Approximating the Number of Relevant Variables in a Parity Implies Proper Learning

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

Bounds for the Number of Tests in Non-Adaptive Randomized Algorithms for Group Testing

We study the group testing problem with non-adaptive randomized algorithms. Several models have been discussed in the literature to determine how to randomly choose the tests. For a model ${\cal M}$, let $m_{\cal M}(n,d)$ be the minimum number of tests required to detect at most $d$ defectives within $n$ items, with success probability at least $1-\delta$, for some constant $\delta$. In this paper, we study the measures $$c_{\cal M}(d)=\lim_{n\to \infty} \frac{m_{\cal M}(n,d)}{\ln n} \mbox{ and } c_{\cal M}=\lim_{d\to \infty} \frac{c_{\cal M}(d)}{d}.$$ In the literature, the analyses of such models only give upper bounds for $c_{\cal M}(d)$ and $c_{\cal M}$, and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for $c_{\cal M}(d)$ and $c_{\cal M}$ for all the known models~${\cal M}$.