RieszNet and ForestRiesz: Automatic Debiased Machine Learning with Neural Nets and Random Forests
Authors: Victor Quintas-Martinez
Victor Chernozhukov
Vasilis Syrgkanis
Whitney Newey
Venue: ICLR 2022 Submitted (2022)

Abstract: Many causal and policy effects of interest are defined by linear functionals of high-dimensional or non-parametric regression functions. $\sqrt{n}$-consistent and asymptotically normal estimation of the object of interest requires debiasing to reduce the effects of regularization and/or model selection on the object of interest. Debiasing is typically achieved by adding a correction term to the plug-in estimator of the functional, that is derived based on a functional-specific theoretical derivation of what is known as the influence function and which leads to properties such as double robustness and Neyman orthogonality. We instead implement an automatic debiasing procedure based on automatically learning the Riesz representation of the linear functional using Neural Nets and Random Forests. Our method solely requires value query oracle access to the linear functional. We propose a multi-tasking Neural Net debiasing method with stochastic gradient descent minimization of a combined Reisz representer and regression loss, while sharing representation layers for the two functions. We also propose a random forest method which learns a locally linear representation of the Reisz function. Even though our methodology applies to arbitrary functionals, we experimentally find that it beats state of the art performance of the prior neural net based estimator of Shi et al. (2019) for the case of the average treatment effect functional. We also evaluate our method on the more challenging problem of estimating average marginal effects with continuous treatments, using semi-synthetic data of gasoline price changes on gasoline demand.

Under review as a conference paper at ICLR 2022
RIESZNET AND FORESTRIESZ: AUTOMATIC DEBIASED MACHINE LEARNING WITH NEURAL NETS AND RANDOM FORESTS
Anonymous authors Paper under double-blind review
ABSTRACT
Many causal and policy effects of interest are defined by linear functionals of high- dimensional non-parametrie regression functions. V7-cconistent and asymp- totically normal estimation of the object of interest requires debiasing to reduce the effects of regularization and/or model selection the object of interest. De- biasing is typically achieved by adding a correction term to the plug-in estimator of the functional, that is derived based a functional-specific theoretical deriva- tion of what is known as the influence function and which leads to properties such as double robustness and Neyman orthogonality. We instead implement an automatic debiasing procedure based on automatically learning the Riesz repre- sentation of the linear functional using Neural Nets and Random Forests. Our method solely requires value query oracle access to the linear functional. We pro- pose a multi-tasking Neural Net debiasing method with stochastic gradient descent minimization of combined Riesz representer and regression loss, while sharing representation layers for the two functions. We also propose a Random Forest method which learns locally linear representation of the Riesz function. Even though our methodology applies to arbitrary functionals, we experimentally find that it beats state of the art performance of the prior neural net based estimator of [Shi et al. (019)) for the case of the average treatment effect functional. We also evaluate our method on the more challenging problem estimating average marginal effects with continuous treatments, using semi-synthetic data of gasoline price changes on gasoline demand.
INTRODUCTION
A large number of problems in causal inference, off-policy evaluation and optimization and inter- pretable machine learning can viewed as estimating the average value of a moment function that depends unknown regression function:
8n E (W: go))), where :
= (Y, 2) and go (2) = E Z].
In most cases, Y will be the outcome of interest, and inputs Z (T, X) will include binary or continuous treatment T and covariates X. Prototypical examples include the estimation of average treatment effects, average policy effects, average derivatives and incremental policy effects.
Example (Average treatment effect). Here Z = (T, X) where T is a binary treatment indicator, and X are covariates. The object of interest and associated moment function are:
0g E(go(1, X)), m(W = x g(0,
If potential outcomes are conditionally independent of treatment T given covariates X, then this object is the average treatment effect (Rosenbaum & Rubin 1983).
Example 2 (Average policy effect). In the context of offline policy evaluation and optimization, our goal to optimize over space of assignment policies X - {0, 1), when having access to observational data collected by some unknown treatment policy. The policy value can also be formulated as the average of a linear moment:
0o (X)(((XX)))
go (0, X)) ) + go(0, m(W:g) x(X) -(0nxxxxxx(x))))) x)) g(0,
long-line of prior work has considered doubly-robuse approaches to optimizing over a space of candidate policies from observational data (see e.g. Dudik et al. 2017 Athey & Wager 2027)

