In this paper, we propose an adaptive stopping rule for kernel-based gradient descent (KGD) algorithms. We introduce the empirical effective dimension to quantify the increments of iterations in KGD and derive an implementable early stopping strategy. We analyze the performance of the adaptive stopping rule in the framework of learning theory. Using the recently developed integral operator approach, we rigorously prove the optimality of the adaptive stopping rule in terms of showing the optimal learning rates for KGD equipped with this rule. Furthermore, a sharp bound on the number of iterations in KGD equipped with the proposed early stopping rule is also given to demonstrate its computational advantage.

翻译：本文提出了一种针对基于核的梯度下降算法的自适应停止准则。我们引入经验有效维度来量化KGD算法中迭代的增量，并推导出一种可实现的早期停止策略。我们在学习理论的框架下分析了自适应停止准则的性能。利用最近发展的积分算子方法，我们严格证明了该自适应停止准则的最优性，展示了采用该准则的KGD算法能达到最优学习速率。此外，我们还给出了采用所提出的早期停止准则的KGD算法迭代次数的锐界，以证明其计算优势。