GloptiNets: Scalable Non-Convex Optimization with Certificates

TL;DR

GloptiNets introduces a scalable non-convex optimization method with certificates that leverages Fourier spectrum decay, enabling efficient solutions for high-dimensional polynomial problems using neural network-inspired techniques.

Contribution

The paper proposes a novel Fourier-based approach for non-convex optimization with certificates, improving scalability and performance over existing algebraic methods.

Findings

Outperforms state-of-the-art methods like Lasserre's hierarchy on certain polynomial problems.

Utilizes GPU parallelism for enhanced scalability.

Effectively handles high-dimensional polynomials with thousands of coefficients.

Abstract

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.