From average case complexity to improper learning complexity

The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are efficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing lower bounds fall short of the best known algorithms. The biggest challenge in proving complexity results is to establish hardness of {\em improper learning} (a.k.a. representation independent learning).The difficulty in proving lower bounds for improper learning is that the standard reductions from $\mathbf{NP}$-hard problems do not seem to apply in this context. There is essentially only one known approach to proving lower bounds on improper learning. It was initiated in (Kearns and Valiant 89) and relies on cryptographic assumptions. We introduce a new technique for proving hardness of improper learning, based on reductions from problems that are hard on average. We put forward a (fairly strong) generalization of Feige's assumption (Feige 02) about the complexity of refuting random constraint satisfaction problems. Combining this assumption with our new technique yields far reaching implications. In particular, 1. Learning $\mathrm{DNF}$'s is hard. 2. Agnostically learning halfspaces with a constant approximation ratio is hard. 3. Learning an intersection of $\omega(1)$ halfspaces is hard.Comment: 34 page

Moment-Matching Polynomials

We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem and deviate significantly from known Fourier-based methods, which require the underlying distribution to have some product structure. Our main application is the first polynomial-time algorithm for agnostically learning any function of a constant number of halfspaces with respect to any log-concave distribution (for any constant accuracy parameter). This result was not known even for the case of learning the intersection of two halfspaces without noise. Additionally, we show that in the "smoothed-analysis" setting, the above results hold with respect to distributions that have sub-exponential tails, a property satisfied by many natural and well-studied distributions in machine learning. Given that our algorithms can be implemented using Support Vector Machines (SVMs) with a polynomial kernel, these results give a rigorous theoretical explanation as to why many kernel methods work so well in practice

The Average Sensitivity of an Intersection of Half Spaces

We prove new bounds on the average sensitivity of the indicator function of an intersection of $k$ halfspaces. In particular, we prove the optimal bound of $O(\sqrt{n\log(k)})$. This generalizes a result of Nazarov, who proved the analogous result in the Gaussian case, and improves upon a result of Harsha, Klivans and Meka. Furthermore, our result has implications for the runtime required to learn intersections of halfspaces