergence compared to a scalar stepsize. As a byproduct of our general results, we emphasize the importance of selecting the compression mechanism and the matrix stepsize in a layer-wise manner, taking advantage of model structure. Moreover, we provide theoretical guarantees for free compression, by designing specific layer-wise compressors for the non-convex matrix smooth objectives. Our findings are supported with empirical evidence.

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 Top-k. 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 (Richtárik 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 (Richtárik et al, 2023). Finally, we validate our theoretical findings with suitable experiments.

The interest in federated learning has surged in recent research due to its unique ability to train a global model using privacy-secured information held locally on each client. This paper pays particular attention to the issue of client-side model heterogeneity, a pervasive challenge in the practical implementation of FL that escalates its complexity. Assuming a scenario where each client possesses varied memory storage, processing capabilities and network bandwidth - a phenomenon referred to as system heterogeneity - there is a pressing need to customize a unique model for each client. In response to this, we present an effective and adaptable federated framework FedP3, representing Federated Personalized and Privacy-friendly network Pruning, tailored for model heterogeneity scenarios. Our proposed methodology can incorporate and adapt well-established techniques to its specific instances. We offer a theoretical interpretation of FedP3 and its locally differential-private variant, DP-FedP3, and theoretically validate their efficiencies.

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 naively, 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. In addition, we also develop new methods for composite and distributed variational inequalities and analyze the high-probability convergence of these methods.

Modern advancements in large-scale machine learning would be impossible without the paradigm of data-parallel distributed computing. Since distributed computing with large-scale models imparts excessive pressure on communication channels, significant recent research has been directed toward co-designing communication compression strategies and training algorithms with the goal of reducing communication costs. While pure data parallelism allows better data scaling, it suffers from poor model scaling properties. Indeed, compute nodes are severely limited by memory constraints, preventing further increases in model size. For this reason, the latest achievements in training giant neural network models also rely on some form of model parallelism. In this work, we take a closer theoretical look at Independent Subnetwork Training (IST), which is a recently proposed and highly effective technique for solving the aforementioned problems. We identify fundamental differences between IST and alternative approaches, such as distributed methods with compressed communication, and provide a precise analysis of its optimization performance on a quadratic model.

Effective communication between the server and workers plays a key role in distributed optimization. In this paper, we focus on optimizing 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 analysis 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.

We propose and study several server-extrapolation strategies for enhancing the theoretical and empirical convergence properties of the popular federated learning optimizer FedProx [Li et al., 2020]. While it has long been known that some form of extrapolation can help in the practice of FL, only a handful of works provide any theoretical guarantees. The phenomenon seems elusive, and our current theoretical understanding remains severely incomplete. In our work, we focus on smooth convex or strongly convex problems in the interpolation regime. In particular, we propose Extrapolated FedProx (FedExProx), and study three extrapolation strategies: a constant strategy (depending on various smoothness parameters and the number of participating devices), and two smoothness-adaptive strategies; one based on the notion of gradient diversity (FedExProx-GraDS), and the other one based on the stochastic Polyak stepsize (FedExProx-StoPS). Our theory is corroborated with carefully constructed numerical experiments.

There has been significant interest in "extreme" compression of large language models (LLMs), i.e. to 1-2 bits per parameter, which allows such models to be executed efficiently on resource-constrained devices. Existing work focused on improved one-shot quantization techniques and weight representations; yet, purely post-training approaches are reaching diminishing returns in terms of the accuracy-vs-bit-width trade-off. State-of-the-art quantization methods such as QuIP# and AQLM include fine-tuning (part of) the compressed parameters over a limited amount of calibration data; however, such fine-tuning techniques over compressed weights often make exclusive use of straight-through estimators (STE), whose performance is not well-understood in this setting. In this work, we question the use of STE for extreme LLM compression, showing that it can be sub-optimal, and perform a systematic study of quantization-aware fine-tuning strategies for LLMs. We propose PV-Tuning - a representation-agnostic framework that generalizes and improves upon existing fine-tuning strategies, and provides convergence guarantees in restricted cases. On the practical side, when used for 1-2 bit vector quantization, PV-Tuning outperforms prior techniques for highly-performant models such as Llama and Mistral. Using PV-Tuning, we achieve the first Pareto-optimal quantization for Llama-2 family models at 2 bits per parameter.

Distributed learning has emerged as a leading paradigm for training large machine learning models. However, in real-world scenarios, participants may be unreliable or malicious, posing a significant challenge to the integrity and accuracy of the trained models. Byzantine fault tolerance mechanisms have been proposed to address these issues, but they often assume full participation from all clients, which is not always practical due to the unavailability of some clients or communication constraints. In our work, we propose the first distributed method with client sampling and provable tolerance to Byzantine workers. The key idea behind the developed method is the use of gradient clipping to control stochastic gradient differences in recursive variance reduction. This allows us to bound the potential harm caused by Byzantine workers, even during iterations when all sampled clients are Byzantine. Furthermore, we incorporate communication compression into the method to enhance communication efficiency. Under general assumptions, we prove convergence rates for the proposed method that match the existing state-of-the-art (SOTA) theoretical results. We also propose a heuristic on how to adjust any Byzantine-robust method to a partial participation scenario via clipping.

We propose a new variant of the Adam optimizer called MicroAdam that specifically minimizes memory overheads, while maintaining theoretical convergence guarantees. We achieve this by compressing the gradient information before it is fed into the optimizer state, thereby reducing its memory footprint significantly. We control the resulting compression error via a novel instance of the classical *error feedback* mechanism from distributed optimization in which *the error correction information is itself compressed* to allow for practical memory gains. We prove that the resulting approach maintains theoretical convergence guarantees competitive to those of AMSGrad, while providing good practical performance. Specifically, we show that MicroAdam can be implemented efficiently on GPUs: on both million-scale (BERT) and billion-scale (LLaMA) models, MicroAdam provides practical convergence competitive to that of the uncompressed Adam baseline, with lower memory usage and similar running time. Our code is available at https://github.com/IST-DASLab/MicroAdam.

Gradient compression is a popular technique for improving communication complexity of stochastic first-order methods in distributed training of machine learning models. However, the existing works consider only with-replacement sampling of stochastic gradients. In contrast, it is well-known in practice and recently confirmed in theory that stochastic methods based on without-replacement sampling, e.g., Random Reshuffling (RR) method, perform better than ones that sample the gradients with-replacement. In this work, we close this gap in the literature and provide the first analysis of methods with gradient compression and without-replacement sampling. We first develop a distributed variant of random reshuffling with gradient compression (Q-RR), and show how to reduce the variance coming from gradient quantization through the use of control iterates. Next, to have a better fit to Federated Learning applications, we incorporate local computation and propose a variant of Q-RR called Q-NASTYA. Q-NASTYA uses local gradient steps and different local and global stepsizes. Next, we show how to reduce compression variance in this setting as well. Finally, we prove the convergence results for the proposed methods and outline several settings in which they improve upon existing algorithms.

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.

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.

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 by constants. After that, we developed a new nearly optimal method, Fragile SGD, and a new optimal method, Amelie SGD, that converge with 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.

The celebrated FedAvg algorithm of McMahan et al. (2017) is based on three components: client sampling (CS), data sampling (DS) and local training (LT). While the first two are reasonably well understood, the third component, whose role is to reduce the number of communication rounds needed to train the model, resisted all attempts at a satisfactory theoretical explanation. Malinovsky et al. (2022) identified four distinct generations of LT methods based on the quality of the provided theoretical communication complexity guarantees. Despite a lot of progress in this area, none of the existing works were able to show that it is theoretically better to employ multiple local gradient-type steps (i.e., to engage in LT) than to rely on a single local gradient-type step only in the important heterogeneous data regime. In a recent breakthrough embodied in their ProxSkip method and its theoretical analysis, Mishchenko et al. (2022) showed that LT indeed leads to provable communication acceleration for arbitrarily heterogeneous data, thus jump-starting the 5th generation of LT methods. However, while these latest generation LT methods are compatible with DS, none of them support CS. We resolve this open problem in the affirmative. In order to do so, we had to base our algorithmic development on new algorithmic and theoretical foundations.

