Near-Optimal Averaging Samplers and Matrix Samplers

TL;DR

This paper introduces the first efficient averaging sampler with near-optimal randomness and sample complexity, extending the concept to matrix samplers and applications in randomness extractors, list-decodable codes, and normed vector spaces.

Contribution

It presents the first efficient averaging sampler with asymptotically optimal randomness and near-optimal sample complexity, and generalizes the concept to matrix samplers and other normed spaces.

Findings

Achieves asymptotically optimal randomness complexity.

Provides near-optimal sample complexity with a small polynomial factor.

Extends the framework to matrix samplers and applications in extractors and coding theory.

Abstract

We present the first efficient averaging sampler that achieves asymptotically optimal randomness complexity and near-optimal sample complexity. For any $\delta < \varepsilon$ and any constant $\alpha > 0$, our sampler uses $m + O(\log (1 / \delta))$ random bits to output $t = O((\frac{1}{\varepsilon^2} \log \frac{1}{\delta})^{1 + \alpha})$ samples $Z_1, \dots, Z_t \in \{0, 1\}^m$ such that for any function $f: \{0, 1\}^m \to [0, 1]$, \[ \Pr\left[\left|\frac{1}{t}\sum_{i=1}^t f(Z_i) - \mathbb{E}[f]\right| \leq \varepsilon\right] \geq 1 - \delta. \] The randomness complexity is optimal up to a constant factor, and the sample complexity is optimal up to the $O((\frac{1}{\varepsilon^2} \log \frac{1}{\delta})^{\alpha})$ factor. Our technique generalizes to matrix samplers. A matrix sampler is defined similarly, except that $f: \{0, 1\}^m \to \mathbb{C}^{d \times d}$ and the absolute value is replaced by the spectral norm. Our matrix sampler achieves randomness complexity $m + \widetilde O (\log(d / \delta))$ and sample complexity $ O((\frac{1}{\varepsilon^2} \log \frac{d}{\delta})^{1 + \alpha})$ for any constant $\alpha > 0$, both near-optimal with only a logarithmic factor in randomness complexity and an additional $\alpha$ exponent on the sample complexity. We use known connections with randomness extractors and list-decodable codes to give applications to these objects. Specifically, we give the first extractor construction with optimal seed length up to an arbitrarily small constant factor above 1, when the min-entropy $k = \beta n$ for a large enough constant $\beta < 1$. Finally, we generalize the definition of averaging sampler to any normed vector space.