# Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and
Instruments

Rahul Singh
MIT Economics  Rahul Singh is a PhD Candidate in Economics and Statistics, MIT
Department of Economics, Cambridge MA 02143 (e-mail: rahul.singh@mit.edu). I
am particularly grateful to Olivia Foley for sharing medical expertise that is
critical for the empirical application. I thank Alberto Abadie, Ben Deaner,
Anna Mikusheva, Whitney Newey, and Vasilis Syrgkanis for helpful comments. I
thank Douglas Almond, Kenneth Chay, and David Lee for building the data set,
as well as Matias Cattaneo for sharing it with their permission. I am grateful
to the Jerry A Hausman Graduate Dissertation Fellowship for financial support.

(Original draft: December 18, 2020. This draft: September 1, 2022.)

###### Abstract

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.

Keywords: potential outcome, reproducing kernel Hilbert space, dose response

##  1 Introduction

Selection on observables is the popular assumption in causal inference that
the assignment of treatment $D$ is as good as random after conditioning on
covariates $X$. It is a strong causal assumption which is often violated even
in laboratory settings. Negative controls, widely used in laboratory science,
guard against unobserved confounding. The idea is to check for spurious
relationships that would only be nonzero in the presence of an unobserved
confounder $U$–an approach sometimes called falsification or specificity
testing. Consider two auxiliary variables: a negative control treatment $Z$
(which a priori has no effect on the actual outcome $Y$), and a negative
control outcome $W$ (which a priori is not affected by the actual treatment
$D$). [Miao and Tchetgen Tchetgen, 2018] and [Deaner, 2018] carefully
formalize a learning problem in which negative controls $(Z,W)$ can not only
check for the presence of unobserved confounding $U$, but also recover the
causal relationship of interest.

As a concrete example, consider the empirical strategy of [Lousdal et al.,
2020]. The goal is to measure the effect of mammography screening $D$ on death
from breast cancer $Y$. The set of covariates $X$ includes marriage status,
number of children, age at first birth, years of education, annual income, and
hormone drug use. The authors used negative controls to document that, even
after taking into account the covariates $X$, unobserved confounding $U$
drives spurious correlations. Specifically, dental care participation $Z$
decreases the likelihood of death from breast cancer $Y$ in the dataset.
Mammography screening $D$ decreases the likelihood of death from other causes
$W$ in the dataset. The authors conclude that unobserved confounding
contaminates treatment effect estimation in this setting.

In the present work, I propose a family of nonparametric algorithms based on
kernel ridge regression that use negative controls to not only detect but also
adjust for unobserved confounding. I consider treatment effects of the
population, of subpopulations, and of alternative populations with alternative
covariate distributions. Moreover, I allow for treatments, covariates, and
negative controls that may be discrete or continuous, and low, high, or
infinite dimensional. Due to the intuitive nature of negative controls, I use
such terminology throughout the paper. In recent work, [Tchetgen Tchetgen et
al., 2020] refer to negative controls as proxy variables in order to emphasize
that they may arise in not only experimental but also observational settings.
Due to the formal resemblance between negative controls and instrumental
variables, the new statistical results I provide also apply to nonparametric
instrumental variable regression (NPIV). Altogether, I provide conceptual,
algorithmic, and statistical contributions.

Conceptual. I unify a variety of learning problems with unobserved confounding
into one general nonparametric learning problem. In semiparametric causal
inference, treatment $D$ is restricted to be binary. I consider nonparametric
causal inference, allowing the treatment $D$ to be binary, discrete, or
continuous. It appears that this is the first work on dose response curves and
heterogeneous treatment effects using negative controls. I provide a template
for future epidemiology research to estimate dose response curves and
heterogeneous treatment effects from medical records despite unobserved
confounding.

Algorithmic. I propose a family of novel estimators with closed form solutions
that are straightforward to implement by matrix operations. To do so, I assume
that the true causal relationship is a function in a reproducing kernel
Hilbert space (RKHS), which is a popular nonparametric setting in machine
learning. The hyperparameters are ridge regression penalties and kernel
hyperparameters. For the former, I derive the closed form solution for leave-
one-out cross validation. The latter have well known heuristics. I evaluate
the estimators in simulations against alternative estimators that ignore
unobserved confounding.

Statistical. I prove uniform ($\sup$ norm) consistency with finite sample
rates. A uniform guarantee encodes caution about worst case scenarios when
informing policy decisions. The finite sample rates of convergence do not
directly depend on the data dimension but rather the smoothness of the true
causal relationship. An important intermediate result is finite sample
analysis of NPIV in $\sup$ norm. Of independent interest, I relate assumptions
required in ill posed inverse problems–existence and completeness–to the RKHS
setting. This characterization appears to be absent from previous work on NPIV
in the RKHS.

To illustrate how the proposed estimators are useful, I conduct a case study.
Estimating the effect of cigarette smoking on infant birth weight is
challenging for several reasons. First, pregnant women are classified as a
vulnerable population, so they are typically excluded from clinical trials;
observational data are the only option. Second, pregnancy induces many
physiological changes, so medical knowledge predicts different dose response
curves for women who are pregnant compared to women who are not pregnant.
Third, medical records exclude an unobserved confounder known to be crucial
for maternal-fetal health: household income. I argue that medical records
include variables that satisfy the properties of negative controls for
unobserved income, and discuss what issues may arise if there are additional
unobserved confounders. I provide preliminary results and outline directions
for future work on this important topic.

The structure of the paper is as follows. Section 2 describes related work.
Section 3 formalizes the learning problem. Section 4 proposes the new
algorithms. Section 5 proves uniform consistency. Section 6 conducts
simulation experiments and estimates the dose response curve of cigarette
smoking on infant birth weight, adjusting for unobserved confounding due to
household income. Section 7 concludes.

##  2 Related work

I view dose response curves and heterogeneous treatment effects as
reweightings of a structural function defined by an ill posed inverse problem.
As such, I extend the partial means framework [Newey, 1994]. Existing work on
partial means considers consumer surplus [Newey, 1994] and certain causal
parameters [Singh et al., 2020] to be reweightings of a regression function.
By contrast, I consider causal parameters that are reweightings of a
structural function. My uniform analysis therefore generalizes uniform
analysis in previous work. To express causal parameters in this way, I
generalize identification theorems for treatment effects that use negative
controls [Miao et al., 2018, Miao and Tchetgen Tchetgen, 2018, Deaner, 2018,
Tchetgen Tchetgen et al., 2020].

Early work on negative controls emphasized their role in detection of
unobserved confounding. As early as the 1950s, epidemiologists proposed the
principle of causal specificity as a diagnostic tool [Berkson, 1958,
Yerushalmy and Palmer, 1959, Hill, 1965]. Subsequent work formalized these
concepts [Rosenbaum, 1989, Weiss, 2002, Lipsitch et al., 2010]. A more recent
literature emphasizes the role of negative controls in adjustment for
unobserved confounding. Many papers eliminate the bias from unobserved
confounding by imposing additional structure: linearity and normality [Gagnon-
Bartsch and Speed, 2012, Wang et al., 2017]; joint normality [Kuroki and
Pearl, 2014]; rank preservation of individual potential outcomes [Tchetgen
Tchetgen, 2014]; or monotonicity of confounding effects [Sofer et al., 2016].
I generalize identification results that relax such additional structure.

In econometrics, closely related strategies adjust for unobserved confounding
in dynamic settings: difference-in-difference [Card, 1990, Meyer, 1995,
Abadie, 2005], and panel proxy control [Deaner, 2018]. Traditional difference-
in-difference analysis requires strong assumptions such as linearity and
additive separability of confounding. [Athey and Imbens, 2006] present a more
general approach, called changes-in-changes, articulated in terms of a
nonseparable, nonlinear structural model. A key assumption is monotonicity of
confounding effects. Importantly, the model I present allows nonlinearity
without requiring monotonicity of confounding effects. The panel proxy control
approach is also articulated in terms of a nonseparable, nonlinear structural
model, and its static special case closely resembles the negative control
model. [Deaner, 2018] presents a series estimator as well as innovative
strategies to handle ill posedness and completeness for both static and
dynamic settings. It is straightforward to use the techniques developed in
this paper to derive an RKHS estimator for the panel proxy setting. See [Sofer
et al., 2016] and [Deaner, 2018] for explicit comparisons of negative control
with difference-in-difference and panel proxy control, respectively.

I contribute to a growing literature that adapts RKHS methods to treatment
effect estimation. [Nie and Wager, 2021] propose an RKHS estimator of
heterogeneous treatment effects under selection on observables and prove mean
square error rates. I pursue a more general definition of heterogeneous
treatment effects conditional on some interpretable subvector $V\subset X$
[Abrevaya et al., 2015] and allow for unobserved confounding. [Singh et al.,
2019] present an RKHS approach for nonparametric instrumental variable
regression and prove mean square error rates. [Singh et al., 2020] present an
RKHS approach for treatment effects identified by selection on observables and
prove uniform rates. I unify the RKHS constructions in both works in order to
handle both ill posedness and reweighting. My work is complementary in that I
consider a new causal setting. My uniform analysis applies to not only
negative control treatment effects but also nonparametric instrumental
variable regression, providing alternative results under fewer assumptions
than [Singh et al., 2019]. I build on fundamental statistical contributions
from [Smale and Zhou, 2005, Smale and Zhou, 2007].

The original draft of this paper was circulated on arXiv in 2020, and it was
the first to bring RKHS analysis to the negative control setting.
Subsequently, several works have proposed alternative RKHS estimators. [Kallus
et al., 2021, Ghassami et al., 2021] study the semiparametric problem rather
than the nonparametric problem considered here. Both works propose doubly
robust estimators that combine nuisance functions estimated by a minimax
procedure. [Chernozhukov et al., 2021] provide abstract conditions to
translate learning theory rates into semiparametric inference when treatment
is binary. I summarize the connection to semiparametrics in Appendix A. The
emphasis of this work is uniform consistency for the nonparametric case.
Uniform nonparametric inference remains an open question for future research.
In follow up work, [Mastouri et al., 2021] focus on only the dose response
curve and analyze excess risk for alternative estimators.

##  3 Learning problem

###  3.1 Treatment effects

Treatment effects are statements about counterfactual outcomes given
hypothetical interventions. Though we observe outcome $Y$, we seek to infer
means of counterfactual outcomes $\\{Y^{(d)}\\}$, where $Y^{(d)}$ is the
potential outcome given the hypothetical intervention $D=d$. The treatment
effect literature aims to measure a rich variety of treatment effects, which I
quote from [Singh et al., 2020, Definition 3.1].

######  Definition 1 (Treatment effects).

I define the following treatment effects.

  1. 1.

Dose response: $\theta_{0}^{ATE}(d):=\mathbb{E}[Y^{(d)}]$ is the
counterfactual mean outcome given intervention $D=d$ for the entire
population.

  2. 2.

Dose response with distribution shift:
$\theta_{0}^{DS}(d,\tilde{\mathbb{P}}):=\mathbb{E}_{\tilde{\mathbb{P}}}[Y^{(d)}]$
is the counterfactual mean outcome given intervention $D=d$ for an alternative
population with data distribution $\tilde{\mathbb{P}}$ (elaborated in
Assumption 3).

  3. 3.

Conditional dose response:
$\theta_{0}^{ATT}(d,d^{\prime}):=\mathbb{E}[Y^{(d^{\prime})}|D=d]$ is the
counterfactual mean outcome given intervention $D=d^{\prime}$ for the
subpopulation who actually received treatment $D=d$.

  4. 4.

Heterogeneous treatment effect:
$\theta_{0}^{CATE}(d,v):=\mathbb{E}[Y^{(d)}|V=v]$ is the counterfactual mean
outcome given intervention $D=d$ for the subpopulation with covariate value
$V=v$.

The superscipt of each nonparametic treatment effect corresponds to its
semiparametric analogue. If treatment is binary, then average treatment effect
(ATE) is $\mathbb{E}[Y^{(1)}-Y^{(0)}]$; average treatment effect with
distribution shift (DS) is $\mathbb{E}_{\tilde{\mathbb{P}}}[Y^{(1)}-Y^{(0)}]$;
average treatment on the treated (ATT) is $\mathbb{E}[Y^{(1)}-Y^{(0)}|D=1]$;
and conditional average treatment effect (CATE) is
$\mathbb{E}[Y^{(1)}-Y^{(0)}|V=v]$. Rather than differences of potential
outcomes indexed by binary treatment, I analyze potential outcomes indexed by
discrete or continuous treatment.

$\theta_{0}^{ATE}(d)$ has many names: dose response curve, continuous
treatment effect, and average structural function. If treatment is binary,
then $\theta_{0}^{ATE}(d)$ is a vector in $\mathbb{R}^{2}$ and the learning
problem is semiparametric. If the treatment is discrete or continuous, then
$\theta_{0}^{ATE}(d)$ is a function and the learning problem is nonparametric.
$\theta_{0}^{DS}(d,\tilde{\mathbb{P}})$ is a closely related variant that
handles the scenario where the covariate distribution has shifted. This
variant may be called distribution shift, covariate shift, policy effect, or
transfer learning.

Both $\theta_{0}^{ATT}(d,d^{\prime})$ and $\theta_{0}^{CATE}(d,v)$ involve
conditioning on a particular subpopulation. If treatment is binary, then
$\theta_{0}^{ATT}(d,d^{\prime})$ is a matrix in $\mathbb{R}^{2\times 2}$ and
the learning problem is semiparametric. If the treatment is discrete or
continuous, then $\theta_{0}^{ATT}(d,d^{\prime})$ is a surface and the
learning problem is nonparametric. Likewise for $\theta_{0}^{CATE}(d,v)$.
$\theta_{0}^{ATT}(d,d^{\prime})$ is called the conditional dose response, and
$\theta_{0}^{CATE}(d,v)$ is called the heterogeneous treatment effect. The
possibility for $D$ to be discrete or continuous and for $V$ to be a
particular covariate, rather than the full set of covariates required for
identification, is more general than the typical heterogeneous treatment
effect [Nie and Wager, 2021]. For $\theta_{0}^{CATE}$, I slightly abuse
notation by denoting the complete set of identifying covariates as $(V,X)$.

###  3.2 Negative control identification

In pioneering work, [Tchetgen Tchetgen et al., 2020] propose a potential
outcome model in which treatment effects can be measured from outcomes $Y$,
treatments $D$, and covariates $(V,X)$ despite unobserved confounding $U$. The
technique involves two auxiliary variables: negative control treatment $Z$,
and negative control outcome $W$. In this model, potential outcomes
$\\{Y^{(d,z)}\\}$ and potential negative control outcomes $\\{W^{(d,z)}\\}$
are initially indexed by both the treatment value $D=d$ and the negative
control treatment value $Z=z$. The identification strategy requires prior
knowledge of how the unobserved confounder, which may be a vector, relates to
the observed variables. The validity of negative controls as articulated in
Assumptions 1 and 2 is relative to a conjectured unobserved confounder.

######  Assumption 1 (Negative controls).

Assume

  1. 1.

No interference: if $D=d$ and $Z=z$ then $Y=Y^{(d,z)}$ and $W=W^{(d,z)}$.

  2. 2.

Latent exchangeability:
$\\{Y^{(d,z)}\\},\\{W^{(d,z)}\\}\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}D,Z|U,X$.

  3. 3.

Overlap: if $f(u,x)>0$ then $f(d,z|u,x)>0$, where $f(u,x)$ and $f(d,z|u,x)$
are densities.

  4. 4.

Negative control treatment and outcome: $Y^{(d,z)}=Y^{(d)}$ and $W^{(d,z)}=W$.

For $\theta_{0}^{CATE}$, replace $X$ with $(V,X)$.

No interference is also called consistency or the stable unit treatment value
assumption in causal inference, and it rules out network effects. Latent
exchangeability states that conditional on covariates $X$ and unobserved
confounder $U$, treatment assignment and negative control treatment assignment
are as good as random. Latent exchangeability relaxes the classic assumption
of conditional exchangeability in which $U=\varnothing$, i.e. in which there
is no unobserved confounder. Overlap ensures that there is no confounder-
covariate stratum such that treatment and negative control treatment have a
restricted support; for any stratum, any value of treatment or negative
control treatment can occur.

In a graphical causal model, there could be many sets of observed variables
that could serve as covariates $X$ and many sets of unobserved variables that
could serve as the unobserved confounder $U$ based on these initial criteria.
The subsequent criteria provide guidance in how to choose $(X,U)$. We will see
that the set of covariates $X$ should be chosen to block as much unobserved
confounding as possible, because the variation in unobserved confounding that
remains must be tied to variation in negative controls.