Communication overhead is well known to be a key bottleneck in large scale distributed learning, and a particularly successful class of methods which help to overcome this bottleneck is based on the idea of communication compression. Some of the most practically effective gradient compressors, such as TopK, are biased, which causes convergence issues unless one employs a well designed error compensation/feedback mechanism. Error compensation is therefore a fundamental technique in the distributed learning literature. In a recent development, Qian et al (NeurIPS 2021) showed that the error-compensation mechanism can be combined with acceleration/momentum, which is another key and highly successful optimization technique. In particular, they developed the error-compensated loop-less Katyusha (ECLK) method, and proved an accelerated linear rate in the strongly convex case. However, the dependence of their rate on the compressor parameter does not match the best dependence obtainable in the non-accelerated error-compensated methods. Our work addresses this problem. We propose several new accelerated error-compensated methods using the catalyst acceleration technique, and obtain results that match the best dependence on the compressor parameter in non-accelerated error-compensated methods up to logarithmic terms.

Stein Variational Gradient Descent (SVGD) is an important alternative to the Langevin-type algorithms for sampling from probability distributions of the form $\pi(x) \propto \exp(-V(x))$. In the existing theory of Langevin-type algorithms and SVGD, the potential function $V$ is often assumed to be $L$-smooth. However, this restrictive condition excludes a large class of potential functions such as polynomials of degree greater than $2$. Our paper studies the convergence of the SVGD algorithm for distributions with $(L_0,L_1)$-smooth potentials. This relaxed smoothness assumption was introduced by Zhang et al. [2019a] for the analysis of gradient clipping algorithms. With the help of trajectory-independent auxiliary conditions, we provide a descent lemma establishing that the algorithm decreases the KL divergence at each iteration and prove a complexity bound for SVGD in the population limit in terms of the Stein Fisher information.

In this work we focus our attention on distributed optimization problems in the context where the communication time between the server and the workers is non-negligible. We obtain novel methods supporting bidirectional compression (both from the server to the workers and vice versa) that enjoy new state-of-the-art theoretical communication complexity for convex and nonconvex problems. Our bounds are the first that manage to decouple the variance/error coming from the workers-to-server and server-to-workers compression, transforming a multiplicative dependence to an additive one. Moreover, in the convex regime, we obtain the first bounds that match the theoretical communication complexity of gradient descent. Even in this convex regime, our algorithms work with biased gradient estimators, which is non-standard and requires new proof techniques that may be of independent interest. Finally, our theoretical results are corroborated through suitable experiments.

During the 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 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 $\alpha$-th moment for $\alpha \in (1,2]$ 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.

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 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.

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 form the method, and as such, our proof technique is of independent interest in the study of nonconvex stochastic optimization enriched with Polyak's momentum.

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 $\widetilde{\Theta}\left(K \times \left(\frac{L}{\alpha \mu} + \frac{L_{\max} \omega}{n \mu} + \omega\right)\right)$ (Gruntkowska et al., 2023) to $\widetilde{\Theta}(K \times (\sqrt{\frac{L (\omega + 1)}{\alpha \mu}} + \sqrt{\frac{L_{\max} \omega^2}{n \mu}} + \frac{1}{\alpha} + \omega))$ in the $\mu$--strongly-convex setting, where $L$ and $L_{\max}$ are smoothness constants, $n$ is \# of workers, $\omega$ and $\alpha$ 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 method (AGD). We obtain similar improvements in the general convex regime as well. Finally, our theoretical findings are corroborated by experimental evidence.

Parallelization is a popular strategy for improving the performance of methods. 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.

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.

Virtually all state-of-the-art methods for training supervised machine learning models are variants of Stochastic Gradient Descent (SGD), enhanced with a number of additional tricks, such as minibatching, momentum, and adaptive stepsizes. However, one of the most basic questions in the design of successful SGD methods, one that is orthogonal to the aforementioned tricks, is the choice of the next training data point to be learning from. Standard variants of SGD employ a sampling with replacement strategy, which means that the next training data point is sampled from the entire data set, often independently of all previous samples. While standard SGD is well understood theoretically, virtually all widely used machine learning software is based on sampling without replacement as this is often empirically superior. That is, the training data is randomly shuffled/permuted, either only once at the beginning, strategy known as random shuffling (RS), or before every epoch, strategy known as random reshuffling (RR), and training proceeds in the data order dictated by the shuffling. RS and RR strategies have for a long time remained beyond the reach of theoretical analysis that would satisfactorily explain their success. However, very recently, Mishchenko et al. [2020] provided tight sublinear convergence rates through a novel analysis, and showed that these strategies can improve upon standard SGD in certain regimes. Inspired by these results, we seek to further improve the rates of shuffling-based methods. In particular, we show that it is possible to enhance them with a variance reduction mechanism, obtaining linear convergence rates. To the best of our knowledge, our linear convergence rates are the best for any method based on sampling without replacement.

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 $O\left( \delta L \exp[-\frac{\mu K}{\delta L}] + \frac{(C + \delta D)}{K\mu}\right)$, where $\delta\ge1$ is a compression parameter which grows when more compression is applied, $L$ and $\mu$ 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.

Federated Learning (FL) is an increasingly popular machine learning paradigm in which multiple nodes try to collaboratively learn under privacy, communication and multiple heterogeneity constraints. A persistent problem in federated learning is that it is not clear what the optimization objective should be: the standard average risk minimization of supervised learning is inadequate in handling several major constraints specific to federated learning, such as communication adaptivity and personalization control. We identify several key desiderata in frameworks for federated learning and introduce a new framework, FLIX, that takes into account the unique challenges brought by federated learning. FLIX has a standard finite-sum form, which enables practitioners to tap into the immense wealth of existing (potentially non-local) methods for distributed optimization. Through a smart initialization that does not require any communication, FLIX does not require the use of local steps but is still provably capable of performing dissimilarity regularization on par with local methods. We give several algorithms for solving the FLIX formulation efficiently under communication constraints. Finally, we corroborate our theoretical results with extensive experimentation.

Recent advances in distributed optimization have shown that Newton-type methods with proper communication compression mechanisms can guarantee fast local rates and low communication cost compared to first order methods. We discover that the communication cost of these methods can be further reduced, sometimes dramatically so, with a surprisingly simple trick: Basis Learn (BL). The idea is to transform the usual representation of the local Hessians via a change of basis in the space of matrices and apply compression tools to the new representation. To demonstrate the potential of using custom bases, we design a new Newton-type method (BL1), which reduces communication cost via both BL technique and bidirectional compression mechanism. Furthermore, we present two alternative extensions (BL2 and BL3) to partial participation to accommodate federated learning applications. We prove local linear and superlinear rates independent of the condition number. Finally, we support our claims with numerical experiments by comparing several first and second order methods.

Optimization problems under affine constraints appear in various areas of machine learning. We consider the task of minimizing a smooth strongly convex function F(x) under the affine constraint Kx = b, with an oracle providing evaluations of the gradient of F and multiplications by K and its transpose. We provide lower bounds on the number of gradient computations and matrix multiplications to achieve a given accuracy. Then we propose an accelerated primal-dual algorithm achieving these lower bounds. Our algorithm is the first optimal algorithm for this class of problems.

Random Reshuffling (RR), also known as Stochastic Gradient Descent (SGD) without replacement, is a popular and theoretically grounded method for finite-sum minimization. We propose two new algorithms: Proximal and Federated Random Reshuffling (ProxRR and FedRR). The first algorithm, ProxRR, solves composite finite-sum minimization problems in which the objective is the sum of a (potentially non-smooth) convex regularizer and an average of $n$ smooth objectives. ProxRR evaluates the proximal operator once per epoch only. When the proximal operator is expensive to compute, this small difference makes ProxRR up to $n$ times faster than algorithms that evaluate the proximal operator in every iteration, such as proximal (stochastic) gradient descent. We give examples of practical optimization tasks where the proximal operator is difficult to compute and ProxRR has a clear advantage. One such task is federated or distributed optimization, where the evaluation of the proximal operator corresponds to communication across the network. We obtain our second algorithm, FedRR, as a special case of ProxRR applied to federated optimization, and prove it has a smaller communication footprint than either distributed gradient descent or Local SGD. Our theory covers both constant and decreasing stepsizes, and allows for importance resampling schemes that can improve conditioning, which may be of independent interest. Our theory covers both convex and nonconvex regimes. Finally, we corroborate our results with experiments on real data sets.

