Skip to main content
Article thumbnail
Location of Repository

Learning Halfspaces Under Log-Concave Densities: Polynomial Approximations and Moment Matching

By Daniel M. Kane, Adam Klivans and Raghu Meka


We give 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 PAC learning the intersection of two halfspaces. We give two very different proofs of this result. The first develops a theory of polynomial approximation for log-concave measures and constructs a low-degree ℓ1 polynomial approximator for sufficiently smooth functions. The second uses techniques related to the classical moment problem to obtain sandwiching polynomials. Both approaches deviate significantly from known Fourier-based methods, where essentially all previous work required the underlying distribution to have some product structure. 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

Topics: Log-concave distributions, smoothed analysis, halfspaces, agnostic learning, Fourier analysis
Year: 2013
OAI identifier: oai:CiteSeerX.psu:
Provided by: CiteSeerX
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • (external link)
  • (external link)
  • Suggested articles

    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.