The negative control treatment condition imposes that the negative control
treatment $Z$ only affects the outcome $Y$ via actual treatment $D$. It is
identical to the exclusion restriction assumed for instrumental variables
[Angrist et al., 1996]. The negative control outcome condition imposes that
the negative control outcome $W$ is unaffected by the treatment $D$ and
negative control treatment $Z$. It is an even stronger exclusion restriction.
Altogether, Assumption 1 formalizes the intuition that if there are spurious
correlations then there is unobserved confounding. It also implies
$Y\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}Z|D,U,X$ and
$W\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}D,Z|U,X$, which
are weaker conditions used in the identification argument [Miao et al., 2018].

$X$$W$$Z$$D$$Y$$U$

Figure 1: Negative control DAG

Figure 1 visualizes a representative directed acyclic graph (DAG). Despite
access to covariates $X$, unobserved confounding $U$ has unblocked paths to
treatment $D$ and outcome $Y$. In the DAG, we also see the proxy
interpretation of this learning problem. Covariates $X$, negative control
treatment $Z$, and negative control outcome $W$ are all imperfect proxies for
a set of control variables that would block unobserved confounding. Covariates
$X$ are proxies that induce treatment and outcome; negative control treatment
$Z$ is a proxy that induces treatment only; and negative control outcome $W$
is a proxy that induces outcome only.

Next, I quote a high level technical condition, which I will later verify for
the RKHS setting. Define the regression
$\gamma_{0}(d,x,z):=\mathbb{E}[Y|D=d,X=x,Z=z]$.

######  Assumption 2 (Confounding bridge).

Assume

  1. 1.

Existence: there exists a solution $h_{0}$ to the operator equation

$\gamma_{0}(d,x,z)=\mathbb{E}[h(D,X,W)|D=d,X=x,Z=z].$

  2. 2.

Completeness: for any function $f$,

$\mathbb{E}[f(U)|D=d,X=x,Z=z]=0\quad\forall(d,x,z)\iff f(U)=0.$

I call $h_{0}$ the confounding bridge, following [Miao and Tchetgen Tchetgen,
2018]. Here, we see the formal resemblance to the nonparametric instrumental
variable regression problem (NPIV) [Newey and Powell, 2003]. In the language
of NPIV, the LHS $\gamma_{0}(d,x,z)$ is the reduced form, while the RHS is a
composition of a stage 1 conditional expectation operator and stage 2
structural function $h_{0}$. In the language of functional analysis, the
operator equation is a Fredholm integral equation of the first kind. Solving
this operator equation for $h_{0}$ involves inverting a linear operator with
infinite dimensional domain; it is an ill posed problem. Indeed, existence
will require conditions on the spectrum of the conditional expectation
operator formalized in Appendix B. Completeness is a technical condition from
the NPIV literature. Taking $f=f_{1}-f_{2}$, it states that the observed
variables $(D,X,Z)$ have sufficiently rich variation in the sense that
different functions of unobserved confounding $f_{1}(U)$ and $f_{2}(U)$ lead
to different projections onto $(D,X,Z)$; if they lead to the same projections,
then $f_{1}=f_{2}$.

There is a subtle yet fundamental connection between the existence of the
confounding bridge and the relevance of negative controls.

######  Proposition 1 (Relevance).

Suppose Assumption 1 holds and $\gamma_{0}$ varies in $z$, i.e. there exist
$(z,z^{\prime})$ such that $\gamma_{0}(d,x,z)\neq\gamma_{0}(d,x,z^{\prime})$.

  1. 1.

If the negative control treatment is irrelevant to the unobserved confounder
in the sense that
$Z\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}U|D,X$, then no
confounding bridge exists.

  2. 2.

If the negative control outcome is irrelevant to the unobserved confounder in
the sense that
$W\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}U|X$, then no
confounding bridge exists.

See Appendix B for the proof, as well as further discussion of the cases in
which $(U,Z,W)$ are discrete or continuous. Assumption 2 formalizes the
converse intuition of Assumption 1: if there is unobserved confounding, then
there are spurious correlations. In order to use $(Z,W)$ to adjust for
unobserved confounding $U$, it must be the case that $(Z,W)$ can detect $U$
well enough. In practice, an analyst must collaborate with domain experts in
order to assess (i) what are the sources of unobserved confounding, (ii)
whether the negative control exclusion restrictions hold, and (iii) whether
the negative controls are relevant. In summary, the key assumptions of
negative control identification apply to settings where there are spurious
correlations if and only if there is unobserved confounding. When domain
knowledge is insufficient to verify these assumptions, then the strategy of
negative controls is inappropriate.

To handle $\theta_{0}^{DS}$, I generalize a standard assumption in transfer
learning.

######  Assumption 3 (Distribution shift).

Assume

  1. 1.

The difference in population distributions $\mathbb{P}$ and
$\tilde{\mathbb{P}}$ is only in the marginal distribution of treatments,
negative control treatments, and covariates:

$\tilde{\mathbb{P}}(Y,W,D,X,Z)=\mathbb{P}(Y,W|D,X,Z)\tilde{\mathbb{P}}(D,X,Z).$

  2. 2.

$\tilde{\mathbb{P}}(D,X,Z)$ is absolutely continuous with respect to
$\mathbb{P}(D,X,Z)$.

######  Proposition 2 (Invariance of confounding bridge).

Under Assumptions 2 and 3, the confounding bridge $h_{0}$ remains the same
across the different populations $\mathbb{P}$ and $\tilde{\mathbb{P}}$.

See Appendix C for the proof. It appears that Assumption 3 and Proposition 2
are the first formalization of distribution shift in negative control and NPIV
settings.

In summary, I place three assumptions: availability of negative controls
(Assumption 1); existence and completeness of the confounding bridge
(Assumption 2); and invariance of the confounding bridge for transfer learning
(Assumption 3). Formally, the theorem that uses these assumptions to express
treatment effects in terms of data is known as an identification result. I
present the main identification result below, extending the powerful insights
of [Miao et al., 2018, Miao and Tchetgen Tchetgen, 2018, Deaner, 2018,
Tchetgen Tchetgen et al., 2020] to additional treatment effects beyond
$\theta_{0}^{ATE}(d)$.

######  Theorem 1 (Identification of treatment effects).

If Assumptions 1 and 2 hold then

  1. 1.

$\theta_{0}^{ATE}(d)=\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(x,w)$.

  2. 2.

If in addition Assumption 3 holds, then
$\theta_{0}^{DS}(d,\tilde{\mathbb{P}})=\int
h_{0}(d,x,w)\mathrm{d}\tilde{\mathbb{P}}(x,w)$.

  3. 3.

$\theta_{0}^{ATT}(d,d^{\prime})=\int
h_{0}(d^{\prime},x,w)\mathrm{d}\mathbb{P}(x,w|d)$.

  4. 4.

$\theta_{0}^{CATE}(d,v)=\int h_{0}(d,v,x,w)\mathrm{d}\mathbb{P}(x,w|v)$.

See Appendix C for the proof. In Theorem 1, we see how negative controls
$(Z,W)$ allow us to adjust for unobserved confounding $U$ and to thereby
recover the treatment effect of interest. Each treatment effect is a
reweighting of the confounding bridge $h_{0}$ defined in Assumption 2 with
respect to some distribution $\mathbb{Q}$. In this sense, each treatment
effect is an example of the same general nonparametric learning problem: $\int
h_{0}(d,x,w)\mathrm{d}\mathbb{Q}$, where $\mathbb{Q}$ may be an unconditional
distribution such as $\mathbb{P}(x,w)$ or a conditional distribution such as
$\mathbb{P}(x,w|d)$.

###  3.3 RKHS background

Until this point, I have placed only causal assumptions formalized in
Assumptions 1, 2, and 3. For computational and analytical tractability, I now
place additional structure on the learning problem: I assume key quantities
are elements of a reproducing kernel Hilbert space (RKHS). The RKHS is a
canonical setting in machine learning, and it is a space of smooth functions
that generalizes the Sobolev space. For a broad statistical audience, I
organize ideas from RKHS learning theory that underpin the algorithm
derivation (Section 4) and consistency guarantee (Section 5) to follow.

Kernel and feature map. I begin with basic kernel and feature map notation.
Consider the RKHS $\mathcal{H}$ which consist of functions of the form
$f:\mathcal{A}\rightarrow\mathbb{R}$, where $\mathcal{A}$ is a Polish space
(defined formally below). An RKHS $\mathcal{H}$ is characterized by its
feature map $\phi(a)$, which can be interpreted as the dictionary of basis
functions for the RKHS in the sense that, for any $f\in\mathcal{H}$,
$f(a)=\langle f,\phi(a)\rangle_{\mathcal{H}}$. The kernel
$k:\mathcal{A}\times\mathcal{A}\rightarrow\mathbb{R}$ is a positive definite,
symmetric, and continuous function such that
$k(a,a^{\prime})=\langle\phi(a),\phi(a^{\prime})\rangle_{\mathcal{H}}$; it is
the inner product of features, so it encodes the geometry of the RKHS.
Alternatively, one may define the kernel first, then define the feature map as
$\phi:\mathcal{A}\rightarrow\mathcal{H}$, $a\mapsto k(a,\cdot)$. The feature
map perspective is helpful for theory, but the kernel perspective is helpful
for practice, since $k(a,a^{\prime})$ is a scalar that can be computed.
Ultimately, I will reduce the algorithm to kernel evaluations.

Kernel mean embedding. We have seen how the feature map helps to evaluate a
function. A related object, called the kernel mean embedding, helps to take
the expectation of a function. Suppose we wish to calculate
$\mathbb{E}[f(A)]$. The idea of kernel mean embedding is to write

$\mathbb{E}[f(A)]=\int f(a)\mathrm{d}\mathbb{P}(a)=\int\langle
f,\phi(a)\rangle_{\mathcal{H}}\mathrm{d}\mathbb{P}(a)=\left\langle
f,\int\phi(a)\mathrm{d}\mathbb{P}(a)\right\rangle_{\mathcal{H}}=\langle
f,\mu\rangle_{\mathcal{H}}$

where the exchange of expectation and inner product requires weak regularity
conditions (defined formally below). The object
$\mu:=\int\phi(a)\mathrm{d}\mathbb{P}(a)$ is called the mean embedding of the
distribution $\mathbb{P}(a)$. A kernel is characteristic when the mapping
$\mathbb{P}(a)\mapsto\mu$ is injective. The geometry of the RKHS implies that,
to calculate the expectation of a function, it suffices to take the product of
the function and the mean embedding of the corresponding distribution. This
idea extends to conditional distributions over a subset of the arguments of
the function. I will use this idea extensively when deriving the algorithm.

Tensor product. What if a function is defined over multiple variables, e.g.
$f:\mathcal{A}\times\mathcal{B}\rightarrow\mathbb{R}$? A natural approach is
to define an RKHS $\mathcal{H}$ for such functions as the combination of RKHSs
$\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$ that contain
functions of the form $f_{1}:\mathcal{A}\rightarrow\mathbb{R}$ and
$f_{2}:\mathcal{B}\rightarrow\mathbb{R}$, respectively. Denote the individual
feature maps by $\phi_{\mathcal{A}}(a)$ and $\phi_{\mathcal{B}}(b)$, then
define the tensor product feature map
$\phi(a,b)=\phi_{\mathcal{A}}(a)\otimes\phi_{\mathcal{B}}(b)$ for the tensor
product RKHS $\mathcal{H}$. The tensor product is a generalization of the
outer product; formally, $[a\otimes b]c=a\langle b,c\rangle$. Then, for any
$f\in\mathcal{H}$, $f(a,b)=\langle
f,\phi_{\mathcal{A}}(a)\otimes\phi_{\mathcal{B}}(b)\rangle_{\mathcal{H}}$. It
turns out that the kernel of this RKHS is simply the product of the kernels of
the individual RKHSs:
$k(a,b;a^{\prime},b^{\prime})=k_{\mathcal{A}}(a,a^{\prime})\cdot
k_{\mathcal{B}}(b,b^{\prime})$. As such, $k(a,b;a^{\prime},b^{\prime})$ is a
scalar that can be computed. In this work, I will extensively use tensor
product constructions. For this reason, the algorithm statements will have the
symbol $\odot$ for the elementwise product of objects that contain kernel
evaluations.

RKHS for operators. So far, I have defined RKHSs for functions of one or more
variables. RKHSs also exist for operators. I denote by
$\mathcal{L}_{2}(\mathcal{H}_{\mathcal{A}},\mathcal{H}_{\mathcal{B}})$ the
space of Hilbert-Schmidt operators of the form
$E:\mathcal{H}_{\mathcal{A}}\rightarrow\mathcal{H}_{\mathcal{B}}$. It turns
out that this space is an RKHS in its own right. The operators of interest are
conditional expectation operators, which correspond to conditional mean
embeddings. For example, consider the goal of calculating
$\mathbb{E}[f(a)|B=b]$. As before, one can express

$\int f(a)\mathrm{d}\mathbb{P}(a|b)=\int\langle
f,\phi_{\mathcal{A}}(a)\rangle_{\mathcal{H}_{\mathcal{A}}}\mathrm{d}\mathbb{P}(a|b)=\left\langle
f,\int\phi_{\mathcal{A}}(a)\mathrm{d}\mathbb{P}(a|b)\right\rangle_{\mathcal{H}_{\mathcal{A}}}=\langle
f,\mu_{a}(b)\rangle_{\mathcal{H}_{\mathcal{A}}}$

where $\mu_{a}(b):=\int\phi_{\mathcal{A}}(a)\mathrm{d}\mathbb{P}(a|b)$ is
called the conditional mean embedding of the distribution $\mathbb{P}(a|b)$.
Observe that

$\mu_{a}(b)=\int\phi(a)\mathrm{d}\mathbb{P}(a|b)=[E\phi(\cdot)](b)=[E^{*}\phi(b)](\cdot)$

where $E^{*}:\mathcal{H}_{\mathcal{B}}\rightarrow\mathcal{H}_{\mathcal{A}}$ is
the adjoint of
$E:\mathcal{H}_{\mathcal{A}}\rightarrow\mathcal{H}_{\mathcal{B}}$, and both
are conditional expectation operators. Formally, the operators
$E:f(\cdot)\mapsto\mathbb{E}[f(A)|B=\cdot]$ and
$E^{*}:g(\cdot)\mapsto\mathbb{E}[g(B)|A=\cdot]$ encode the same information as
the conditional mean embedding $\mu_{a}(b)$. This relationship will facilitate
estimation and analysis.

Closed form solution. The final piece of RKHS machinery necessary for the
algorithm derivation (Section 4) is the so-called kernel trick. We have seen
how a kernel evaluation is, in the end, simply a scalar. So, if an analyst can
express an algorithm exclusively in terms of kernel evaluations, then the
algorithm can be easily computed. A virtue of kernel methods is that they tend
to have closed form solutions in terms of kernel evaluations. Conceptually, an
RKHS algorithm $\hat{f}(a)=\langle\hat{f},\phi(a)\rangle_{\mathcal{H}}$
involves a possibly nonlinear feature map $\phi(\cdot)$ applied to the data,
so such an algorithm maintains computational simplicity while allowing for
rich nonlinearity. Consider, for example, the kernel ridge regression

$\hat{f}=\operatorname*{\arg\\!\min}_{f\in\mathcal{H}}\frac{1}{n}\sum_{i=1}^{n}\\{y_{i}-\langle
f,\phi(a_{i})\rangle_{\mathcal{H}}\\}^{2}+\lambda\|f\|^{2}_{\mathcal{H}}$ (1)

with regularization hyperparameter $\lambda>0$. Its closed form solution is

$\hat{f}(a)=\langle\hat{f},\phi(a)\rangle_{\mathcal{H}}=\mathbf{Y}^{\top}(\mathbf{K}_{AA}+n\lambda\mathbf{I})^{-1}\mathbf{K}_{Aa}$
(2)

where $\mathbf{Y}\in\mathbb{R}^{n}$ is the vector of outcomes with $i$-th
entry $y_{i}$, $\mathbf{K}_{AA}\in\mathbb{R}^{n\times n}$ is the kernel matrix
with $(i,j)$-th entry $k(a_{i},a_{j})$, and $\mathbf{K}_{Aa}\in\mathbb{R}^{n}$
is the evaluation vector with $i$-th entry $k(a_{i},a)$. The algorithms I
propose generalize kernel ridge regression. Sometimes, instead of regressing
the outcome $Y$ on features $\phi(A)$, I will regress the outcome $Y$ on mean
embeddings $\mu_{a}(B)$. At other times, I will regress one collection of
features $\phi(A)$ on another collection of features $\phi(B)$. These
generalizations haves losses and closed form solutions that generalize those
of kernel ridge regression. As such, they are simple combinations of kernel
matrices and evaluation vectors despite being nonparametric.

