[Paper Review] Inequalities and tail bounds for elementary symmetric polynomial with applications

Parikshit Gopalan, Amir Yehudayoff|arXiv (Cornell University)|Feb 14, 2014

Complexity and Algorithms in Graphs8 references3 citations

TL;DR

This paper establishes new inequalities for elementary symmetric polynomials in real variables, showing that if $ S_k $ and $ S_{k+1} $ are small relative to $ S_{k-1} $, then all higher-degree symmetric polynomials $ S_\ell $ for $ \ell > k $ are also small. This leads to tight tail bounds under $ k $-wise independence, enabling a simpler, modular analysis of pseudorandom generators and min-wise independent hash functions with improved seed-length dependence on $ n $, achieving $ O(\log \log n) $ seed length in the inverse-polynomial error regime.

ABSTRACT

We study the extent of independence needed to approximate the product of bounded random variables in expectation, a natural question that has applications in pseudorandomness and min-wise independent hashing. For random variables whose absolute value is bounded by $1$, we give an error bound of the form $σ^{Ω(k)}$ where $k$ is the amount of independence and $σ^2$ is the total variance of the sum. Previously known bounds only applied in more restricted settings, and were quanitively weaker. We use this to give a simpler and more modular analysis of a construction of min-wise independent hash functions and pseudorandom generators for combinatorial rectangles due to Gopalan et al., which also slightly improves their seed-length. Our proof relies on a new analytic inequality for the elementary symmetric polynomials $S_k(x)$ for $x \in \mathbb{R}^n$ which we believe to be of independent interest. We show that if $|S_k(x)|,|S_{k+1}(x)|$ are small relative to $|S_{k-1}(x)|$ for some $k>0$ then $|S_\ell(x)|$ is also small for all $\ell > k$. From these, we derive tail bounds for the elementary symmetric polynomials when the inputs are only $k$-wise independent.

Motivation & Objective

• To establish a new analytic inequality for elementary symmetric polynomials $ S_k(x) $ in real vectors, showing that smallness of $ S_k $ and $ S_{k+1} $ relative to $ S_{k-1} $ implies smallness of all higher-degree symmetric polynomials.
• To derive tight tail bounds for elementary symmetric polynomials when inputs are only $ k $-wise independent, improving upon prior bounds in the literature.
• To apply these bounds to simplify and improve the analysis of pseudorandom generators for combinatorial rectangles and min-wise independent hash functions.
• To reduce the seed-length of pseudorandom generators to $ O(\log \log n) $, nearly matching known lower bounds, in the inverse-polynomial error regime.
• To provide a modular and intuitive framework for analyzing constructions in pseudorandomness and hashing, particularly for the $ GMR+12 $ generator.

Proposed method

• Derive a new inequality showing that if $ |S_k(x)|, |S_{k+1}(x)| \<lt \varepsilon \cdot |S_{k-1}(x)| $, then $ |S_\ell(x)| \ll \varepsilon \cdot |S_{k-1}(x)| $ for all $ \ell > k $, using properties of elementary symmetric polynomials.
• Use the inequality to bound the $ L_2 $-norm of the $ k $-th elementary symmetric polynomial under $ k $-wise independent inputs, leveraging moment bounds and concentration inequalities.
• Apply a hybrid argument to compare the output distribution of a pseudorandom generator with the uniform distribution, bounding the statistical distance between consecutive hybrids.
• Use Holder's inequality and tail bounds to control the error introduced by rare events in the hybrid construction, ensuring error remains within $ \delta $.
• Construct a pseudorandom generator for combinatorial rectangles via a recursive composition of hash functions, where each stage uses $ k $-wise independent functions with $ k = O(\log(1/\delta)) $.
• Leverage the symmetric polynomial inequality to bound the deviation of the product of bounded random variables under limited independence, leading to a $ (c_1\sigma)^{c_2k} $ error bound.

Experimental results

Research questions

• RQ1 Under what conditions on the elementary symmetric polynomials $ S_k(x) $, $ S_{k+1}(x) $, and $ S_{k-1}(x) $ can we guarantee that all higher-degree symmetric polynomials $ S_\ell(x) $ for $ \ell > k $ are also small?
• RQ2 Can we derive tighter tail bounds for elementary symmetric polynomials when inputs are only $ k $-wise independent, rather than fully independent?
• RQ3 How can the new symmetric polynomial inequality be used to improve the analysis of pseudorandom generators for combinatorial rectangles?
• RQ4 What is the minimal seed-length required for pseudorandom generators that achieve inverse-polynomial error in the $ [0,1] $-bounded, low-variance setting?
• RQ5 Can the new inequality be extended to complex-valued inputs, and how does it compare to existing results in the literature?

Key findings

• The paper proves a new inequality: if $ |S_k(x)| $ and $ |S_{k+1}(x)| $ are small relative to $ |S_{k-1}(x)| $, then $ |S_\ell(x)| $ is also small for all $ \ell > k $, which strengthens and generalizes prior results.
• For $ k $-wise independent random variables $ X_i \in [-1,1] $ with total variance $ \sigma^2 $, the error in approximating $ \mathbb{E}[\prod X_i] $ is bounded by $ (c_1\sigma)^{c_2k} $, with absolute constants $ c_1 > 1 $, $ 0 < c_2 < 1 $.
• When $ \sigma < 1/(2c_1) $, $ k = O(\log(1/\delta)/\log(1/\sigma)) $-wise independence suffices for $ \delta $-error approximation of the product expectation.
• In the $ [0,1] $-valued, low-variance regime with $ \sigma^2 < n^{-\varepsilon} $, $ O(1) $-wise independence suffices for inverse-polynomial error, improving over previous $ O(\log n) $-wise bounds.
• The new analysis simplifies and improves the seed-length of the $ GMR+12 $ pseudorandom generator for combinatorial rectangles, reducing it to $ O(\log \log n) $, which nearly matches the $ \Omega(\log \log n) $ lower bound.
• The method provides a modular and intuitive framework for analyzing min-wise independent hashing and pseudorandom generators, with applications to $ [0,1] $-bounded, low-variance settings.