Improved Kernel Alignment Regret Bound for Online Kernel Learning
This work provides incremental improvements in theoretical bounds for online kernel learning, benefiting researchers in machine learning optimization.
The paper tackles the problem of improving regret bounds and computational complexity for online kernel learning with the Hinge loss, achieving a regret of O(sqrt(mathcal{A}_T)) with O(ln^2 T) complexity under exponential eigenvalue decay, and extending to batch learning with an excess risk bound of O((1/T) sqrt(mathbb{E}[mathcal{A}_T])), improving previous O(1/sqrt{T}) bounds.
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.