Spectral view. The statistical guarantees of Section 5 require the spectral
view of the RKHS $\mathcal{H}$. The spectral view is more challenging, but it
is the only way to articular RKHS learning theory. Let $\mathbb{L}_{2}$ denote
the space of square integrable functions mapping from $\mathcal{A}$ to
$\mathbb{R}$ with respect to measure $\mathbb{P}$. For a fixed kernel $k$,
define the convolution operator $L:\mathbb{L}_{2}\rightarrow\mathbb{L}_{2}$,
$f\mapsto\int k(a,\cdot)f(a)\mathrm{d}\mathbb{P}(a)$. By the spectral theorem,
we can express the operator $L$ in terms of its countable eigenvalues
$\\{\eta_{j}\\}$ and eigenfunctions $\\{\varphi_{j}\\}$:
$Lf=\sum_{j=1}^{\infty}\eta_{j}\langle f,\varphi_{j}\rangle\cdot\varphi_{j}$.
Without loss of generality, $\\{\eta_{j}\\}$ is a weakly decreasing sequence
and $\\{\varphi_{j}\\}$ forms an orthonormal basis of $\mathbb{L}_{2}$. With
this spectral notation, we are ready to formalize the sense in which the RKHS
$\mathcal{H}$ is a smooth subset of $\mathbb{L}_{2}$. Since
$\\{\varphi_{j}\\}$ forms an orthonormal basis of $\mathbb{L}_{2}$, any
$f,g\in\mathbb{L}_{2}$ can be expressed as
$f=\sum_{j=1}^{\infty}f_{j}\varphi_{j}$ and
$g=\sum_{j=1}^{\infty}g_{j}\varphi_{j}$. By [Cucker and Smale, 2002, Theorem
4], $\mathbb{L}_{2}$ and the RKHS $\mathcal{H}$ can be explicitly represented
as

$\displaystyle\mathbb{L}_{2}$
$\displaystyle=\left\\{f=\sum_{j=1}^{\infty}f_{j}\varphi_{j}:\;\sum_{j=1}^{\infty}f_{j}^{2}<\infty\right\\},\quad\langle
f,g\rangle_{\mathbb{L}_{2}}=\sum_{j=1}^{\infty}f_{j}g_{j}$
$\displaystyle\mathcal{H}$
$\displaystyle=\left\\{f=\sum_{j=1}^{\infty}f_{j}\varphi_{j}:\;\sum_{j=1}^{\infty}\frac{f_{j}^{2}}{\eta_{j}}<\infty\right\\},\quad\langle
f,g\rangle_{\mathcal{H}}=\sum_{j=1}^{\infty}\frac{f_{j}g_{j}}{\eta_{j}}.$

The RKHS $\mathcal{H}$ is the subset of $\mathbb{L}_{2}$ for which higher
order terms in the series $\\{\varphi_{j}\\}$ have a smaller contribution. In
the RKHS, there is a penalty on higher order coefficients, and the magnitude
of the penalty corresponds to how small the eigenvalue is.

Main assumption. Finally, I articulate the main approximation assumption of
this paper. Formally, I assume that a statistical target $f_{0}$ satisfies

$f_{0}\in\mathcal{H}^{c}:=\left\\{f=\sum_{j=1}^{\infty}f_{j}\varphi_{j}:\;\sum_{j=1}^{\infty}\frac{f_{j}^{2}}{\eta^{c}_{j}}<\infty\right\\},\quad
c\in(1,2].$ (3)

For $c=1$, we see that $\mathcal{H}^{1}=\mathcal{H}$; I am simply assuming
that $f_{0}$ is correctly specified as an element of the RKHS. For $c>1$, I am
assuming that $f_{0}$ is in the interior of the RKHS. This assumption is
called the source condition in statistical learning theory and econometrics
[Smale and Zhou, 2007, Caponnetto and De Vito, 2007, Carrasco et al., 2007].
As we will see, a larger value of $c$ corresponds to a smoother target $f_{0}$
and a faster uniform rate. I allow $c$ to be as large as $c=2$, which is the
highest degree of smoothness to which kernel ridge estimators can adapt. In
Appendix D, I compare this main approximation assumption with alternative
approximation assumptions in the negative control and NPIV literatures.

Special case: Sobolev space. The abstract approximation condition is easy to
interpret in the context of Sobolev spaces. Denote by $\mathbb{H}_{2}^{s}$ the
Sobolev space of functions of the form $f:\mathcal{A}\rightarrow\mathbb{R}$
with $\mathcal{A}\subset\mathbb{R}^{p}$. The parameter $s$ denotes how many
derivatives of $f$ are square integrable. The Sobolev space
$\mathbb{H}_{2}^{s}$ is an RKHS if and only if $s>p/2$ [Berlinet and Thomas-
Agnan, 2011, Theorem 132]. Its kernel is known as the Matérn kernel. Suppose
we take $\mathcal{H}=\mathbb{H}_{2}^{s}$ with $s>p/2$ as the RKHS for
estimation. If the true target $f_{0}$ is in $\mathbb{H}_{2}^{s_{0}}$, then
$c=s_{0}/s$ [Fischer and Steinwart, 2020]. In the notation of (3),
$\mathbb{H}_{2}^{s_{0}}=[\mathbb{H}_{2}^{s}]^{c}$. Clearly $c>1$ means that
the target $f_{0}$ is in the interior of $\mathbb{H}_{2}^{s}$. Rates in terms
of $c$ adapt to the smoothness of $f_{0}$ and are invariant to dimension, as
long as $s_{0}>s>p/2$.

##  4 Algorithm

###  4.1 RKHS construction

I provide a new RKHS construction for negative control treatment effect
estimation, generalizing and unifying the constructions in [Singh et al.,
2019, Singh et al., 2020]. In my construction, I define RKHSs for treatment
$D$, negative controls $(Z,W)$, and covariates $(V,X)$. For example, for
treatment $D$ define the RKHS $\mathcal{H}_{\mathcal{D}}$ with feature map
$\phi_{\mathcal{D}}(d)$ and kernel
$k_{\mathcal{D}}(d,d^{\prime})=\langle\phi_{\mathcal{D}}(d),\phi_{\mathcal{D}}(d^{\prime})\rangle_{\mathcal{H}_{\mathcal{D}}}$.
Formally, $\phi_{\mathcal{D}}:\mathcal{D}\rightarrow\mathcal{H}_{\mathcal{D}}$
and $k_{\mathcal{D}}:\mathcal{D}\times\mathcal{D}\rightarrow\mathbb{R}$. To
lighten notation, I suppress subscripts when arguments are provided, e.g. I
write $\phi(d)=\phi_{\mathcal{D}}(d)$.

From these individual RKHSs, I construct a tensor product RKHS $\mathcal{H}$
for the confounding bridge $h_{0}$. For clarity of exposition, I initially
focus on the case without $V$, i.e. excluding $\theta_{0}^{CATE}$. I assume
the confounding bridge $h_{0}$ is an element of the RKHS with tensor product
feature map $\phi(d,x,w):=\phi(d)\otimes\phi(x)\otimes\phi(w)$, i.e.
$h_{0}\in\mathcal{H}:=\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{W}}$.
As before, the feature map can be interpreted as the dictionary of basis
functions since $h_{0}(d,x,w)=\langle
h_{0},\phi(d,x,w)\rangle_{\mathcal{H}}=\langle
h_{0},\phi(d)\otimes\phi(x)\otimes\phi(w)\rangle_{\mathcal{H}}$. In Appendix
B, I discuss how an analogous assumption for the regression $\gamma_{0}$
relates to Assumption 2.

This tensor product RKHS construction plays a central role in deriving simple
representations of treatment effects and hence deriving simple estimators,
under weak regularity conditions. The RKHS aspect allows for the technique of
kernel mean embedding: an analyst can reweight a function $h_{0}(d,x,w)$
according to some counterfactual distribution by reweighting its feature map
$\phi(d,x,w)$ by that distribution. The tensor product aspect ensures
separability of the different variables in the feature map; since
$\phi(d,x,w)=\phi(d)\otimes\phi(x)\otimes\phi(w)$, the reweighting can apply
to the specific variables $\phi(x)\otimes\phi(w)$. Using these properties, I
represent the causal quantities defined in Definition 1 and identified in
Theorem 1 in a more tractable form. To begin, I state the regularity
conditions.

######  Assumption 4 (RKHS regularity conditions).

Assume

  1. 1.

$k_{\mathcal{D}}$, $k_{\mathcal{X}}$, $k_{\mathcal{W}}$, and $k_{\mathcal{Z}}$
are continuous and bounded:

$\sup_{d\in\mathcal{D}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\leq\kappa_{d},\;\sup_{x\in\mathcal{X}}\|\phi(x)\|_{\mathcal{H}_{\mathcal{X}}}\leq\kappa_{x},\;\sup_{w\in\mathcal{W}}\|\phi(w)\|_{\mathcal{H}_{\mathcal{W}}}\leq\kappa_{w},\;\sup_{z\in\mathcal{Z}}\|\phi(z)\|_{\mathcal{H}_{\mathcal{Z}}}\leq\kappa_{z};$

  2. 2.

$\phi(d)$, $\phi(x)$, $\phi(w)$, and $\phi(z)$ are measurable;

  3. 3.

$k_{\mathcal{X}}$ and $k_{\mathcal{W}}$ are characteristic.

For $\theta_{0}^{CATE}$, extend the stated assumptions from $X$ to $(V,X)$.

Continuity, boundedness, and measurability are weak conditions satisfied by
commonly used kernels. The characteristic property is a regularity condition
for embedding a distribution in an RKHS, and it is satisfied by commonly used
kernels as well [Sriperumbudur et al., 2010]. Formally, $k_{\mathcal{X}}$ is
characteristic if and only if, for all Borel probability measures
$\mathbb{Q}$, the mapping $\mathbb{Q}\mapsto\int\phi(x)\mathrm{d}\mathbb{Q}$
is injective. I explain the role of the characteristic property below.

######  Theorem 2 (Representation via kernel mean embedding).

Suppose the conditions of Theorem 1 hold. Further suppose Assumption 4 holds
and $h_{0}\in\mathcal{H}$. Then

$\gamma_{0}(d,x,z)=\langle
h_{0},\phi(d)\otimes\phi(x)\otimes\mu_{w}(d,x,z)\rangle_{\mathcal{H}}\text{
where }\mu_{w}(d,x,z):=\int\phi(w)\mathrm{d}\mathbb{P}(w|d,x,z).$

Moreover

  1. 1.

$\theta_{0}^{ATE}(d)=\langle h_{0},\phi(d)\otimes\mu\rangle_{\mathcal{H}}$
where $\mu:=\int[\phi(x)\otimes\phi(w)]\mathrm{d}\mathbb{P}(x,w)$;

  2. 2.

$\theta_{0}^{DS}(d,\tilde{\mathbb{P}})=\langle
h_{0},\phi(d)\otimes\nu\rangle_{\mathcal{H}}$ where
$\nu:=\int[\phi(x)\otimes\phi(w)]\mathrm{d}\tilde{\mathbb{P}}(x,w)$;

  3. 3.

$\theta_{0}^{ATT}(d,d^{\prime})=\langle
h_{0},\phi(d^{\prime})\otimes\mu(d)\rangle_{\mathcal{H}}$ where
$\mu(d):=\int[\phi(x)\otimes\phi(w)]\mathrm{d}\mathbb{P}(x,w|d)$;

  4. 4.

$\theta_{0}^{CATE}(d,v)=\langle
h_{0},\phi(d)\otimes\phi(v)\otimes\mu(v)\rangle_{\mathcal{H}}$ where
$\mu(v):=\int[\phi(x)\otimes\phi(w)]\mathrm{d}\mathbb{P}(x,w|v)$.

I present the proof in Appendix E. Whereas the expressions in Theorem 1 are
reweightings of the confounding bridge $h_{0}$, the expressions in Theorem 2
are inner products of $h_{0}$. The quantity $\mu_{w}(d,x,z)$ encodes the
conditional distribution $\mathbb{P}(w|d,x,z)$ from the integral equation that
defines the confounding bridge $h_{0}$. The quantities $\mu,\nu,\mu(d),\mu(v)$
embed various reweighting distributions: $\mathbb{P}(x,w)$,
$\tilde{\mathbb{P}}(x,w)$, $\mathbb{P}(x,w|d)$, and $\mathbb{P}(x,w|v)$,
respectively. In general, the quantity
$\int[\phi(x)\otimes\phi(w)]\mathrm{d}\mathbb{Q}$ encodes the distribution
$\mathbb{Q}$ as a function in
$\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{W}}$. The
characteristic property for $k_{\mathcal{X}}$ and $k_{\mathcal{W}}$ ensures
that the mapping
$\mathbb{Q}\mapsto\int[\phi(x)\otimes\phi(w)]\mathrm{d}\mathbb{Q}$ is
injective, so that the RKHS representation of the reweighting distribution
$\mathbb{Q}$ is unique.

These representations abstractly suggest estimators. For example, for
$\theta_{0}^{ATE}(d)$ the estimator should be of the form
$\hat{\theta}^{ATE}(d)=\langle\hat{h},\phi(d)\otimes\hat{\mu}\rangle_{\mathcal{H}}$,
where $\hat{h}$ is an estimator of the confounding bridge $h_{0}$ and
$\hat{\mu}$ is an estimator of the mean embedding $\mu$. I propose a
regularized kernel estimator of the confounding bridge function in the spirit
of two stage least squares (2SLS): first project $\phi(W)$ onto
$\phi(D)\otimes\phi(X)\otimes\phi(Z)$ to obtain $\hat{\mu}_{w}(D,X,Z)$, then
project $Y$ onto $\phi(D)\otimes\phi(X)\otimes\hat{\mu}_{w}(D,X,Z)$ to obtain
$\hat{h}$. I estimate unconditional mean embeddings $\hat{\mu},\hat{\nu}$ with
simple averages, and I estimate conditional mean embeddings
$\hat{\mu}(d),\hat{\mu}(v)$ with projections similar to
$\hat{\mu}_{w}(d,x,z)$.111When $V$ is discrete, a simple average suffices; see
the discussion in Appendix J.

###  4.2 Generalized regression loss

It is not obvious that
$\hat{\theta}^{ATE}(d)=\langle\hat{h},\phi(d)\otimes\hat{\mu}\rangle_{\mathcal{H}}$
has a closed form expression in terms of kernel matrices. In this section, I
state the generalized regression losses that define the estimator. These
losses generalize the loss in (1). Along the way, I discuss the connection to
2SLS and provide intuition for these techniques. In the next section, I
formally prove that a closed form expression exists and then solve for it,
generalizing the expression in (2).

Similar to 2SLS, I estimate the confounding bridge $\hat{h}$ in two stages. In
the first stage, I estimate the conditional mean embedding
$\hat{\mu}_{w}(d,x,z)$. Let $n$ be the number of observations of
$(d_{i},x_{i},w_{i},z_{i})$ used to estimate the conditional mean embedding
$\hat{\mu}_{w}(d,x,z)$ with regularization parameter $\lambda$. The
generalized regression loss for the regression of $\phi(W)$ on $\phi(D,X,Z)$
is

$\hat{E}=\operatorname*{\arg\\!\min}_{E\in\mathcal{L}_{2}(\mathcal{H}_{\mathcal{W}},\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}})}\frac{1}{n}\sum_{i=1}^{n}\|\phi(w_{i})-E^{*}\phi(d_{i},x_{i},z_{i})\|^{2}_{\mathcal{H}_{\mathcal{W}}}+\lambda\|E\|^{2}_{\mathcal{L}_{2}(\mathcal{H}_{\mathcal{W}},\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}})}$

so that $\hat{\mu}_{w}(d,x,z)=\hat{E}^{*}\phi(d,x,z)$. In this notation,
$E^{*}$ is the adjoint operator of $E$, and
$\mathcal{L}_{2}(\mathcal{H}_{\mathcal{W}},\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}})$
is an RKHS whose elements are operators of the form
$E:\mathcal{H}_{\mathcal{W}}\rightarrow\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}}$.

In the second stage, I estimate the confounding bridge $\hat{h}$. Let $m$ be
the number of observations of
$(\dot{y}_{i},\dot{d}_{i},\dot{x}_{i},\dot{z}_{i})$ used to estimate the
confounding bridge $\hat{h}$ with regularization parameter $\xi$. This
notation allows the analyst to use different quantities of observations
$(n,m)$ to estimate $\hat{\mu}_{w}(d,x,z)$ and $\hat{h}$, or to reuse the same
observations. To estimate $\hat{h}$, I regress $Y$ on
$\hat{\mu}(D,X,Z):=\phi(D)\otimes\phi(X)\otimes\hat{\mu}_{w}(D,X,Z)$. The
generalized regression loss for the regression of $Y$ on $\hat{\mu}(D,X,Z)$ is

$\hat{h}=\operatorname*{\arg\\!\min}_{h\in\mathcal{H}}\frac{1}{m}\sum_{i=1}^{m}\left\\{\dot{y}_{i}-\langle
h,\hat{\mu}(\dot{d}_{i},\dot{x}_{i},\dot{z}_{i})\rangle_{\mathcal{H}}\right\\}^{2}+\xi\|h\|^{2}_{\mathcal{H}}.$

Finally, when estimating $\hat{\theta}^{ATT}$ and $\hat{\theta}^{CATE}$, one
must also estimate the conditional mean embeddings $\hat{\mu}(d)$ and
$\hat{\mu}(v)$. The losses for these conditional mean embeddings mirror the
loss for the conditional mean embedding $\hat{\mu}_{w}(d,x,z)$.

