Fast and strong convergence of online learning algorithms

• Published in: Advances in computational mathematics Vol. 45; no. 5-6; pp. 2745 - 2770
• Main Authors: Guo, Zheng-Chu, Shi, Lei
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2019; Springer Nature B.V
• Subjects: Algorithms; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Convergence; Distance learning; Fine structure; Hilbert space; Learning theory; Machine learning; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Regularization; Visualization; 62J02; Strong convergence in an RKHS; 68T05; Learning theory; Online learning; 68Q32; 62L20; Capacity dependent error analysis
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09707-8

In this paper, we study the online learning algorithm without explicit regularization terms. This algorithm is essentially a stochastic gradient descent scheme in a reproducing kernel Hilbert space (RKHS). The polynomially decaying step size in each iteration can play a role of regularization to ensure the generalization ability of online learning algorithm. We develop a novel capacity dependent analysis on the performance of the last iterate of online learning algorithm. This answers an open problem in learning theory. The contribution of this paper is twofold. First, our novel capacity dependent analysis can lead to sharp convergence rate in the standard mean square distance which improves the results in the literature. Second, we establish, for the first time, the strong convergence of the last iterate with polynomially decaying step sizes in the RKHS norm. We demonstrate that the theoretical analysis established in this paper fully exploits the fine structure of the underlying RKHS, and thus can lead to sharp error estimates of online learning algorithm.