We introduce ProxSkip—a surprisingly simple and provably efficient method for minimizing the sum of a smooth ($f$) and an expensive nonsmooth proximable ($\psi$) function. The canonical approach to solving such problems is via the proximal gradient descent (ProxGD) algorithm, which is based on the evaluation of the gradient of $f$ and the prox operator of $\psi$ in each iteration. In this work we are specifically interested in the regime in which the evaluation of prox is costly relative to the evaluation of the gradient, which is the case in many applications. ProxSkip allows for the expensive prox operator to be skipped in most iterations: while its iteration complexity is $\mathcal{O}(\kappa \log \nicefrac{1}{\varepsilon})$, where $\kappa$ is the condition number of $f$, the number of prox evaluations is $\mathcal{O}(\sqrt{\kappa} \log \nicefrac{1}{\varepsilon})$ only. Our main motivation comes from federated learning, where evaluation of the gradient operator corresponds to taking a local GD step independently on all devices, and evaluation of prox corresponds to (expensive) communication in the form of gradient averaging. In this context, ProxSkip offers an effective acceleration of communication complexity. Unlike other local gradient-type methods, such as FedAvg, SCAFFOLD, S-Local-GD and FedLin, whose theoretical communication complexity is worse than, or at best matching, that of vanilla GD in the heterogeneous data regime, we obtain a provable and large improvement without any heterogeneity-bounding assumptions.

We propose and study a new class of gradient compressors for communication-efficient training—three point compressors (3PC)—as well as efficient distributed nonconvex optimization algorithms that can take advantage of them. Unlike most established approaches, which rely on a static compressor choice (e.g., TopK), our class allows the compressors to evolve throughout the training process, with the aim of improving the theoretical communication complexity and practical efficiency of the underlying methods. We show that our general approach can recover the recently proposed state-of-the-art error feedback mechanism EF21 (Richtárik et al, 2021) and its theoretical properties as a special case, but also leads to a number of new efficient methods. Notably, our approach allows us to improve upon the state-of-the-art in the algorithmic and theoretical foundations of the lazy aggregation literature (Liu et al, 2017; Lan et al, 2017). As a by-product that may be of independent interest, we provide a new and fundamental link between the lazy aggregation and error feedback literature. A special feature of our work is that we do not require the compressors to be unbiased.

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.

Stein Variational Gradient Descent (SVGD) is an algorithm for sampling from a target density which is known up to a multiplicative constant. Although SVGD is a popular algorithm in practice, its theoretical study is limited to a few recent works. We study the convergence of SVGD in the population limit, (i.e., with an infinite number of particles) to sample from a non-logconcave target distribution satisfying Talagrand’s inequality T1. We first establish the convergence of the algorithm. Then, we establish a dimension-dependent complexity bound in terms of the Kernelized Stein Discrepancy (KSD). Unlike existing works, we do not assume that the KSD is bounded along the trajectory of the algorithm. Our approach relies on interpreting SVGD as a gradient descent over a space of probability measures.

Variational inequalities in general and saddle point problems in particular are increasingly relevant in machine learning applications, including adversarial learning, GANs, transport and robust optimization. With increasing data and problem sizes necessary to train high performing models across various applications, we need to rely on parallel and distributed computing. However, in distributed training, communication among the compute nodes is a key bottleneck during training, and this problem is exacerbated for high dimensional and over-parameterized models. Due to these considerations, it is important to equip existing methods with strategies that would allow to reduce the volume of transmitted information during training while obtaining a model of comparable quality. In this paper, we present the first theoretically grounded distributed methods for solving variational inequalities and saddle point problems using compressed communication: MASHA1 and MASHA2. Our theory and methods allow for the use of both unbiased (such as Rand$k$; MASHA1) and contractive (such as Top$k$; MASHA2) compressors. New algorithms support bidirectional compressions, and also can be modified for stochastic setting with batches and for federated learning with partial participation of clients. We empirically validated our conclusions using two experimental setups: a standard bilinear min-max problem, and large-scale distributed adversarial training of transformers.

In distributed or federated optimization and learning, communication between the different computing units is often the bottleneck and gradient compression is widely used to reduce the number of bits sent within each communication round of iterative methods. There are two classes of compression operators and separate algorithms making use of them. In the case of unbiased random compressors with bounded variance (e.g., rand-k), the DIANA algorithm of Mishchenko et al. (2019), which implements a variance reduction technique for handling the variance introduced by compression, is the current state of the art. In the case of biased and contractive compressors (e.g., top-k), the EF21 algorithm of Richtárik et al. (2021), which instead implements an error-feedback mechanism, is the current state of the art. These two classes of compression schemes and algorithms are distinct, with different analyses and proof techniques. In this paper, we unify them into a single framework and propose a new algorithm, recovering DIANA and EF21 as particular cases. Our general approach works with a new, larger class of compressors, which has two parameters, the bias and the variance, and includes unbiased and biased compressors as particular cases. This allows us to inherit the best of the two worlds: like EF21 and unlike DIANA, biased compressors, like top-k, whose good performance in practice is recognized, can be used. And like DIANA and unlike EF21, independent randomness at the compressors allows to mitigate the effects of compression, with the convergence rate improving when the number of parallel workers is large. This is the first time that an algorithm with all these features is proposed. We prove its linear convergence under certain conditions. Our approach takes a step towards better understanding of two so-far distinct worlds of communication-efficient distributed learning.

In this paper, we present the first stepsize schedule for Newton method resulting in fast global and local convergence guarantees. In particular, we a) prove an $\mathcal O \left( 1/{k^2} \right)$ global rate, which matches the state-of-the-art global rate of cubically regularized Newton method of Polyak and Nesterov (2006) and of regularized Newton method of Mishchenko (2021), and the later variant of Doikov and Nesterov (2021), b) prove a local quadratic rate, which matches the best-known local rate of second-order methods, and c) our stepsize formula is simple, explicit, and does not require solving any subproblem. Our convergence proofs hold under affine-invariant assumptions closely related to the notion of self-concordance. Finally, our method has competitive performance when compared to existing baselines which share the same fast global convergence guarantees.

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.

In this paper we study the convex-concave saddle-point problem $\min_x \max_y f(x) + y^\top\mathbf{A}x - g(y)$, where $f(x)$ and $g(y)$ are smooth and convex functions. We propose an Accelerated Primal-Dual Gradient Method (APDG) for solving this problem, achieving (i) an optimal linear convergence rate in the strongly-convex-strongly-concave regime, matching the lower complexity bound (Zhang et al., 2021), and (ii) an accelerated linear convergence rate in the case when only one of the functions $f(x)$ and $g(y)$ is strongly convex or even none of them are. Finally, we obtain a linearly convergent algorithm for the general smooth and convex-concave saddle point problem $\min_x \max_y F(x,y)$ without the requirement of strong convexity or strong concavity.

We study distributed optimization methods based on the {\em local training (LT)} paradigm, i.e., methods which achieve communication efficiency by performing richer local gradient-based training on the clients before (expensive) parameter averaging is allowed to take place. While these methods were first proposed about a decade ago, and form the algorithmic backbone of federated learning, there is an enormous gap between their practical performance, and our theoretical understanding. Looking back at the progress of the field, we {\em identify 5 generations of LT methods}: 1) heuristic, 2) homogeneous, 3) sublinear, 4) linear, and 5) accelerated. The 5${}^{\rm th}$ generation was initiated by the ProxSkip method of Mishchenko et al. (2022), whose analysis provided the first theoretical confirmation that LT is a communication acceleration mechanism. Inspired by this recent progress, we contribute to the 5${}^{\rm th}$ generation of LT methods by showing that it is possible to enhance ProxSkip further using {\em variance reduction}. While all previous theoretical results for LT methods ignore the cost of local work altogether, and are framed purely in terms of the number of communication rounds, we construct a method that can be substantially faster in terms of the {\em total training time} than the state-of-the-art method ProxSkip in theory and practice in the regime when local computation is sufficiently expensive. We characterize this threshold theoretically, and confirm our theoretical predictions with empirical results. Our treatment of variance reduction is generic, and can work with a large number of variance reduction techniques, which may lead to future applications in the future. Finally, we corroborate our theoretical results with carefully engineered proof-of-concept experiments.

Inspired by a recent breakthrough of Mishchenko et al. [2022], who for the first time showed that local gradient steps can lead to provable communication acceleration, we propose an alternative algorithm which obtains the same communication acceleration as their method (ProxSkip). Our approach is very different, however: it is based on the celebrated method of Chambolle and Pock [2011], with several nontrivial modifications: i) we allow for an inexact computation of the prox operator of a certain smooth strongly convex function via a suitable gradient-based method (e.g., GD or Fast GD), ii) we perform a careful modification of the dual update step in order to retain linear convergence. Our general results offer the new state-of-the-art rates for the class of strongly convex-concave saddle-point problems with bilinear coupling characterized by the absence of smoothness in the dual function. When applied to federated learning, we obtain a theoretically better alternative to ProxSkip: our method requires fewer local steps ($\mathcal{O}(\kappa^{1/3})$ or $\mathcal{O}(\kappa^{1/4})$, compared to $\mathcal{O}(\kappa^{1/2})$ of ProxSkip), and performs a deterministic number of local steps instead. Like ProxSkip, our method can be applied to optimization over a connected network, and we obtain theoretical improvements here as well.