###  4.3 Closed form

I present a closed form solution for the confounding bridge estimator,
generalizing kernel instrumental variable regression [Singh et al., 2019,
Algorithm 1] to my extended RKHS construction.

######  Algorithm 1 (Estimation of confounding bridge).

Let $\odot$ mean elementwise product. Then

$\displaystyle\mathbf{A}$
$\displaystyle=\mathbf{K}_{DD}\odot\mathbf{K}_{XX}\odot\mathbf{K}_{ZZ}\in\mathbb{R}^{n\times
n},\quad\dot{\mathbf{A}}=\mathbf{K}_{D\dot{D}}\odot\mathbf{K}_{X\dot{X}}\odot\mathbf{K}_{Z\dot{Z}}\in\mathbb{R}^{n\times
m},$ $\displaystyle\mathbf{B}$
$\displaystyle=(\mathbf{A}+n\lambda\mathbf{I})^{-1}\dot{\mathbf{A}}\in\mathbb{R}^{n\times
m},\quad\quad\quad\mathbf{M}=\mathbf{K}_{\dot{D}\dot{D}}\odot\mathbf{K}_{\dot{X}\dot{X}}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{WW}\mathbf{B}\\}\in\mathbb{R}^{m\times
m},$ $\displaystyle\boldsymbol{\hat{\alpha}}$
$\displaystyle=(\mathbf{M}\mathbf{M}^{\top}+m\xi\mathbf{M})^{-1}\mathbf{M}\dot{\mathbf{Y}}\in\mathbb{R}^{m},\quad\hat{h}(d,x,w)=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot\mathbf{K}_{\dot{X}x}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{Ww}\\}]\in\mathbb{R}$

where $(\lambda,\xi)$ are ridge penalty hyperparameters.

See Appendix E for the derivation, which begins with an original proof that
such an $\boldsymbol{\hat{\alpha}}\in\mathbb{R}^{m}$ even exists. The
elementwise products arise because tensor product RKHSs correspond to product
kernels. For example, the kernel of
$\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}}$
is
$k(d,x,z;d^{\prime},x^{\prime},z^{\prime})=k_{\mathcal{D}}(d,d^{\prime})k_{\mathcal{X}}(x,x^{\prime})k_{\mathcal{Z}}(z,z^{\prime})$
so its kernel matrix is
$\mathbf{K}_{DD}\odot\mathbf{K}_{XX}\odot\mathbf{K}_{ZZ}$. $\mathbf{M}$ is
essentially the kernel matrix for the conditional mean embedding
$\hat{\mu}(d,x,z)=\phi(d)\otimes\phi(x)\otimes\hat{\mu}_{w}(d,x,z)$.
Interpreting these expressions, $\boldsymbol{\hat{\alpha}}$ is clearly the
regularized empirical projection of $Y$ onto
$\phi(D)\otimes\phi(X)\otimes\hat{\mu}_{w}(D,X,Z)$ in the spirit of 2SLS.

Next, I present closed form solutions for treatment effects, e.g.
$\hat{\theta}^{ATE}(d)=\langle\hat{h},\phi(d)\otimes\hat{\mu}\rangle_{\mathcal{H}}$,
building on $\hat{h}$ from Algorithm 1. Whereas [Singh et al., 2020, Algorithm
3.1] estimate treatment effects assuming selection on observables, Algorithm 2
estimates treatment effects assuming access to negative controls. For
$\theta_{0}^{DS}$, let $\tilde{n}$ be the number of observations of
$(\tilde{x}_{i},\tilde{w}_{i})$ drawn from population $\tilde{\mathbb{P}}$.

######  Algorithm 2 (Estimation of treatment effects).

Treatment effect estimators have the closed form solutions

  1. 1.

$\hat{\theta}^{ATE}(d)=n^{-1}\sum_{i=1}^{n}\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot\mathbf{K}_{\dot{X}x_{i}}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{Ww_{i}}\\}]$

  2. 2.

$\hat{\theta}^{DS}(d,\tilde{\mathbb{P}})=\tilde{n}^{-1}\sum_{i=1}^{\tilde{n}}\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot\mathbf{K}_{\dot{X}\tilde{x}_{i}}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{W\tilde{w}_{i}}\\}]$

  3. 3.

$\hat{\theta}^{ATT}(d,d^{\prime})=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d^{\prime}}\odot\\{[\mathbf{K}_{\dot{X}X}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{WW}\\}](\mathbf{K}_{DD}+n\lambda_{1}\mathbf{I})^{-1}\mathbf{K}_{Dd}\\}]$

  4. 4.

$\hat{\theta}^{CATE}(d,v)=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot\mathbf{K}_{\dot{V}v}\odot\\{[\mathbf{K}_{\dot{X}X}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{WW}\\}](\mathbf{K}_{VV}+n\lambda_{2}\mathbf{I})^{-1}\mathbf{K}_{Vv}\\}]$

where $(\lambda_{1},\lambda_{2})$ are ridge regression penalty
hyperparameters. In $\hat{\theta}^{CATE}(d,v)$, $\boldsymbol{\hat{\alpha}}$ is
the coefficient for the confounding bridge that includes $V$.

See Appendix E for the derivation. I give theoretical values for the
regularization parameters that balance bias and variance in Section 5 below.
In particular, I specify $(\lambda,\xi)$ in Theorem 3 and
$(\lambda_{1},\lambda_{2})$ in Theorem 4. In Appendix F, I propose a practical
tuning procedure based on the closed form solution of leave-one-out cross
validation (LOOCV) to empirically balance bias and variance, and I discuss the
time complexity.

###  4.4 Summary

To fix ideas, I summarize the end-to-end procedure for
$\hat{\theta}^{ATE}(d)$. For simplicity, I suppose that the analyst re-uses
observations in the two stages of confounding bridge estimation. I provide
additional discussion in Appendix E for researchers who are new to kernel
methods.

######  Algorithm 3 (End-to-end details).

Given $n$ observations of outcome $Y$, treatment $D$, covariates $X$, negative
control outcome $W$, and negative control treatment $Z$,

  1. 1.

Specify the kernels
$k_{\mathcal{D}},k_{\mathcal{X}},k_{\mathcal{W}},k_{\mathcal{Z}}$.

    1. (a)

For multivariate objects, e.g. $X=(X_{1},...,X_{p})$, use the product of
scalar kernels

$k_{\mathcal{X}}(x,x^{\prime})=\prod_{j=1}^{p}k_{\mathcal{X}_{1}}(x_{1},x_{1}^{\prime})\cdot...\cdot
k_{\mathcal{X}_{p}}(x_{p},x_{p}^{\prime}).$

    2. (b)

Tune the scalar kernel hyperparameters. For example, if the treatment kernel
$k_{D}$ is chosen as the Gaussian kernel, a standard heuristic is to use the
median interpoint distance among observed treatment values.

    3. (c)

Compute the kernel matrices, e.g. $\mathbf{K}_{DD}\in\mathbb{R}^{n\times n}$
with $(i,j)$-th entry $k_{\mathcal{D}}(d_{i},d_{j})$.

  2. 2.

Specify the regularization hyperparameters $(\lambda,\xi)$.

    1. (a)

For $\lambda$, I derive the closed form solution of LOOCV in Appendix F.

    2. (b)

The same procedure applies to $\xi$, plugging in the chosen value of
$\lambda$.

  3. 3.

Estimate the confounding bridge $\hat{h}$ in two stages, using
$(\lambda,\xi)$.

    1. (a)

Estimate the distribution in the integral equation $\hat{\mathbb{P}}(w|d,x,z)$
via its mean embedding $\hat{\mu}_{w}(d,x,z)$ with regularization $\lambda$ as

$[\hat{\mu}_{w}(d,x,z)](w)=\mathbf{K}_{wW}(\mathbf{K}_{DD}\odot\mathbf{K}_{XX}\odot\mathbf{K}_{ZZ}+n\lambda\mathbf{I})^{-1}[\mathbf{K}_{Dd}\odot\mathbf{K}_{Xx}\odot\mathbf{K}_{Zz}].$

    2. (b)

Regress $Y$ onto $\phi(D)\otimes\phi(X)\otimes\hat{\mu}_{w}(D,X,Z)$ with
regularization $\xi$. It turns out that
$\hat{h}(d,x,w)=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{Dd}\odot\mathbf{K}_{Xx}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{Ww}\\}]\in\mathbb{R}$
where

$\displaystyle\mathbf{A}$
$\displaystyle=\mathbf{K}_{DD}\odot\mathbf{K}_{XX}\odot\mathbf{K}_{ZZ}\in\mathbb{R}^{n\times
n},\quad\mathbf{B}=(\mathbf{A}+n\lambda\mathbf{I})^{-1}\mathbf{A}\in\mathbb{R}^{n\times
n},$ $\displaystyle\mathbf{M}$
$\displaystyle=\mathbf{K}_{DD}\odot\mathbf{K}_{XX}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{WW}\mathbf{B}\\}\in\mathbb{R}^{n\times
n},\quad\boldsymbol{\hat{\alpha}}=(\mathbf{M}\mathbf{M}^{\top}+n\xi\mathbf{M})^{-1}\mathbf{M}\mathbf{Y}\in\mathbb{R}^{n}.$

  4. 4.

Estimate the counterfactual distribution $\hat{\mathbb{P}}(x,w)$ via its mean
embedding $\hat{\mu}$ as

$[\hat{\mu}](x,w)=\frac{1}{n}\sum_{i=1}^{n}k_{\mathcal{X}}(x_{i},x)k_{\mathcal{W}}(w_{i},w)\in\mathbb{R}.$

  5. 5.

Estimate the dose response $\hat{\theta}^{ATE}(d)$ by combining $\hat{h}$ and
$\hat{\mu}$ according to
$\hat{\theta}^{ATE}(d)=\langle\hat{h},\phi(d)\otimes\hat{\mu}\rangle_{\mathcal{H}}$.
To do so, match the common arguments $(x,w)$ of $\hat{h}$ and $\hat{\mu}$. In
summary,

$\hat{\theta}^{ATE}(d)=\frac{1}{n}\sum_{i=1}^{n}\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{Dd}\odot\mathbf{K}_{Xx_{i}}\odot\\{\mathbf{B}^{\top}\mathbf{K}_{Ww_{i}}\\}]\in\mathbb{R}.$

##  5 Consistency

To define the learning problem in Section 3, I placed three assumptions:
availability of negative controls (Assumption 1); existence and completeness
of the confounding bridge (Assumption 2); and invariance of the confounding
bridge for transfer learning (Assumption 3). To construct an algorithm in
Section 4, I assumed RKHS regularity (Assumption 4). To guarantee uniform
consistency in this section, I place three final assumptions: original space
regularity (Assumption 5); smoothness of conditional expectation operators
(Assumption 6); and smoothness of the confounding bridge (Assumption 7). I
first prove uniform consistency of the confounding bridge, then uniform
consistency of treatment effects. As before, for $\theta_{0}^{CATE}$ I extend
the stated assumptions from $X$ to $(V,X)$.

###  5.1 Confounding bridge

I require weak regularity conditions on the original spaces of the outcome
$Y$, treatment $D$, covariates $(V,X)$, and negative controls $(W,Z)$.

######  Assumption 5 (Original space regularity conditions).

Assume

  1. 1.

$Y\in\mathcal{Y}\subset\mathbb{R}$ is bounded, i.e. there exists $C<\infty$
such that $|Y|\leq C$ almost surely.

  2. 2.

$\mathcal{D}$, $\mathcal{X}$, $\mathcal{W}$, and $\mathcal{Z}$ are Polish
spaces.

To simplify notation and analysis, I require that the outcome $Y\in\mathbb{R}$
is a bounded scalar. More generally, $\mathcal{Y}$ could be a separable
Hilbert space. I preserve generality for treatment, covariates, and negative
controls. A Polish space is a separable and completely metrizable topological
space. Random variables with support in a Polish space may be discrete or
continuous and low, high, or infinite dimensional. As such, I allow for
treatment, covariates, and negative controls that could even be texts, graphs,
or images.

Next, I place a smoothness condition in the sense of (3) for the conditional
mean embedding $\mu_{w}(d,x,z)$. In anticipation of later analysis, I
articulate this assumption abstractly. Consider the abstract conditional mean
embedding $\mu_{a}(b):=\int\phi(a)\mathrm{d}\mathbb{P}(a|b)$ where
$a\in\mathcal{A}_{j}$ and $b\in\mathcal{B}_{j}$. I will ultimately consider
the three different conditional mean embeddings $\mu_{w}(d,x,z)$, $\mu(d)$,
and $\mu(v)$ indexed by $j\in\\{0,1,2\\}$. As previewed in Section 3, the
conditional expectation operator
$E_{j}:\mathcal{H}_{\mathcal{A}_{j}}\rightarrow\mathcal{H}_{\mathcal{B}_{j}}$,
$f(\cdot)\mapsto\mathbb{E}[f(A_{j})|B_{j}=\cdot]$ encodes the same information
as $\mu_{a}(b)$. In particular,

$\mu_{a}(b)=\int\phi(a)\mathrm{d}\mathbb{P}(a|b)=[E_{j}\phi(\cdot)](b)=[E_{j}^{*}\phi(b)](\cdot),\quad
a\in\mathcal{A}_{j},\quad b\in\mathcal{B}_{j}$

where
$E_{j}^{*}:\mathcal{H}_{\mathcal{B}_{j}}\rightarrow\mathcal{H}_{\mathcal{A}_{j}}$,
$g(\cdot)\mapsto\mathbb{E}[g(B_{j})|A_{j}=\cdot]$ is the adjoint of $E_{j}$. I
denote the space of Hilbert-Schmidt operators between
$\mathcal{H}_{\mathcal{A}_{j}}$ and $\mathcal{H}_{\mathcal{B}_{j}}$ by
$\mathcal{L}_{2}(\mathcal{H}_{\mathcal{A}_{j}},\mathcal{H}_{\mathcal{B}_{j}})$,
which is an RKHS in its own right.

######  Assumption 6 (Smooth conditional expectation operator).

Assume
$E_{j}\in[\mathcal{L}_{2}(\mathcal{H}_{\mathcal{A}_{j}},\mathcal{H}_{\mathcal{B}_{j}})]^{{}^{c_{j}}}$.

To specialize the assumption, all one has to do is specify $\mathcal{A}_{j}$
and $\mathcal{B}_{j}$. For example, for $\mu_{w}(d,x,z)$,
$\mathcal{A}_{0}=\mathcal{W}$ and
$\mathcal{B}_{0}=\mathcal{D}\times\mathcal{X}\times\mathcal{Z}$. By assuming
smoothness of $E_{0}$, I assume smoothness of $\mu_{w}(d,x,z)$. I explicitly
specialize the assumption in Appendices G and H.

The final assumption that I place in order to prove uniform consistency of the
confounding bridge estimator $\hat{h}$ is that the confounding bridge $h_{0}$
is smooth in the sense of (3). Recall that the features for the RKHS
$\mathcal{H}$ are $\phi(d,x,w)=\phi(d)\otimes\phi(x)\otimes\phi(w)$. Recall
from Theorem 2 that we must solve the integral equation
$\gamma_{0}(d,x,z)=\langle h_{0},\mu(d,x,z)\rangle_{\mathcal{H}}$ where
$\mu(d,x,z)=\phi(d)\otimes\phi(x)\otimes\mu_{w}(d,x,z)$ is a mean embedding.
By construction,

$\langle\mu(d,x,z),\mu(d^{\prime},x^{\prime},z^{\prime})\rangle_{\mathcal{H}}=k_{\mathcal{D}}(d,d^{\prime})k_{\mathcal{X}}(x,x^{\prime})\int
k_{\mathcal{W}}(w,w^{\prime})\mathrm{d}\mathbb{P}(w|d,x,z)\mathrm{d}\mathbb{P}(w^{\prime}|d^{\prime},x^{\prime},z^{\prime}),$

which we may interpret as the inner product of a space
$\mathcal{H}_{\mu}\subset\mathcal{H}$. In words, the kernel of
$\mathcal{H}_{\mu}$ is simply the kernel of $\mathcal{H}$ after integrating
out $(w,w^{\prime})$ according to the conditional distribution
$\mathbb{P}(w|d,x,z)$ from the integral equation. Formally,
$\mathcal{H}_{\mu}$ can be viewed as an RKHS of functions evaluated on mean
embeddings instead of features [Szabó et al., 2016].