---

Under review as a conference paper at ICLR 2022
Example (Average marginal effect / derivative). Here Z (T, where T is a continuously distributed policy variable of interest, X are covariates, and:
m( 0g (g( [X) ôt
8a- =E
ōt
This object is essentially the average slope in the partial dependence plot frequently used in work on interpretable machine learning (see e.g. ZZhao & Hastie 2027 Friedmmnn 2007] Molnar 2020)
Example 4 (Incremental policy effects). Here Z X),, where T is continuously distributed policy variable of interest, X are covariates, and π - [-1, 1] is an incremental policy of infinitesimally increasing or descreasing the treatment from its baseline value (see e.g. Athey & Wager 2021 The incremental value of such an infinitesimal policy change takes the form:
0o 10=EP ( [g0(T,X)]
m(W:g) - xx()) at
Such incremental policy effects can also be useful within the context of policy gradient algorithms in deep reinforcement learning. so as to take gradient steps towards better policy, and debiasing techniques have already been used in that context (Grathwohl et al. 2017).
Even though the non- non-parametric regression function typically estimable only at slower than para- metric rates, one can often achieve parametric rates for the average moment function. However, this typically not achieved by simply pluging a non-parametric regression estimate into the moment formula and averaging. but requires debiasing approaches to reduce the effects of regularization when learning the -on-ammmitiicc regression.
Typical debiasing techniques are tailored to the moment of interest. In this work present auto- matic debiasing techniques that use the representation power of neural nets and random forests and which only require oracle access to the moment of interest. Our resulting average moment estima- tors are typically consistent at parametric /n rates and are asymptotically normal, allowing for the construction of confidence intervals with approximately nominal coverage. The latter is essential in social science applications and can also prove useful in policy learning applications, so as to quan- tify the uncertainty of different policies and implement automated policy optimization algorithms which require uncertainty bounds (e.g. algorithms that use optimism in the face of uncertainty).
Relative to previous works in the automatically debiased ML (Auto-DML) literature, the contribu- tion of this paper is twofold. On the one hand, we provide the first practical implementation of Auto-DML using neural networks (RieszNet) and random forests (ForestRiesz). As such, we com- plement the theoretical guarantees of Chernnzhuuu et al. (2021) for generic machine learners. On the other hand, we show that our methods perform better than existing benchmarks and that infer- ence based asymptotic confidence intervals obtains coverage close to nominal in two settings of great relevance in applied research: the average treatment effect of binary treatment and the average marginal effect (derivative) of continuous treatment.
The rest of the paper is structured as follows. Section 2 provides some background on estimation of average moments of regression functions. In Section 2.1 we describe the form of the debiasing term, and in Section 2.2 we explain how it can be automatically estimated. Sections 3 and introduuc our proposed estimators: RieszNet and ForestRiesz, respectively. Finally, in Section we present our experimental results.
2
ESTIMATION OF AVERAGE MOMENTS OF REGRESSION FUNCTIONS
2.1 DEBIASING THE AVERAGE MOMENT
We focus on problems where there exists function on with finite second moment such that:
[[m(W; g)] Eloo(Z))(Z)),, for all g with E[g(Z)² < co.
By the Riesz representation theorem, existence of such an (x) is equivalent to E[m(W: g)] being mean-square continuous linear functional of g. We will refer to this (x the Riesz representer (RR). Existence of the RR is equivalent to the semiparametric variance bound for 0g being finite, as in Newey (1994) and Hirshberg and Wager (2018).
2

