The stochastic Ravine accelerated gradient method with general extrapolation coefficients

Hedy Attouch (1) , Jalal M. Fadili (2, 3, 4, 5) , Vyacheslav Kungurtsev (6)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 Equipe Image - Laboratoire GREYC - UMR6072
3 NU - Normandie Université
4 ENSICAEN - École Nationale Supérieure d'Ingénieurs de Caen
5 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen
6 CTU - Czech Technical University in Prague

In a real Hilbert space domain setting, we study the convergence properties of the stochastic Ravine accelerated gradient method for convex differentiable optimization. We consider the general form of this algorithm where the extrapolation coefficients can vary with each itera- tion, and where the evaluation of the gradient is subject to random errors. This general treatment models a breadth of practical algorithms and numerical implementations. We show that, under a proper tuning of the extrapolation parameters, and when the error variance associated with the gradient evaluations or the step-size sequences vanish sufficiently fast, the Ravine method provides fast convergence of the values both in expectation and almost surely. We also improve the con- vergence rates from O(·) to o(·) in expectation and almost sure sense. Moreover, we show almost sure summability property of the gradients, which implies the fast convergence of the gradients towards zero. This property reflects the fact that the high-resolution ODE of the Ravine method includes a Hessian-driven damping term. When the space is also separable, our analysis allows also to establish almost sure weak convergence of the sequence of iterates provided by the algorithm. We finally specialize the analysis to consider different parameter choices, including vanishing and constant (heavy ball method with friction) damping parameter, and present a comprehensive land- scape of the tradeoffs in speed and accuracy associated with these parameter choices and statistical properties on the sequence of errors in the gradient computations. We provide a thorough discus- sion of the similarities and differences with the Nesterov accelerated gradient which satisfies similar asymptotic convergence rates.

Keywords: Ravine method; Nesterov accelerated gradient method; general extrapolation coefficient; Hessian driven damping; convergence rates; Lyapunov analysis Mathematics Subject Classification (2000) 37N40; stochastic errors

Domains: Optimization and Control [math.OC]

Hedy Attouch, Jalal M. Fadili, Vyacheslav Kungurtsev. The stochastic Ravine accelerated gradient method with general extrapolation coefficients. 2024. ⟨hal-04506457v1⟩