[论文解读] Balancing Covariates in Randomized Experiments with the Gram-Schmidt Walk Design

Venkat, Prayaag|arXiv (Cornell University)|Nov 8, 2019

Advanced Causal Inference Techniques参考文献 42被引用 13

一句话总结

本文提出了Gram-Schmidt Walk（GSW）设计，一种新颖的实验设计方法，通过可调参数在协变量平衡与稳健性之间取得平衡。该方法在渐近意义上实现近乎完美的协变量平衡，同时保持有限样本下的稳健性，从而实现保守置信区间的渐近有效推断。

ABSTRACT

In this paper, we initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices: given an input matrix A ∈ ℝ^{m × n}, output a value that is a lower bound on disc(A) = min_{x ∈ {± 1}ⁿ} ‖Ax‖_∞ for every A, but is close to the typical value of disc(A) with high probability over the choice of a random A. This problem is important because of its connections to conjecturally-hard average-case problems such as negatively-spiked PCA [Afonso S. Bandeira et al., 2020], the number-balancing problem [Gamarnik and Kızıldağ, 2021] and refuting random constraint satisfaction problems [Prasad Raghavendra et al., 2017]. We give the first polynomial-time algorithms with non-trivial guarantees for two main settings. First, when the entries of A are i.i.d. standard Gaussians, it is known that disc(A) = Θ (√n2^{-n/m}) with high probability [Karthekeyan Chandrasekaran and Santosh S. Vempala, 2014; Aubin et al., 2019; Paxton Turner et al., 2020] and that super-constant levels of the Sum-of-Squares SDP hierarchy fail to certify anything better than disc(A) ≥ 0 when m < n - o(n) [Mrinalkanti Ghosh et al., 2020]. In contrast, our algorithm certifies that disc(A) ≥ exp(-O(n²/m)) with high probability. As an application, this formally refutes a conjecture of Bandeira, Kunisky, and Wein [Afonso S. Bandeira et al., 2020] on the computational hardness of the detection problem in the negatively-spiked Wishart model. Second, we consider the integer partitioning problem: given n uniformly random b-bit integers a₁, …, a_n, certify the non-existence of a perfect partition, i.e. certify that disc(A) ≥ 1 for A = (a₁, …, a_n). Under the scaling b = α n, it is known that the probability of the existence of a perfect partition undergoes a phase transition from 1 to 0 at α = 1 [Christian Borgs et al., 2001]; our algorithm certifies the non-existence of perfect partitions for some α = O(n). We also give efficient non-deterministic algorithms with significantly improved guarantees, raising the possibility that the landscape of these certification problems closely resembles that of e.g. the problem of refuting random 3SAT formulas in the unsatisfiable regime. Our algorithms involve a reduction to the Shortest Vector Problem and employ the Lenstra-Lenstra-Lovász algorithm.

研究动机与目标

• 解决实验设计中协变量平衡与稳健性之间的根本权衡问题。
• 通过最坏情况均方误差形式化稳健性，通过协变量的线性函数形式化平衡。
• 开发一种设计，使研究者可通过单一参数调节平衡与稳健性之间的权衡。
• 确保渐近正态性，并提供保守的方差估计量以实现有效推断。
• 证明该设计在大样本中可摆脱平衡-稳健性权衡，实现完美平衡的同时稳健性损失逐渐减小。

提出的方法

• 将设计表述为分布差异问题，将实验设计转化为算法差异。
• 将Gram–Schmidt Walk算法（Bansal等，2019）适配用于生成随机处理分配分布。
• 使用稳健性参数ϕ来限制Horvitz–Thompson估计量的最坏情况均方误差。
• 确保在该设计下所有协变量的线性函数同时达到平衡，其界由岭回归损失导出。
• 利用差异理论推导有限样本下均方误差和抽样分布尾部的界。
• 提供保守且一致的方差估计量，以实现渐近有效的置信区间。

实验结果

研究问题

• RQ1 实验设计中的平衡-稳健性权衡能否被正式量化并加以调控？
• RQ2 是否存在一种单一设计，可在多种协变量配置下同时实现高协变量平衡与充分稳健性？
• RQ3 在GSW设计下，Horvitz–Thompson估计量的渐近行为如何？
• RQ4 GSW设计是否能实现具有保守置信区间的渐近有效推断？
• RQ5 在平衡性、稳健性及均方误差方面，GSW设计与重抽样设计和配对设计相比表现如何？

主要发现

• Gram–Schmidt Walk设计将Horvitz–Thompson估计量的均方误差限制在潜在结果对协变量进行隐式岭回归的损失范围内。
• 在大样本中，GSW设计渐近地实现了所有协变量线性函数的完美平衡，从而有效摆脱了平衡-稳健性权衡。
• 当n = 2960时，ϕ = 0.01的GSW设计的平衡算子范数约为重抽样设计的1/57，配对设计的1/41。
• 在ϕ = 0.01的GSW设计下，结果D的均方误差比配对设计小超过31倍。
• GSW设计（ϕ = 0.01）的主要置信区间宽度比伯努利分布和完全随机化设计窄约10–14%，且当n ≥ 296时，覆盖概率超过名义水平。
• 保守的方差估计量确保了渐近有效的置信区间；基于有限样本边界的替代区间具有更高覆盖概率，但区间宽度更大。