Self-normalization techniques for streaming confident regression

We consider, in a generic streaming regression setting, the problem of building a confidence interval (and distribution) on the next observation based on past observed data. The observations given to the learner are of the form (x, y) with y = f (x) + ξ, where x can have arbitrary dependency on the past observations, f is unknown and the noise ξ is sub-Gaussian conditionally on the past observations. Further, the observations are assumed to come from some external filtering process making the number of observations itself a random stopping time. In this challenging scenario that captures a large class of processes with non-anticipative dependencies, we study the ordinary, ridge, and kernel least-squares estimates and provide confidence intervals based on self-normalized vector-valued martingale techniques, applied to the estimation of the mean and of the variance. We then discuss how these adaptive confidence intervals can be used in order to detect a possible model mismatch as well as to estimate the future (self-information, quadratic, or transportation) loss of the learner at a next step