Title: 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 finite-dimensional 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 self-normalized inequalities we build
upper and lower bound estimates for the variance, leading to Bersntein-like
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.