arXiv subjects

Peter Richtárik

Publications and source records attributed to Peter Richtárik.

At least 163 records · Page 9Linked to original sources

arXiv2022

A Stochastic Decoupling Method for Minimizing the Sum of Smooth and Non-Smooth Functions

We consider the problem of minimizing the sum of three convex functions: i) a smooth function $f$ in the form of an expectation or a finite average, ii) a non-smooth function $g$ in the form of a finite average of proximable functions $g_j$, and iii) a proximable regularizer $R$. We design a variance-reduced method which is able to progressively learn the proximal operator of $g$ via the computation of the proximal operator of a single randomly selected function $g_j$ in each iteration only. Our method can provably and efficiently accommodate many strategies for the estimation of the gradient of $f$, including via standard and variance-reduced stochastic estimation, effectively decoupling the smooth part of the problem from the non-smooth part. We prove a number of iteration complexity results, including a general ${\cal O}(1/t)$ rate, ${\cal O}(1/t^2)$ rate in the case of strongly convex smooth $f$, and several linear rates in special cases, including accelerated linear rate. For example, our method achieves a linear rate for the problem of minimizing a strongly convex function $f$ subject to linear constraints under no assumption on the constraints beyond consistency. When combined with SGD or SAGA estimators for the gradient of $f$, this leads to a very efficient method for empirical risk minimization. Our method generalizes several existing algorithms, including forward-backward splitting, Douglas-Rachford splitting, proximal SGD, proximal SAGA, SDCA, randomized Kaczmarz and Point-SAGA. However, our method leads to many new specific methods in special cases; for instance, we obtain the first randomized variant of the Dykstra's method for projection onto the intersection of closed convex sets.

math.OC↗

arXiv2022

IntSGD: Adaptive Floatless Compression of Stochastic Gradients

We propose a family of adaptive integer compression operators for distributed Stochastic Gradient Descent (SGD) that do not communicate a single float. This is achieved by multiplying floating-point vectors with a number known to every device and then rounding to integers. In contrast to the prior work on integer compression for SwitchML by Sapio et al. (2021), our IntSGD method is provably convergent and computationally cheaper as it estimates the scaling of vectors adaptively. Our theory shows that the iteration complexity of IntSGD matches that of SGD up to constant factors for both convex and non-convex, smooth and non-smooth functions, with and without overparameterization. Moreover, our algorithm can also be tailored for the popular all-reduce primitive and shows promising empirical performance.

cs.LG↗

arXiv2022

Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling

In this paper we study the convex-concave saddle-point problem $\min_x \max_y f(x) + y^T \mathbf{A} x - g(y)$, 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.

math.OC↗

arXiv2022

FLIX: A Simple and Communication-Efficient Alternative to Local Methods in Federated Learning

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.

cs.LG↗

arXiv2022

3PC: Three Point Compressors for Communication-Efficient Distributed Training and a Better Theory for Lazy Aggregation

We propose and study a new class of gradient communication mechanisms 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., Top-$K$), our class allows the compressors to {\em 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 {\em lazy aggregation} literature (Chen et al., 2018). 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.

cs.LG↗

arXiv2022

AI-SARAH: Adaptive and Implicit Stochastic Recursive Gradient Methods

We present AI-SARAH, a practical variant of SARAH. As a variant of SARAH, this algorithm employs the stochastic recursive gradient yet adjusts step-size based on local geometry. AI-SARAH implicitly computes step-size and efficiently estimates local Lipschitz smoothness of stochastic functions. It is fully adaptive, tune-free, straightforward to implement, and computationally efficient. We provide technical insight and intuitive illustrations on its design and convergence. We conduct extensive empirical analysis and demonstrate its strong performance compared with its classical counterparts and other state-of-the-art first-order methods in solving convex machine learning problems.

cs.LG↗

arXiv2022

Distributed Proximal Splitting Algorithms with Rates and Acceleration

We analyze several generic proximal splitting algorithms well suited for large-scale convex nonsmooth optimization. We derive sublinear and linear convergence results with new rates on the function value suboptimality or distance to the solution, as well as new accelerated versions, using varying stepsizes. In addition, we propose distributed variants of these algorithms, which can be accelerated as well. While most existing results are ergodic, our nonergodic results significantly broaden our understanding of primal-dual optimization algorithms.

math.OC↗

arXiv2022

Server-Side Stepsizes and Sampling Without Replacement Provably Help in Federated Optimization

We present a theoretical study of server-side optimization in federated learning. Our results are the first to show that the widely popular heuristic of scaling the client updates with an extra parameter is very useful in the context of Federated Averaging (FedAvg) with local passes over the client data. Each local pass is performed without replacement using Random Reshuffling, which is a key reason we can show improved complexities. In particular, we prove that whenever the local stepsizes are small, and the update direction is given by FedAvg in conjunction with Random Reshuffling over all clients, one can take a big leap in the obtained direction and improve rates for convex, strongly convex, and non-convex objectives. In particular, in non-convex regime we get an enhancement of the rate of convergence from $\mathcal{O}\left(\varepsilon^{-3}\right)$ to $\mathcal{O}\left(\varepsilon^{-2}\right)$. This result is new even for Random Reshuffling performed on a single node. In contrast, if the local stepsizes are large, we prove that the noise of client sampling can be controlled by using a small server-side stepsize. To the best of our knowledge, this is the first time that local steps provably help to overcome the communication bottleneck. Together, our results on the advantage of large and small server-side stepsizes give a formal justification for the practice of adaptive server-side optimization in federated learning. Moreover, we consider a variant of our algorithm that supports partial client participation, which makes the method more practical.