######  Assumption 7 (Smooth confounding bridge).

Assume $h_{0}\in\mathcal{H}_{\mu}^{c}$.

As a technical aside, an analyst may introduce additional nonlinearity by
enriching the model and enriching this assumption. A richer model would
instead allow $\gamma_{0}(d,x,z)=H_{0}\mu(d,x,z)$ where
$H_{0}:\mathcal{H}\rightarrow\mathbb{R}$ is a nonlinear mapping in a richer
RKHS. In such case, the smoothness assumption would be placed on $H_{0}$
rather than $h_{0}$ and a notion of Hölder continuity is required [Szabó et
al., 2016, Table 1]. For clarity, I omit this complexity.

Under these conditions, I arrive at the first main result: uniform consistency
of the confounding bridge. This result appears to be the first finite sample
uniform analysis of nonparametric instrumental variable regression in the
RKHS. By allowing $n\neq m$, I allow the possibility of asymmetric sample
splitting and the use of observations from different data sets. The finite
sample rates are expressed in terms of $(n,m)$. The parameter $a>0$
characterizes the ratio between the sample sizes.

######  Theorem 3 (Consistency of confounding bridge).

Suppose Assumptions 2, 4, 5, 6 with $\mathcal{A}_{0}=\mathcal{W}$ and
$\mathcal{B}_{0}=\mathcal{D}\times\mathcal{X}\times\mathcal{Z}$, and 7 hold.
Set $\lambda=n^{-\frac{1}{c_{0}+1}}$ and $n=m^{\frac{a(c_{0}+1)}{c_{0}-1}}$
where $a>0$.

  1. 1.

If $a\leq(c+3)/(c+1)$ then
$\|\hat{h}-h_{0}\|_{\infty}=O_{p}(m^{-\frac{1}{2}\frac{a(c-1)}{c+3}})$ with
$\xi=m^{-\frac{a}{c+3}}$.

  2. 2.

If $a\geq(c+3)/(c+1)$ then
$\|\hat{h}-h_{0}\|_{\infty}=O_{p}(m^{-\frac{1}{2}\frac{c-1}{c+1}})$ with
$\xi=m^{-\frac{1}{c+1}}$.

See Appendix G for exact finite sample rates. At $a=(c+3)/(c+1)<2$, the
convergence rate $m^{-\frac{1}{2}\frac{c-1}{c+1}}$ attains the rate of single
stage kernel ridge regression with respect to $m$ [Smale and Zhou, 2007]. This
rate is calibrated by $c$, the smoothness of confounding bridge. However, this
rate requires the ratio between sample sizes to be
$n=m^{\frac{c+3}{c+1}\cdot\frac{(c_{0}+1)}{c_{0}-1}}$, implying $n\gg m$. In
practice, the analyst often uses the same observations to estimate
$\hat{\mu}_{w}(d,x,z)$ and $\hat{h}$.

######  Corollary 1 (Reusing samples).

If samples are reused to estimate $\hat{\mu}_{w}(d,x,z)$ and $\hat{h}$, then
$n=m$, $a=(c_{0}-1)/(c_{0}+1)<1$, $\xi=n^{-\frac{c_{0}-1}{(c_{0}+1)(c+3)}}$,
and
$\|\hat{h}-h_{0}\|_{\infty}=O_{p}(n^{-\frac{1}{2}\frac{c_{0}-1}{c_{0}+1}\frac{c-1}{c+3}})$.

This rate adapts to the smoothness $c_{0}$ of the conditional distribution as
well as the smoothness $c$ of the confounding bridge. The slow rate reflects
the challenge of a uniform norm guarantee in an ill posed inverse problem. A
faster rate could be possible under further assumptions on the spectral decay
of the RKHS, which I leave to future work.

###  5.2 Treatment effects

Recall from Theorem 2 that $\theta_{0}^{ATT}$ and $\theta_{0}^{CATE}$ contain
conditional mean embeddings $\mu(d)$ and $\mu(v)$, respectively. I estimate
these conditional mean embeddings by regularized projections in Algorithm 2.
To control bias introduced by regularization, I place smoothness conditions in
the sense of (3) for $\mu(d)$ and $\mu(v)$ as well. As previewed in the
discussion about $\mu_{w}(d,x,z)$ and $E_{0}$, a conditional mean embedding
corresponds to a conditional expectation operator. As before, all one has to
do is specify $\mathcal{A}_{j}$ and $\mathcal{B}_{j}$ to specialize the
assumption. For $\mu(d)$, $\mathcal{A}_{1}=\mathcal{X}\times\mathcal{W}$ and
$\mathcal{B}_{1}=\mathcal{D}$; for $\mu(v)$,
$\mathcal{A}_{2}=\mathcal{X}\times\mathcal{W}$ and
$\mathcal{B}_{2}=\mathcal{V}$.

Under these conditions, I arrive at the second main result: uniform
consistency of the treatment effect estimators. For simplicity, I assume the
analyst reuses samples, so that $n=m$. Recall $\tilde{n}$ is the number of
observations drawn from the alternative population $\tilde{\mathbb{P}}$.

######  Theorem 4 (Consistency of treatment effects).

Suppose Assumption 1 holds, as well as the conditions of Theorem 3. Set
$(\lambda,\lambda_{1},\lambda_{2})=(n^{-\frac{1}{c_{0}+1}},n^{-\frac{1}{c_{1}+1}},n^{-\frac{1}{c_{2}+1}})$
and $\xi=n^{-\frac{c_{0}-1}{(c_{0}+1)(c+3)}}$.

  1. 1.

Then

$\|\hat{\theta}^{ATE}-\theta_{0}^{ATE}\|_{\infty}=O_{p}\left(n^{-\frac{1}{2}\frac{c_{0}-1}{c_{0}+1}\frac{c-1}{c+3}}\right).$

  2. 2.

If in addition Assumption 3 holds, then

$\|\hat{\theta}^{DS}(\cdot,\tilde{\mathbb{P}})-\theta_{0}^{DS}(\cdot,\tilde{\mathbb{P}})\|_{\infty}=O_{p}\left(n^{-\frac{1}{2}\frac{c_{0}-1}{c_{0}+1}\frac{c-1}{c+3}}+\tilde{n}^{-\frac{1}{2}}\right).$

  3. 3.

If in addition Assumption 6 holds with
$\mathcal{A}_{1}=\mathcal{X}\times\mathcal{W}$ and
$\mathcal{B}_{1}=\mathcal{D}$, then

$\|\hat{\theta}^{ATT}-\theta_{0}^{ATT}\|_{\infty}=O_{p}\left(n^{-\frac{1}{2}\frac{c_{0}-1}{c_{0}+1}\frac{c-1}{c+3}}+n^{-\frac{1}{2}\frac{c_{1}-1}{c_{1}+1}}\right).$

  4. 4.

If in addition Assumption 6 holds with
$\mathcal{A}_{2}=\mathcal{X}\times\mathcal{W}$ and
$\mathcal{B}_{2}=\mathcal{V}$, then

$\|\hat{\theta}^{CATE}-\theta_{0}^{CATE}\|_{\infty}=O_{p}\left(n^{-\frac{1}{2}\frac{c_{0}-1}{c_{0}+1}\frac{c-1}{c+3}}+n^{-\frac{1}{2}\frac{c_{2}-1}{c_{2}+1}}\right).$

See Appendix H for exact finite sample rates. Inspecting the rates, we see
that each one is a sum of the rate for the confounding bridge from Corollary 1
with the rate for the appropriate mean embedding estimation procedure. The
rates adapt to the smoothness parameters $(c,c_{0},c_{1},c_{2})$ of the
confounding bridge $h_{0}$ and the conditional expectation operators $E_{0}$,
$E_{1}$, and $E_{2}$. Equivalently, the rates adapt to the smoothness of the
confounding bridge $h_{0}$ and the conditional distributions
$\mathbb{P}(w|d,x,z)$, $\mathbb{P}(x,w|d)$, and $\mathbb{P}(x,w|v)$.

The goal of this project is to propose dose response and heterogeneous
treatment effect estimators to ultimately inform policy and medical decisions.
For this reason, I prove a uniform guarantee that strictly controls error for
any level of treatment, rather than a mean square guarantee that controls
error for the average level of treatment. Uniform guarantees come at the cost
of slower rates. In negative control treatment effect estimation, the ill
posedness of the confounding bridge learning problem compounds this
phenomenon. Theorem 4 appears to be the first finite sample analysis for
nonparametric negative control treatment effects, and it holds under weak
assumptions. Obtaining faster rates, perhaps via further assumptions on
spectral decay, is an important direction for future work.

##  6 Simulation and application

###  6.1 Simulations

I evaluate the empirical performance of the new estimators. I focus on dose
response with negative controls, and consider various designs with varying
sample sizes. Specifically, I compare the new algorithm that uses negative
controls (`N.C.`) with an existing RKHS algorithm for nonparametric treatment
effects (`T.E.`) [Singh et al., 2020] that ignores unobserved confounding and
instead classifies negative controls as additional covariates. Whereas the new
algorithm involves reweighting a confounding bridge, the previous algorithm
involves reweighting a regression. For each design, sample size, and
algorithm, I implement 100 simulations and calculate mean square error (MSE)
with respect to the true counterfactual function.

(a) Quadratic

(b) Sigmoid

(c) Peaked

Figure 2:  Simulation results for various designs

Specifically, I adapt the continuous treatment effect design proposed by
[Colangelo and Lee, 2020]. Whereas the original setting studied by [Colangelo
and Lee, 2020] has no unobserved confounding, my modification does have
unobserved confounding. The goal is to learn the counterfactual function
$\theta_{0}^{ATE}(d)$, which may be a quadratic, sigmoid, or peaked function.
A single observation consists of the tuple $(Y,W,D,Z,X)$ for outcome, negative
control outcome, treatment, negative control treatment, and covariates.
$(Y,D)$ are continuous scalars. In the baseline experiment,
$X\in\mathbb{R}^{5}$ and $(Z,W)\in\mathbb{R}$.

To explore the role of sample size, I consider
$n\in\\{100,500,1000,5000,10000\\}$. To explore the role of dimension, I focus
on the quadratic design, fix sample size at $n=1000$, and then vary
$dim(X)\in\\{1,5,10,50,100\\}$, $dim(Z)\in\\{1,5,10\\}$, or
$dim(W)\in\\{1,5,10\\}$. This range of sample sizes and dimensions is common
in epidemiology research. Figures 2 and 3 visualize results. Across designs,
sample sizes, and dimensions, the use of negative controls to adjust for
unobserved confounding improves performance. The improvement is generally
increasing in $n$ and $dim(Z)$ but decreasing in $dim(X)$ and $dim(W)$.
Intuitively, $(X,W)$ are the variables used in the reweighting step, which is
common across the two estimators `N.C.` and `T.E.`; as this step becomes
relatively more important, the estimators become more similar. See Appendix I
for implementation details as well as additional simulations that confirm: (i)
robustness to tuning; (ii) improvement when treatment is discrete; and (iii)
inefficiency in the absence of unobserved confounding.

(a) Covariate

(b) N.C. treatment

(c) N.C. outcome

Figure 3:  Simulation results for various dimensions

###  6.2 Dose response of cigarette smoking

Estimating the effect of cigarette smoking on infant birth weight is
challenging for several reasons. First, pregnant women are classified as a
vulnerable population, so they are typically excluded from clinical trials of
any kind. When the treatment of interest causes harm, ethical considerations
preclude randomization. Therefore observational data are the only option.
Second, pregnancy induces many physiological changes, so medical knowledge
predicts different dose response curves for women who are pregnant compared to
women who are not pregnant. For example, plasma volume increases 35%, cardiac
output increases 40%, and glomerular filtration rate (a measure of kidney
function) increases 50% during pregnancy [Cunningham et al., 2014]. Therefore
the shape of the dose response curve for pregnant women is an unknown,
nonparametric quantity. Third, medical records exclude an unobserved
confounder known to be crucial for maternal-fetal health: household income
[Joseph et al., 2007].

In this section, I argue that medical records include variables that satisfy
the properties of negative controls for unobserved income. I provide
preliminary results and outline directions for future work on this topic.
Finally, I discuss what issues may arise if there are additional unobserved
confounders. The purpose of this case study is to illustrate how the proposed
estimators may be useful in epidemiology research, though the findings are not
conclusive.

$X$$W$ptbirth order,sex,Rh$Z$ptedu-cation$D$smoking$Y$birth weight$U$income

Figure 4: Smoking DAG

I estimate the dose response curve of cigarette smoking on infant birth weight
using a data set of singleton births in the state of Pennsylvania between 1989
and 1991 assembled by [Almond et al., 2005] and subsequently analyzed by
[Cattaneo, 2010]. I focus on Pennsylvania because smoking data are available
for over 95% of mothers. I focus on singleton births because multiple
gestations reflect a variety of factors and result in different fetal growth
trajectories. 21% of women report smoking during pregnancy, and I subset to
this sample. I consider the subpopulations of (a) nonhispanic white women who
smoke ($n=73,834$), (b) nonhispanic black women who smoke ($n=17,625$), and
(c) hispanic women who smoke ($n=2,152$). Formally, I estimate
$\theta_{0}^{CATE}(d,v)$ where $D\in\mathbb{R}$ is the number of cigarettes
smoked per day, and $V$ concatenates mother’s race $V_{1}\in\\{\text{white,
black, hispanic}\\}$ and mother’s smoking status $V_{2}\in\\{0,1\\}$. See
Appendix J for further discussion.

The classification of variables extensively relies on domain knowledge, so I
sought the expertise of physicians from the Department of Obstetrics,
Gynecology & Reproductive Biology at Harvard Medical School. Together, we
arrived at the classification given in Appendix J, based on a canonical
textbook [Cunningham et al., 2014]. Figure 4 illustrates the model.
Demographics, alcohol consumption, prenatal care, existing medical conditions,
county, and year serve as covariates $X$ since they may be associated with
both smoking $D$ and birth weight $Y$.

Education serves as a negative control treatment $Z$ because it reflects
unobserved confounding due to household income $U$ but has no direct medical
effect on birth weight $Y$. Formally, we require
$Z\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}Y|D,U,X$:
education is independent of birth weight after conditioning on smoking,
income, and observed covariates. Prenatal care and weight gain are the
observed covariates that, along with smoking and income, justify the
conditional independence between education and birth weight.

Infant birth order and sex serve as a negative control outcomes $W$ because
family size reflects household income $U$ but is not directly caused by
smoking $D$ or education $Z$. Formally, we require
$W\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}D,Z|U,X$:
family size is independent of smoking and education after conditioning on
income and observed covariates. Age and marriage status are the observed
covariates that, along with income, justify the conditional independence
between education and family size. We also include Rh sensitization as a
negative control outcome because it is one of the few medical conditions not
affected by smoking (it is caused by blood type).

(a) White smoking mothers

(b) Black smoking mothers

Figure 5:  Effect of cigarette smoking on birth weight for different
subpopulations; $\theta_{0}^{CATE}(d,v)$ where $D\in\mathbb{R}$ is the number
of cigarettes smoked per day, and $V$ concatenates mother’s race
$V_{1}\in\\{\text{white, black, hispanic}\\}$ and mother’s smoking status
$V_{2}\in\\{0,1\\}$.

I implement both the new algorithm (`N.C.`) and an existing RKHS algorithm for
continuous treatment effect (`T.E`) [Singh et al., 2020] that ignores
unobserved confounding. For the method that ignores unobserved confounding, I
classify negative controls as additional covariates. Figures 5 and 6 visualize
results for white, black, and hispanic smoking mothers. The effect of
cigarettes smoked per day $D$ on birth weight in grams $Y$ is generally
negative with similar shapes across subpopulations. The counterfactual birth
weights for black and hispanic mothers are lower than for white mothers when
the number of cigarettes is high. The main finding is that using negative
controls leads to higher dose response curves. Under the stated causal
assumptions, the gap between `N.C.` and `T.E.` is the magnitude of unobserved
confounding due to income. These preliminary results support the clinical
hypothesis that poverty is an unmeasured confounder that affects infant birth
weight. Unobserved poverty may substantially mislead observational studies
that fail to account for it. See Appendix J for implementation details.

(c) Hispanic smoking mothers Figure 6: Effect of cigarette smoking on birth
weight for different subpopulations; $\theta_{0}^{CATE}(d,v)$ where
$D\in\mathbb{R}$ is the number of cigarettes smoked per day, and $V$
concatenates mother’s race $V_{1}\in\\{\text{white, black, hispanic}\\}$ and
mother’s smoking status $V_{2}\in\\{0,1\\}$.

