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 $\exp({o(n/\log n)})$.

翻译：考虑在随机分类噪声存在下，通过随机均匀标注样本访问奇偶函数的模型。本文证明，近似奇偶函数中相关变量的数量与适定学习奇偶函数具有同等计算难度。具体而言，令$\gamma:{\mathbb R}^+\to {\mathbb R}^+$（其中$\gamma(x) \ge x$）为任意严格递增函数。首先，我们证明：对于任意能在多项式时间内返回奇偶函数$f$相关变量数目$d(f)$的$\gamma$近似值$D$（即满足$\gamma^{-1}(d(f)) \leq D \leq \gamma(d(f))$）的算法，均可被用于多项式时间内构造解决长期未决的$k(n)$稀疏奇偶函数（即具有$k(n)\le n$个相关变量的奇偶函数）多项式时间学习问题，其中$k(n) = \omega_n(1)$。其次，我们证明：对于任意能在$T(n)$时间内返回奇偶函数$f$相关变量数目$d(f)$的$\gamma$近似值的算法，均可被用于在多项式时间内构造出$poly(\Gamma(n))T(\Gamma(n)^2)$时间的奇偶函数适定学习算法，其中$\Gamma(x)=\gamma(\gamma(x))$。若$T(\Gamma(n)^2)=\exp({o(n/\log n)})$成立，该结果将解决另一个长期悬而未决的问题：在随机分类噪声存在下以$\exp({o(n/\log n)})$时间实现奇偶函数的适定学习。