Preserving Randomness for Adaptive Algorithms

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 $F \colon \{0, 1\}^n \to \{-1, 1\}$ 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).

Suppose you have a randomized algorithm that estimates some numeric quantity, and you plan to use it $k$ times. Suppose the estimation algorithm makes $n$ coin tosses. This situation seems to call for $nk$ coin tosses in total. In this paper, we show that you can actually get away with only making about $n + k$ coin tosses if you use some tricks.