We consider the problem of approximating the regression function from noisy vector-valued data by an online learning algorithm using an appropriate reproducing kernel Hilbert space (RKHS) as prior. In an online algorithm, i.i.d. samples become available one by one by a random process and are successively processed to build approximations to the regression function. We are interested in the asymptotic performance of such online approximation algorithms and show that the expected squared error in the RKHS norm can be bounded by $C^2 (m+1)^{-s/(2+s)}$, where $m$ is the current number of processed data, the parameter $0<s\leq 1$ expresses an additional smoothness assumption on the regression function and the constant $C$ depends on the variance of the input noise, the smoothness of the regression function and further parameters of the algorithm.

翻译：我们考虑通过在线学习算法从含噪向量值数据中近似回归函数的问题，该算法采用适当的再生核希尔伯特空间（RKHS）作为先验。在线算法中，独立同分布样本通过随机过程逐个获得，并依次处理以构建回归函数的近似。我们关注此类在线近似算法的渐近性能，并证明在RKHS范数下的期望平方误差可被限定为$C^2 (m+1)^{-s/(2+s)}$，其中$m$为当前已处理数据数量，参数$0<s\leq 1$表示对回归函数的额外光滑性假设，常数$C$取决于输入噪声方差、回归函数光滑性及算法的其他参数。