Improved Kernel Alignment Regret Bound for Online Kernel Learning

TL;DR

This paper presents an improved online kernel learning algorithm with better regret bounds and computational complexity, especially under certain eigenvalue decay conditions, and extends these results to batch learning with enhanced excess risk bounds.

Contribution

The paper introduces a new algorithm that achieves superior regret bounds and computational efficiency compared to previous methods, depending on eigenvalue decay rates.

Findings

Regret bound of O(√A_T) with exponential eigenvalue decay

Regret bound of O((A_T T)^{1/4}) in general case

Enhanced excess risk bound of O(1/T)√E[A_T] in batch learning

Abstract

In this paper, we improve the kernel alignment regret bound for online kernel learning in the regime of the Hinge loss function. Previous algorithm achieves a regret of $O((\mathcal{A}_TT\ln{T})^{\frac{1}{4}})$ at a computational complexity (space and per-round time) of $O(\sqrt{\mathcal{A}_TT\ln{T}})$, where $\mathcal{A}_T$ is called \textit{kernel alignment}. We propose an algorithm whose regret bound and computational complexity are better than previous results. Our results depend on the decay rate of eigenvalues of the kernel matrix. If the eigenvalues of the kernel matrix decay exponentially, then our algorithm enjoys a regret of $O(\sqrt{\mathcal{A}_T})$ at a computational complexity of $O(\ln^2{T})$. Otherwise, our algorithm enjoys a regret of $O((\mathcal{A}_TT)^{\frac{1}{4}})$ at a computational complexity of $O(\sqrt{\mathcal{A}_TT})$. We extend our algorithm to batch learning and obtain a $O(\frac{1}{T}\sqrt{\mathbb{E}[\mathcal{A}_T]})$ excess risk bound which improves the previous $O(1/\sqrt{T})$ bound.