Approximation Analysis of Kaczmarz Type Online Schemes and Fourier Analysis of Some Learning Algorithms Involving Sample Pair-based Loss Functions

Description

Learning theory provides mathematical foundations of many algorithms for analyzing and processing big data which is an important task in various fields of science and engineering. Kernel-based online schemes can be used for learning function features or data structures from samples in efficient ways. This project aims at approximation analysis of kernel-based online schemes and Fourier analysis of some learning algorithms involving sample pair-based loss functions. We shall first conduct error analysis for randomized Kaczmarz algorithms by developing an online learning approach. A unified gradient descent framework will be analyzed and confidence-based error bounds will be derived for kernel-based online learning algorithms. Then we shall study minimum error entropy regularization schemes by Fourier transform and error decomposition techniques. Minimizers of Renyi's entropy with piecewise independent distributions will be analyzed by techniques from wavelet analysis. Finally characterizations of minimizers of the generalization error associated with some ranking loss functions involving sample pairs will be given by means of methods from multivariate approximation theory and reproducing kernel Hilbert spaces generated by radial basis functions.