arXiv subjects

Mathieu Even

Publications and source records attributed to Mathieu Even.

24 records · Page 2Linked to original sources

arXiv2023

Asynchronous SGD Beats Minibatch SGD Under Arbitrary Delays

The existing analysis of asynchronous stochastic gradient descent (SGD) degrades dramatically when any delay is large, giving the impression that performance depends primarily on the delay. On the contrary, we prove much better guarantees for the same asynchronous SGD algorithm regardless of the delays in the gradients, depending instead just on the number of parallel devices used to implement the algorithm. Our guarantees are strictly better than the existing analyses, and we also argue that asynchronous SGD outperforms synchronous minibatch SGD in the settings we consider. For our analysis, we introduce a novel recursion based on "virtual iterates" and delay-adaptive stepsizes, which allow us to derive state-of-the-art guarantees for both convex and non-convex objectives.

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arXiv2022

Asynchronous speedup in decentralized optimization

In decentralized optimization, nodes of a communication network each possess a local objective function, and communicate using gossip-based methods in order to minimize the average of these per-node functions. While synchronous algorithms are heavily impacted by a few slow nodes or edges in the graph (the \emph{straggler problem}), their asynchronous counterparts are notoriously harder to parametrize. Indeed, their convergence properties for networks with heterogeneous communication and computation delays have defied analysis so far. In this paper, we use a \emph{ continuized} framework to analyze asynchronous algorithms in networks with delays. Our approach yields a precise characterization of convergence time and of its dependency on heterogeneous delays in the network. Our continuized framework benefits from the best of both continuous and discrete worlds: the algorithms it applies to are based on event-driven updates. They are thus essentially discrete and hence readily implementable. Yet their analysis is essentially in continuous time, relying in part on the theory of delayed ODEs. Our algorithms moreover achieve an \emph{asynchronous speedup}: their rate of convergence is controlled by the eigengap of the network graph weighted by local delays, instead of the network-wide worst-case delay as in previous analyses. Our methods thus enjoy improved robustness to stragglers.

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arXiv2022

Sample Optimality and All-for-all Strategies in Personalized Federated and Collaborative Learning

In personalized Federated Learning, each member of a potentially large set of agents aims to train a model minimizing its loss function averaged over its local data distribution. We study this problem under the lens of stochastic optimization. Specifically, we introduce information-theoretic lower bounds on the number of samples required from all agents to approximately minimize the generalization error of a fixed agent. We then provide strategies matching these lower bounds, in the all-for-one and all-for-all settings where respectively one or all agents desire to minimize their own local function. Our strategies are based on a gradient filtering approach: provided prior knowledge on some notions of distances or discrepancies between local data distributions or functions, a given agent filters and aggregates stochastic gradients received from other agents, in order to achieve an optimal bias-variance trade-off.

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arXiv2021

A Continuized View on Nesterov Acceleration for Stochastic Gradient Descent and Randomized Gossip

We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; and a discretization of the continuized process can be computed exactly with convergence rates similar to those of Nesterov original acceleration. We show that the discretization has the same structure as Nesterov acceleration, but with random parameters. We provide continuized Nesterov acceleration under deterministic as well as stochastic gradients, with either additive or multiplicative noise. Finally, using our continuized framework and expressing the gossip averaging problem as the stochastic minimization of a certain energy function, we provide the first rigorous acceleration of asynchronous gossip algorithms.

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arXiv2021

Fast Stochastic Bregman Gradient Methods: Sharp Analysis and Variance Reduction

We study the problem of minimizing a relatively-smooth convex function using stochastic Bregman gradient methods. We first prove the convergence of Bregman Stochastic Gradient Descent (BSGD) to a region that depends on the noise (magnitude of the gradients) at the optimum. In particular, BSGD with a constant step-size converges to the exact minimizer when this noise is zero (\emph{interpolation} setting, in which the data is fit perfectly). Otherwise, when the objective has a finite sum structure, we show that variance reduction can be used to counter the effect of noise. In particular, fast convergence to the exact minimizer can be obtained under additional regularity assumptions on the Bregman reference function. We illustrate the effectiveness of our approach on two key applications of relative smoothness: tomographic reconstruction with Poisson noise and statistical preconditioning for distributed optimization.

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arXiv2021

Asynchrony and Acceleration in Gossip Algorithms

This paper considers the minimization of a sum of smooth and strongly convex functions dispatched over the nodes of a communication network. Previous works on the subject either focus on synchronous algorithms, which can be heavily slowed down by a few slow nodes (the straggler problem), or consider a model of asynchronous operation (Boyd et al., 2006) in which adjacent nodes communicate at the instants of Poisson point processes. We have two main contributions. 1) We propose CACDM (a Continuously Accelerated Coordinate Dual Method), and for the Poisson model of asynchronous operation, we prove CACDM to converge to optimality at an accelerated convergence rate in the sense of Nesterov et Stich, 2017. In contrast, previously proposed asynchronous algorithms have not been proven to achieve such accelerated rate. While CACDM is based on discrete updates, the proof of its convergence crucially depends on a continuous time analysis. 2) We introduce a new communication scheme based on Loss-Networks, that is programmable in a fully asynchronous and decentralized way, unlike the Poisson model of asynchronous operation that does not capture essential aspects of asynchrony such as non-instantaneous communications and computations. Under this Loss-Network model of asynchrony, we establish for CDM (a Coordinate Dual Method) a rate of convergence in terms of the eigengap of the Laplacian of the graph weighted by local effective delays. We believe this eigengap to be a fundamental bottleneck for convergence rates of asynchronous optimization. Finally, we verify empirically that CACDM enjoys an accelerated convergence rate in the Loss-Network model of asynchrony.

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