#### SURF: A Simple, Universal, Robust, Fast Distribution Learning Algorithm

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

#### Authors

*Yi Hao, Ayush Jain, Alon Orlitsky, Vaishakh Ravindrakumar*

#### Abstract

Sample- and computationally-efficient distribution estimation is a fundamental tenet in statistics and machine learning. We present $\SURF$, an algorithm for approximating distributions by piecewise polynomials. $\SURF$ is:
simple, replacing prior complex optimization techniques by straight-forward empirical probability approximation of each potential polynomial piece through simple empirical-probability interpolation, and using plain divide-and-conquer to merge the pieces; universal, as well-known polynomial-approximation results imply that it accurately approximates a large class of common distributions;
robust to distribution mis-specification as for any degree $d \le 8$, it estimates any distribution to an $\ell_1$ distance $< 3$ times that of the nearest degree-$d$ piecewise polynomial, improving known factor upper bounds of 3 for single polynomials and 15 for polynomials with arbitrarily many pieces;
fast, using optimal sample complexity, running in near sample-linear time, and if given sorted samples it may be parallelized to run in sub-linear time.
In experiments, $\SURF$ outperforms state-of-the art algorithms.