GloptiNets: Scalable Non-Convex Optimization with Certificates

Gaspard Beugnot (1, 2, 3) , Julien Mairal (4) , Alessandro Rudi (1, 2, 3)

1 PSL - Université Paris Sciences et Lettres
2 DI-ENS - Département d'informatique - ENS Paris
3 SIERRA - Statistical Machine Learning and Parsimony
4 Thoth - Apprentissage de modèles à partir de données massives

We present a novel approach to non-convex optimization with certificates, which handles smooth functions on the hypercube or on the torus. Unlike traditional methods that rely on algebraic properties, our algorithm exploits the regularity of the target function intrinsic in the decay of its Fourier spectrum. By defining a tractable family of models, we allow at the same time to obtain precise certificates and to leverage the advanced and powerful computational techniques developed to optimize neural networks. In this way the scalability of our approach is naturally enhanced by parallel computing with GPUs. Our approach, when applied to the case of polynomials of moderate dimensions but with thousands of coefficients, outperforms the state-of-the-art optimization methods with certificates, as the ones based on Lasserre's hierarchy, addressing problems intractable for the competitors.

Mots clés: Non-convex optimization; Kernel Sum-of-Squares; Polynomial optimization

Domaines: Optimisation et contrôle [math.OC]; Apprentissage [cs.LG]

Gaspard Beugnot, Julien Mairal, Alessandro Rudi. GloptiNets: Scalable Non-Convex Optimization with Certificates. NeurIPS 2023 - 37th Conference on Neural Information Processing Systems, Dec 2023, New Orleans, United States. pp.1-21. ⟨hal-04138843v2⟩