---

Under review as a conference paper at ICLR 2022
This RR exists in each of Examples | to For instance, in Example | the RR is an( (Tpo(()) - (1 (1/ (o())) where Po (X) = Pr(T = X) is the propensity score and in Example integration by parts gives okp (Z)
(X / fo where fo(Z) is the joint probability density function (pdf) of 7 and Z. In general. the RR involves (unknown) nonpara- metric functions of the data, like the propensity score or the density fo (Z) and its derivative.
The RR a crucial object in the debiased ML literature, since it allows us to construct debiasing term for the moment function m (W; g) (see Chernozhukov al. 2018a| for details). The debiasing term in this case takes the form aY))(Y 9(Z)). To see that, consider the score m(W; g) +
(Y g(Z)) 0. It satisfies the following property:
[[m(W::) + aZ
- Po1 = -E((a(Z) - (o())) - go ())]
This is sometimes known double robustness, since the score will be zero in expectation when ei- ther (Z) (z( or g = 9o(Z) are correctly specified. An estimator of 0g can be constructed from this score and first-stage estimators ā and â. Let [.] denote the empirical expectation over
sample of size n, i.e. [ Σ Zi. We consider:
0 = E, [m(W:g) + (00(YY--(2))))
(1)
What double robustness means in practice is that the bias of this estimator will vanish at rate equal to the product of the mean-square convergence rates of ā and g. Therefore, in cases where the regression function do can be estimated very well, the rate requirements on ā will be less strict, and vice versa. More notably, whenever the product of the mean-square convergence rates of and 9 is larger than vn, the estimator 0 will be asymptotically normal at the parametric rate vn, as proven formally Theorem of Chernozhukov et al. 2021)
The regression estimator I and the RR estimator ā may use samples different than the i-th, which constitutes cross-fitting. Cross-fitting reduces bias from using the i-th sample estimating a and g- Also 9 and a may use different samples, which constitutes double cross-fitting (see e.g. Newey & Robins 2018, for the benefits of double cross-fitting in reducing the requirements on the quality of the regression and RR estimates).
2.2 RIESZ REPRESENTER AS MINIMIZER OF STOCHASTIC Loss
The theoretical foundation for this paper is the recent work of Chernozkuuo et al. (2021), who show that one can view the Riesz representer as the minimizer of the loss function:
ag arg min 2[a(Z)² 2m(W:a))
over some sufficiently flexible hypothesis space A and hence consider an empirical estimate of the Riesz representer by minimizing the corresponding empirical loss within some hypothesis space A:
á = arg min E., [a( am)
(2)
aEA,
The benefits of estimating the RR using this loss are twofold: (i) we do not need to derive an analytic form of the RR of the object of interest, (ii) we are trading-off bias and variance for the actual RR, since the loss is asymptotically equivalent to the square loss [[(( plug-in Riesz estimators that first solve some classification, regression or density estimation problem and then plug the resulting estimate into the analytic RR formula. This approach can lead to finite sample instabilities, for instance, in the case of binary treatment effects, when the propensity scores are close to or and they appear in the denominator (2018b) optimized the loss function in equation over linear Riesz functions with growing feature map, while Chernozhukov et al. (200) allowed for the estimation of the RR in arbitrary function spaces, but proposed computationally harder minimax loss formulation.
ax (2, as opposed to
the RR. Prior work by Chernozhukov et al.)
From a theoretical standpoint, Chernozzukov et al. (2021) also provide fast statistical estimation rates. Let || |l2 denote the l, norm of a function of a random input, i.. ||a]|| = /E]a(Z)3]. We also let || oo denote the loo norm, i.e. ||a||>o max (a(z))-
3

