Product distribution learning with imperfect advice

TL;DR

This paper presents an efficient algorithm for learning unknown product distributions on the Boolean hypercube using advice about a known distribution, reducing sample complexity under certain mean vector proximity conditions.

Contribution

It introduces a novel learning algorithm that leverages advice about a known distribution to improve sample efficiency in distribution learning.

Findings

Sample complexity is reduced to d^{1-\u03b7}/\u03b5^2 with advice.

Algorithm works without prior bounds on mean vector differences.

Achieves close distribution approximation with fewer samples.

Abstract

Given i.i.d.~samples from an unknown distribution $P$, the goal of distribution learning is to recover the parameters of a distribution that is close to $P$. When $P$ belongs to the class of product distributions on the Boolean hypercube $\{0,1\}^d$, it is known that $\Omega(d/\varepsilon^2)$ samples are necessary to learn $P$ within total variation (TV) distance $\varepsilon$. We revisit this problem when the learner is also given as advice the parameters of a product distribution $Q$. We show that there is an efficient algorithm to learn $P$ within TV distance $\varepsilon$ that has sample complexity $\tilde{O}(d^{1-\eta}/\varepsilon^2)$, if $\|\mathbf{p} - \mathbf{q}\|_1 < \varepsilon d^{0.5 - \Omega(\eta)}$. Here, $\mathbf{p}$ and $\mathbf{q}$ are the mean vectors of $P$ and $Q$ respectively, and no bound on $\|\mathbf{p} - \mathbf{q}\|_1$ is known to the algorithm a priori.