An unanticipated result is that the dose response curves appear nonmonotonic;
estimated counterfactual birth weight increases before it decreases. This
phenomenon prevails across subpopulations, and it can be seen in not only
`N.C.` but also `T.E.` and the raw data. We propose two conjectures based on
the data and domain knowledge. Both of these conjectures are ways in which the
data generating process may violate the causal assumptions in Assumption 1.

First, it may be that measurement error contaminates observations. In the raw
data, it appears that when the number of cigarettes was between one and 10 it
may have been rounded up to 10. Indeed, Figures 5 and 6 document substantial
point masses at multiples of 10. This phenomenon would violate our causal
model, since it would mean that when the true treatment value $d$ was less
than 10, we observe $D=10$, $Z=z$, yet $Y=Y^{(d,z)}$. In such case, estimates
of the dose response for $d<10$ may be unreliable. How to account for
measurement error in negative control estimation remains an open question.

Second, it could be that another unobserved confounder exists, is not detected
by the negative controls, and disproportionately affects women who reported
smoking less than 10 cigarettes. Previous studies suggest that rural-urban
classification, poverty, and psychosocial stress are possible confounders
[Hobel et al., 2008]. In our analysis, we account for rural-urban
classification as an observed covariate and we account for poverty via
negative controls, but we did not find plausible negative controls for stress
in this data set; see Appendix J for further discussion. Indeed, psychosocial
stress is notoriously difficult to measure, and it may cause both smoking and
low birth weight. We pose for future work a further analysis that adjusts for
unobserved confounding due to both income and stress.

##  7 Conclusion

I propose a new family of nonparametric algorithms for learning treatment
effects with negative controls. The estimators are easily implemented and
uniformly consistent. As a contribution to the negative control literature, I
propose methods to estimate dose response curves and heterogeneous treatment
effects under the assumption that treatment effects are smooth. As a
contribution to the kernel methods literature, I show how the RKHS is well
suited to causal inference in the presence of unobserved confounding. As a
contribution to maternal-fetal medicine, I propose a toolkit for estimating
dose response curves for pregnant women from medical records despite
unobserved confounding. The results suggest that RKHS methods may be an
effective bridge between epidemiology and machine learning.

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##  Appendix A Semiparametric inference

While the main contribution of this paper is uniform consistency in
nonparametric settings, in this section I present complementary results for
semiparametric settings. In particular, I articulate conditions that imply
Gaussian approximation and valid confidence intervals for the setting with
binary treatment, appealing to the semiparametric theory of [Cui et al., 2020,
Kallus et al., 2021, Ghassami et al., 2021, Chernozhukov et al., 2021].

To lighten notation, fix the treatment value $d^{\prime}\in\\{0,1\\}$ and
denote $\theta_{0}=\theta_{0}^{ATE}(d^{\prime})$. Observe that for each
treatment value,

$\theta_{0}=\int h_{0}(d^{\prime},x,w)\mathrm{d}\mathbb{P}(x,w)=\int
h_{0}(d,x,w)\tau_{0}(d,x,w)\mathrm{d}\mathbb{P}(d,x,w)$

where by standard propensity score arguments

$\tau_{0}(d,x,w)=\frac{1\\{d=d^{\prime}\\}}{\pi_{0}(x,w)},\quad\pi_{0}(x,w)=\mathbb{P}(D=d^{\prime}|X=x,W=w).$

Just as the primary confounding bridge $h_{0}$ is defined as the solution to
the operator equation

$\gamma_{0}(d,x,z)=\mathbb{E}[h(D,X,W)|D=d,X=x,Z=z]$

one may define a secondary confounding bridge $\alpha_{0}$ as the solution to
the operator equation

$\tau_{0}(d,x,w)=\mathbb{E}[\alpha(D,X,Z)|D=d,X=x,W=w].$

######  Proposition 3 (Secondary confounding bridge).

Suppose Assumptions 1 and 2 hold. In addition, suppose there exists a solution
$\alpha_{0}$ to the operator equation

$\tau_{0}(d,x,w)=\mathbb{E}[\alpha(D,X,Z)|D=d,X=x,W=w].$

Then

$\theta_{0}=\int y\alpha_{0}(d,x,z)\mathrm{d}\mathbb{P}(y,d,x,z).$

###### Proof.

The result follows from the law of iterated expectations and the definitions
of the primary and secondary confounding bridges. Write

$\displaystyle\theta_{0}$ $\displaystyle=\int
h_{0}(d^{\prime},x,w)\mathrm{d}\mathbb{P}(x,w)$ $\displaystyle=\int
h_{0}(d,x,w)\tau_{0}(d,x,w)\mathrm{d}\mathbb{P}(d,x,w)$ $\displaystyle=\int
h_{0}(d,x,w)\mathbb{E}[\alpha_{0}(D,X,Z)|D=d,X=x,W=w]\mathrm{d}\mathbb{P}(d,x,w)$
$\displaystyle=\int
h_{0}(d,x,w)\alpha_{0}(d,x,z)\mathrm{d}\mathbb{P}(d,z,x,w)$
$\displaystyle=\int\mathbb{E}[h_{0}(D,X,W)|D=d,X=x,Z=z]\alpha_{0}(d,x,z)\mathrm{d}\mathbb{P}(d,x,z)$
$\displaystyle=\int\gamma_{0}(d,x,z)\alpha_{0}(d,x,z)\mathrm{d}\mathbb{P}(d,x,z)$
$\displaystyle=\int y\alpha_{0}(d,x,z)\mathrm{d}\mathbb{P}(y,d,z,x).$

∎

The secondary confounding bridge formulation in Proposition 3 generalizes the
familiar inverse propensity weight formulation for treatment effects. Instead
of using the propensity score $\pi_{0}$ encoded by $\tau_{0}$, the formulation
uses a secondary confounding bridge $\alpha_{0}$ defined as the solution to an
ill posed inverse problem that involves the propensity score. Analogously, the
primary confounding bridge formulation in Theorem 1 generalizes the familiar
g-formula for treatment effects, using the confounding bridge $h_{0}$ rather
than the regression $\gamma_{0}$.

Both the primary and secondary confounding bridges $(h_{0},\alpha_{0})$ appear
in the semiparametrically efficient asymptotic variance, so it is natural to
incorporate them into semiparametric estimation and inference [Cui et al.,
2020]. The targeted and debiased machine learning literatures provide a meta
algorithm to do so, as well as rate conditions that suffice for Gaussian
approximation. I quote the meta algorithm and rate conditions, adapting them
to the treatment effect $\theta_{0}$ identified by negative controls.

######  Algorithm 4 (Debiased machine learning).

Given a sample $(Y_{i},W_{i},D_{i},X_{i},Z_{i})$ $(i=1,...,n)$, partition the
sample into folds $(I_{\ell})$ $(\ell=1,...,L)$. Denote by $I_{\ell}^{c}$ the
complement of $I_{\ell}$.

  1. 1.

For each fold $\ell$, estimate $\hat{h}_{\ell}$ and $\hat{\alpha}_{\ell}$ from
observations in $I_{\ell}^{c}$.

  2. 2.

Estimate $\hat{\theta}=n^{-1}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}[\hat{h}_{\ell}(d,X_{i},W_{i})+\hat{\alpha}_{\ell}(D_{i},X_{i},Z_{i})\\{Y_{i}-\hat{h}_{\ell}(D_{i},X_{i},W_{i})\\}]$.

  3. 3.

Estimate its $95$% confidence interval as $\hat{\theta}\pm
1.96\hat{\sigma}n^{-1/2}$, where

$\hat{\sigma}^{2}=n^{-1}\sum_{\ell=1}^{L}\sum_{i\in
I_{\ell}}[\hat{h}_{\ell}(d,X_{i},W_{i})+\hat{\alpha}_{\ell}(D_{i},X_{i},Z_{i})\\{Y_{i}-\hat{h}_{\ell}(D_{i},X_{i},W_{i})\\}-\hat{\theta}]^{2}.$

Towards a formal statement of the inference result, define the following
moments:

$\sigma^{2}=\mathbb{E}[\psi_{0}(Y,W,D,Z,X)^{2}],\quad\chi^{3}=\mathbb{E}[|\psi_{0}(Y,W,D,Z,X)|^{3}],\quad\omega^{4}=\mathbb{E}[\psi_{0}(Y,W,D,Z,X)^{4}],$

where

$\psi_{0}(Y,W,D,Z,X)=h_{0}(d,X,W)+\alpha_{0}(D,X,Z)\\{Y-h_{0}(D,X,W)\\}-\theta_{0}$

is the asymptotic influence of each observation. Next, I define mean square
error.

######  Definition 2 (Mean square error).

Write the mean square error $\mathcal{R}(\hat{h}_{\ell})$ and the projected
mean square error $\mathcal{P}(\hat{h}_{\ell})$ of $\hat{h}_{\ell}$ trained on
observations indexed by $I^{c}_{\ell}$ as

$\displaystyle\mathcal{R}(\hat{h}_{\ell})$
$\displaystyle=\mathbb{E}[\\{\hat{h}_{\ell}(D,X,W)-h_{0}(D,X,W)\\}^{2}\mid
I^{c}_{\ell}]$ $\displaystyle\mathcal{P}(\hat{h}_{\ell})$
$\displaystyle=\mathbb{E}([\mathbb{E}\\{\hat{h}_{\ell}(D,X,W)-h_{0}(D,X,W)\mid
D,X,Z,I^{c}_{\ell}\\}]^{2}\mid I^{c}_{\ell}).$

Likewise define $\mathcal{R}(\hat{\alpha}_{\ell})$ and
$\mathcal{P}(\hat{\alpha}_{\ell})$:

$\displaystyle\mathcal{R}(\hat{\alpha}_{\ell})$
$\displaystyle=\mathbb{E}[\\{\hat{\alpha}_{\ell}(D,X,Z)-\alpha_{0}(D,X,Z)\\}^{2}\mid
I^{c}_{\ell}]$ $\displaystyle\mathcal{P}(\hat{\alpha}_{\ell})$
$\displaystyle=\mathbb{E}([\mathbb{E}\\{\hat{\alpha}_{\ell}(D,X,Z)-\alpha_{0}(D,X,Z)\mid
D,X,W,I^{c}_{\ell}\\}]^{2}\mid I^{c}_{\ell}).$

Sufficiently fast rates of mean square error and projected mean square error
imply Gaussian approximation. The following lemma summarizes the rate
conditions.

######  Lemma 1 (Semiparametric inference; Corollary 5.1 of [Chernozhukov et
al., 2021]).

Assume the propensity score $\pi_{0}(x,w)$ is bounded away from zero and one
and the following regularity conditions hold for some absolute constant
$C<\infty$:

$\mathbb{E}[\\{Y-h_{0}(D,X,W)\\}^{2}\mid D,X,W]\leq
C,\quad\|\alpha_{0}\|_{\infty}\leq C,\quad\|\hat{\alpha}_{\ell}\|_{\infty}\leq
C.$

Further assume the following learning rate conditions, as $n\rightarrow\infty$

  1. 1.

$\left\\{\left(\chi/\sigma\right)^{3}+\omega^{2}\right\\}n^{-1/2}=o(1)$;

  2. 2.

$\\{\mathcal{R}(\hat{h}_{\ell})\\}^{1/2}=o_{p}(1)$ and
$\\{\mathcal{R}(\hat{\alpha}_{\ell})\\}^{1/2}=o_{p}(1)$;

  3. 3.

$\\{n\mathcal{R}(\hat{h}_{\ell})\mathcal{R}(\hat{\alpha}_{\ell})\\}^{1/2}\wedge\\{n\mathcal{P}(\hat{h}_{\ell})\mathcal{R}(\hat{\alpha}_{\ell})\\}^{1/2}\wedge\\{n\mathcal{R}(\hat{h}_{\ell})\mathcal{P}(\hat{\alpha}_{\ell})\\}^{1/2}=o_{p}(1)$.

Then the estimator $\hat{\theta}$ in Algorithm 4 is consistent and
asymptotically Gaussian, and the confidence interval in Algorithm 4 includes
$\theta_{0}$ with probability approaching the nominal level. Formally,

$\hat{\theta}\overset{p}{\rightarrow}\theta_{0},\quad\sigma^{-1}n^{1/2}(\hat{\theta}-\theta_{0})\overset{d}{\rightarrow}\mathcal{N}(0,1),\quad\mathbb{P}\left\\{\theta_{0}\in\left(\hat{\theta}\pm
1.96\hat{\sigma}n^{-1/2}\right)\right\\}\rightarrow 0.95.$

In summary, the algorithmic techniques developed in this paper may be combined
with semiparametric theory as long as the rate conditions in Lemma 1 are
satisfied. Theorem 3 in the main text provides a $\sup$ norm guarantee for the
primary confounding bridge $\hat{h}$, which implies slow rates of
$\mathcal{P}(\hat{h}_{\ell})$ and $\mathcal{R}(\hat{h}_{\ell})$. [Singh et
al., 2019, Theorem 4] directly provides fast rates of
$\mathcal{P}(\hat{h}_{\ell})$. [Kallus et al., 2021, Theorems 4 and 8] and
[Ghassami et al., 2021, Theorem 5 and Lemma 1] provide fast rates of
$\mathcal{P}(\hat{\alpha}_{\ell})$ and slow rates of
$\mathcal{R}(\hat{\alpha}_{\ell})$ for minimax kernel estimators of the
secondary confounding bridge. An interesting direction for future work would
be to develop a kernel ridge regression estimator for the secondary
confounding bridge.

##  Appendix B Relevance and existence

In this appendix, I revisit the high level conditions in Assumption 2. These
conditions are standard in the negative control and instrumental variable
literatures. I begin by illustrating how irrelevance of negative controls
violates existence. Then I characterize existence and completeness in the
RKHS. This characterization appears to be absent from previous work on
nonparametric instrumental variable regression in the RKHS.

###  B.1 Relevance

Existence of the confounding bridge is fundamentally connected to the
relevance of negative controls, as articulated in Proposition 1. The heart of
the argument is as follows.

######  Lemma 2 (Rephrasing existence).

Suppose Assumption 1 holds. The existence condition in Assumption 2 holds if
and only if there exists a solution $h_{0}$ to the operator equation

$\gamma_{0}(d,x,z)=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u|d,x,z).$

###### Proof.

As in [Miao et al., 2018, Theorem 1], Assumption 1 implies

$\displaystyle
W\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}D,Z|U,X.$

Hence

$\displaystyle\mathbb{P}(w|d,x,z)$
$\displaystyle=\int\mathbb{P}(w,u|d,x,z)\mathrm{d}u$
$\displaystyle=\int\mathbb{P}(w|u,d,x,z)\mathrm{d}\mathbb{P}(u|d,x,z)$
$\displaystyle=\int\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u|d,x,z).$

Therefore

$\displaystyle\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|d,x,z)$
$\displaystyle=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u|d,x,z).$

∎

For interpretation, consider the special case where $(U,Z,W)$ are discrete
with finite supports $(\mathcal{U},\mathcal{Z},\mathcal{W})$, respectively.
Then for any fixed $(\bar{d},\bar{x})$, the following representations are
possible [Miao et al., 2018, Shi et al., 2020]:

$\gamma_{0}(\bar{d},\bar{x},z)\in\mathbb{R}^{1\times|\mathcal{Z}|},\quad
h_{0}(\bar{d},\bar{x},w)\in\mathbb{R}^{1\times|\mathcal{W}|},\quad\mathbb{P}(w|u,\bar{x})\in\mathbb{R}^{|\mathcal{W}|\times|\mathcal{U}|},\quad\mathbb{P}(u|\bar{d},\bar{x},z)\in\mathbb{R}^{|\mathcal{U}|\times|\mathcal{Z}|}.$

For this special case, it is clear from the expression in Lemma 2

$\gamma_{0}(\bar{d},\bar{x},z)=\int
h_{0}(\bar{d},\bar{x},w)\mathrm{d}\mathbb{P}(w|u,\bar{x})\mathrm{d}\mathbb{P}(u|\bar{d},\bar{x},z)$

that there exists a solution $h_{0}$ when $|\mathcal{W}|\geq|\mathcal{U}|$,
$|\mathcal{Z}|\geq|\mathcal{U}|$, and both matrices $\mathbb{P}(w|u,\bar{x})$
and $\mathbb{P}(u|\bar{d},\bar{x},z)$ are full rank for any fixed values
$(\bar{d},\bar{x})$ [Shi et al., 2020, Lemma 1].
$|\mathcal{W}|\geq|\mathcal{U}|$ and $|\mathcal{Z}|\geq|\mathcal{U}|$ mean
that the negative controls are expressive enough relative to the unobserved
confounder. Full rank $\mathbb{P}(w|u,\bar{x})$ and
$\mathbb{P}(u|\bar{d},\bar{x},z)$ mean that the conditional distributions are
non-degenerate; variation in the negative controls is relevant for recovering
variation in the unobserved confounder. When $|\mathcal{W}|>|\mathcal{U}|$ or
$|\mathcal{Z}|>|\mathcal{U}|$, the solution $h_{0}$ is not unique.
Nonetheless, the completeness condition ensures that treatment effects are
point identified.222In particular, completeness is deployed in the proof of
Theorem 1 to argue that, despite the fact that $h_{0}$ may be non-unique, the
partial mean $\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x)$ is unique.

