arXiv subjects

Konstantin Mishchenko

Publications and source records attributed to Konstantin Mishchenko.

At least 37 records · Page 2Linked to original sources

arXiv2022

Dualize, Split, Randomize: Toward Fast Nonsmooth Optimization Algorithms

We consider minimizing the sum of three convex functions, where the first one F is smooth, the second one is nonsmooth and proximable and the third one is the composition of a nonsmooth proximable function with a linear operator L. This template problem has many applications, for instance, in image processing and machine learning. First, we propose a new primal-dual algorithm, which we call PDDY, for this problem. It is constructed by applying Davis-Yin splitting to a monotone inclusion in a primal-dual product space, where the operators are monotone under a specific metric depending on L. We show that three existing algorithms (the two forms of the Condat-Vu algorithm and the PD3O algorithm) have the same structure, so that PDDY is the fourth missing link in this self-consistent class of primal-dual algorithms. This representation eases the convergence analysis: it allows us to derive sublinear convergence rates in general, and linear convergence results in presence of strong convexity. Moreover, within our broad and flexible analysis framework, we propose new stochastic generalizations of the algorithms, in which a variance-reduced random estimate of the gradient of F is used, instead of the true gradient. Furthermore, we obtain, as a special case of PDDY, a linearly converging algorithm for the minimization of a strongly convex function F under a linear constraint; we discuss its important application to decentralized optimization.

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arXiv2022

Tighter Theory for Local SGD on Identical and Heterogeneous Data

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.

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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.

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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.

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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.

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arXiv2021

On Seven Fundamental Optimization Challenges in Machine Learning

Many recent successes of machine learning went hand in hand with advances in optimization. The exchange of ideas between these fields has worked both ways, with machine learning building on standard optimization procedures such as gradient descent, as well as with new directions in the optimization theory stemming from machine learning applications. In this thesis, we discuss new developments in optimization inspired by the needs and practice of machine learning, federated learning, and data science. In particular, we consider seven key challenges of mathematical optimization and develop a solution to each. Our first contribution is the resolution of a key open problem in Federated Learning: we establish the first theoretical guarantees for the famous Local SGD algorithm in the heterogeneous data regime. As the second challenge, we close the gap between the upper and lower bounds for the theory of two algorithms known as Random Reshuffling (RR) and Shuffle-Once that are widely used in practice, and set as the default data selection strategies for SGD in modern machine learning software. Our third contribution can be seen as a combination of our new theory for proximal RR and Local SGD yielding a new algorithm, which we call FedRR. Unlike Local SGD, FedRR can provably beat gradient descent in communication complexity in the heterogeneous data regime. The fourth challenge is related to the class of adaptive methods. In particular, we present the first parameter-free stepsize rule for gradient descent that provably works for any locally smooth convex objective. The fifth challenge is the development of an algorithm for distributed optimization with quantized updates that preserves linear convergence of gradient descent. Finally, in our sixth and seventh challenges, we develop new VR mechanisms applicable to the non-smooth setting based on proximal operators.

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arXiv2021

Random Reshuffling: Simple Analysis with Vast Improvements

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 $κ^2$ to $κ$ (resp. from $κ$ to $\sqrtκ$) 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.

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arXiv2021

Proximal and Federated Random Reshuffling

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 Reshuffing (ProxRR and FedRR). The first algorithm, ProxRR, solves composite convex 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. We obtain the second algorithm, FedRR, as a special case of ProxRR applied to a reformulation of distributed problems with either homogeneous or heterogeneous data. We study the algorithms' convergence properties with constant and decreasing stepsizes, and show that they have considerable advantages over Proximal and Local SGD. In particular, our methods have superior complexities and 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. We give examples of practical optimization tasks where the proximal operator is difficult to compute and ProxRR has a clear advantage. Finally, we corroborate our results with experiments on real data sets.

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arXiv2020

Adaptive Gradient Descent without Descent

We present a strikingly simple proof that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. No need for functional values, no line search, no information about the function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on the smoothness in a neighborhood of a solution. Given that the problem is convex, our method converges even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including logistic regression and matrix factorization.

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arXiv2020

Revisiting Stochastic Extragradient

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 et al., 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 bilinear 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.

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arXiv2020

First Analysis of Local GD on Heterogeneous Data

We provide the first convergence analysis of local gradient descent for minimizing the average of smooth and convex but otherwise arbitrary functions. Problems of this form and local gradient descent as a solution method are of importance in federated learning, where each function is based on private data stored by a user on a mobile device, and the data of different users can be arbitrarily heterogeneous. We show that in a low accuracy regime, the method has the same communication complexity as gradient descent.

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arXiv2019

A Distributed Flexible Delay-tolerant Proximal Gradient Algorithm

We develop and analyze an asynchronous algorithm for distributed convex optimization when the objective writes a sum of smooth functions, local to each worker, and a non-smooth function. Unlike many existing methods, our distributed algorithm is adjustable to various levels of communication cost, delays, machines computational power, and functions smoothness. A unique feature is that the stepsizes do not depend on communication delays nor number of machines, which is highly desirable for scalability. We prove that the algorithm converges linearly in the strongly convex case, and provide guarantees of convergence for the non-strongly convex case. The obtained rates are the same as the vanilla proximal gradient algorithm over some introduced epoch sequence that subsumes the delays of the system. We provide numerical results on large-scale machine learning problems to demonstrate the merits of the proposed method.

