Tame functions are a class of nonsmooth, nonconvex functions, which feature 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 consider approximating tame functions with piecewise polynomial functions. 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 mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.

翻译：驯服函数是一类非光滑、非凸函数，广泛存在于多种应用场景中：包括使用所有常见激活函数的深度神经网络训练中遇到的函数、混合整数规划的值函数，或小分子的波函数。我们考虑用分段多项式函数逼近驯服函数。我们在任意全维立方体上，对给定分段数的分段多项式函数逼近驯服函数的逼近质量给出了界。我们还首次提出了分段多项式回归的混合整数规划模型。结合这些方法，可用于估计驯服函数。我们展示了具有前景的计算结果。