---

Under review as a conference paper at ICLR 2022
Theorem 1 (Chernozuuko al. (2021)). Let on be upper bound on the critical radius (Wain- wright 2019 of the function spaces:
{2 ( (o(2))ooz)) ao(z))
An, 7 E [0,1]}
and (w+ + 1 (m(w: a)) - mw; oo)) a E An: E [0, 1]}: and suppose that for all I in the spaces of Equation 3
|| 1. Suppose, furthermore, that m.
satisfies the mean-ssuured contimuity property:
a'11 for some M 2 Then for some universal constant C, we have that w.p. 5:
Ya, a' E A: B[(m(W::)) - m(W;a S M ||a -oce--1111
| ool1 C (68, (6. M+ M Mlg((1/)
+ inf ||c, a.EA
1 - 0o112
(4)
The critical radius is a quantity that has been analyzed for several function spaces of interest, such as high-dimen -igh-dimensional linear functions with bounded norms, neural networks and shallow regression trees, many times showing that 6, = Od,, n -1/2), where d, are various notions of dimension of the hypothesis space (see e.g. [Chernozuukov al. 202I for concrete rates). Theorem | can be applied to provide fast statistical estimation guarantees for the corresponding function spaces, albeit this result does not give guidelines on how to optimize over such function spaces practice. In our work, we provide heuristics for taking this theorem to practice for the case of neural networks and random forests and propose several practical enhancements.
RIESZNET: TARGETED REGULARIZATION AND MULTI-TASKING
Our design of the RieszNet architecture starts by showing the following lemma:
Lemma 1. In order to estimate the average moment of the regression function go - E Z suffices to estimate regression fanctions of the form go(Z) = ho (oo ()), where (A(A) E and A (Z) is the evaluation of the Riesz representer at sample. In other words, it suffices to estimate a regression function that solely conditions on the value of the Riesz representer.
Proof. It is easy verify that:
0o (W: = E/ooooooo (()(aa(()))()) Elm(W;; (W; ho o an)]
E[E[Y A ao( (aho(ao(()))
This property is a generalization of the observation that, in the case of average treatment effect es- timation, it suffices to condition the propensity value of each sample and the treatment value (Rosenbaum & Rubin 1983). In the case the average treatment effect moment, these two quan- tities suffice reproduce the Riesz representer. The aforementioned observation generalizes this well-known fact in causal estimation, which was also invoked in the prior work of Shi et al. (2019).
Lemma D alllos us argue that, when estimating the regression function, it suffices to features that are predictive of the Riesz representer. This leads to a multi-tasking neural network architecture, which generalization of that of Shi al. (0199 to arbitrary linear functionals and moment functions.
We consider deep neural representation of the RR of the form: ao (Z; 3, wl:x) (B, fi (Z; wis)), )) where f1 (X; wi*s) is the final feature representation layer of an arbitrary deep neural architecture with & hidden layers and weights wi:k- The goal of the Riesz estimate is to minimize the Riesz loss:
RRloss(B, wi:s) == E,, (a(Z; B, w 2 m(W: a(: B, [1+1))
(5)
In the limit, the representation layer (z; wi:k) will contain sufficient information to represent the true RR. Thus, conditioning this layer construct the regression function, suffices to get consistent estimate. Hence, even if these features are completely driven by predicting the RR, they will be seper-set that is required by Lemma
Based on this observation, we will represent the regression function with a deep neural network, starting from the final layer of the Riesz representer, i.e. g( w1:4) f2( (Z; wi:k); ((441))))
4

