V. Feldman and H. Lee and R. Servedio.

Much work has been done on learning various classes of ``simple" monotone functions under the uniform distribution. In this paper we give the first unconditional lower bounds for learning problems of this sort by showing that polynomial-time algorithms cannot learn shallow monotone Boolean formulas under the uniform distribution in the well-studied Statistical Query (SQ) model.

We introduce a new approach to understanding the learnability of ``simple" monotone functions that is based on a recent characterization of Strong SQ learnability by \citet{Simon-2007} %Simon \cite{Simon-2007}. Using the characterization we first show that depth-3 monotone formulas of size $n^{o(1)}$ cannot be learned by any polynomial-time SQ algorithm to accuracy $1 - 1/(\log n)^{\Omega(1)}.$ We then build on this result to show that depth-4 monotone formulas of size $n^{o(1)}$ cannot be learned even to a certain ${\frac 1 2} + o(1)$ accuracy in polynomial time. This improved hardness is achieved using a general technique that we introduce for amplifying the hardness of ``mildly hard'' learning problems in either the PAC or SQ framework. This hardness amplification for learning builds on the ideas in the work of \citet{ODonnell-2002} on hardness amplification for approximating functions using small circuits, and is applicable to a number of other contexts.

Finally, we demonstrate that our approach can also be used to reduce the well-known open problem of learning juntas to learning of depth-3 monotone formulas.

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