[论文解读] Approximate Model Counting: Is SAT Oracle More Powerful Than NP Oracle?

Chakraborty, Diptarka, Chakraborty, Sourav+2 more|arXiv (Cornell University)|Jan 1, 2023

SARS-CoV-2 detection and testing被引用 3

一句话总结

本文全面综述了Fano不等式及其变体在统计估计中建立与算法无关的不可能性结果的应用。它展示了如何结合互信息界和约简技术，使Fano不等式在离散与连续设置下均能导出紧致的极小极大下界，涵盖群组测试、图模型选择、稀疏回归、密度估计和凸优化等问题。

ABSTRACT

Given a Boolean formula ϕ over n variables, the problem of model counting is to compute the number of solutions of ϕ. Model counting is a fundamental problem in computer science with wide-ranging applications in domains such as quantified information leakage, probabilistic reasoning, network reliability, neural network verification, and more. Owing to the #P-hardness of the problems, Stockmeyer initiated the study of the complexity of approximate counting. Stockmeyer showed that log n calls to an NP oracle are necessary and sufficient to achieve (ε,δ) guarantees. The hashing-based framework proposed by Stockmeyer has been very influential in designing practical counters over the past decade, wherein the SAT solver substitutes the NP oracle calls in practice. It is well known that an NP oracle does not fully capture the behavior of SAT solvers, as SAT solvers are also designed to provide satisfying assignments when a formula is satisfiable, without additional overhead. Accordingly, the notion of SAT oracle has been proposed to capture the behavior of SAT solver wherein given a Boolean formula, an SAT oracle returns a satisfying assignment if the formula is satisfiable or returns unsatisfiable otherwise. Since the practical state-of-the-art approximate counting techniques use SAT solvers, a natural question is whether an SAT oracle is more powerful than an NP oracle in the context of approximate model counting. The primary contribution of this work is to study the relative power of the NP oracle and SAT oracle in the context of approximate model counting. The previous techniques proposed in the context of an NP oracle are weak to provide strong bounds in the context of SAT oracle since, in contrast to an NP oracle that provides only one bit of information, a SAT oracle can provide n bits of information. We therefore develop a new methodology to achieve the main result: a SAT oracle is no more powerful than an NP oracle in the context of approximate model counting.

研究动机与目标

• 通过Fano不等式提供一个统一的框架，用于推导统计估计中与算法无关的不可能性结果。
• 将Fano不等式扩展至近似恢复情形，使在放宽精度要求的问题中也能获得下界。
• 通过覆盖论证将连续估计问题约简为离散假设检验，弥合离散与连续估计问题之间的鸿沟。
• 展示Fano不等式在多样化统计问题中的实用性，包括高维和噪声环境。
• 利用信息论工具，为稀疏线性回归、密度估计和凸优化等关键问题建立极小极大下界。

提出的方法

• 通过约简框架将统计估计问题转化为多假设检验。
• 应用Fano不等式以边界假设检验中的错误概率，将其与参数和观测之间的互信息联系起来。
• 利用数据处理不等式和张量化方法，边界高维或i.i.d.设置下的互信息。
• 采用基于KL散度的互信息边界分析噪声或参数化模型。
• 通过覆盖论证对连续参数空间进行离散化，应用局部极小极大方法处理连续估计问题。
• 通过引入距离阈值，将Fano不等式适配于近似恢复，使在估计误差有界的情形下也能获得边界。

实验结果

研究问题

• RQ1 如何将Fano不等式扩展以处理近似恢复而非精确参数识别？
• RQ2 在高维统计模型中，Fano不等式在何种最小条件下能导出紧致的极小极大下界？
• RQ3 在多大程度上可将连续估计问题约简为离散假设检验以进行不可能性分析？
• RQ4 通过数据处理和张量化方法获得的互信息边界在多大程度上提升了基于Fano的下界紧致性？
• RQ5 在哪些场景下——如稀疏回归或凸优化——Fano不等式能导出非渐近且信息论最优的下界？

主要发现

• 结合近似恢复的Fano不等式导出一个下界，其涉及在给定距离内参数集合最大大小的对数，从而在放宽的估计场景中获得边界。
• 对于非自适应设计的群组测试，本文推导出精确恢复的极小极大下界为k log(p/k)阶，与已知可实现结果一致。
• 在稀疏线性回归中，当噪声服从次高斯分布时，极小极大风险至少为s log(p/s)阶，其中s为p维空间中的s稀疏向量。
• 在Hölder类上的密度估计中，极小极大风险的下界由平滑度和维度决定，与已知的非渐近结果一致。
• 在具有噪声预言机访问的凸优化中，本文建立了达到ϵ-最优性所需查询次数的下界，表明对于M个候选函数，Ω(log M)次查询是必要的。
• 约简框架可将连续极小极大估计问题转化为离散假设检验，且在许多情况下，所得下界紧致至常数因子。