[论文解读] Anti-concentration for polynomials of independent random variables

Alman, Josh, Hoi H. Nguyen+1 more|arXiv (Cornell University)|Jul 3, 2015

Complexity and Algorithms in Graphs参考文献 22被引用 10

一句话总结

本文为独立随机变量的d次全多项式建立了近乎最优的反集中不等式，扩展了经典的Littlewood-Offord结果。它引入了一种基于秩的框架，并证明对于Rademacher、p-偏置以及一般独立变量，该多项式落入任意长度为1的区间的概率被有界，其衰减率约为r^{-1/(4d+1)}，其中r为多项式的秩，衰减率中包含对数和指数因子。该结果解决了复杂性理论中的一个长期挑战，并为随机图和布尔函数逼近提供了紧致界。

ABSTRACT

Probabilistic polynomials over commutative rings offer a powerful way of representing Boolean functions. Although many degree lower bounds for such representations have been proved, sparsity lower bounds (counting the number of monomials in the polynomials) have not been so common. Sparsity upper bounds are of great interest for potential algorithmic applications, since sparse probabilistic polynomials are the key technical tool behind the best known algorithms for many core problems, including dense All-Pairs Shortest Paths, and the existence of sparser polynomials would lead to breakthrough algorithms for these problems. In this paper, we prove several strong lower bounds on the sparsity of probabilistic and approximate polynomials computing Boolean functions when 0 means "false". Our main result is that the AND of n ORs of c log n variables requires probabilistic polynomials (over any commutative ring which isn't too large) of sparsity n^Ω(log c) to achieve even 1/4 error. The lower bound is tight, and it rules out a large class of polynomial-method approaches for refuting the APSP and SETH conjectures via matrix multiplication. Our other results include: - Every probabilistic polynomial (over a commutative ring) for the disjointness function on two n-bit vectors requires exponential sparsity in order to achieve exponentially low error. - A generic lower bound that any function requiring probabilistic polynomials of degree d must require probabilistic polynomials of sparsity Ω(2^d). - Building on earlier work, we consider the probabilistic rank of Boolean functions which generalizes the notion of sparsity for probabilistic polynomials, and prove separations of probabilistic rank and probabilistic sparsity. Some of our results and lemmas are basis independent. For example, over any basis {a,b} for true and false where a ≠ b, and any commutative ring R, the AND function on n variables has no probabilistic R-polynomial with 2^o(n) sparsity, o(n) degree, and 1/2^o(n) error simultaneously. This AND lower bound is our main technical lemma used in the above lower bounds.

研究动机与目标

• 将经典的Littlewood-Offord反集中不等式推广至任意次数的独立随机变量的高阶多项式。
• 解决复杂性理论中关于计算奇偶函数下界的长期开放问题。
• 为随机图中固定图的副本数推导出紧致的反集中界。
• 统一并改进先前针对一般分布（包括Rademacher、p-偏置以及非同分布）多项式反集中结果。
• 提供一个框架，其衰减速率几乎达到固定d时猜想的最优衰减速率r^{-1/2}，仅相差次多项式因子。

提出的方法

• 将多项式的秩定义为系数绝对值至少为1的互不相交d元组的最大数量。
• 使用条件化与约化技术，通过辅助伯努利变量将p-偏置和一般分布转化为类似Rademacher的设定。
• 应用一种类似Chernoff的界，证明以高概率，约化后的多项式其秩至少为原秩乘以与分布相关的因子。
• 以Rademacher变量的主结果（定理1.6）作为基础情形，通过测度变换与条件化将结果推广至p-偏置和一般分布。
• 通过迭代条件化与多项式限制技术证明复杂性理论中的下界，将问题约化为有界受限多项式逼近OR函数的概率。
• 递归应用基于秩的反集中界，当多项式次数相对于相关变量数过小时，导出矛盾。

实验结果

研究问题

• RQ1 能否将Littlewood-Offord对线性形式的反集中界推广至任意次数的多项式？
• RQ2 对于d次多项式，其小球概率P(P ∈ I)在秩r下的最优衰减速率为何？
• RQ3 能否利用反集中不等式在复杂性理论中建立计算奇偶函数的近乎最优下界？
• RQ4 多项式在p-偏置及一般非i.i.d.分布下的反集中行为如何变化？
• RQ5 在给定多项式秩r的条件下，d次多项式取值落在长度为1的区间内的概率的最紧可能界为何？

主要发现

• 对于Rademacher变量，本文证明对任意长度为1的区间I，有P(P ∈ I) ≤ min{B d^{4/3} √log r / r^{1/(4d+1)}, exp(B d^2 (log log r)^2) / √r}，其中r为多项式的秩。
• 该界近乎最优：对于固定d，其衰减速率与猜想的r^{-1/2}一致，仅相差次多项式因子exp(B d^2 (log log r)^2)。
• 对于p-偏置分布，界为P(P ∈ I) ≤ min{B d^{4/3} (log ˜r)^{1/2} / ˜r^{1/(4d+1)}, exp(B d^2 (log log ˜r)^2) / √˜r}，其中˜r = 2^d α^d r且α = min{p, 1−p}。
• 对于满足正则性条件的一般独立变量，界为P(P ∈ I) ≤ min{B d^{4/3} (log ˜r)^{1/2} / ˜r^{1/(4d+1)}, exp(B d^2 (log log ˜r)^2) / √˜r}，其中˜r = (pϵ)^d r。
• 结果表明，对于稀疏多项式（r = Θ(n)），小球概率为O(n^{-1/2 + o(1)}），优于先前O(n^{-1/2 + c/2d})的界。
• 本文证明在特定分布下，ϵ-逼近OR函数所需的最小次数为Ω((log log n)/(log log log n))，解决了Razborov与Viola提出的一个挑战。