Preserving Randomness for Adaptive Algorithms

TL;DR

This paper presents a method to run randomized estimation algorithms on multiple adaptive inputs with significantly fewer random bits, combining pseudorandom generators and output modifications, with near-optimal randomness use in certain cases.

Contribution

It introduces a novel approach that reduces randomness complexity for adaptive algorithms, using a variant of the INW pseudorandom generator and output shifting/rounding techniques.

Findings

Reduces randomness from nk to n + O(k log(d+1)) for adaptive inputs.

Proves output modification is necessary for this setting.

Provides a near-optimal randomness complexity when dimension d is small.

Abstract

Suppose $\mathsf{Est}$ is a randomized estimation algorithm that uses $n$ random bits and outputs values in $\mathbb{R}^d$. We show how to execute $\mathsf{Est}$ on $k$ adaptively chosen inputs using only $n + O(k \log(d + 1))$ random bits instead of the trivial $nk$ (at the cost of mild increases in the error and failure probability). Our algorithm combines a variant of the INW pseudorandom generator (STOC '94) with a new scheme for shifting and rounding the outputs of $\mathsf{Est}$. We prove that modifying the outputs of $\mathsf{Est}$ is necessary in this setting, and furthermore, our algorithm's randomness complexity is near-optimal in the case $d \leq O(1)$. As an application, we give a randomness-efficient version of the Goldreich-Levin algorithm; our algorithm finds all Fourier coefficients with absolute value at least $\theta$ of a function $F: \{0, 1\}^n \to \{-1, 1\}$ using $O(n \log n) \cdot \text{poly}(1/\theta)$ queries to $F$ and $O(n)$ random bits (independent of $\theta$), improving previous work by Bshouty et al. (JCSS '04).