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

Mathieu Even (1, 2) , Raphaël Berthier (3) , Francis Bach (3) , Nicolas Flammarion (4) , Pierre Gaillard (5) , Hadrien Hendrikx (2, 3) , Laurent Massoulié (2, 6) , Adrien Taylor (3)

1 DI-ENS - Département d'informatique - ENS Paris
2 DYOGENE - Dynamics of Geometric Networks
3 SIERRA - Statistical Machine Learning and Parsimony
4 EPFL - Ecole Polytechnique Fédérale de Lausanne
5 Thoth - Apprentissage de modèles à partir de données massives
6 MSR - INRIA - Microsoft Research - Inria Joint Centre

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.

Domaines: Optimisation et contrôle [math.OC]; Probabilités [math.PR]; Machine Learning [stat.ML]; Calcul parallèle, distribué et partagé [cs.DC]; Système multi-agents [cs.MA]

Mathieu Even, Raphaël Berthier, Francis Bach, Nicolas Flammarion, Pierre Gaillard, et al.. A Continuized View on Nesterov Acceleration for Stochastic Gradient Descent and Randomized Gossip. NeurIPS 2021 - 35th Conference on Neural Information Processing Systems, Dec 2021, Sydney (virtual), Australia. pp.1-32. ⟨hal-03405165⟩