  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.