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arXiv2019

Stochastic Newton and Cubic Newton Methods with Simple Local Linear-Quadratic Rates

We present two new remarkably simple stochastic second-order methods for minimizing the average of a very large number of sufficiently smooth and strongly convex functions. The first is a stochastic variant of Newton's method (SN), and the second is a stochastic variant of cubically regularized Newton's method (SCN). We establish local linear-quadratic convergence results. Unlike existing stochastic variants of second order methods, which require the evaluation of a large number of gradients and/or Hessians in each iteration to guarantee convergence, our methods do not have this shortcoming. For instance, the simplest variants of our methods in each iteration need to compute the gradient and Hessian of a {\em single} randomly selected function only. In contrast to most existing stochastic Newton and quasi-Newton methods, our approach guarantees local convergence faster than with first-order oracle and adapts to the problem's curvature. Interestingly, our method is not unbiased, so our theory provides new intuition for designing new stochastic methods.

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arXiv2019

Sinkhorn Algorithm as a Special Case of Stochastic Mirror Descent

We present a new perspective on the celebrated Sinkhorn algorithm by showing that is a special case of incremental/stochastic mirror descent. In order to see this, one should simply plug Kullback-Leibler divergence in both mirror map and the objective function. Since the problem has unbounded domain, the objective function is neither smooth nor it has bounded gradients. However, one can still approach the problem using the notion of relative smoothness, obtaining that the stochastic objective is 1-relative smooth. The discovered equivalence allows us to propose 1) new methods for optimal transport, 2) an extension of Sinkhorn algorithm beyond two constraints.

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arXiv2019

A Self-supervised Approach to Hierarchical Forecasting with Applications to Groupwise Synthetic Controls

When forecasting time series with a hierarchical structure, the existing state of the art is to forecast each time series independently, and, in a post-treatment step, to reconcile the time series in a way that respects the hierarchy (Hyndman et al., 2011; Wickramasuriya et al., 2018). We propose a new loss function that can be incorporated into any maximum likelihood objective with hierarchical data, resulting in reconciled estimates with confidence intervals that correctly account for additional uncertainty due to imperfect reconciliation. We evaluate our method using a non-linear model and synthetic data on a counterfactual forecasting problem, where we have access to the ground truth and contemporaneous covariates, and show that we largely improve over the existing state-of-the-art method.

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arXiv2019

99% of Distributed Optimization is a Waste of Time: The Issue and How to Fix it

Many popular distributed optimization methods for training machine learning models 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. It is also well known that many such methods, including SGD, SAGA, and accelerated SGD for over-parameterized models, do not scale well with the number of parallel workers. In this paper we observe that the above template is fundamentally inefficient in that too much data is unnecessarily communicated by the workers, 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.

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arXiv2019

MISO is Making a Comeback With Better Proofs and Rates

MISO, also known as Finito, was one of the first stochastic variance reduced methods discovered, yet its popularity is fairly low. Its initial analysis was significantly limited by the so-called Big Data assumption. Although the assumption was lifted in subsequent work using negative momentum, this introduced a new parameter and required knowledge of strong convexity and smoothness constants, which is rarely possible in practice. We rehabilitate the method by introducing a new variant that needs only smoothness constant and does not have any extra parameters. Furthermore, when removing the strong convexity constant from the stepsize, we present a new analysis of the method, which no longer uses the assumption that every component is strongly convex. This allows us to also obtain so far unknown nonconvex convergence of MISO. To make the proposed method efficient in practice, we derive minibatching bounds with arbitrary uniform sampling that lead to linear speedup when the expected minibatch size is in a certain range. Our numerical experiments show that MISO is a serious competitor to SAGA and SVRG and sometimes outperforms them on real datasets.

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arXiv2019

Stochastic Distributed Learning with Gradient Quantization and Variance Reduction

We consider distributed optimization where the objective function is spread among different devices, each sending incremental model updates to a central server. To alleviate the communication bottleneck, recent work proposed various schemes to compress (e.g.\ quantize or sparsify) the gradients, thereby introducing additional variance $ω\geq 1$ that might slow down convergence. For strongly convex functions with condition number $κ$ distributed among $n$ machines, we (i) give a scheme that converges in $\mathcal{O}((κ+ κ\fracω{n} + ω)$ $\log (1/ε))$ steps to a neighborhood of the optimal solution. For objective functions with a finite-sum structure, each worker having less than $m$ components, we (ii) present novel variance reduced schemes that converge in $\mathcal{O}((κ+ κ\fracω{n} + ω+ m)\log(1/ε))$ steps to arbitrary accuracy $ε> 0$. These are the first methods that achieve linear convergence for arbitrary quantized updates. We also (iii) give analysis for the weakly convex and non-convex cases and (iv) verify in experiments that our novel variance reduced schemes are more efficient than the baselines.

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