We consider approximating so-called tame functions, a class of nonsmooth, nonconvex functions, with piecewise polynomial functions. Tame functions appear in a wide range of applications: functions encountered in the training of deep neural networks with all common activations, value functions of mixed-integer programs, or wave functions of small molecules. We bound the quality of approximation of a tame function by a piecewise polynomial function with a given number of segments on any full-dimensional cube. We also present the first ever mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.

翻译：本文研究了用分段多项式函数逼近一类非光滑、非凸函数——所谓驯顺函数的理论方法。驯顺函数广泛应用于多种场景：包括采用常见激活函数的深度神经网络训练中出现的函数、混合整数规划的值函数，以及小分子的波函数。我们给出了在任意满维立方体上，使用给定段数的分段多项式函数逼近驯顺函数的逼近质量界，并首次提出分段多项式回归的混合整数规划建模方法。这两项成果相结合可用于驯顺函数估计，并展示了具有潜力的计算效果。