To address the high communication costs of distributed machine learning, a large body of work has been devoted in recent years to designing various compression strategies, such as sparsification and quantization, and optimization algorithms capable of using them. Recently, Safaryan et al. (2021) pioneered a dramatically different compression design approach: they first use the local training data to form local smoothness matrices and then propose to design a compressor capable of exploiting the smoothness information contained therein. While this novel approach leads to substantial savings in communication, it is limited to sparsification as it crucially depends on the linearity of the compression operator. In this work, we generalize their smoothness-aware compression strategy to arbitrary unbiased compression operators, which also include sparsification. Specializing our results to stochastic quantization, we guarantee significant savings in communication complexity compared to standard quantization. In particular, we prove that block quantization with $n$ blocks theoretically outperforms single block quantization, leading to a reduction in communication complexity by an $\mathcal{O}(n)$ factor, where $n$ is the number of nodes in the distributed system. Finally, we provide extensive numerical evidence with convex optimization problems that our smoothness-aware quantization strategies outperform existing quantization schemes as well as the aforementioned smoothness-aware sparsification strategies with respect to three evaluation metrics: the number of iterations, the total amount of bits communicated, and wall-clock time.

Communication efficiency has been widely recognized as the bottleneck for large-scale decentralized machine learning applications in multi-agent or federated environments. To tackle the communication bottleneck, there have been many efforts to design communication-compressed algorithms for decentralized nonconvex optimization, where the clients are only allowed to communicate a small amount of quantized information (aka bits) with their neighbors over a predefined graph topology. Despite significant efforts, the state-of-the-art algorithm in the nonconvex setting still suffers from a slower rate of convergence $O((G/T)^{2/3})$ compared with their uncompressed counterpart, where $G$ measures the data heterogeneity across different clients, and $T$ is the number of communication rounds. This paper proposes BEER, which adopts communication compression with gradient tracking, and shows it converges at a faster rate of $O(1/T)$. This significantly improves over the state-of-the-art rate, by matching the rate without compression even under arbitrary data heterogeneity. Numerical experiments are also provided to corroborate our theory and confirm the practical superiority of beer in the data heterogeneous regime.

Communication is one of the key bottlenecks in the distributed training of large-scale machine learning models, and lossy compression of exchanged information, such as stochastic gradients or models, is one of the most effective instruments to alleviate this issue. Among the most studied compression techniques is the class of unbiased compression operators with variance bounded by a multiple of the square norm of the vector we wish to compress. By design, this variance may remain high, and only diminishes if the input vector approaches zero. However, unless the model being trained is overparameterized, there is no a-priori reason for the vectors we wish to compress to approach zero during the iterations of classical methods such as distributed compressed {\sf SGD}, which has adverse effects on the convergence speed. Due to this issue, several more elaborate and seemingly very different algorithms have been proposed recently, with the goal of circumventing this issue. These methods are based on the idea of compressing the {\em difference} between the vector we would normally wish to compress and some auxiliary vector which changes throughout the iterative process. In this work we take a step back, and develop a unified framework for studying such methods, conceptually, and theoretically. Our framework incorporates methods compressing both gradients and models, using unbiased and biased compressors, and sheds light on the construction of the auxiliary vectors. Furthermore, our general framework can lead to the improvement of several existing algorithms, and can produce new algorithms. Finally, we performed several numerical experiments which illustrate and support our theoretical findings.

We present a unified framework for analyzing local SGD methods in the convex and strongly convex regimes for distributed/federated training of supervised machine learning models. We recover several known methods as a special case of our general framework, including Local SGD/FedAvg, SCAFFOLD, and several variants of SGD not originally designed for federated learning. Our framework covers both the identical and heterogeneous data settings, supports both random and deterministic number of local steps, and can work with a wide array of local stochastic gradient estimators, including shifted estimators which are able to adjust the fixed points of local iterations for faster convergence. As an application of our framework, we develop multiple novel FL optimizers which are superior to existing methods. In particular, we develop the first linearly converging local SGD method which does not require any data homogeneity or other strong assumptions.

Bayesian optimization (BO) is a data-efficient approach to automatically tune the hyperparameters of machine learning models. In practice, one frequently has to solve similar hyperparameter tuning problems sequentially. For example, one might have to tune a type of neural network learned across a series of different classification problems. Recent work on multi-task BO exploits knowledge gained from previous hyperparameter tuning tasks to speed up a new tuning task. However, previous approaches do not account for the fact that BO is a sequential decision making procedure. Hence, there is in general a mismatch between the number of evaluations collected in the current tuning task compared to the number of evaluations accumulated in all previously completed tasks. In this work, we enable multi-task BO to compensate for this mismatch, such that the transfer learning procedure is able to handle different data regimes in a principled way. We propose a new multi-task BO method that learns a set of ordered, non-linear basis functions of increasing complexity via nested drop-out and automatic relevance determination. Experiments on a variety of hyperparameter tuning problems show that our method improves the sample efficiency of recently published multi-task BO methods.

Decentralized optimization methods enable on-device training of machine learning models without a central coordinator. In many scenarios communication between devices is energy demanding and time consuming and forms the bottleneck of the entire system. We propose a new randomized first-order method which tackles the communication bottleneck by applying randomized compression operators to the communicated messages. By combining our scheme with a new variance reduction technique that progressively throughout the iterations reduces the adverse effect of the injected quantization noise, we obtain a scheme that converges linearly on strongly convex decentralized problems while using compressed communication only. We prove that our method can solve the problems without any increase in the number of communications compared to the baseline which does not perform any communication compression while still allowing for a significant compression factor which depends on the conditioning of the problem and the topology of the network. We confirm our theoretical findings in numerical experiments.

We develop and analyze MARINA: a new communication efficient method for non-convex distributed learning over heterogeneous datasets. MARINA employs a novel communication compression strategy based on the compression of gradient differences that is reminiscent of but different from the strategy employed in the DIANA method of Mishchenko et al. (2019). Unlike virtually all competing distributed first-order methods, including DIANA, ours is based on a carefully designed biased gradient estimator, which is the key to its superior theoretical and practical performance. The communication complexity bounds we prove for MARINA are evidently better than those of all previous first-order methods. Further, we develop and analyze two variants of MARINA: VR-MARINA and PP-MARINA. The first method is designed for the case when the local loss functions owned by clients are either of a finite sum or of an expectation form, and the second method allows for a partial participation of clients {–} a feature important in federated learning. All our methods are superior to previous state-of-the-art methods in terms of oracle/communication complexity. Finally, we provide a convergence analysis of all methods for problems satisfying the Polyak-{Ł}ojasiewicz condition.

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.

We propose ADOM – an accelerated method for smooth and strongly convex decentralized optimization over time-varying networks. ADOM uses a dual oracle, i.e., we assume access to the gradient of the Fenchel conjugate of the individual loss functions. Up to a constant factor, which depends on the network structure only, its communication complexity is the same as that of accelerated Nesterov gradient method. To the best of our knowledge, only the algorithm of Rogozin et al. (2019) has a convergence rate with similar properties. However, their algorithm converges under the very restrictive assumption that the number of network changes can not be greater than a tiny percentage of the number of iterations. This assumption is hard to satisfy in practice, as the network topology changes usually can not be controlled. In contrast, ADOM merely requires the network to stay connected throughout time.

In this paper, we propose a novel stochastic gradient estimator—ProbAbilistic Gradient Estimator (PAGE)—for nonconvex optimization. PAGE is easy to implement as it is designed via a small adjustment to vanilla SGD: in each iteration, PAGE uses the vanilla minibatch SGD update with probability $p_t$ or reuses the previous gradient with a small adjustment, at a much lower computational cost, with probability $1-p_t$. We give a simple formula for the optimal choice of $p_t$. Moreover, we prove the first tight lower bound $\Omega(n+\frac{\sqrt{n}}{\epsilon^2})$ for nonconvex finite-sum problems, which also leads to a tight lower bound $\Omega(b+\frac{\sqrt{b}}{\epsilon^2})$ for nonconvex online problems, where $b:= \min\{\frac{\sigma^2}{\epsilon^2}, n\}$. Then, we show that PAGE obtains the optimal convergence results $O(n+\frac{\sqrt{n}}{\epsilon^2})$ (finite-sum) and $O(b+\frac{\sqrt{b}}{\epsilon^2})$ (online) matching our lower bounds for both nonconvex finite-sum and online problems. Besides, we also show that for nonconvex functions satisfying the Polyak-Ł{ojasiewicz} (PL) condition, PAGE can automatically switch to a faster linear convergence rate $O(\cdot\log \frac{1}{\epsilon})$. Finally, we conduct several deep learning experiments (e.g., LeNet, VGG, ResNet) on real datasets in PyTorch showing that PAGE not only converges much faster than SGD in training but also achieves the higher test accuracy, validating the optimal theoretical results and confirming the practical superiority of PAGE.

