  Consider the model where we can access a parity function through random
uniform labeled examples in the presence of random classification noise. In
this paper, we show that approximating the number of relevant variables in the
parity function is as hard as properly learning parities.
  More specifically, let $\gamma:{\mathbb R}^+\to {\mathbb R}^+$, where
$\gamma(x) \ge x$, be any strictly increasing function. In our first result, we
show that from any polynomial-time algorithm that returns a
$\gamma$-approximation, $D$ (i.e., $\gamma^{-1}(d(f)) \leq D \leq
\gamma(d(f))$), of the number of relevant variables~$d(f)$ for any parity $f$,
we can, in polynomial time, construct a solution to the long-standing open
problem of polynomial-time learning $k(n)$-sparse parities (parities with
$k(n)\le n$ relevant variables), where $k(n) = \omega_n(1)$.
  In our second result, we show that from any $T(n)$-time algorithm that, for
any parity $f$, returns a $\gamma$-approximation of the number of relevant
variables $d(f)$ of $f$, we can, in polynomial time, construct a
$poly(\Gamma(n))T(\Gamma(n)^2)$-time algorithm that properly learns parities,
where $\Gamma(x)=\gamma(\gamma(x))$.
  If $T(\Gamma(n)^2)=\exp({o(n/\log n)})$, this would resolve another
long-standing open problem of properly learning parities in the presence of
random classification noise in time $\exp({o(n/\log n)})$.