---

Under review as a conference paper at ICLR 2022
with d - k additional hidden layers and weights w(++ 1)ad- The regression is simply trying to mini- mize the square loss:
REGloss(W1.d) == E, [(Y - o(Z; w1
(6)
Note that the parameters of the common layers also enter the regression loss, and hence even if the RR function is a constant, their gradient information will primarily be driven by the regression loss and will reduce variance by explaining more of the output Y.
Finally, we will add a regularization term that is the analogue of the targeted regularization intro- duced by [Shi al. (2019). In fact, the intuition behind the following regularization term dates back to the early work of Bang & Robins (2005). Bang & Robins (2005) observed that one can show double robustness of a plug-in estimator in the case of estimation of average effects, if one simply adds the inverse propensity as regression variable, in a linear manner, and does not penalize its coefficient. This idea generalizes to Riesz functions and general moments. In particular, if we add the RR as an extra input to our regression problem in a linear manner, i.e. learn regression function of the form: ((Z) = (Z; wi.a) + c. c.ag (),, where is an un on-penalized parameter. Then note that we minimize the square loss with respect to widd and then the resulting estimate, will satisfy the property (due to the first order condition with respect to e), that:
En [[Y [(Z w1:a) c oZ) (Z)) = 0
Then note that the debiasing correction in the doubly-robust moment formulation is identically equal to zero when we use the regression function g. since: E, [(Y g( - ag (Z)] = 0. Thus the plug- in estimate of the average moment is equivalent the doubly-robust estimate, when uses the regression model g. since:
E, (m: = E,, (m] (Z:g)1 + E,, [Y - -(()) - [2)]
A similar intuition underlines the TMLE framework. However, in that framework, the parameter 6 is not simultaneously optimized together with the regression parameters w, but rather in a post- processing step: first an arbitrary regression model g is fitted (via any regression approach) and subsequently the preliminary g is corrected by solving linear regression problem between the residuals Y - g() and the Riesz representer a(2), to estimate coefficient i.e. minimizing the square loss:
E, [Y I(Z) c aZ))))
Then the corrected regression model g + c (Z) is used in plug-in manner to estimate the average moment. For an overview of these variants of doubly-robust estimators see (Tran et al. (2019). In that respect, our Riesz estimation approach can be viewed as automating the process of identifying the least favorable parametric sub-model required by the TMLE framework and which is typically done case-by-case basis and based on analytical derivations of the efficient influence function and contributes to the recent line of work on such automated TMLE (Carone et al. 2019).
In this work, similar Shi et al. (2099) take intermediate avenue, where the correction regression loss from the TMLE post-procsssing step is added as a regularization term, leading to the overall loss that is optimized by multi tullii-askkng deep architecture:
min RRloss(3, wi:k) + X REGloss(w)..) Baoraae
+ A2 [Y g(Z: wi:d) + ae: w1. + R(3, wi:d)
where is any regularization penalty on the parameters of the neural network, which crucially does not take input. Minimizing the neural network parameters of the loss defined in Equation 7 using stochastic first order methods constitutes our RieszNet estimation method for the average moment of regression function. Note that in the extreme case when X 0, then the second loss is equivalent to the one-step approach of Bang & Robins (2005), while as λ2 goes to zero the parameters Wi.d are primarily optimized based on the square loss, and hence the e estimated given a fixed regression function g, thereby mimicking the two-step approach of the TMLE framework.
4 FORESTRIESZ: LOCALLY LINEAR RIESZ ESTIMATION
One approach to constructing a tree that approximates the solution to the Riesz loss minimization problem is to simply use the Riesz loss as a criterion function when finding an optimal split among
5

---

Under review as a conference paper at ICLR 2022
all variables Z. However, note that this approach introduces large discontinuity in the treatment variable T, which is part of Z. Such discontinuous in T function spaces will typically not satisfy the mean-squared continuity property. Furthermore, since the moment function typically evaluates the function input at multiple treatment points, the critical radius of the resulting function space m runs the risk of being extremely large and hence the estimation enror not converging to zero. Moreover, unlike the case of regression forest, it is not clear what the "local node" solution will be if we are allowed to split on the treatment variable, since the local minimization problem can be ill- -ll-posed.
As a concrete case, consider the example of an average treatment effect of binary treatment. One could potentially minimize one of the two treatments. that case the local node solution to the Riesz loss minimization problem is not well-defined.
the Riesz loss by constructing child nodes that contain no samples from
For this reason, we consider an alternative formulation, where the tree is only allowed to split on variables other than the treatment, i.e. the variables X. Then consider representation that locally linear with respect to some pre-defined feature map (T, X) E IRd (e.g. a polynomial (T,T,), where B (X) is series). Then the Riesz function is represented in the form: a(Z) - 3(X) a non- con-paamettric (potentially discontinuous) function estimated based on the tree splits and 6 X) smooth feature map. In that case, by the linearity of the moment, the Riesz loss takes the form: 2
min 8
where we use the short-hand notation m(W; () = (m(W; .1),.... nwW oa)). Since 3(.) is al- lowed to be fully con-parametric, we can equivalently formulate the above minimization problem as satisfying the local first-order conditions conditional on each target x, i.e.:
((z) solves (()(Z))(Z)
m(W; の) X 2]
This problem falls the class of problems defined via solutions moment equations. Hence, we can apply the recent framework of Generalized Random Forests Athey et al. (2019) to solve this local moment problem via random forests.
That is exactly the approach we take in this work. We note that we depart from the exact algorithm presented in Athey et al.) (2019) that we slightly modify the criterion function to not solely max- imize the heterogeneity of the resulting local estimates from split (as in Athhe et al. 2019). but rather to exactly minimize the Riesz loss criterion. The two criteria are slightly different. In par- ticular, when we consider the splitting of a root node into two child nodes, then (Athey et al. (2019) Our criterion penalizes splits where the
chooses a split that maximizes N₁ (x)) + (X)2 local jacobian matrix:
(child)
Σ o(Z.)(Z,))
cchild|
#Echild
is not ill-posed (where |child| denotes the number of samples in a child node). In particular, note that the local solution at every leaf is of the form:
A(child) = ((chidd)= (cchild
(childd) = Jchind Exechua m(W;:)
1
(10)
and the average Riesz loss after a split is proportional to:
Σ
cciild| ((childy J(chil))((chhll)))
(1.)dE(1.2)
Hence, minimizing the Riesz loss is equivalent to maximizing the negative of the above quan- tity. Note that the heterogeneity criterion of (Athey et al. (2019) would simply maximize Echide(1.2) |child| ((child)' However, we note that the consistency results of Athey et al. (2019) do not depend on the exact criterion that is used and solely depend on the splits heing sufficiently random and balanced. Hence, they casily extend to the criterion that use here.
ignoring the ill-posedness of the local Jacobian matrix.
Finally, we note that our forest approach is also amenable to multi-tasking, since we can add to the moment equations the extra set of moment equations that correspond to the regression problem i.e. simply E - (xx) | X = x] = O and invoking a Generalized Random Forest for the super-set of these equations and the Riesz loss moment equations. This leads to multi-tasking forest approach that learns a single forest to represent both the regression function and the Riesz function, to be used for subsequent debiasing of the average moment.