cs.LG↗

arXiv2022

MARINA: Faster Non-Convex Distributed Learning with Compression

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-Lojasiewicz condition.

cs.LG↗

arXiv2021

CANITA: Faster Rates for Distributed Convex Optimization with Communication Compression

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 (Nesterov, 1983, 2004) and Adam (Kingma and Ba, 2014). In order to combine the benefits of communication compression and convergence acceleration, we propose a \emph{compressed and accelerated} gradient method based on ANITA (Li, 2021) for distributed optimization, which we call CANITA. Our CANITA achieves the \emph{first accelerated rate} $O\bigg(\sqrt{\Big(1+\sqrt{\frac{ω^3}{n}}\Big)\frac{L}ε} + ω\big(\frac{1}ε\big)^{\frac{1}{3}}\bigg)$, which improves upon the state-of-the-art non-accelerated rate $O\left((1+\fracω{n})\frac{L}ε + \frac{ω^2+ω}{ω+n}\frac{1}ε\right)$ of DIANA (Khaled et al., 2020) for distributed general convex problems, where $ε$ is the target error, $L$ is the smooth parameter of the objective, $n$ is the number of machines/devices, and $ω$ is the compression parameter (larger $ω$ means more compression can be applied, and no compression implies $ω=0$). Our results show that as long as the number of devices $n$ is large (often true in distributed/federated learning), or the compression $ω$ is not very high, CANITA achieves the faster convergence rate $O\Big(\sqrt{\frac{L}ε}\Big)$, i.e., the number of communication rounds is $O\Big(\sqrt{\frac{L}ε}\Big)$ (vs. $O\big(\frac{L}ε\big)$ 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).

cs.LG↗

arXiv2021

Basis Matters: Better Communication-Efficient Second Order Methods for Federated Learning

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: {\em 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 {\em 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.

cs.LG↗

arXiv2021

ZeroSARAH: Efficient Nonconvex Finite-Sum Optimization with Zero Full Gradient Computation

We propose ZeroSARAH -- a novel variant of the variance-reduced method SARAH (Nguyen et al., 2017) -- for minimizing the average of a large number of nonconvex functions $\frac{1}{n}\sum_{i=1}^{n}f_i(x)$. To the best of our knowledge, in this nonconvex finite-sum regime, all existing variance-reduced methods, including SARAH, SVRG, SAGA and their variants, need to compute the full gradient over all $n$ data samples at the initial point $x^0$, and then periodically compute the full gradient once every few iterations (for SVRG, SARAH and their variants). Note that SVRG, SAGA and their variants typically achieve weaker convergence results than variants of SARAH: $n^{2/3}/ε^2$ vs. $n^{1/2}/ε^2$. Thus we focus on the variant of SARAH. The proposed ZeroSARAH and its distributed variant D-ZeroSARAH are the \emph{first} variance-reduced algorithms which \emph{do not require any full gradient computations}, not even for the initial point. Moreover, for both standard and distributed settings, we show that ZeroSARAH and D-ZeroSARAH obtain new state-of-the-art convergence results, which can improve the previous best-known result (given by e.g., SPIDER, SARAH, and PAGE) in certain regimes. Avoiding any full gradient computations (which are time-consuming steps) is important in many applications as the number of data samples $n$ usually is very large. Especially in the distributed setting, periodic computation of full gradient over all data samples needs to periodically synchronize all clients/devices/machines, which may be impossible or unaffordable. Thus, we expect that ZeroSARAH/D-ZeroSARAH will have a practical impact in distributed and federated learning where full device participation is impractical.

cs.LG↗

arXiv2021

Permutation Compressors for Provably Faster Distributed Nonconvex Optimization

We study the MARINA method of Gorbunov et al (2021) -- the current state-of-the-art distributed non-convex optimization method in terms of theoretical communication complexity. Theoretical superiority of this method can be largely attributed to two sources: the use of a carefully engineered biased stochastic gradient estimator, which leads to a reduction in the number of communication rounds, and the reliance on {\em independent} stochastic communication compression operators, which leads to a reduction in the number of transmitted bits within each communication round. In this paper we i) extend the theory of MARINA to support a much wider class of potentially {\em correlated} compressors, extending the reach of the method beyond the classical independent compressors setting, ii) show that a new quantity, for which we coin the name {\em Hessian variance}, allows us to significantly refine the original analysis of MARINA without any additional assumptions, and iii) identify a special class of correlated compressors based on the idea of {\em random permutations}, for which we coin the term Perm$K$, the use of which leads to $O(\sqrt{n})$ (resp. $O(1 + d/\sqrt{n})$) improvement in the theoretical communication complexity of MARINA in the low Hessian variance regime when $d\geq n$ (resp. $d \leq n$), where $n$ is the number of workers and $d$ is the number of parameters describing the model we are learning. We corroborate our theoretical results with carefully engineered synthetic experiments with minimizing the average of nonconvex quadratics, and on autoencoder training with the MNIST dataset.

