过参数化与泛化误差：加权三角插值
Overparameterization and generalization error: weighted trigonometric interpolation
摘要：
受超参数化场景中学习后的深度神经网络令人惊讶的良好泛化特性以及相关的双下降现象的启发，本文分析了超参数化线性学习问题中平滑度与低泛化误差之间的关系。我们研究了一个随机傅里叶级数模型，其中的任务是从等距样本估计未知的傅里叶系数。我们推导出精确表达式的泛化误差的平原和加权最小二乘估计。我们精确地展示了如何偏向光滑插值，在加权三角插值的形式，可以导致较小的泛化误差在overparameterized政权相比，underparameterized政权。这提供了对过度参数化的力量的洞察，这在现代机器学习中很常见。
英文摘要：
Motivated by surprisingly good generalization properties of learned deep neural networks in overparameterized scenarios and by the related double descent phenomenon, this paper analyzes the relation between smoothness and low generalization error in an overparameterized linear learning problem. We study a random Fourier series model, where the task is to estimate the unknown Fourier coefficients from equidistant samples. We derive exact expressions for the generalization error of both plain and weighted least squares estimators. We show precisely how a bias towards smooth interpolants, in the form of weighted trigonometric interpolation, can lead to smaller generalization error in the overparameterized regime compared to the underparameterized regime. This provides insight into the power of overparameterization, which is common in modern machine learning.
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