Various gradient compression schemes have been proposed to mitigate the communication cost in distributed training of large scale machine learning models. Sign-based methods, such as signSGD (Bernstein et al., 2018), have recently been gaining popularity because of their simple compression rule and connection to adaptive gradient methods, like ADAM. In this paper, we analyze sign-based methods for non-convex optimization in three key settings: (i) standard single node, (ii) parallel with shared data and (iii) distributed with partitioned data. For single machine case, we generalize the previous analysis of signSGD relying on intuitive bounds on success probabilities and allowing even biased estimators. Furthermore, we extend the analysis to parallel setting within a parameter server framework, where exponentially fast noise reduction is guaranteed with respect to number of nodes, maintaining $1$-bit compression in both directions and using small mini-batch sizes. Next, we identify a fundamental issue with signSGD to converge in distributed environment. To resolve this issue, we propose a new sign-based method, {\em Stochastic Sign Descent with Momentum (SSDM)}, which converges under standard bounded variance assumption with the optimal asymptotic rate. We validate several aspects of our theoretical findings with numerical experiments.

We consider the task of minimizing the sum of smooth and strongly convex functions stored in a decentralized manner across the nodes of a communication network whose links are allowed to change in time. We solve two fundamental problems for this task. First, we establish {\em the first lower bounds} on the number of decentralized communication rounds and the number of local computations required to find an $\epsilon$-accurate solution. Second, we design two {\em optimal algorithms} that attain these lower bounds: (i) a variant of the recently proposed algorithm ADOM (Kovalev et al, 2021) enhanced via a multi-consensus subroutine, which is optimal in the case when access to the dual gradients is assumed, and (ii) a novel algorithm, called ADOM+, which is optimal in the case when access to the primal gradients is assumed. We corroborate the theoretical efficiency of these algorithms by performing an experimental comparison with existing state-of-the-art methods.

Due to the high communication cost in distributed and federated learning, methods relying on compressed communication are becoming increasingly popular. Besides, the best theoretically and practically performing gradient-type methods invariably rely on some form of acceleration/momentum to reduce the number of communications (faster convergence), e.g., Nesterov's accelerated gradient descent [31, 32] and Adam [14]. In order to combine the benefits of communication compression and convergence acceleration, we propose a \emph{compressed and accelerated} gradient method based on ANITA [20] for distributed optimization, which we call CANITA. Our CANITA achieves the \emph{first accelerated rate} $O\bigg(\sqrt{\Big(1+\sqrt{\frac{\omega^3}{n}}\Big)\frac{L}{\epsilon}} + \omega\big(\frac{1}{\epsilon}\big)^{\frac{1}{3}}\bigg)$, which improves upon the state-of-the-art non-accelerated rate $O\left((1+\frac{\omega}{n})\frac{L}{\epsilon} + \frac{\omega^2+\omega}{\omega+n}\frac{1}{\epsilon}\right)$ of DIANA [12] for distributed general convex problems, where $\epsilon$ is the target error, $L$ is the smooth parameter of the objective, $n$ is the number of machines/devices, and $\omega$ is the compression parameter (larger $\omega$ means more compression can be applied, and no compression implies $\omega=0$). Our results show that as long as the number of devices $n$ is large (often true in distributed/federated learning), or the compression $\omega$ is not very high, CANITA achieves the faster convergence rate $O\Big(\sqrt{\frac{L}{\epsilon}}\Big)$, i.e., the number of communication rounds is $O\Big(\sqrt{\frac{L}{\epsilon}}\Big)$ (vs. $O\big(\frac{L}{\epsilon}\big)$ achieved by previous works). As a result, CANITA enjoys the advantages of both compression (compressed communication in each round) and acceleration (much fewer communication rounds).

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.

Error feedback (EF), also known as error compensation, is an immensely popular convergence stabilization mechanism in the context of distributed training of supervised machine learning models enhanced by the use of contractive communication compression mechanisms, such as Top-$k$. First proposed by Seide et al [2014] as a heuristic, EF resisted any theoretical understanding until recently [Stich et al., 2018, Alistarh et al., 2018]. While these early breakthroughs were followed by a steady stream of works offering various improvements and generalizations, the current theoretical understanding of EF is still very limited. Indeed, to the best of our knowledge, all existing analyses either i) apply to the single node setting only, ii) rely on very strong and often unreasonable assumptions, such as global boundedness of the gradients, or iterate-dependent assumptions that cannot be checked a-priori and may not hold in practice, or iii) circumvent these issues via the introduction of additional unbiased compressors, which increase the communication cost. In this work we fix all these deficiencies by proposing and analyzing a new EF mechanism, which we call EF21, which consistently and substantially outperforms EF in practice. Moreover, our theoretical analysis relies on standard assumptions only, works in the distributed heterogeneous data setting, and leads to better and more meaningful rates. In particular, we prove that EF21 enjoys a fast $\mathcal{O}(1/T)$ convergence rate for smooth nonconvex problems, beating the previous bound of $\mathcal{O}(1/T^{2/3})$, which was shown under a strong bounded gradients assumption. We further improve this to a fast linear rate for Polyak-Lojasiewicz functions, which is the first linear convergence result for an error feedback method not relying on unbiased compressors. Since EF has a large number of applications where it reigns supreme, we believe that our 2021 variant, EF21, will have a large impact on the practice of communication efficient distributed learning.

Large scale distributed optimization has become the default tool for the training of supervised machine learning models with a large number of parameters and training data. Recent advancements in the field provide several mechanisms for speeding up the training, including {\em compressed communication}, {\em variance reduction} and {\em acceleration}. However, none of these methods is capable of exploiting the inherently rich data-dependent smoothness structure of the local losses beyond standard smoothness constants. In this paper, we argue that when training supervised models, {\em smoothness matrices}---information-rich generalizations of the ubiquitous smoothness constants---can and should be exploited for further dramatic gains, both in theory and practice. In order to further alleviate the communication burden inherent in distributed optimization, we propose a novel communication sparsification strategy that can take full advantage of the smoothness matrices associated with local losses. To showcase the power of this tool, we describe how our sparsification technique can be adapted to three distributed optimization algorithms---DCGD, DIANA and ADIANA---yielding significant savings in terms of communication complexity. The new methods always outperform the baselines, often dramatically so.

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.

In this work we present a new framework for the analysis and design of randomized gossip algorithms for solving the average consensus problem. We show how classical randomized iterative methods for solving linear systems can be interpreted as gossip algorithms when applied to special systems encoding the underlying network and explain in detail their decentralized nature. Our general framework recovers a comprehensive array of well-known gossip algorithms as special cases, including the pairwise randomized gossip algorithm and path averaging gossip, and allows for the development of provably faster variants. The flexibility of the new approach enables the design of a number of new specific gossip methods. For instance, we propose and analyze novelblockand the first provablyacceleratedrandomized gossip protocols, anddualrandomized gossip algorithms. From a numerical analysis viewpoint, our work is the first that explores in depth the decentralized nature of randomized iterative methods for linear systems and proposes them as methods for solving the average consensus problem. We evaluate the performance of the proposed gossip protocols by performing extensive experimental testing on typical wireless network topologies.

We consider the problem of unconstrained minimization of a smooth objective function in ℝn in a setting where only function evaluations are possible. While importance sampling is one of the most popular techniques used by machine learning practitioners to accelerate the convergence of their models when applicable, there is not much existing theory for this acceleration in the derivative-free setting. In this paper, we propose the first derivative free optimization method with importance sampling and derive new improved complexity results on non-convex, convex and strongly convex functions. We conduct extensive experiments on various synthetic and real LIBSVM datasets confirming our theoretical results. We test our method on a collection of continuous control tasks on MuJoCo environments with varying difficulty. Experiments show that our algorithm is practical for high dimensional continuous control problems where importance sampling results in a significant sample complexity improvement.

We provide a new analysis of local SGD, removing unnecessary assumptions and elaborating on the difference between two data regimes: identical and heterogeneous. In both cases, we improve the existing theory and provide values of the optimal stepsize and optimal number of local iterations. Our bounds are based on a new notion of variance that is specific to local SGD methods with different data. The tightness of our results is guaranteed by recovering known statements when we plug $H=1$, where $H$ is the number of local steps. The empirical evidence further validates the severe impact of data heterogeneity on the performance of local SGD.

