S. Gratton

S. Gratton, A. Kopanicakova, Ph. L. Toint

A class of multi-level algorithms for unconstrained nonlinear optimization is presented which does not require the evaluation of the objective function. The class contains the momentum-less AdaGrad method as a particular (single-level) instance. The choice of avoiding the evaluation of the objective function is intended to make the algorithms of the class less sensitive to noise, while the multi-level feature aims at reducing their computational cost. The evaluation complexity of these algorithms is analyzed and their behaviour in the presence of noise is then illustrated in the context of training deep neural networks for supervised learning applications.

Marc Salvadó-Benasco, Aymane Kssim, Alexander Heinlein et al.

A multi-preconditioned LBFGS (MP-LBFGS) algorithm is introduced for training finite-basis physics-informed neural networks (FBPINNs). The algorithm is motivated by the nonlinear additive Schwarz method and exploits the domain-decomposition-inspired additive architecture of FBPINNs, in which local neural networks are defined on subdomains, thereby localizing the network representation. Parallel, subdomain-local quasi-Newton corrections are then constructed on the corresponding local parts of the architecture. A key feature is a novel nonlinear multi-preconditioning mechanism, in which subdomain corrections are optimally combined through the solution of a low-dimensional subspace minimization problem. Numerical experiments indicate that MP-LBFGS can improve convergence speed, as well as model accuracy over standard LBFGS while incurring lower communication overhead.

Serge Gratton, Alena Kopaničáková, Philippe Toint

Two OFFO (Objective-Function Free Optimization) noise tolerant algorithms are presented that handle bound constraints, inexact gradients and use second-order information when available.The first is a multi-level method exploiting a hierarchical description of the problem and the second is a domain-decomposition method covering the standard addditive Schwarz decompositions. Both are generalizations of the first-order AdaGrad algorithm for unconstrained optimization. Because these algorithms share a common theoretical framework, a single convergence/complexity theory is provided which covers them both. Its main result is that, with high probability, both methods need at most $O(ε^{-2})$ iterations and noisy gradient evaluations to compute an $ε$-approximate first-order critical point of the bound-constrained problem. Extensive numerical experiments are discussed on applications ranging from PDE-based problems to deep neural network training, illustrating their remarkable computational efficiency.