Overparameterization and Generalization Error: Weighted Trigonometric Interpolation

Xie, Yuege; Chou, Hung-Hsu; Rauhut, Holger und Ward, Rachel (2022): Overparameterization and Generalization Error: Weighted Trigonometric Interpolation. In: SIAM Journal on Mathematics of Data Science, Vol. 4, No. 2: pp. 885-908

Abstract

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 toward 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.

• Item Type: Journal article
• Keywords: overparameterization; generalization error; weighted optimization; smoothness
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Chair of Mathematics of Information Processing
• Subjects: 500 Science > 510 Mathematics
• ISSN: 2577-0187
• Language: English