Almost Optimal Pseudorandom Generators for Spherical Caps: Extended Abstract

Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere, aka spherical caps, which besides being interesting and basic geometric objects, also arise frequently in the analysis of various randomized algorithms (e.g., randomized rounding). We give an explicit PRG which fools spherical caps within error ε and has an almost optimal seed-length of O(log n + log(1/ε) ⋅ log log(1/ε)). For an inverse-polynomially growing error ε, our generator has a seed-length optimal up to a factor of O( log log (n)). The most efficient PRG previously known (due to Kane 2012) requires a seed-length of Ω(log3/2(n)) in this setting. We also obtain similar constructions to fool halfspaces with respect to the Gaussian distribution. Our construction and analysis are significantly different from previous works on PRGs for halfspaces and build on the iterative dimension reduction ideas of Kane et. al. 2011 and Celis et. al. 2013, the classical moment problem from probability theory and explicit constructions of approximate orthogonal designs based on the seminal work of Bourgain and Gamburd 2011 on expansion in Lie groups.