Exact learning of linear combinations of monotone terms from function value queries

Abstract

We investigate the exact learning of real-valued functions on {0,l}ⁿ represented by a weighted sum of a number of monotone terms using queries for the values of the target function at assignments of the learner's choice. When all coefficients are nonnegative, we show that these functions are learnable with at most (n — ⌊log₂ k⌋+1)k queries, where n is the number of variables and k is the number of terms in the target. We also prove a lower bound of $\Omega \left( {\tfrac{{nk}}{{\log _2 k}}} \right)$ on the number of queries necessary for learning this class, so no algorithm can reduce the number of queries dramatically. In the general case, namely, when the coefficients vary over the reals, we show the number of queries required for exact learning of k-term subclass is upper bounded by q(n, ⌊log₂ k⌋+1) and is lower bounded by q(n,⌊log₂ k⌋), where $q\left( {n,l} \right) = \sum\nolimits_{i = 0}^l {\left( {\begin{array}{*{20}c}n \\i \\\end{array} } \right)}$. The latter bounds are shown by generalizing Roth and Benedek's technique for analyzing the learning problem for k-sparse multivariate polynomials over GF(2) [8].