cs.LG↗

arXiv2021

Error Compensated Loopless SVRG, Quartz, and SDCA for Distributed Optimization

The communication of gradients is a key bottleneck in distributed training of large scale machine learning models. In order to reduce the communication cost, gradient compression (e.g., sparsification and quantization) and error compensation techniques are often used. In this paper, we propose and study three new efficient methods in this space: error compensated loopless SVRG method (EC-LSVRG), error compensated Quartz (EC-Quartz), and error compensated SDCA (EC-SDCA). Our method is capable of working with any contraction compressor (e.g., TopK compressor), and we perform analysis for convex optimization problems in the composite case and smooth case for EC-LSVRG. We prove linear convergence rates for both cases and show that in the smooth case the rate has a better dependence on the parameter associated with the contraction compressor. Further, we show that in the smooth case, and under some certain conditions, error compensated loopless SVRG has the same convergence rate as the vanilla loopless SVRG method. Then we show that the convergence rates of EC-Quartz and EC-SDCA in the composite case are as good as EC-LSVRG in the smooth case. Finally, numerical experiments are presented to illustrate the efficiency of our methods.

math.OC↗

arXiv2021

Doubly Adaptive Scaled Algorithm for Machine Learning Using Second-Order Information

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

cs.LG↗

arXiv2021

FedPAGE: A Fast Local Stochastic Gradient Method for Communication-Efficient Federated Learning

Federated Averaging (FedAvg, also known as Local-SGD) (McMahan et al., 2017) is a classical federated learning algorithm in which clients run multiple local SGD steps before communicating their update to an orchestrating server. We propose a new federated learning algorithm, FedPAGE, able to further reduce the communication complexity by utilizing the recent optimal PAGE method (Li et al., 2021) instead of plain SGD in FedAvg. We show that FedPAGE uses much fewer communication rounds than previous local methods for both federated convex and nonconvex optimization. Concretely, 1) in the convex setting, the number of communication rounds of FedPAGE is $O(\frac{N^{3/4}}{Sε})$, improving the best-known result $O(\frac{N}{Sε})$ of SCAFFOLD (Karimireddy et al.,2020) by a factor of $N^{1/4}$, where $N$ is the total number of clients (usually is very large in federated learning), $S$ is the sampled subset of clients in each communication round, and $ε$ is the target error; 2) in the nonconvex setting, the number of communication rounds of FedPAGE is $O(\frac{\sqrt{N}+S}{Sε^2})$, improving the best-known result $O(\frac{N^{2/3}}{S^{2/3}ε^2})$ of SCAFFOLD (Karimireddy et al.,2020) by a factor of $N^{1/6}S^{1/3}$, if the sampled clients $S\leq \sqrt{N}$. Note that in both settings, the communication cost for each round is the same for both FedPAGE and SCAFFOLD. As a result, FedPAGE achieves new state-of-the-art results in terms of communication complexity for both federated convex and nonconvex optimization.

cs.LG↗

arXiv2021

Quasi-Newton Methods for Machine Learning: Forget the Past, Just Sample

We present two sampled quasi-Newton methods (sampled LBFGS and sampled LSR1) for solving empirical risk minimization problems that arise in machine learning. Contrary to the classical variants of these methods that sequentially build Hessian or inverse Hessian approximations as the optimization progresses, our proposed methods sample points randomly around the current iterate at every iteration to produce these approximations. As a result, the approximations constructed make use of more reliable (recent and local) information, and do not depend on past iterate information that could be significantly stale. Our proposed algorithms are efficient in terms of accessed data points (epochs) and have enough concurrency to take advantage of parallel/distributed computing environments. We provide convergence guarantees for our proposed methods. Numerical tests on a toy classification problem as well as on popular benchmarking binary classification and neural network training tasks reveal that the methods outperform their classical variants.

math.OC↗

arXiv2021

PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization

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 $Ω(n+\frac{\sqrt{n}}{ε^2})$ for nonconvex finite-sum problems, which also leads to a tight lower bound $Ω(b+\frac{\sqrt{b}}{ε^2})$ for nonconvex online problems, where $b:= \min\{\frac{σ^2}{ε^2}, n\}$. Then, we show that PAGE obtains the optimal convergence results $O(n+\frac{\sqrt{n}}{ε^2})$ (finite-sum) and $O(b+\frac{\sqrt{b}}{ε^2})$ (online) matching our lower bounds for both nonconvex finite-sum and online problems. Besides, we also show that for nonconvex functions satisfying the Polyak-Łojasiewicz (PL) condition, PAGE can automatically switch to a faster linear convergence rate $O(\cdot\log \frac{1}ε)$. 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.

cs.LG↗