On the Learnability of Monotone Functions

Abstract

A longstanding lacuna in the field of computational learning theory is the learnability of succinctly representable monotone Boolean functions, i.e., functions that preserve the given order of the input. This thesis makes significant progress towards understanding both the possibilities and the limitations of learning various classes of monotone functions by carefully considering the complexity measures used to evaluate them. We show that Boolean functions computed by polynomial-size monotone circuits are hard to learn assuming the existence of one-way functions. Having shown the hardness of learning general polynomial-size monotone circuits, we show that the class of Boolean functions computed by polynomial-size depth-3 monotone circuits are hard to learn using statistical queries. As a counterpoint, we give a statistical query learning algorithm that can learn random polynomial-size depth-2 monotone circuits (i.e., monotone DNF formulas). As a preliminary step towards a fully polynomial-time, proper learning algorithm for learning polynomial-size monotone decision trees, we also show the relationship between the average depth of a monotone decision tree, its average sensitivity, and its variance. Finally, we return to monotone DNF formulas, and we show that they are teachable (a different model of learning) in the average case. We also show that non-monotone DNF formulas, juntas, and sparse GF2 formulas are teachable in the average case