Improved guarantees for agnostic learning of disjunctions

Given some arbitrary distribution D over {0, 1}ⁿ and arbitrary target function c^*, the problem of agnostic learning of disjunctions is to achieve an error rate comparable to the error OPT _disj of the best disjunction with respect to (D, c ^*). Achieving error O(n · OPT_disj) + ε is trivial, and Winnow [13] achieves error O(r · OPT_disj) + ε, where r is the number of relevant variables in the best disjunction. In recent work, Peleg [14] shows how to achieve a bound of Õ (√n · OPT_disj) + ε in polynomial time. In this paper we improve on Peleg's bound, giving a polynomial-time algorithm achieving a bound of O(n^1/3+^α · OPT_disj) + ε for any constant α > 0. The heart of the algorithm is a method for weak-learning when OPT_disj = O(1/n¹/^3+α), which can then be fed into existing agnostic boosting procedures to achieve the desired guarantee.