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Improved parallel derandomization via finite automata with applications

Jeff Giliberti, David G. Harris

TL;DR

The paper tackles deterministic parallel derandomization by constructing small-support distributions that fool polynomial-space statistical tests implemented by finite automata. It introduces a unifying FOOL framework built atop a work-efficient REDUCE subroutine, leveraging lattice-approximation to sparsify drivestreams while tracking an aggregate, Lipschitz-based error that bounds the final-weight discrepancy. The authors show substantial improvements in processor complexity and scalability, with concrete results for Gale-Berlekamp Switching Game and approximate MAX-CUT via SDP rounding, aided by optimizations like state-space truncation and FFT-based transition computation. This work advances practical deterministic parallel derandomization, offering near-seed-length guarantees and broad applicability beyond binary alphabets, while aligning theoretical efficiency with implementable parallel strategies.

Abstract

A central approach to algorithmic derandomization is the construction of small-support probability distributions that "fool" randomized algorithms, often enabling efficient parallel (NC) implementations. An abstraction of this idea is fooling polynomial-space statistical tests computed via finite automata (Sivakumar 2002); this encompasses a wide range of properties including $k$-wise independence and sums of random variables. We present new parallel algorithms to fool finite-state automata, with significantly reduced processor complexity. Briefly, our approach is to iteratively sparsify distributions using a work-efficient lattice rounding routine and maintain accuracy by tracking an aggregate weighted error that is determined by the Lipschitz value of the statistical tests being fooled. We illustrate with improved applications to the Gale-Berlekamp Switching Game and to approximate MAX-CUT via SDP rounding. These involve further several optimizations, including truncating the state space of the automata and using FFT-based convolutions to compute transition probabilities efficiently.

Improved parallel derandomization via finite automata with applications

TL;DR

The paper tackles deterministic parallel derandomization by constructing small-support distributions that fool polynomial-space statistical tests implemented by finite automata. It introduces a unifying FOOL framework built atop a work-efficient REDUCE subroutine, leveraging lattice-approximation to sparsify drivestreams while tracking an aggregate, Lipschitz-based error that bounds the final-weight discrepancy. The authors show substantial improvements in processor complexity and scalability, with concrete results for Gale-Berlekamp Switching Game and approximate MAX-CUT via SDP rounding, aided by optimizations like state-space truncation and FFT-based transition computation. This work advances practical deterministic parallel derandomization, offering near-seed-length guarantees and broad applicability beyond binary alphabets, while aligning theoretical efficiency with implementable parallel strategies.

Abstract

A central approach to algorithmic derandomization is the construction of small-support probability distributions that "fool" randomized algorithms, often enabling efficient parallel (NC) implementations. An abstraction of this idea is fooling polynomial-space statistical tests computed via finite automata (Sivakumar 2002); this encompasses a wide range of properties including -wise independence and sums of random variables. We present new parallel algorithms to fool finite-state automata, with significantly reduced processor complexity. Briefly, our approach is to iteratively sparsify distributions using a work-efficient lattice rounding routine and maintain accuracy by tracking an aggregate weighted error that is determined by the Lipschitz value of the statistical tests being fooled. We illustrate with improved applications to the Gale-Berlekamp Switching Game and to approximate MAX-CUT via SDP rounding. These involve further several optimizations, including truncating the state space of the automata and using FFT-based convolutions to compute transition probabilities efficiently.

Paper Structure

This paper contains 24 sections, 23 theorems, 68 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Technical approach
  4. Distribution error tracking
  5. REDUCE subroutine
  6. Further optimizations
  7. Preliminaries
  8. Basic definitions for automata
  9. Measurement of distribution error
  10. Comparison with other automaton models
  11. $\bullet$ Nonuniform probability spaces
  12. $\bullet$ Multiple automata
  13. $\bullet$ Leveled automata
  14. Lattice approximation
  15. Overview of algorithms and data structures
  16. ...and 9 more sections

Key Result

Theorem 1.1

Consider a probability space $\Omega$ on $n$ binary random variables and a collection of $\ell$ statistical tests over $\Omega$, each with $\eta$ possible states and Lipschitz value $\lambda$. There is a deterministic algorithm to generate a distribution $D$ of size $\varepsilon^{-2} \mathop{\mathrm

Theorems & Definitions (43)

  • Theorem 1.1: Simplified
  • Theorem 1.2: Gale-Berlekamp Switching Game
  • Theorem 1.3: MAX-CUT Approximation
  • Definition 2.1: Lipschitz value/confusion of a weight functions
  • Theorem 2.2
  • Definition 2.3: Lattice Approximation Problem (LAP)
  • Theorem 2.4: Theorem 1.3 of GG23
  • Proposition 2.5
  • proof
  • proof
  • ...and 33 more