Next, I interpret the special case in which $(U,Z,W)$ are continuous with
supports $(\mathcal{U},\mathcal{Z},\mathcal{W})$, respectively. I uncover the
sense in which the existence assumption may be quite stringent. As before, fix
$(\bar{d},\bar{x})$. Now, fix $(z,z^{\prime})$ such that
$\gamma_{0}(\bar{d},\bar{x},z)\neq\gamma_{0}(\bar{d},\bar{x},z^{\prime})$. If,
for these choices,
$\mathbb{P}(u|\bar{d},\bar{x},z)=\mathbb{P}(u|\bar{d},\bar{x},z^{\prime})$,
then by Lemma 2 the confounding bridge does not exist. In other words, to
violate Assumption 2, there simply needs to exist some $(\bar{d},\bar{x})$
stratum such that $\gamma_{0}$ takes on different values at $z$ versus
$z^{\prime}$ yet the conditional densities of unobserved confounding coincide.

Finally, I prove the result given in the main text.

###### Proof of Proposition 1.

With Lemma 2, I prove each claim separately.

  1. 1.

$Z\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}U|D,X$ means
that $\mathbb{P}(u|d,x,z)=\mathbb{P}(u|d,x)$. Towards a contradiction, suppose
$h_{0}$ exists. Then by Lemma 2,

$\gamma_{0}(d,x,z)=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u|d,x).$

The RHS does not depend on $z$, implying
$\gamma_{0}(d,x,z)=\gamma_{0}(d,x,z^{\prime})$, which violates the hypothesis
that $\gamma_{0}$ varies in $z$.

  2. 2.

$W\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}U|X$ means that
$\mathbb{P}(w|u,x)=\mathbb{P}(w|x)$. Towards a contradiction, suppose $h_{0}$
exists. Then by Lemma 2,

$\gamma_{0}(d,x,z)=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|x)\mathrm{d}\mathbb{P}(u|d,x,z)=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|x).$

The RHS does not depend on $z$, implying
$\gamma_{0}(d,x,z)=\gamma_{0}(d,x,z^{\prime})$, which violates the hypothesis
that $\gamma_{0}$ varies in $z$.

∎

###  B.2 Existence

Next I revisit the existence condition. In particular, I present (i) a lemma
to characterize existence in ill posed inverse problems; (ii) a proposition
verifying existence in the nonparametric negative control problem; and (iii) a
proposition verifying existence in the RKHS negative control problem.

####  B.2.1 Picard’s criterion

######  Lemma 3 (Picard’s criterion; Theorem 15.18 of [Kress, 1989]).

Let $K:H_{1}\rightarrow H_{2}$ be a compact operator from the Hilbert space
$H_{1}$ to the Hilbert space $H_{2}$, with singular value decomposition
$(e^{1}_{k},\eta_{k},e_{k}^{2})^{\infty}_{k=1}$. Given the function $g\in
H_{2}$, the equation $Kf=g$ has a solution if and only if

  1. 1.

Inclusion: $g\in\mathcal{N}(K^{*})^{\perp}$ i.e. $g$ is an element of the
orthogonal complement to the null space of the adjoint operator $K^{*}$;

  2. 2.

Penalized square summability: $\sum_{k=1}^{\infty}\eta_{k}^{-2}\langle
g,e_{k}^{2}\rangle_{H_{2}}<\infty$.

Note that $e_{k}^{1}\in H_{1}$, $e_{k}^{2}\in H_{2}$, and
$K^{*}:H_{2}\rightarrow H_{1}$ by construction. Various works use this
technical lemma to prove existence of the confounding bridge [Miao et al.,
2018, Deaner, 2018]. Inclusion essentially means that $g$ is in the
appropriate row space. Penalized square summability means that $K^{-}g$ has a
finite norm since $g$ is not too aligned with the right singular functions of
$K$ relative to the spectral decay.

In what follows, I provide conditions in $\mathbb{L}_{2}$ and subsequently
conditions in the RKHS that verify the abstract conditions in Lemma 3. The
RKHS conditions are stronger versions of the $\mathbb{L}_{2}$ conditions,
analogous to how the RKHS is a subset of $\mathbb{L}_{2}$ as we have seen in
Section 3.

####  B.2.2 Previous work

I quote a proposition verifying existence of the confounding bridge in the
negative control problem. To begin, fix the values $(\bar{d},\bar{x})$. Given
$(\bar{d},\bar{x})$, denote by $\mathbb{L}_{2}(\mathbb{P}(w|\bar{d},\bar{x}))$
the space of functions $g:\mathcal{W}\rightarrow\mathbb{R}$ that are square
integrable with respect to the conditional distribution
$\mathbb{P}(w|\bar{d},\bar{x})$, i.e. $\int
g(w)^{2}\mathrm{d}\mathbb{P}(w|\bar{d},\bar{x})<\infty$. Likewise for
$\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$. Then define the operator

$E_{\bar{d},\bar{x}}:\mathbb{L}_{2}(\mathbb{P}(w|\bar{d},\bar{x}))\rightarrow\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x})),\quad
f(\cdot)\mapsto\mathbb{E}[f(W)|D=\bar{d},X=\bar{x},Z=\cdot].$

Matching symbols with Lemma 3,
$H_{1}=\mathbb{L}_{2}(\mathbb{P}(w|\bar{d},\bar{x}))$,
$H_{2}=\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$, and the adjoint
operator is

$E^{*}_{\bar{d},\bar{x}}:\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))\rightarrow\mathbb{L}_{2}(\mathbb{P}(w|\bar{d},\bar{x})),\quad
g(\cdot)\mapsto\mathbb{E}[g(Z)|D=\bar{d},X=\bar{x},W=\cdot].$

######  Proposition 4 (Existence in $\mathbb{L}_{2}$ for fixed
$(\bar{d},\bar{x})$; Proposition 1 of [Miao et al., 2018]).

Fix $(\bar{d},\bar{x})$. Denote by $f(w|\bar{d},\bar{x},z)$ and
$f(z|\bar{d},\bar{x},w)$ the densities of $\mathbb{P}(w|\bar{d},\bar{x},z)$
and $\mathbb{P}(z|\bar{d},\bar{x},w)$. Suppose

  1. 1.

Regularity: $\int
f(w|\bar{d},\bar{x},z)f(z|\bar{d},\bar{x},w)\mathrm{d}w\mathrm{d}z<\infty$;

  2. 2.

Completeness: for any function
$g\in\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$,

$\mathbb{E}[g(Z)|D=\bar{d},X=\bar{x},W=w]=0\quad\forall w\iff g(Z)=0;$

  3. 3.

Correct specification:
$\gamma_{0}(\bar{d},\bar{x},\cdot)\in\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$;

  4. 4.

Penalized square summability:
$\sum_{k=1}^{\infty}\eta_{k}^{-2}\langle\gamma_{0}(\bar{d},\bar{x},\cdot),e^{z}_{k}\rangle_{\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))}<\infty$,
where the singular value decomposition exists due to the regularity condition.

Then for any fixed $(\bar{d},\bar{x})$, there exists some
$f_{\bar{d},\bar{x}}\in\mathbb{L}_{2}(\mathbb{P}(w|\bar{d},\bar{x}))$ such
that

$\gamma_{0}(\bar{d},\bar{x},z)=[E_{\bar{d},\bar{x}}f_{\bar{d},\bar{x}}](z)=\mathbb{E}[f_{\bar{d},\bar{x}}(W)|D=\bar{d},X=\bar{x},Z=z].$

Regularity pertains to the conditional distribution $\mathbb{P}(w|d,x,z)$ in
the integral operator equation. It ensures compactness of the corresponding
conditional expectation operator in order to appeal to Lemma 3. Completeness
is a technical condition from the NPIV literature which, together with correct
specification of $\gamma_{0}$, implies the inclusion condition of Lemma 3.
Penalized square summability is identical to Lemma 3.

###### Proof.

I verify the conditions of Lemma 3. Rewriting the regularity condition,

$\displaystyle\int
f(w|\bar{d},\bar{x},z)f(z|\bar{d},\bar{x},w)\mathrm{d}w\mathrm{d}z$
$\displaystyle=\int\frac{f(z,w|\bar{d},\bar{x})^{2}}{f(w|\bar{d},\bar{x})f(z|\bar{d},\bar{x})}\mathrm{d}w\mathrm{d}z$

which is a sufficient condition for compactness of $E_{\bar{d},\bar{x}}$
[Carrasco et al., 2007, Example 2.3]. By compactness, the singular value
decomposition $(e_{k}^{w},\eta_{k},e_{k}^{z})_{k=1}^{\infty}$ exists, where
$e_{k}^{w}\in\mathbb{L}_{2}(\mathbb{P}(w|\bar{d},\bar{x}))$ and
$e_{k}^{z}\in\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$.

Next, I verify inclusion by appealing to completeness and correct
specification. Towards this end, I first argue that
$\mathcal{N}(E_{\bar{d},\bar{x}}^{*})^{\perp}=\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$.
It suffices to show $\mathcal{N}(E_{\bar{d},\bar{x}}^{*})=0$, i.e. the null
space only consists of the zero function. Consider any
$g\in\mathcal{N}(E_{\bar{d},\bar{x}}^{*})$. By definition of the null space,

$\displaystyle 0(\cdot)$ $\displaystyle=[E_{\bar{d},\bar{x}}^{*}g](\cdot)$
$\displaystyle=\mathbb{E}[g(Z)|D=\bar{d},X=\bar{x},W=\cdot].$

By completeness, I conclude that $\mathcal{N}(E_{\bar{d},\bar{x}}^{*})=0$.
Therefore
$\mathcal{N}(E_{\bar{d},\bar{x}}^{*})^{\perp}=\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$.
Finally, correct specification implies
$\gamma_{0}(\bar{d},\bar{x},\cdot)\in\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$.

Penalized square summability is immediate. ∎

######  Corollary 2 (Existence in $\mathbb{L}_{2}$; Proposition 1 of [Miao et
al., 2018]).

If the conditions of Proposition 4 hold for each $(\bar{d},\bar{x})$, then
there exists a confounding bridge $h_{0}$ such that

$\gamma_{0}(d,x,z)=\mathbb{E}[h_{0}(D,X,W)|D=d,X=x,Z=z].$

###### Proof.

For each $(\bar{d},\bar{x})$, set
$h_{0}(\bar{d},\bar{x},w)=f_{\bar{d},\bar{x}}(w)$. ∎

####  B.2.3 RKHS

In $\mathbb{L}_{2}$, I introduced the function spaces
$\mathbb{L}_{2}(\mathbb{P}(w|\bar{d},\bar{x}))$ and
$\mathbb{L}_{2}(\mathbb{P}(z|\bar{d},\bar{x}))$ which are induced by
conditioning on $(\bar{d},\bar{x})$. Now, I introduce analogous function
spaces $\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$ and
$\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$. In particular, I demonstrate
that these function spaces are RKHSs and characterize their kernels. For now,
I simply state their kernels. Later in this section, I will demonstrate how
these kernels relate to the tensor product RKHS construction.

Recall that the kernels for $\mathcal{H}_{\mathcal{W}}$ and
$\mathcal{H}_{\mathcal{Z}}$ are $k(w,w^{\prime})$ and $k(z,z^{\prime})$,
respectively. The kernels for $\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$
and $\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$ are

$k_{\bar{d},\bar{x}}(w,w^{\prime})=k(\bar{d},\bar{d})k(\bar{x},\bar{x})k(w,w^{\prime}),\quad
k_{\bar{d},\bar{x}}(z,z^{\prime})=k(\bar{d},\bar{d})k(\bar{x},\bar{x})k(z,z^{\prime}).$

Define the scalar $c_{\bar{d},\bar{x}}=k(\bar{d},\bar{d})k(\bar{x},\bar{x})$.
Then

$k_{\bar{d},\bar{x}}(w,w^{\prime})=c_{\bar{d},\bar{x}}\cdot
k(w,w^{\prime}),\quad
k_{\bar{d},\bar{x}}(z,z^{\prime})=c_{\bar{d},\bar{x}}\cdot k(z,z^{\prime})$

and it is clear that these induced kernels are simply rescaled versions of the
original kernels $k(w,w^{\prime})$ and $k(z,z^{\prime})$ according to the
conditioned value $(\bar{d},\bar{x})$, so they remain positive definite and
hence valid. Then define the operator

$E_{\bar{d},\bar{x}}:\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}\rightarrow\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}},\quad
f(\cdot)\mapsto\mathbb{E}[f(W)|D=\bar{d},X=\bar{x},Z=\cdot].$

Matching symbols with Lemma 3,
$H_{1}=\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$,
$H_{2}=\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$, and the adjoint operator
is

$E^{*}_{\bar{d},\bar{x}}:\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}\rightarrow\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}},\quad
g(\cdot)\mapsto\mathbb{E}[g(Z)|D=\bar{d},X=\bar{x},W=\cdot].$

######  Proposition 5 (Existence in the RKHS for fixed $(\bar{d},\bar{x})$).

Suppose Assumption 4 holds, as well as

  1. 1.

Regularity:
$E_{\bar{d},\bar{x}}\in\mathcal{L}_{2}(\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}},\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}})$,
i.e. the space of Hilbert-Schmidt operators from
$\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$ to
$\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$;

  2. 2.

Completeness:
$closure(span\\{\mu^{\bar{d},\bar{x}}_{z}(\bar{d},\bar{x},w)\\}_{w\in\mathcal{W}})=\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$;

  3. 3.

Correct specification:
$\gamma_{0}(\bar{d},\bar{x},\cdot)\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$;

  4. 4.

Penalized square summability:
$\sum_{k=1}^{\infty}\eta_{k}^{-2}\langle\gamma_{0}(\bar{d},\bar{x},\cdot),e_{k}^{z}\rangle^{2}_{\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}}<\infty$.

Then for any fixed $(\bar{d},\bar{x})$, there exists some
$f_{\bar{d},\bar{x}}\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$ such that

$\gamma_{0}(\bar{d},\bar{x},z)=[E_{\bar{d},\bar{x}}f_{\bar{d},\bar{x}}](z)=\mathbb{E}[f_{\bar{d},\bar{x}}(W)|D=\bar{d},X=\bar{x},Z=z].$

I name the conditions of Proposition 5 to match the conditions of Proposition
4, which are standard in the literature. Regularity is again a condition on
the smoothness of the distribution $\mathbb{P}(w|d,x,z)$. Completeness becomes
a condition on how well the encodings of conditional distributions of $Z$ can
recover the full space. Correct specification is with respect to the RKHS
rather than $\mathbb{L}_{2}$. Penalized square summability is as before,
though using the appropriate inner product.

###### Proof.

Hilbert-Schmidt operators are compact [Carrasco et al., 2007, Theorem 2.32],
verifying the compactness requirement. By compactness, the singular value
decomposition $(e_{k}^{w},\eta_{k},e_{k}^{z})_{k=1}^{\infty}$ exists, where
$e_{k}^{w}\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$ and
$e_{k}^{z}\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$.

