On Gossip Algorithms for Machine Learning with Pairwise Objectives
Authors: Igor Colin, Aurélien Bellet, Stephan Clémençon, Joseph Salmon
Organizations: LTCI, S2A, IP Paris · LTCI, Télécom Paris, Institut Polytechnique de Paris · PREMEDICAL · PreMeDICaL team, Inria, Idesp, Inserm, Université de Montpellier · LTCI, IDS, S2A, IP Paris · IROKO, UM · IMAG, IROKO, Univ Montpellier, Inria, CNRS
In the IoT era, information is more and more frequently picked up by connected smart sensors with increasing, though limited, storage, communication and computation abilities. Whether due to privacy constraints or to the structure of the distributed system, the development of statistical learning methods dedicated to data that are shared over a network is now a major issue. Gossip-based algorithms have been developed for the purpose of solving a wide variety of statistical learning tasks, ranging from data aggregation over sensor networks to decentralized multi-agent optimization. Whereas the vast majority of contributions consider situations where the function to be estimated or optimized is a basic average of individual observations, it is the goal of this article to investigate the case where the latter is of pairwise nature, taking the form of a U -statistic of degree two. Motivated by various problems such as similarity learning, ranking or clustering for instance, we revisit gossip algorithms specifically designed for pairwise objective functions and provide a comprehensive theoretical framework for their convergence. This analysis fills a gap in the literature by establishing conditions under which these methods succeed, and by identifying the graph properties that critically affect their efficiency. In particular, a refined analysis of the convergence upper and lower bounds is performed.
Figures & tables
Figure 1: Logistic AUC. Solid lines are averages and filled area are standard deviations.
Asynchronous primal-dual methods for decentralized non-smooth convex optimization often require each node to maintain O(d) auxiliary variables, where d is its degree. This dependence on degree increases memory requirements and can amplify the effects of stale information, especially in dense networks. Motivated by the challenge of frugal memory management in decentralized learning, we introduce Goal-PD, an asynchronous gossip-based primal-dual algorithm that maintains only two variables per node, regardless of the node's degree. We establish almost-sure convergence of Goal-PD to a minimizer of the underlying optimization problem, and prove linear convergence when the objective functions are piecewise linear-quadratic. For decentralized mean estimation, we show that pairwise averaging is a special case of Goal-PD, which establishes a direct link between the proposed primal-dual framework and classical gossip. Experiments on synthetic and real datasets over various network topologies, with non-smooth objectives including median estimation, show that Goal-PD converges faster than existing asynchronous baselines while requiring significantly less memory by design.
Anna van Elst, Olivier Fercoq, Igor Colin +1
LTCI, Télécom Paris, Institut Polytechnique de Paris
We propose a decentralized privacy-preserving learning algorithm in which each agent holds a single private sample and a shared model. Samples are learned sequentially, and each update must preserve the endpoint mappings at previously learned samples while protecting private data. This gives each agent three roles: (i) a learner that updates the model parameters, (ii) a teacher whose sample is learned at the current iteration, and (iii) a protected agent whose sample has already been learned. We build on Tuning without Forgetting (TwF) method to preserve previously learned mappings and show that TwF provides an indistinguishability guarantee for the learner whenever the set of protected agents contains another sample with the same label. For the teacher, we formulate a minimax optimal control problem that models the differential privacy noise as a worst-case disturbance to prevent performance loss while maintaining the same level of privacy for the gradient. For the protected agents, we compute the projections locally and aggregate them using a private push-sum gossip protocol. We prove geometric convergence of the decentralized gossip algorithm and of the distributed projection for TwF.
Estimating the generative mechanism of large-scale networks is a fundamental challenge in statistical machine learning. It requires the identification of the latent connectivity structure, which is in general an NP-hard combinatorial problem due to the absence of canonical node labels. We address this challenge by allowing for probabilistic couplings, thereby relaxing the assignment problem. Our estimation framework can be formulated as a semi-relaxed Gromov-Wasserstein objective and provides a low-dimensional representation of the generative structure. We solve this via a block-coordinate conditional gradient algorithm. Despite the relaxation, the resulting solution is typically deterministic: in fact, we show that the optimality gap between the relaxed solution and the deterministic assignment vanishes at rate O(1/n), where n is the number of nodes. This allows for tractable recovery of the underlying model and enables rigorous statistical analysis: we establish consistency and minimax-optimal convergence rates for both stochastic block models and Holder-smooth graphons. Our implementation scales efficiently with n, as demonstrated on both synthetic and real-world datasets.
Charles Dufour, Ulysse Naepels, Leonardo V. Santoro
EPFL, Institute of Mathematics Lausanne, Switzerland