TY - GEN

T1 - Almost optimal pseudorandom generators for spherical caps

AU - Kothari, Pravesh K.

AU - Meka, Raghu

PY - 2015/6/14

Y1 - 2015/6/14

N2 - 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 e and has an almost optimal seed-length of O(logn + log(1/∈) ·loglog(1/∈)). For an inverse-polynomially growing error ∈, our generator has a seed-length optimal up to a factor of O(loglog (n)). The most efficient PRG previously known (due to Kane [31]) 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 [45, 9], the classical moment problem from probability theory and explicit constructions of approximate orthogonal designs based on the seminal work of Bourgain and Gamburd [6] on expansion in Lie groups.

AB - 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 e and has an almost optimal seed-length of O(logn + log(1/∈) ·loglog(1/∈)). For an inverse-polynomially growing error ∈, our generator has a seed-length optimal up to a factor of O(loglog (n)). The most efficient PRG previously known (due to Kane [31]) 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 [45, 9], the classical moment problem from probability theory and explicit constructions of approximate orthogonal designs based on the seminal work of Bourgain and Gamburd [6] on expansion in Lie groups.

KW - Halfspaces

KW - Orthogonal designs

KW - Pseudorandom Generators

UR - http://www.scopus.com/inward/record.url?scp=84958756824&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958756824&partnerID=8YFLogxK

U2 - 10.1145/2746539.2746611

DO - 10.1145/2746539.2746611

M3 - Conference contribution

AN - SCOPUS:84958756824

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 247

EP - 256

BT - STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

T2 - 47th Annual ACM Symposium on Theory of Computing, STOC 2015

Y2 - 14 June 2015 through 17 June 2015

ER -