In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent (SGD) which so far have required different intuitions, convergence analyses, have different applications, and which have been developed separately in various communities. We show that our framework includes methods with and without the following tricks, and their combinations: variance reduction, importance sampling, mini-batch sampling, quantization, and coordinate sub-sampling. As a by-product, we obtain the first unified theory of SGD and randomized coordinate descent (RCD) methods, the first unified theory of variance reduced and non-variance-reduced SGD methods, and the first unified theory of quantized and non-quantized methods. A key to our approach is a parametric assumption on the iterates and stochastic gradients. In a single theorem we establish a linear convergence result under this assumption and strong-quasi convexity of the loss function. Whenever we recover an existing method as a special case, our theorem gives the best known complexity result. Our approach can be used to motivate the development of new useful methods, and offers pre-proved convergence guarantees. To illustrate the strength of our approach, we develop five new variants of SGD, and through numerical experiments demonstrate some of their properties.

We fix a fundamental issue in the stochastic extragradient method by providing a new sampling strategy that is motivated by approximating implicit updates. Since the existing stochastic extragradient algorithm, called Mirror-Prox, of (Juditsky, 2011) diverges on a simple bilinear problem when the domain is not bounded, we prove guarantees for solving variational inequality that go beyond existing settings. Furthermore, we illustrate numerically that the proposed variant converges faster than many other methods on several convex-concave saddle-point problems. We also discuss how extragradient can be applied to training Generative Adversarial Networks (GANs) and how it compares to other methods. Our experiments on GANs demonstrate that the introduced approach may make the training faster in terms of data passes, while its higher iteration complexity makes the advantage smaller.

The stochastic variance-reduced gradient method (SVRG) and its accelerated variant (Katyusha) have attracted enormous attention in the machine learning community in the last few years due to their superior theoretical properties and empirical behaviour on training supervised machine learning models via the empirical risk minimization paradigm. A key structural element in both of these methods is the inclusion of an outer loop at the beginning of which a full pass over the training data is made in order to compute the exact gradient, which is then used in an inner loop to construct a variance-reduced estimator of the gradient using new stochastic gradient information. In this work, we design loopless variants of both of these methods. In particular, we remove the outer loop and replace its function by a coin flip performed in each iteration designed to trigger, with a small probability, the computation of the gradient. We prove that the new methods enjoy the same superior theoretical convergence properties as the original methods. For loopless SVRG, the same rate is obtained for a large interval of coin flip probabilities, including the probability $\frac{1}{n}$, where $n$ is the number of functions. This is the first result where a variant of SVRG is shown to converge with the same rate without the need for the algorithm to know the condition number, which is often unknown or hard to estimate correctly. We demonstrate through numerical experiments that the loopless methods can have superior and more robust practical behavior.

In this paper, we propose a new randomized second-order optimization algorithm—Stochastic Subspace Cubic Newton (SSCN)—for minimizing a high dimensional convex function $f$. Our method can be seen both as a \emph{stochastic} extension of the cubically-regularized Newton method of Nesterov and Polyak (2006), and a \emph{second-order} enhancement of stochastic subspace descent of Kozak et al. (2019). We prove that as we vary the minibatch size, the global convergence rate of SSCN interpolates between the rate of stochastic coordinate descent (CD) and the rate of cubic regularized Newton, thus giving new insights into the connection between first and second-order methods. Remarkably, the local convergence rate of SSCN matches the rate of stochastic subspace descent applied to the problem of minimizing the quadratic function $\frac12 (x-x^*)^\top \nabla^2f(x^*)(x-x^*)$, where $x^*$ is the minimizer of $f$, and hence depends on the properties of $f$ at the optimum only. Our numerical experiments show that SSCN outperforms non-accelerated first-order CD algorithms while being competitive to their accelerated variants.

We propose an accelerated version of stochastic variance reduced coordinate descent – ASVRCD. As other variance reduced coordinate descent methods such as SEGA or SVRCD, our method can deal with problems that include a non-separable and non-smooth regularizer, while accessing a random block of partial derivatives in each iteration only. However, ASVRCD incorporates Nesterov’s momentum, which offers favorable iteration complexity guarantees over both SEGA and SVRCD. As a by-product of our theory, we show that a variant of Katyusha (Allen-Zhu, 2017) is a specific case of ASVRCD, recovering the optimal oracle complexity for the finite sum objective.

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.

Most algorithms for solving optimization problems or finding saddle points of convex-concave functions are fixed-point algorithms. In this work we consider the generic problem of finding a fixed point of an average of operators, or an approximation thereof, in a distributed setting. Our work is motivated by the needs of federated learning. In this context, each local operator models the computations done locally on a mobile device. We investigate two strategies to achieve such a consensus: one based on a fixed number of local steps, and the other based on randomized computations. In both cases, the goal is to limit communication of the locally-computed variables, which is often the bottleneck in distributed frameworks. We perform convergence analysis of both methods and conduct a number of experiments highlighting the benefits of our approach.

In this paper, we propose a unified analysis of variants of distributed SGD with arbitrary compressions and delayed updates. Our framework is general enough to cover different variants of quantized SGD, Error-Compensated SGD (EC-SGD), and SGD with delayed updates (D-SGD). Via single theorem, we derive the complexity results for all the methods that fit our framework. For the existing methods, this theorem gives the best-known complexity results. Moreover, using our general scheme, we develop new variants of SGD that combine variance reduction or arbitrary sampling with error feedback and quantization and derive the convergence rates for these methods beating the state-of-the-art results. In order to illustrate the strength of our framework, we develop 16 new methods that fit this. In particular, we propose the first method called EC-SGD-DIANA that is based on error-feedback for biased compression operator and quantization of gradient differences and prove the convergence guarantees showing that EC-SGD-DIANA converges to the exact optimum asymptotically in expectation with constant learning rate for both convex and strongly convex objectives when workers compute full gradients of their loss functions. Moreover, for the case when the loss function of the worker has the form of finite sum, we modified the method and got a new one called EC-LSVRG-DIANA which is the first distributed stochastic method with error feedback and variance reduction that converges to the exact optimum asymptotically in expectation with constant learning rate.

In this work, we consider the optimization formulation of personalized federated learning recently introduced by Hanzely & Richtarik (2020) which was shown to give an alternative explanation to the workings of local SGD methods. Our first contribution is establishing the first lower bounds for this formulation, for both the communication complexity and the local oracle complexity. Our second contribution is the design of several optimal methods matching these lower bounds in almost all regimes. These are the first provably optimal methods for personalized federated learning. Our optimal methods include an accelerated variant of FedProx, and an accelerated variance-reduced version of FedAvg/Local SGD. We demonstrate the practical superiority of our methods through extensive numerical experiments.

We consider the task of decentralized minimization of the sum of smooth strongly convex functions stored across the nodes of a network. For this problem, lower bounds on the number of gradient computations and the number of communication rounds required to achieve $\varepsilon$ accuracy have recently been proven. We propose two new algorithms for this decentralized optimization problem and equip them with complexity guarantees. We show that our first method is optimal both in terms of the number of communication rounds and in terms of the number of gradient computations. Unlike existing optimal algorithms, our algorithm does not rely on the expensive evaluation of dual gradients. Our second algorithm is optimal in terms of the number of communication rounds, without a logarithmic factor. Our approach relies on viewing the two proposed algorithms as accelerated variants of the Forward Backward algorithm to solve monotone inclusions associated with the decentralized optimization problem. We also verify the efficacy of our methods against state-of-the-art algorithms through numerical experiments.

Random Reshuffling (RR) is an algorithm for minimizing finite-sum functions that utilizes iterative gradient descent steps in conjunction with data reshuffling. Often contrasted with its sibling Stochastic Gradient Descent (SGD), RR is usually faster in practice and enjoys significant popularity in convex and non-convex optimization. The convergence rate of RR has attracted substantial attention recently and, for strongly convex and smooth functions, it was shown to converge faster than SGD if 1) the stepsize is small, 2) the gradients are bounded, and 3) the number of epochs is large. We remove these 3 assumptions, improve the dependence on the condition number from $\kappa^2$ to $\kappa$ (resp.\ from $\kappa$ to $\sqrt{\kappa}$) and, in addition, show that RR has a different type of variance. We argue through theory and experiments that the new variance type gives an additional justification of the superior performance of RR. To go beyond strong convexity, we present several results for non-strongly convex and non-convex objectives. We show that in all cases, our theory improves upon existing literature. Finally, we prove fast convergence of the Shuffle-Once (SO) algorithm, which shuffles the data only once, at the beginning of the optimization process. Our theory for strongly convex objectives tightly matches the known lower bounds for both RR and SO and substantiates the common practical heuristic of shuffling once or only a few times. As a byproduct of our analysis, we also get new results for the Incremental Gradient algorithm (IG), which does not shuffle the data at all.

