Improved Parallel Derandomization via Finite Automata with Applications

2024-11-27

arXiv.org

AI Summary

This work addresses parallel derandomization by constructing low-support probability distributions that “fool” polynomial-space statistical tests modeled as finite automata, enabling NC-level efficient parallel algorithms. We propose an iterative sparsification framework based on lattice rounding, introducing a novel Lipschitz-weighted error tracking mechanism, coupled with automaton state truncation and FFT-accelerated convolution computation. Our approach is the first to reduce the processor complexity of fooling finite automata to the NC-computable range. We validate the method on two canonical problems: the Gale–Berlekamp switching game and SDP-based rounding for MAX-CUT approximation. Experimental results demonstrate that our algorithm maintains high-fidelity statistical fooling while achieving substantial gains in parallel efficiency—reducing processor complexity significantly without increasing total work, which remains asymptotically optimal.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Fool finite-state automata with reduced processor complexity
• Improve parallel derandomization via lattice rounding
• Enhance applications like Gale-Berlekamp game and MAX-CUT

Innovation

• Parallel algorithms for fooling finite-state automata
• Work-efficient lattice rounding for distribution sparsification
• FFT-based convolutions for efficient transition probabilities