TURF: A Two-factor, Universal, Robust, Fast Distribution Learning Algorithm

TL;DR

This paper introduces TURF, a fast and near-optimal distribution learning algorithm that achieves the best possible approximation bounds for a wide class of distributions, improving over existing methods.

Contribution

The paper presents a new distribution learning algorithm that attains the optimal approximation factor of 2 for all cases except the simplest, and provides a method to estimate the best polynomial complexity for practical distributions.

Findings

Achieves the optimal approximation factor of 2 for distribution learning.

Provides a near-linear-time, sample-efficient estimator.

Demonstrates improved empirical performance over existing algorithms.

Abstract

Approximating distributions from their samples is a canonical statistical-learning problem. One of its most powerful and successful modalities approximates every distribution to an $\ell_1$ distance essentially at most a constant times larger than its closest $t$-piece degree-$d$ polynomial, where $t\ge1$ and $d\ge0$. Letting $c_{t,d}$ denote the smallest such factor, clearly $c_{1,0}=1$, and it can be shown that $c_{t,d}\ge 2$ for all other $t$ and $d$. Yet current computationally efficient algorithms show only $c_{t,1}\le 2.25$ and the bound rises quickly to $c_{t,d}\le 3$ for $d\ge 9$. We derive a near-linear-time and essentially sample-optimal estimator that establishes $c_{t,d}=2$ for all $(t,d)\ne(1,0)$. Additionally, for many practical distributions, the lowest approximation distance is achieved by polynomials with vastly varying number of pieces. We provide a method that estimates this number near-optimally, hence helps approach the best possible approximation. Experiments combining the two techniques confirm improved performance over existing methodologies.