基于格的深度神经网络：正则性与定制正则化
Lattice-based Deep Neural Networks: Regularity and Tailored Regularization
AI总结：
本文提出利用格规则作为深度神经网络的训练点，通过限制网络参数以匹配目标函数的正则性特征，从而获得更好的理论泛化误差界限。
摘要：
本文综述了将格规则应用于深度神经网络（DNNs）的研究，格规则是一类准蒙特卡洛方法。它们在高维积分和函数逼近的各种情境中已显示出有效性。由于其非常简单的公式，格规则极容易实现——只需一个良好的整数生成向量，其长度与问题的维数相匹配。近年来，DNNs的应用和理论研究涌现出大量研究活动。我们回顾了最近关于使用格规则作为DNNs训练点的研究，其中我们得到了DNNs的显式正则性界限。通过限制网络参数以匹配目标函数的正则性特征，我们证明了使用定制格训练点的DNNs可以实现良好的理论泛化误差界限，隐含常数与输入维度无关。我们还通过数值实验表明，使用我们定制的正则化训练的DNNs在标准ℓ₂正则化下表现显著更好。
英文摘要：
This survey article is concerned with the application of lattice rules to Deep Neural Networks (DNNs), lattice rules being a family of quasi-Monte Carlo methods. They have demonstrated effectiveness in various contexts for high-dimensional integration and function approximation. They are extremely easy to implement thanks to their very simple formulation -- all that is required is a good integer generating vector of length matching the dimensionality of the problem. In recent years there has been a burst of research activities on the application and theory of DNNs. We review our recent article on using lattice rules as training points for DNNs with a smooth activation function, where we obtained explicit regularity bounds of the DNNs. By imposing restrictions on the network parameters to match the regularity features of the target function, we prove that DNNs with tailored lattice training points can achieve good theoretical generalization error bounds, with implied constants independent of the input dimension. We also demonstrate numerically that DNNs trained with our tailored regularization perform significantly better than with standard $\ell_2$ regularization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
