GloptiNets: Scalable Non-Convex Optimization with Certificates
This provides a scalable solution for non-convex optimization with certificates, leveraging GPU parallelism to address problems previously intractable for competitors.
The paper tackles non-convex optimization with certificates for smooth functions on hypercubes or tori, using Fourier spectrum decay to define tractable models that outperform state-of-the-art methods like Lasserre's hierarchy, handling polynomials with thousands of coefficients in moderate dimensions.
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.