We consider the task of sampling with respect to a log concave probability distribution. The potential of the target distribution is assumed to be composite, i.e., written as the sum of a smooth convex term, and a nonsmooth convex term possibly taking infinite values. The target distribution can be seen as a minimizer of the Kullback-Leibler divergence defined on the Wasserstein space (i.e., the space of probability measures). In the first part of this paper, we establish a strong duality result for this minimization problem. In the second part of this paper, we use the duality gap arising from the first part to study the complexity of the Proximal Stochastic Gradient Langevin Algorithm (PSGLA), which can be seen as a generalization of the Projected Langevin Algorithm. Our approach relies on viewing PSGLA as a primal dual algorithm and covers many cases where the target distribution is not fully supported. In particular, we show that if the potential is strongly convex, the complexity of PSGLA is $\cO(1/\varepsilon^2)$ in terms of the 2-Wasserstein distance. In contrast, the complexity of the Projected Langevin Algorithm is $\cO(1/\varepsilon^{12})$ in terms of total variation when the potential is convex.

In this paper we discuss sparsification of worker-to-server communication in large distributed systems. We improve upon algorithms that fit the following template: a local gradient estimate is computed independently by each worker, then communicated to a master, which subsequently performs averaging. The average is broadcast back to the workers, which use it to perform a gradient-type step to update the local version of the model. We observe that the above template is fundamentally inefficient in that too much data is unnecessarily communicated from the workers to the server, which slows down the overall system. We propose a fix based on a new update-sparsification method we develop in this work, which we suggest be used on top of existing methods. Namely, we develop a new variant of parallel block coordinate descent based on independent sparsification of the local gradient estimates before communication. We demonstrate that with only $m/n$ blocks sent by each of $n$ workers, where $m$ is the total number of parameter blocks, the theoretical iteration complexity of the underlying distributed methods is essentially unaffected. As an illustration, this means that when $n=100$ parallel workers are used, the communication of 99% blocks is redundant, and hence a waste of time. Our theoretical claims are supported through extensive numerical experiments which demonstrate an almost perfect match with our theory on a number of synthetic and real datasets.

Stochastic optimization lies at the heart of machine learning, and its cornerstone is stochastic gradient descent (SGD), a method introduced over 60 years ago. The last eight years have seen an exciting new development: variance reduction for stochastic optimization methods. These variance-reduced (VR) methods excel in settings where more than one pass through the training data is allowed, achieving a faster convergence than SGD in theory and practice. These speedups underline the surge of interest in VR methods and the fast-growing body of work on this topic. This review covers the key principles and main developments behind VR methods for optimization with finite data sets and is aimed at nonexpert readers. We focus mainly on the convex setting and leave pointers to readers interested in extensions for minimizing nonconvex functions.

Robust principal component analysis (RPCA) is a well-studied problem whose goal is to decompose a matrix into the sum of low-rank and sparse components. In this paper, we propose a nonconvex feasibility reformulation of RPCA problem and apply an alternating projection method to solve it. To the best of our knowledge, this is the first paper proposing a method that solves RPCA problem without considering any objective function, convex relaxation, or surrogate convex constraints. We demonstrate through extensive numerical experiments on a variety of applications, including shadow removal, background estimation, face detection, and galaxy evolution, that our approach matches and often significantly outperforms current state-of-the-art in various ways.

Accelerated coordinate descent is a widely popular optimization algorithm due to its efficiency on large-dimensional problems. It achieves state-of-the-art complexity on an important class of empirical risk minimization problems. In this paper we design and analyze an accelerated coordinate descent (\texttt{ACD}) method which in each iteration updates a random subset of coordinates according to an arbitrary but fixed probability law, which is a parameter of the method. While mini-batch variants of \texttt{ACD} are more popular and relevant in practice, there is no importance sampling for \texttt{ACD} that outperforms the standard uniform mini-batch sampling. Through insights enabled by our general analysis, we design new importance sampling for mini-batch \texttt{ACD} which significantly outperforms previous state-of-the-art minibatch \texttt{ACD} in practice. We prove a rate that is at most $\mathcal{O}(\sqrt{\tau})$ times worse than the rate of minibatch \texttt{ACD} with uniform sampling, but can be $\mathcal{O}(n/\tau)$ times better, where $\tau$ is the minibatch size. Since in modern supervised learning training systems it is standard practice to choose $\tau \ll n$, and often $\tau=\mathcal{O}(1)$, our method can lead to dramatic speedups. Lastly, we obtain similar results for minibatch nonaccelerated \texttt{CD} as well, achieving improvements on previous best rates.

In this work we present novel provably accelerated gossip algorithms for solving the average consensus problem. The proposed protocols are inspired from the recently developed accelerated variants of the randomized Kaczmarz method - a popular method for solving linear systems. In each gossip iteration all nodes of the network update their values but only a pair of them exchange their private information. Numerical experiments on popular wireless sensor networks showing the benefits of our protocols are also presented.

We provide the first importance sampling variants of variance reduced algorithms for empirical risk minimization with non-convex loss functions. In particular, we analyze non-convex versions of \texttt{SVRG}, \texttt{SAGA} and \texttt{SARAH}. Our methods have the capacity to speed up the training process by an order of magnitude compared to the state of the art on real datasets. Moreover, we also improve upon current mini-batch analysis of these methods by proposing importance sampling for minibatches in this setting. Surprisingly, our approach can in some regimes lead to superlinear speedup with respect to the minibatch size, which is not usually present in stochastic optimization. All the above results follow from a general analysis of the methods which works with arbitrary sampling, i.e., fully general randomized strategy for the selection of subsets of examples to be sampled in each iteration. Finally, we also perform a novel importance sampling analysis of \texttt{SARAH} in the convex setting.

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.

We develop a randomized Newton method capable of solving learning problems with huge dimensional feature spaces, which is a common setting in applications such as medical imaging, genomics and seismology. Our method leverages randomized sketching in a new way, by finding the Newton direction constrained to the space spanned by a random sketch. We develop a simple global linear convergence theory that holds for practically all sketching techniques, which gives the practitioners the freedom to design custom sketching approaches suitable for particular applications. We perform numerical experiments which demonstrate the efficiency of our method as compared to accelerated gradient descent and the full Newton method. Our method can be seen as a refinement and a randomized extension of the results of Karimireddy, Stich, and Jaggi (2019).

We propose a new algorithm---Stochastic Proximal Langevin Algorithm (SPLA)---for sampling from a log concave distribution. Our method is a generalization of the Langevin algorithm to potentials expressed as the sum of one stochastic smooth term and multiple stochastic nonsmooth terms. In each iteration, our splitting technique only requires access to a stochastic gradient of the smooth term and a stochastic proximal operator for each of the nonsmooth terms. We establish nonasymptotic sublinear and linear convergence rates under convexity and strong convexity of the smooth term, respectively, expressed in terms of the KL divergence and Wasserstein distance. We illustrate the efficiency of our sampling technique through numerical simulations on a Bayesian learning task.

We propose a surprisingly simple model to estimate supervised video backgrounds. Our model is based on L1 regression. As existing methods for L1 regression do not scale to high-resolution videos, we propose several simple, fast, and scalable methods including iteratively reweighted least squares, a homotopy method, and stochastic gradient descent to solve the problem. Our extensive implementations of the model and methods show that they match or outperform other state-of-the-art online and batch methods that are both supervised and unsupervised in virtually all quantitative and qualitative measures and in fractions of their execution time.

The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is violated for cases where the objective function is strongly convex. In Bottou et al. (2018), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. We show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime. We then move on to the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime in the case of diminished learning rate. It is well-known that SGD converges if a sequence of learning rates $\{\eta_t\}$ satisfies $\sum_{t=0}^\infty \eta_t \rightarrow \infty$ and $\sum_{t=0}^\infty \eta^2_t < \infty$. We show the convergence of SGD for strongly convex objective function without using bounded gradient assumption when $\{\eta_t\}$ is a diminishing sequence and $\sum_{t=0}^\infty \eta_t \rightarrow \infty$. In other words, we extend the current state-of-the-art class of learning rates satisfying the convergence of SGD.