---

Under review as a conference paper at ICLR 2022
5 EXPERIMENTAL RESULTS
In this section, we evaluate the performance of RieszNet and ForestRiesz two settings that are central in causal and policy estimation: the Average Treatment Effect (ATE) of binary treatment (Example M and the Average Derivative of a continuous treatment (Example 3 Throughout this section, use RieszNet and ForestRiesz leam the regression function Do and RR ogg, and com- pare the following three methods: (i) direct, (ii) Inverse Propensity Score weighting (IPS) and (iii) doubly-robust (DR):
Odrrect = E,, (W::))
Opw (,(ō(
- 0DR = E. (W -
The first method simply plugs in the regression estimate 9 into the moment of interest and averages. The second method the fact that, by the Riesz representation theorem and the tower property of conditional expectations, 8g = Eim go)l = E[an gu 2) Elao (Y) The third, our preferred method, combines both approaches
debiasing device, explained in Section 2
5.1 AVERAGE TREATMENT EFFECT IN THE IHDP DATASET
Following et al. (0119) we evaluate the performance of our estimators for the Average Treatment Effect (ATE) binary treatment on 1000 cemi-synthetic datasets based on the Infant Health and Development Program (IHDP). IHDP was a randomized control trial aimed at studying the effect of home visits and attendance at specialized clinies on future developmental and health outcomes for low birth weight, premature infants (Gross) 1993). We use the NPCT package in R to generate the semi-synthetic datasets under setting "A" (Dorie 2016). Each dataset consists of 747 observations of outcome Y. binary treatment I' and 25 continuous and binary confounders X.
mean absolute error (MAE) over the 1000 semi-synthetic datasets. Our preferred estimator, which uses the doubly-robust (DR) moment function to estimate the ATE, achieves MAE of 0.110 (std. emr. 0.003) and 0.126 (std. err. 0.004) when using RieszNet and ForestRiesz, respectively[ A natural benchmark against which to compare estimators. These use the known form of the Riesz representer for the case the ATE and an estimate of the propensity score Po(X) Pr(T = 1 X) to construct the Riesz representer as:
Auto-DML methods plug-
T
1-T
a X)
(X)
p(X)
The state-of-the-art neural-network-basee plug-in estimator is the Dragonnet of Shi al. (019)) which gives a MAE of O. 0.14 over our 1000 instances of the data. A plug-in estimator where both the regression function and the propensity score are estimated by random forests yields a MAE of 0.389. Hence, automatic debiasing seems a promising alternative to current methods even for causal parameters like the ATE, for which the form of the Riesz representer well-known.
Table 1: RieszNet and ForestRiesz: Mean Absolute Error (MAE) and its standard error over 1000 semi-synthetic datasets based on the IHDP experiment.
(a) RieszNet
(b) ForestRiesz
MAE std. em.
MAE 1: std.
DR
0.110 0.003 0.123 1 0.004
DR
0.126 + 0.004
Direct
Direct
0.197 0.007
IPS
0.122 + 0.037
IPS
0.669 0.004
Benchmark:
Benchmark:
Dragonnet
0.146 + 0.010
RF Plug-in
0.389 0.024
Shi al. 2019
To assess the coverage of our asymptotic confidence intervals in the same setting, we perform an- other experiment in which this time we also redraw the treatment, according to the propensity score setting "True" in the NPCI package. Outcomes are still generated under setting "A."
See Appendix A.1 for the architecture and tuning details used for RieszNet all experiments.

