Accelerated gradient methods with strong convergence to the minimum norm minimizer: a dynamic approach combining time scaling, averaging, and Tikhonov regularization

Hedy Attouch (1) , Zaki Chbani (2) , Hassan Riahi (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 UCA - Université Cadi Ayyad [Marrakech]

In a Hilbert framework, for convex differentiable optimization, we consider accelerated gradient methods obtained by combining temporal scaling and averaging techniques with Tikhonov regularization. We start from the continuous steepest descent dynamic with an additional Tikhonov regularization term whose coefficient vanishes asymptotically. We provide an extensive Lyapunov analysis of this first-order evolution equation. Then we apply to this dynamic the method of time scaling and averaging recently introduced by Attouch, Bot and Nguyen. We thus obtain an inertial dynamic which involves viscous damping associated with Nesterov's method, implicit Hessian damping and Tikhonov regularization. Under an appropriate setting of the parameters, just using Jensen's inequality, without the need for another Lyapunov analysis, we show that the trajectories have at the same time several remarkable properties: they provide a rapid convergence of values, fast convergence of the gradients to zero, and strong convergence to the minimum norm minimizer. These results complete and improve the previous results obtained by the authors.

Mots clés: Accelerated gradient methods; convex optimization; damped inertial dynamics; Hessian driven damping; Tikhonov approximation; time scaling and averaging Mathematics Subject Classification (2010) 37N40; 46N10; 49M30; 65K05; 65K10; 65K15; 65L08; 65L09; 90B50; 90C25

Domaines: Optimisation et contrôle [math.OC]

Hedy Attouch, Zaki Chbani, Hassan Riahi. Accelerated gradient methods with strong convergence to the minimum norm minimizer: a dynamic approach combining time scaling, averaging, and Tikhonov regularization. 2022. ⟨hal-03859665⟩