AKORN: Adaptive Knots generated Online for RegressioN splines

Abstract

In order to attain optimal rates, state-of-the-art algorithms for non-parametric regression require that a hyperparameter be tuned according to the smoothness of the ground truth (Tibshirani, 2014). This amounts to an assumption of oracle access to certain features of the data-generating process. We present a parameter-free algorithm for offline non-parametric regression over $TV_1$-bounded functions. By feeding offline data into an optimal online denoising algorithm styled after (Baby et al., 2021), we are able to use change-points to adaptively select knots that respect the geometry of the underlying ground truth. We call this procedure AKORN (Adaptive Knots gener- ated Online for RegressioN splines). By combining forward and backward passes over the data, we obtain an estimator whose empirical performance is close to Trend Filtering (Kim et al., 2009; Tibshirani, 2014), even when we provide the latter with oracle knowledge of the ground truth’s smoothness.