[Paper Review] Pseudorandomness for concentration bounds and signed majorities

Parikshit Gopalan, Daniel M. Kane+1 more|arXiv (Cornell University)|Nov 17, 2014

Complexity and Algorithms in Graphs18 references3 citations

TL;DR

This paper presents a near-optimal pseudorandom generator that fools halfspaces and signed majorities over {±1}ⁿ with seed length Õ(log(n/ε)), significantly improving upon the prior O(log n + log²(1/ε)) bound. It achieves this by combining Fourier-analytic tools with iterative dimension reduction and gradually increasing independence, enabling near-optimal derandomization of Chernoff-type concentration bounds and majorities with polynomially small error.

ABSTRACT

The problem of constructing pseudorandom generators that fool halfspaces has been studied intensively in recent times. For fooling halfspaces over the hypercube with polynomially small error, the best construction known requires seed-length O(log^2 n) (MekaZ13). Getting the seed-length down to O(log(n)) is a natural challenge in its own right, which needs to be overcome in order to derandomize RL. In this work we make progress towards this goal by obtaining near-optimal generators for two important special cases: 1) We give a near optimal derandomization of the Chernoff bound for independent, uniformly random bits. Specifically, we show how to generate a x in {1,-1}^n using $ ilde{O}(\log (n/ε))$ random bits such that for any unit vector u, matches the sub-Gaussian tail behaviour predicted by the Chernoff bound up to error eps. 2) We construct a generator which fools halfspaces with {0,1,-1} coefficients with error eps with a seed-length of $ ilde{O}(\log(n/ε))$. This includes the important special case of majorities. In both cases, the best previous results required seed-length of $O(\log n + \log^2(1/ε))$. Technically, our work combines new Fourier-analytic tools with the iterative dimension reduction techniques and the gradually increasing independence paradigm of previous works (KaneMN11, CelisRSW13, GopalanMRTV12).

Motivation & Objective

• To construct a pseudorandom generator (PRG) that fools halfspaces with {0,1,-1} coefficients using near-optimal seed length.
• To achieve near-optimal derandomization of the Chernoff bound for independent {±1} random variables.
• To reduce the seed length for fooling halfspaces from O(log n + log²(1/ε)) to Õ(log(n/ε)) for polynomially small error.
• To advance the long-standing open problem of derandomizing RL with O(log n) seed length.

Proposed method

• Combines Fourier-analytic tools with iterative dimension reduction to control the influence of high-degree Fourier coefficients.
• Employs the gradually increasing independence paradigm to simulate sub-Gaussian tail behavior of weighted sums.
• Uses a coordinate-wise product of two PRGs: one for controlling low-frequency components and one for high-frequency components.
• Applies Taylor expansion and analytic bounds on trigonometric products to control error in characteristic functions.
• Leverages the fact that the characteristic function of a sum of independent {±1} variables can be approximated via exponential and polynomial terms.
• Employs a hybrid analysis: for small α, uses concentration of measure; for large α, uses polynomial approximation of the characteristic function.

Experimental results

Research questions

• RQ1 Can we construct a PRG with seed length Õ(log(n/ε)) that fools all halfspaces over {±1}ⁿ with error ε?
• RQ2 Can we achieve near-optimal derandomization of the Chernoff bound for weighted sums of independent {±1} variables?
• RQ3 Can we improve the seed length for fooling halfspaces with {0,1,-1} coefficients beyond O(log n + log²(1/ε))?
• RQ4 Does combining iterative dimension reduction with Fourier-analytic techniques yield tighter error bounds for pseudorandom generators?

Key findings

• The paper constructs an explicit PRG with seed length Õ(log(n/ε)) that fools all halfspaces with {0,1,-1} coefficients to error ε.
• It achieves near-optimal derandomization of the Chernoff bound: for any unit vector u, the inner product u·x matches the sub-Gaussian tail behavior up to error ε using Õ(log(n/ε)) random bits.
• The generator achieves error ε with seed length Õ(log(n/ε)), improving upon the prior O(log n + log²(1/ε)) bound.
• The method enables black-box derandomization of algorithms relying on Chernoff-type tail bounds with only a polynomial slowdown.
• The analysis shows that the characteristic function of the sum can be approximated via a truncated Taylor expansion with controlled error, bounded by moments of the weights.
• The final generator is the coordinate-wise product of two PRGs, each handling different regimes of the characteristic function parameter α.