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تسجيل مستخدم جديدTesting and reconstruction via decision trees
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We study sublinear and local computation algorithms for decision trees, focusing on testing and reconstruction. Our first result is a tester that runs in $mathrm{poly}(log s, 1/varepsilon)cdot nlog n$ time, makes $mathrm{poly}(log s,1/varepsilon)cdot log n$ queries to an unknown function $f$, and: $circ$ Accepts if $f$ is $varepsilon$-close to a size-$s$ decision tree; $circ$ Rejects if $f$ is $Omega(varepsilon)$-far from decision trees of size $s^{tilde{O}((log s)^2/varepsilon^2)}$. Existing testers distinguish size-$s$ decision trees from those that are $varepsilon$-far from from size-$s$ decision trees in $mathrm{poly}(s^s,1/varepsilon)cdot n$ time with $tilde{O}(s/varepsilon)$ queries. We therefore solve an incomparable problem, but achieve doubly-exponential-in-$s$ and exponential-in-$s$ improvements in time and query complexities respectively. We obtain our tester by designing a reconstruction algorithm for decision trees: given query access to a function $f$ that is close to a small decision tree, this algorithm provides fast query access to a small decision tree that is close to $f$. By known relationships, our results yield reconstruction algorithms for numerous other boolean function properties -- Fourier degree, randomized and quantum query complexities, certificate complexity, sensitivity, etc. -- which in turn yield new testers for these properties. Finally, we give a hardness result for testing whether an unknown function is $varepsilon$-close-to or $Omega(varepsilon)$-far-from size-$s$ decision trees. We show that an efficient algorithm for this task would yield an efficient algorithm for properly learning decision trees, a central open problem of learning theory. It has long been known that proper learning algorithms for any class $mathcal{H}$ yield property testers for $mathcal{H}$; this provides an example of a converse.
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