Almost Optimal Pseudorandom Generators for Spherical Caps

TL;DR

This paper presents an explicit pseudorandom generator for spherical caps that nearly matches the optimal seed-length, significantly improving over previous constructions and also extending to Gaussian halfspaces.

Contribution

The authors develop a new PRG for spherical caps with near-optimal seed-length, using novel techniques like iterative dimension reduction and orthogonal designs.

Findings

Achieves seed-length of $O( ext{log} n + ext{log}(1/ extepsilon) ext{loglog}(1/ extepsilon))$

Improves seed-length over previous work by Kane (2012)

Extends constructions to Gaussian halfspaces

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 $\epsilon$ and has an almost optimal seed-length of $O(\log n + \log(1/\epsilon) \cdot \log\log(1/\epsilon))$. For an inverse-polynomially growing error $\epsilon$, 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 $\Omega(\log^{3/2}{(n)})$ 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 \emph{classical moment problem} from probability theory and explicit constructions of \emph{orthogonal designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion in Lie groups.