Randomized coordinate descent (RCD) methods are state-of-the-art algorithms for training linear predictors via minimizing regularized empirical risk. When the number of examples (n) is much larger than the number of features (d), a common strategy is to apply RCD to the dual problem. On the other hand, when the number of features is much larger than the number of examples, it makes sense to apply RCD directly to the primal problem. In this paper we provide the first joint study of these two approaches when applied to L2-regularized linear ERM. First, we show through a rigorous analysis that for dense data, the above intuition is precisely correct. However, we find that for sparse and structured data, primal RCD can significantly outperform dual RCD even if $d << n$, and vice versa, dual RCD can be much faster than primal RCD even if $n << d$. Moreover, we show that, surprisingly, a single sampling strategy minimizes both the (bound on the) number of iterations and the overall expected complexity of RCD. Note that the latter complexity measure also takes into account the average cost of the iterations, which depends on the structure and sparsity of the data, and on the sampling strategy employed. We confirm our theoretical predictions using extensive experiments with both synthetic and real data sets.

We study the problem of minimizing the sum of three convex functions: a differentiable, twice-differentiable and a non-smooth term in a high dimensional setting. To this effect we propose and analyze a randomized block cubic Newton (RBCN) method, which in each iteration builds a model of the objective function formed as the sum of the natural models of its three components: a linear model with a quadratic regularizer for the differentiable term, a quadratic model with a cubic regularizer for the twice differentiable term, and perfect (proximal) model for the nonsmooth term. Our method in each iteration minimizes the model over a random subset of blocks of the search variable. RBCN is the first algorithm with these properties, generalizing several existing methods, matching the best known bounds in all special cases. We establish ${\cal O}(1/\epsilon)$, ${\cal O}(1/\sqrt{\epsilon})$ and ${\cal O}(\log (1/\epsilon))$ rates under different assumptions on the component functions. Lastly, we show numerically that our method outperforms the state-of-the-art on a variety of machine learning problems, including cubically regularized least-squares, logistic regression with constraints, and Poisson regression.

Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime, which results in more relaxed conditions than those in (Bottou et al.,2016). We then move on the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate.

We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the way for many applications in the field of optimization and machine learning. As an application of our general theory, we develop the first accelerated (deterministic and stochastic) quasi-Newton updates. Our updates lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments. Experiments with empirical risk minimization show that our rules can accelerate training of machine learning models.

We propose a novel randomized first order optimization method---SEGA (SkEtched GrAdient method)---which progressively throughout its iterations builds a variance-reduced estimate of the gradient from random linear measurements (sketches) of the gradient provided at each iteration by an oracle. In each iteration, SEGA updates the current estimate of the gradient through a sketch-and-project operation using the information provided by the latest sketch, and this is subsequently used to compute an unbiased estimate of the true gradient through a random relaxation procedure. This unbiased estimate is then used to perform a gradient step. Unlike standard subspace descent methods, such as coordinate descent, SEGA can be used for optimization problems with a non-separable proximal term. We provide a general convergence analysis and prove linear convergence for strongly convex objectives. In the special case of coordinate sketches, SEGA can be enhanced with various techniques such as importance sampling, minibatching and acceleration, and its rate is up to a small constant factor identical to the best-known rate of coordinate descent.

The state-of-the-art methods for solving optimization problems in big dimensions are variants of randomized coordinate descent (RCD). In this paper we introduce a fundamentally new type of acceleration strategy for RCD based on the augmentation of the set of coordinate directions by a few spectral or conjugate directions. As we increase the number of extra directions to be sampled from, the rate of the method improves, and interpolates between the linear rate of RCD and a linear rate independent of the condition number. We develop and analyze also inexact variants of these methods where the spectral and conjugate directions are allowed to be approximate only. We motivate the above development by proving several negative results which highlight the limitations of RCD with importance sampling.

Minibatching is a very well studied and highly popular technique in supervised learning, used by practitioners due to its ability to accelerate training through better utilization of parallel processing power and reduction of stochastic variance. Another popular technique is importance sampling–a strategy for preferential sampling of more important examples also capable of accelerating the training process. However, despite considerable effort by the community in these areas, and due to the inherent technical difficulty of the problem, there is virtually no existing work combining the power of importance sampling with the strength of minibatching. In this paper we propose the first practical importance sampling for minibatches and give simple and rigorous complexity analysis of its performance. We illustrate on synthetic problems that for training data of certain properties, our sampling can lead to several orders of magnitude improvement in training time. We then test the new sampling on several popular data sets, and show that the improvement can reach an order of magnitude.

We propose a novel limited-memory stochastic block BFGS update for incorporating enriched curvature information in stochastic approximation methods. In our method, the estimate of the inverse Hessian matrix that is maintained by it, is updated at each iteration using a sketch of the Hessian, i.e., a randomly generated compressed form of the Hessian. We propose several sketching strategies, present a new quasi-Newton method that uses stochastic block BFGS updates combined with the variance reduction approach SVRG to compute batch stochastic gradients, and prove linear convergence of the resulting method. Numerical tests on large-scale logistic regression problems reveal that our method is more robust and substantially outperforms current state-of-the-art methods.

We propose a new algorithm for minimizing regularized empirical loss: Stochastic Dual Newton Ascent (SDNA). Our method is dual in nature: in each iteration we update a random subset of the dual variables. However, unlike existing methods such as stochastic dual coordinate ascent, SDNA is capable of utilizing all local curvature information contained in the examples, which leads to striking improvements in both theory and practice – sometimes by orders of magnitude. In the special case when an L2-regularizer is used in the primal, the dual problem is a concave quadratic maximization problem plus a separable term. In this regime, SDNA in each step solves a proximal subproblem involving a random principal submatrix of the Hessian of the quadratic function; whence the name of the method.

Accelerated coordinate descent is widely used in optimization due to its cheap per-iteration cost and scalability to large-scale problems. Up to a primal-dual transformation, it is also the same as accelerated stochastic gradient descent that is one of the central methods used in machine learning. In this paper, we improve the best known running time of accelerated coordinate descent by a factor up to \sqrtn. Our improvement is based on a clean, novel non-uniform sampling that selects each coordinate with a probability proportional to the square root of its smoothness parameter. Our proof technique also deviates from the classical estimation sequence technique used in prior work. Our speed-up applies to important problems such as empirical risk minimization and solving linear systems, both in theory and in practice.

In this paper we develop and analyze Hydra: HYbriD cooRdinAte descent method for solving loss minimization problems with big data. We initially partition the coordinates (features) and assign each partition to a different node of a cluster. At every iteration, each node picks a random subset of the coordinates from those it owns, independently from the other computers, and in parallel computes and applies updates to the selected coordinates based on a simple closed-form formula. We give bounds on the number of iterations sufficient to approximately solve the problem with high probability, and show how it depends on the data and on the partitioning. We perform numerical experiments with a LASSO instance described by a 3TB matrix.

This paper introduces AdaSDCA: an adaptive variant of stochastic dual coordinate ascent (SDCA) for solving the regularized empirical risk minimization problems. Our modification consists in allowing the method adaptively change the probability distribution over the dual variables throughout the iterative process. AdaSDCA achieves provably better complexity bound than SDCA with the best fixed probability distribution, known as importance sampling. However, it is of a theoretical character as it is expensive to implement. We also propose AdaSDCA+: a practical variant which in our experiments outperforms existing non-adaptive methods.

Distributed optimization methods for large-scale machine learning suffer from a communication bottleneck. It is difficult to reduce this bottleneck while still efficiently and accurately aggregating partial work from different machines. In this paper, we present a novel generalization of the recent communication-efficient primal-dual framework (COCOA) for distributed optimization. Our framework, COCOA+, allows for additive combination of local updates to the global parameters at each iteration, whereas previous schemes only allow conservative averaging. We give stronger (primal-dual) convergence rate guarantees for both COCOA as well as our new variants, and generalize the theory for both methods to cover non-smooth convex loss functions. We provide an extensive experimental comparison that shows the markedly improved performance of COCOA+ on several real-world distributed datasets, especially when scaling up the number of machines.

We study the problem of minimizing the average of a large number of smooth convex functions penalized with a strongly convex regularizer. We propose and analyze a novel primal-dual method (Quartz) which at every iteration samples and updates a random subset of the dual variables, chosen according to an arbitrary distribution. In contrast to typical analysis, we directly bound the decrease of the primal-dual error (in expectation), without the need to first analyze the dual error. Depending on the choice of the sampling, we obtain efficient serial and mini-batch variants of the method. In the serial case, our bounds match the best known bounds for SDCA (both with uniform and importance sampling). With standard mini-batching, our bounds predict initial data-independent speedup as well as additional data-driven speedup which depends on spectral and sparsity properties of the data.

We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent(SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss.