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.

翻译：我们提出了一种新颖的带证书非凸优化方法，该方法可处理超立方体或环面上的光滑函数。与依赖代数特性的传统方法不同，我们的算法利用了目标函数傅里叶谱衰减所固有的正则性。通过定义一族易于处理的模型，我们既能获得精确的证书，又能利用为优化神经网络而开发的先进强大计算技术。通过这种方式，基于GPU的并行计算自然增强了我们方法的可扩展性。当处理中等维度但包含数千个系数的多项式时，我们的方法在带证书优化方面优于基于Lasserre层次结构等最先进方法，能够解决竞争对手难以处理的问题。