Browse Theses

This thesis investigates the use of cryptographic assumptions to obtain strong lower bounds on the ability of Probably Approximately Correct (PAC) algorithms to learn various concept classes over the Boolean domain. We show that under realistic cryptographic assumptions there is no polynomial time prediction algorithm with membership queries for Boolean formulas, constant depth threshold circuits, and many other natural concept classes. Also, we show that if there exist non-uniform one-way functions, then membership queries won't help with predicting CNF or DNF formulas.

We investigate cryptographic lower bounds on the learnability of Boolean formulas, Boolean circuits, and constant depth threshold circuits on the uniform distribution and other specific distributions. We first show that weakly learning these classes on the uniform distribution is as hard as factoring Blum integers. We formalize the notion of a trivially learnable distribution and extend these hardness results to all non-trivial distributions. Furthermore, we show that a sub-exponential lower bound for factoring implies a lower bound for learning constant depth Boolean circuits on the uniform distribution which is almost tight.

We also show that a realistic assumption not rooted in number theory, namely the intractability of solving the subset sum problem of certain dimensions, can be used to show that Boolean circuits of low depth are not learnable on the uniform and other specific distributions. We observe that, under cryptographic assumptions, all our bounds can be used to establish tradeoffs between the running time and the number of samples necessary to learn.