Negative control is a strategy for learning the causal relationship between treatment and outcome in the presence of unmeasured confounding. The treatment effect can nonetheless be identified if two auxiliary variables are available: a negative control treatment (which has no effect on the actual outcome), and a negative control outcome (which is not affected by the actual treatment). These auxiliary variables can also be viewed as proxies for a traditional set of control variables, and they bear resemblance to instrumental variables. I propose a family of algorithms based on kernel ridge regression for learning nonparametric treatment effects with negative controls. Examples include dose response curves, dose response curves with distribution shift, and heterogeneous treatment effects. Data may be discrete or continuous, and low, high, or infinite dimensional. I prove uniform consistency and provide finite sample rates of convergence. I estimate the dose response curve of cigarette smoking on infant birth weight adjusting for unobserved confounding due to household income, using a data set of singleton births in the state of Pennsylvania between 1989 and 1991.

翻译：消极控制是一种战略,用于在未测到的混乱情况下了解治疗与结果之间的因果关系。但是,如果存在两个辅助变量,治疗效果是可以确定的:负控制治疗(对实际结果没有影响)和负控制结果(对实际治疗没有影响),这些辅助变量也可以被视为传统一系列控制变量的替代物,它们与工具变量相似。我提议以内核脊回归为基础的一套算法,用于学习带有负控制的非对称治疗效果,例如剂量反应曲线、分布变化的剂量反应曲线和多种治疗效果。数据可以是离散的或连续的或连续的,低的、高的或无限的。我证明一致性,并提供有限的趋同率样本。我用1989年至1991年期间宾夕法尼亚州单吨出生的一组数据估算了婴儿出生体重吸烟的剂量反应曲线,根据未观察到的家庭收入进行调整。