2011/03/29 by Pierre Tarrès, Yuan Yao, Tarrès, Pierre +1
Computer Science · Engineering · Mathematics · #Advanced Optimization Algorithms Research #Control Systems and Identification #Distributed Sensor Networks and Detection Algorithms #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1103.5538
openalex publication_date 2011/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, an online learning algorithm is proposed as sequential stochastic approximation of a regularization path converging to the regression function in reproducing kernel Hilbert spaces (RKHSs). We show that it is possible to produce the best known strong (RKHS norm) convergence rate of batch learning, through a careful choice of the gain or step size sequences, depending on regularity assumptions on the regression function. The corresponding weak (mean square distance) convergence rate is optimal in the sense that it reaches the minimax and individual lower rates in the literature. In both cases we deduce almost sure convergence, using Bernstein-type inequalities for martingales in Hilbert spaces. To achieve this we develop a bias-variance decomposition similar to the batch learning setting; the bias consists in the approximation and drift errors along the regularization path, which display the same rates of convergence, and the variance arises from the sample error analysed as a reverse martingale difference sequence. The rates above are obtained by an optimal trade-off between the bias and the variance.