Streaming kernel regression with provably adaptive mean, variance, and regularization
Abstract
We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finitedimensional linear regression with fixed regularization and known variance to the kernel setup with a regularization parameter allowed to be a measurable function of past observations. Then, using appropriate selfnormalized inequalities we build upper and lower bound estimates for the variance, leading to Bersnteinlike concentration bounds. The later is used in order to define the adaptive regularization. The bounds resulting from our technique are valid uniformly over all observation points and all time steps, and are compared against the literature with numerical experiments. Finally, the potential of these tools is illustrated by an application to kernelized bandits, where we revisit the Kernel UCB and Kernel Thompson Sampling procedures, and show the benefits of the novel adaptive kernel tuning strategy.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.00768
 arXiv:
 arXiv:1708.00768
 Bibcode:
 2017arXiv170800768D
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning