We consider the task of estimating functions belonging to a specific class of nonsmooth functions, namely so-called tame functions. These functions appear in a wide range of applications: training deep learning, value functions of mixed-integer programs, or wave functions of small molecules. We show that tame functions are approximable by piecewise polynomials on any full-dimensional cube. We then 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：我们考虑一类特定非光滑函数——即所谓驯函数的估计问题。此类函数广泛应用于深度学习训练、混合整数规划价值函数及小分子波函数等领域。研究表明，驯函数可在任意满维立方体上通过分段多项式进行逼近。我们首次提出分段多项式回归的混合整数规划模型，两者结合可用于驯函数估计，并展示了具有前景的计算结果。