GloptiNets: Scalable Non-Convex Optimization with Certificates

Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track

Bibtex Paper Supplemental


Gaspard Beugnot, Julien Mairal, Alessandro Rudi


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 {\em 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.