Separating distribution-free and mistake-bound learning models over the Boolean domain

Two of the most commonly used models in computational learning theory are the distribution-free model, in which examples are chosen from a fixed but arbitrary distribution, and the absolute mistake-bound model, in which examples are presented in order by an adversary. Over the Boolean domain {0,1}ⁿ, it is known that if the learner is allowed unlimited computational resources, then any concept class learnable in one model is also learnable in the other. In addition, any polynomial-time learning algorithm for a concept class in the mistake-bound model can be transformed into one that learns the class in the distribution-free model. It is shown that if one-way functions exist, then the converse does not hold. The author presents a concept class over {0.1}ⁿ that is learnable in the distribution-free model but is not learnable in the absolute mistake-bound model if one-way functions exist. In addition, the concept class remains hard to learn in the mistake-bound model, even if the learner is allowed a polynomial number of membership queries