Almost Optimal Proper Learning and Testing Polynomials

TL;DR

This paper introduces an almost optimal polynomial-time proper learning algorithm for Boolean sparse multivariate polynomials, achieving sublinear query complexity in 1/ε and nearly linear in s, improving upon previous methods.

Contribution

It presents the first nearly optimal proper learning algorithm with sublinear query complexity in 1/ε for Boolean sparse polynomials, along with an almost tight lower bound and a new nearly optimal testing algorithm.

Findings

Query complexity is sublinear in 1/ε.

Algorithm is nearly linear in the sparsity s.

Provides an almost tight lower bound for query complexity.

Abstract

We give the first almost optimal polynomial-time proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. For $s$-sparse polynomial over $n$ variables and $\epsilon=1/s^\beta$, $\beta>1$, our algorithm makes $$q_U=\left(\frac{s}{\epsilon}\right)^{\frac{\log \beta}{\beta}+O(\frac{1}{\beta})}+ \tilde O\left(s\right)\left(\log\frac{1}{\epsilon}\right)\log n$$ queries. Notice that our query complexity is sublinear in $1/\epsilon$ and almost linear in $s$. All previous algorithms have query complexity at least quadratic in $s$ and linear in $1/\epsilon$. We then prove the almost tight lower bound $$q_L=\left(\frac{s}{\epsilon}\right)^{\frac{\log \beta}{\beta}+\Omega(\frac{1}{\beta})}+ \Omega\left(s\right)\left(\log\frac{1}{\epsilon}\right)\log n,$$ Applying the reduction in~\cite{Bshouty19b} with the above algorithm, we give the first almost optimal polynomial-time tester for $s$-sparse polynomial. Our tester, for $\beta>3.404$, makes $$\tilde O\left(\frac{s}{\epsilon}\right)$$ queries.