Abstract

Given a function f mapping n-variate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of ag-fieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of alln-variate degreedpolynomials which agree withfon a the natural phenomena underlying the input/output relationship tiny but non-negligible fraction, , of the input space. We give a of the function, and the goal of the learner is to come up with a randomized algorithm for solving this task which accessesfas a simple explanation which best fits the examples. Therefore the black box and runs in time polynomial in1;nand exponential in best explanation may account for only part of the phenomena. d, provided is(pd=jFj). For the special case whend=1, In some situations, when the phenomena appears very irregular, we solve this problem for jFj>0. In this case the providing an explanation which fits only part of it is better than nothing. Interestingly, Kearns et. al. did not consider the use of running time of our algorithm is bounded by a polynomial queries (but rather examples drawn from an arbitrary distribu-and exponential ind. Our algorithm generalizes a previously tion) as they were skeptical that queries could be of any help. known algorithm, due to Goldreich and Levin, that solves this We show that queries do seem to help (see below). task for the case whenF=GF(2)(andd=1).

Citations