Weintroducethe"continuized"Nesterovacceleration,aclosevariantofNesterov acceleration whose variables are indexed by a continuous time parameter.

Decentralized optimization is increasingly popular in machine learning for its scalabilityandefficiency.

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; but 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.

Neural Information Processing Systems http://nips.cc/

Abstract--As decentralized AI and edge intelligence become increasingly prevalent, ensuring robustness and trustworthiness in such distributed settings has become a critical issue--especially in the presence of corrupted or adversarial data. Traditional decentralized algorithms are vulnerable to data contamination as they typically rely on simple statistics (e.g., means or sum), motivating the need for more robust statistics. In line with recent work on decentralized estimation of trimmed means and ranks, we develop gossip algorithms for computing a broad class of rank-based statistics, including L-statistics and rank statistics-- both known for their robustness to outliers. We apply our method to perform robust distributed two-sample hypothesis testing, introducing the first gossip algorithm for Wilcoxon rank-sum tests. We provide rigorous convergence guarantees, including the first convergence rate bound for asynchronous gossip-based rank estimation.

In fully decentralized FL, the common hypothesis of a central server is also removed, letting users, represented as nodes in a graph, train the model via peer-to-peer communications along edges.

Gossip algorithms and their accelerated versions have been studied exclusively in discrete time on graphs. In this work, we take a different approach, and consider the scaling limit of gossip algorithms in both large graphs and large number of iterations. These limits lead to well-known partial differential equations (PDEs) with insightful properties. On lattices, we prove that the non-accelerated gossip algorithm of Boyd et al. [2006] converges to the heat equation, and the accelerated Jacobi polynomial iteration of Berthier et al. [2020] converges to the Euler-Poisson-Darboux (EPD) equation - a damped wave equation. Remarkably, with appropriate parameters, the fundamental solution of the EPD equation has the ideal gossip behaviour: a uniform density over an ellipsoid, whose radius increases at a rate proportional to t - the fastest possible rate for locally communicating gossip algorithms. This is in contrast with the heat equation where the density spreads on a typical scale of $\sqrt{t}$. Additionally, we provide simulations demonstrating that the gossip algorithms are accurately approximated by their limiting PDEs.

This paper addresses the problem of robust estimation in gossip algorithms over arbitrary communication graphs. Gossip algorithms are fully decentralized, relying only on local neighbor-to-neighbor communication, making them well-suited for situations where communication is constrained. A fundamental challenge in existing mean-based gossip algorithms is their vulnerability to malicious or corrupted nodes. In this paper, we show that an outlier-robust mean can be computed by globally estimating a robust statistic. More specifically, we propose a novel gossip algorithm for rank estimation, referred to as \textsc{GoRank}, and leverage it to design a gossip procedure dedicated to trimmed mean estimation, coined \textsc{GoTrim}. In addition to a detailed description of the proposed methods, a key contribution of our work is a precise convergence analysis: we establish an $\mathcal{O}(1/t)$ rate for rank estimation and an $\mathcal{O}(1 / {t})$ rate for trimmed mean estimation, where by $t$ is meant the number of iterations. Moreover, we provide a breakdown point analysis of \textsc{GoTrim}. We empirically validate our theoretical results through experiments on diverse network topologies, data distributions and contamination schemes.

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; but 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.