Nearly Optimal Algorithms with Sublinear Computational Complexity for Online Kernel Regression
This addresses a fundamental problem in online learning for researchers and practitioners by providing efficient algorithms with theoretical guarantees, though it is incremental as it builds on prior work to improve computational efficiency.
The paper tackles the trade-off between regret and computational complexity in online kernel regression by proposing two algorithms, AOGD-ALD and NONS-ALD, which achieve nearly optimal regret bounds with sublinear computational complexity, such as O(ln^2 T) for exponential eigenvalue decay.
The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of $O(\sqrt{L(f)})$ and $O(\mathrm{d}_{\mathrm{eff}}(μ)\ln{T})$ at a computational complexity in $O(\ln^2{T})$. If the eigenvalues decay polynomially with degree $p\geq 1$, then our algorithms keep the same regret bounds at a computational complexity in $o(T)$ in the case of $p>4$ and $p\geq 10$, respectively. $L(f)$ is the cumulative losses of $f$ and $\mathrm{d}_{\mathrm{eff}}(μ)$ is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.