Next, I verify inclusion by appealing to completeness and correct
specification. Towards this end, I first argue that
$\mathcal{N}(E_{\bar{d},\bar{x}}^{*})^{\perp}=\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$.
It suffices to show $\mathcal{N}(E_{\bar{d},\bar{x}}^{*})=0$, i.e. the null
space is the zero function. Consider any
$g\in\mathcal{N}(E_{\bar{d},\bar{x}}^{*})$. By definition of the null space,

$\displaystyle 0(\cdot)$ $\displaystyle=[E_{\bar{d},\bar{x}}^{*}g](\cdot)$
$\displaystyle=\mathbb{E}[g(Z)|D=\bar{d},X=\bar{x},W=\cdot].$

In Assumption 4, I impose that the kernels are bounded. This assumption has
several implications. First, the feature maps are Bochner integrable
[Steinwart and Christmann, 2008, Definition A.5.20]. Bochner integrability
permits the exchange of expectation and inner product. Second, the mean
embeddings exist. Third, the induced kernel
$k_{\bar{d},\bar{x}}(z,z^{\prime})$ is also bounded and hence the induced RKHS
$\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$ inherits these favorable
properties. Therefore

$\displaystyle\mathbb{E}[g(Z)|D=\bar{d},X=\bar{x},W=\cdot]$
$\displaystyle=\int g(z)\mathrm{d}\mathbb{P}(z|\bar{d},\bar{x},\cdot)$
$\displaystyle=\int\langle
g,\phi^{\bar{d},\bar{x}}(z)\rangle_{\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}}\mathrm{d}\mathbb{P}(z|\bar{d},\bar{x},\cdot)$
$\displaystyle=\left\langle
g,\int\phi^{\bar{d},\bar{x}}(z)\mathrm{d}\mathbb{P}(z|\bar{d},\bar{x},\cdot)\right\rangle_{\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}}$
$\displaystyle=\langle
g,\mu^{\bar{d},\bar{x}}_{z}(\bar{d},\bar{x},\cdot)\rangle_{\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}}$

where $\phi^{\bar{d},\bar{x}}(z)$ is the feature map of
$\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$ and
$\mu^{\bar{d},\bar{x}}_{z}(\bar{d},\bar{x},w)=\int\phi^{\bar{d},\bar{x}}(w)\mathrm{d}\mathbb{P}(z|\bar{d},\bar{x},w)$
is the conditional mean embedding of $\mathbb{P}(z|\bar{d},\bar{x},w)$. In
summary,

$0(\cdot)=\langle
g,\mu^{\bar{d},\bar{x}}_{z}(\bar{d},\bar{x},\cdot)\rangle_{\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}}.$

By hypothesis,
$g\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}=closure(span\\{\mu^{\bar{d},\bar{x}}_{z}(\bar{d},\bar{z},w)\\}_{w\in\mathcal{W}})$.
Since $\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$ is a Hilbert space,

$\langle
g,\mu^{\bar{d},\bar{x}}_{z}(\bar{d},\bar{x},w)\rangle_{\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}}=0\quad\forall
w\in\mathcal{W}\iff g=0.$

Combining these results, $g\in\mathcal{N}(E_{\bar{d},\bar{x}}^{*})$ implies
$g=0$, so $\mathcal{N}(E_{\bar{d},\bar{x}}^{*})=0$ and
$\mathcal{N}(E_{\bar{d},\bar{x}}^{*})^{\perp}=\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$.
Finally, correct specification implies
$\gamma_{0}(\bar{d},\bar{x},\cdot)\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$.

Penalized square summability is immediate. ∎

######  Corollary 3 (Existence in the RKHS).

If the conditions of Proposition 5 hold for each $(\bar{d},\bar{x})$, then
there exists a confounding bridge $h_{0}$ such that

$\gamma_{0}(d,x,z)=\mathbb{E}[h_{0}(D,X,W)|D=d,X=x,Z=z].$

###### Proof.

For each $(\bar{d},\bar{x})$, set
$h_{0}(\bar{d},\bar{x},w)=f_{\bar{d},\bar{x}}(w)$. ∎

Finally, I relate the spaces $\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$ and
$\mathcal{H}_{\mathcal{Z}}^{\bar{d},\bar{x}}$ with the tensor product
construction in the main text. In particular, I show how the spaces
$\mathcal{H}_{\mathcal{W}}^{\bar{d},\bar{x}}$ and
$\mathcal{H}_{\mathcal{Z}}^{\bar{d},\bar{x}}$ can be induced by a tensor
product construction.

######  Proposition 6 (Relation among kernels).

Suppose Assumption 4 holds.

  1. 1.

If
$\gamma_{0}\in\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}}$
then
$\gamma_{0}(\bar{d},\bar{x},\cdot)\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{Z}}$.

  2. 2.

If
$h_{0}\in\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{W}}$
then
$h_{0}(\bar{d},\bar{x},\cdot)\in\mathcal{H}^{\bar{d},\bar{x}}_{\mathcal{W}}$.

The succinct assumption that that the reduced form belongs to a tensor product
RKHS, i.e.
$\gamma_{0}\in\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}}$,
implies the correct specification condition in Proposition 5 for each value
$(\bar{d},\bar{x})$. In the main text, I simply assume that the confounding
bridge exists and belongs to a tensor product RKHS, i.e.
$h_{0}\in\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{W}}$.
Proposition 6 demonstrates that this assumption is coherent with conditions
that imply existence in Proposition 5. I pose as a question for future
research how to characterize the conditions with which one can prove, rather
than assume, that the confounding bridge is an element of an RKHS.

###### Proof.

I prove each result separately.

  1. 1.

If
$\gamma_{0}\in\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}}$,
then

$\displaystyle\gamma_{0}(\bar{d},\bar{x},z)$
$\displaystyle=\langle\gamma_{0},\phi(\bar{d})\otimes\phi(\bar{x})\otimes\phi(z)\rangle_{\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}}}.$

Moreover,

$\displaystyle k_{\bar{d},\bar{x}}(z,z^{\prime})$
$\displaystyle=k(\bar{d},\bar{d})k(\bar{x},\bar{x})k(z,z^{\prime})$
$\displaystyle=\langle\phi(\bar{d})\otimes\phi(\bar{x})\otimes\phi(z),\phi(\bar{d})\otimes\phi(\bar{x})\otimes\phi(z^{\prime})\rangle_{\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}}}$

so the feature maps coincide.

  2. 2.

If
$h_{0}\in\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{W}}$,
then

$\displaystyle h_{0}(\bar{d},\bar{x},w)$
$\displaystyle=\langle\gamma_{0},\phi(\bar{d})\otimes\phi(\bar{x})\otimes\phi(w)\rangle_{\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{W}}}.$

Moreover,

$\displaystyle k_{\bar{d},\bar{x}}(w,w^{\prime})$
$\displaystyle=k(\bar{d},\bar{d})k(\bar{x},\bar{x})k(w,w^{\prime})$
$\displaystyle=\langle\phi(\bar{d})\otimes\phi(\bar{x})\otimes\phi(w),\phi(\bar{d})\otimes\phi(\bar{x})\otimes\phi(w^{\prime})\rangle_{\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{W}}}$

so the feature maps coincide.

∎

##  Appendix C Identification proof

To lighten notation, I abbreviate conditional expectations and conditional
probabilities. For example, I write

$\gamma_{0}(d,x,z)=\mathbb{E}[Y|D=d,X=x,Z=z]=\mathbb{E}[Y|d,x,z].$

###### Proof of Proposition 2.

$h_{0}$ is defined as a solution to
$\gamma_{0}(d,x,z)=\mathbb{E}[h(D,X,W)|D=d,X=x,Z=z]$, which exists by
Assumption 2. The reduced form $\gamma_{0}(d,x,z)=\mathbb{E}[Y|D=d,X=x,Z=z]$
is the same across populations since $\mathbb{P}(Y|D,X,Z)$ is invariant in
Assumption 3. The conditional expectation operator is also the same across
populations since $\mathbb{P}(W|D,X,Z)$ is invariant in Assumption 3. ∎

######  Proposition 7.

Suppose the conditions of Theorem 1 hold. Then

$\mathbb{E}[Y|d,u,x]=\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x)$

For $\theta_{0}^{CATE}$, replace $x$ with $(v,x)$.

###### Proof.

I generalize [Miao et al., 2018, Theorem 1]. By Assumption 1

$\displaystyle
Y\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}Z|U,D,X$
$\displaystyle
W\raisebox{0.50003pt}{\rotatebox[origin={c}]{90.0}{$\models$}}D,Z|U,X.$

The former implies

$\displaystyle\gamma_{0}(d,x,z)$ $\displaystyle=\mathbb{E}[Y|d,x,z]$
$\displaystyle=\int\mathbb{E}[Y,u|d,x,z]\mathrm{d}u$
$\displaystyle=\int\mathbb{E}[Y|u,d,x,z]\mathrm{d}\mathbb{P}(u|d,x,z)$
$\displaystyle=\int\mathbb{E}[Y|u,d,x]\mathrm{d}\mathbb{P}(u|d,x,z).$

The latter implies

$\displaystyle\mathbb{P}(w|d,x,z)$
$\displaystyle=\int\mathbb{P}(w,u|d,x,z)\mathrm{d}u$
$\displaystyle=\int\mathbb{P}(w|u,d,x,z)\mathrm{d}\mathbb{P}(u|d,x,z)$
$\displaystyle=\int\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u|d,x,z).$

Using these results and existence

$\displaystyle\int\mathbb{E}[Y|u,d,x]\mathrm{d}\mathbb{P}(u|d,x,z)$
$\displaystyle=\gamma_{0}(d,x,z)$ $\displaystyle=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|d,x,z)$ $\displaystyle=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u|d,x,z).$

Hence by completeness

$\mathbb{E}[Y|u,d,x]=\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x).$

∎

###### Proof of Theorem 1.

I prove each result, appealing to Assumption 1 and Proposition 7.

$\displaystyle\theta_{0}^{ATE}(d)$ $\displaystyle=\mathbb{E}[Y^{(d)}]$
$\displaystyle=\int\mathbb{E}[Y^{(d)}|u,x]\mathrm{d}\mathbb{P}(u,x)$
$\displaystyle=\int\mathbb{E}[Y^{(d)}|d,u,x]\mathrm{d}\mathbb{P}(u,x)$
$\displaystyle=\int\mathbb{E}[Y|d,u,x]\mathrm{d}\mathbb{P}(u,x)$
$\displaystyle=\int
h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u,x)$
$\displaystyle=\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w,u,x)$
$\displaystyle=\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w,x);$
$\displaystyle\theta_{0}^{DS}(d,\tilde{\mathbb{P}})$
$\displaystyle=\mathbb{E}_{\tilde{\mathbb{P}}}[Y^{(d)}]$
$\displaystyle=\int\mathbb{E}_{\tilde{\mathbb{P}}}[Y^{(d)}|u,x]\mathrm{d}\tilde{\mathbb{P}}(u,x)$
$\displaystyle=\int\mathbb{E}_{\tilde{\mathbb{P}}}[Y^{(d)}|d,u,x]\mathrm{d}\tilde{\mathbb{P}}(u,x)$
$\displaystyle=\int\mathbb{E}_{\tilde{\mathbb{P}}}[Y|d,u,x]\mathrm{d}\tilde{\mathbb{P}}(u,x)$
$\displaystyle=\int
h_{0}(d,x,w)\mathrm{d}\tilde{\mathbb{P}}(w|u,x)\mathrm{d}\tilde{\mathbb{P}}(u,x)$
$\displaystyle=\int h_{0}(d,x,w)\mathrm{d}\tilde{\mathbb{P}}(w,u,x)$
$\displaystyle=\int h_{0}(d,x,w)\mathrm{d}\tilde{\mathbb{P}}(w,x);$
$\displaystyle\theta_{0}^{ATT}(d,d^{\prime})$
$\displaystyle=\mathbb{E}[Y^{(d^{\prime})}|D=d]$
$\displaystyle=\int\mathbb{E}[Y^{(d^{\prime})}|d,u,x]\mathrm{d}\mathbb{P}(u,x|d)$
$\displaystyle=\int\mathbb{E}[Y^{(d^{\prime})}|d^{\prime},u,x]\mathrm{d}\mathbb{P}(u,x|d)$
$\displaystyle=\int\mathbb{E}[Y|d^{\prime},u,x]\mathrm{d}\mathbb{P}(u,x|d)$
$\displaystyle=\int
h_{0}(d^{\prime},x,w)\mathrm{d}\mathbb{P}(w|u,x)\mathrm{d}\mathbb{P}(u,x|d)$
$\displaystyle=\int
h_{0}(d^{\prime},x,w)\mathrm{d}\mathbb{P}(w|d,u,x)\mathrm{d}\mathbb{P}(u,x|d)$
$\displaystyle=\int h_{0}(d^{\prime},x,w)\mathrm{d}\mathbb{P}(w,u,x|d)$
$\displaystyle=\int h_{0}(d^{\prime},x,w)\mathrm{d}\mathbb{P}(w,x|d);$
$\displaystyle\theta_{0}^{CATE}(d,v)$ $\displaystyle=\mathbb{E}[Y^{(d)}|V=v]$
$\displaystyle=\int\mathbb{E}[Y^{(d)}|u,v,x]\mathrm{d}\mathbb{P}(u,x|v)$
$\displaystyle=\int\mathbb{E}[Y^{(d)}|d,u,v,x]\mathrm{d}\mathbb{P}(u,x|v)$
$\displaystyle=\int\mathbb{E}[Y|d,u,v,x]\mathrm{d}\mathbb{P}(u,x|v)$
$\displaystyle=\int
h_{0}(d,v,x,w)\mathrm{d}\mathbb{P}(w|u,v,x)\mathrm{d}\mathbb{P}(u,x|v)$
$\displaystyle=\int h_{0}(d,v,x,w)\mathrm{d}\mathbb{P}(w,u,x|v)$
$\displaystyle=\int h_{0}(d,v,x,w)\mathrm{d}\mathbb{P}(w,x|v).$

∎

##  Appendix D Discussion of the source condition

In this appendix, I provide further discussion of the smoothness assumption
that drives my analysis: the source condition. The source condition pertains
to estimation of $h_{0}$ rather than existence of $h_{0}$. In this appendix, I
discuss estimation assumptions; see Appendix B for a thorough discussion of
existence assumptions.

###  D.1 Source conditions in this work

A source condition is an approximation assumption that helps to control the
bias from ridge regularization. I place four source conditions in this work,
parametrized by $(c,c_{0},c_{1},c_{2})$. The source condition parametrized by
$c$ quantifies the smoothness of the confounding bridge $h_{0}$. The source
conditions parametrized by $(c_{0},c_{1},c_{2})$ quantify the smoothness of
the conditional distributions $\mathbb{P}(w|d,x,z)$, $\mathbb{P}(x,w|d)$, and
$\mathbb{P}(x,w|v)$. Due to the similarity of $(c_{0},c_{1},c_{2})$, in this
discussion I focus on only $(c,c_{0})$.

To understand the role of $(c,c_{0})$, recall the integral equation

$\gamma_{0}(d,x,z)=\int h_{0}(d,x,w)\mathrm{d}\mathbb{P}(w|d,x,z).$

The source conditions apply to both objects on the RHS. In Section 4, I
provide an RKHS construction to express the integral equation as

$\gamma_{0}(d,x,z)=\langle h_{0},\phi(d)\otimes\phi(x)\otimes
E_{0}^{*}[\phi(d)\otimes\phi(x)\otimes\phi(z)]\rangle_{\mathcal{H}}.$

The conditional expectation operator $E_{0}$ encodes the conditional
distribution $\mathbb{P}(w|d,x,z)$, and I quantify its smoothness by $c_{0}$.
I quantify the smoothness of the confounding bridge $h_{0}$ by $c$. Recall
that $\phi(d)$, $\phi(x)$, and $\phi(z)$ can be interpreted as dictionaries of
basis functions.

I propose an estimator of the confounding bridge $h_{0}$ that proceeds in two
stages similar to two stage least squares (2SLS): (i) estimate the conditional
expectation operator $E_{0}$ with a generalized kernel ridge regression; (ii)
estimate $h_{0}$ with a generalized kernel ridge regression, using the
estimator of $E_{0}$ from the first stage. See Section 4 for discussion of the
estimation procedure. Because both $(E_{0},h_{0})$ involve ridge
regularization, I place the source conditions $(c_{0},c)$ to control the
regularization bias. See Appendix G for the formal arguments by which the
source conditions imply bounds on regularization bias.

The source condition $c_{0}$ placed on $E_{0}$ means that the conditional
expectation operator for $\mathbb{P}(w|d,x,z)$ has rapidly decaying Fourier
coefficients as formalized in Section 3. In particular, $c_{0}$ quantifies how
fast the Fourier coefficients of $E_{0}$ decay relative to the eigenvalues of
the kernel of
$\mathcal{L}_{2}(\mathcal{H}_{\mathcal{W}},\mathcal{H}_{\mathcal{D}}\otimes\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{Z}})$.
Similarly, the source condition $c$ placed on $h_{0}$ means that the
confounding bridge has rapidly decaying Fourier coefficients as formalized in
Section 3. In particular, $c$ quantifies how fast the Fourier coefficients of
$h_{0}$ decay relative to the eigenvalues of the kernel of
$\mathcal{H}_{\mu}$.

###  D.2 Source conditions in ill posed inverse problems

As documented in the main text, the confounding bridge learning problem is
precisely the nonparametric instrumental variable (NPIV) regression problem
where $(D,X)$ are exogenous regressors, $W$ is an endogenous regressor, and
$Z$ is the instrument. Several works in the NPIV literature deploy source
conditions or similar assumptions including [Hall and Horowitz, 2005, Darolles
et al., 2011, Singh et al., 2019]. I demonstrate that the source conditions in