---

Under review as a conference paper at ICLR 2022
The results in Figure based on 100 instances of the dataset, show that the performance of RieszNet and ForestRiesz is excellent in terms of coverage when using the doubly-robust (DR) moment. Con- fidence intervals cover the true parameter 93% and 96% of the time (for a nominal 95%). respec- tively. The DR moment also has the lowest RMSE. On the other hand, the direct method (which does not use the debiasing term) seems to have lower bias for the RieszNet estimator, although in both cases its coverage is very poor. This is because the standard errors without the debiasing term greatly underestimate the true variance of the estimator.
dr. Bies: -0 0,050, RMSE: 0.139 Coverage: 0.9 0.930
Bias: 8.005, Coverage: 0.960
direct. Blas: 030 RMSE-0010 Coverage: 0,890
RMSE:
Coverage:
Blas: -0.750, RMSE: 0.765, Coveragi: 0.240
true
50
direc
direct
frequencs
30-
froquency
10
estimates
estimates
(a) RieszNet
(b) ForestRiesz
Figure 1: RieszNet and ForestRiesz: Bias, RMSE, coverage and distribution of estimates over 100
semi-synthetic datasets based on the IHDP experiment, where we redraw T.
5.2 BHP GASOLINE DEMAND DATA
To evaluate the performance of our estimators for average marginal effects of continuous treatment, we conduct a semi-synthetic experiment based on gasoline demand data from Blundell et al. (2017) [BHP]. The dataset constructed from the 2001 National Household Travel Survey, and contains 3,640 observations at the household level. The outcome of interest Y is (log) gasoline consumption. We want to estimate the effects of changing (log) price T, adjusting for differences in confounders X, including (log) household income, (log) number of drivers, (log) household respondent age, and a battery of geographic controls.
We generate our semi-synthetic data as follows. First, we estimate (xX) == E[T X] and ox( Var(T | X) by a Random Forest of T and (T - xX)) on X. respectively. We then draw 3,640 observations of T ~ (( of f(.). The error term e is drawn from a (O,0 with r² chosen to guarantee that the simulated regression R² matches the one in the true data.
and generate Y X(T + c, for six different choices
The exact form of f in each design is detailed in Appendix A.2 In the "simple f" designs we have constant, homogeneous marginal effect of -0.6 (within the range of estimates in Blundell al. 2012 using the real survey data). In the "complex f" designs, we have regression function that is cubic in T, and where there heterogeneous marginal effects by income (built to average approxi- mately- -0.6). In both cases, we evaluate the performance of the estimators without confounders X, and with confounders entering the regression function linearly and non-linearly.
Table presents the results for the most challenging design: a complex regression function with linear and non-linear confounders (see Tables A1 and A2 in the Appendix for full set of results in all designs). ForestRiesz with the doubly-robust moment combined with the TMLE adjustment (in which we use a corrected regression Z(Z) = g(Z)+ + ā(Z), where e is the OLS coefficient of Y - 7 ( ā(Z)) and cross-fitting seems to have the best performance in cases with many linear and non-lineat confounders. Both simple cross- cross-fitting with multitasking and double cross- fitting yield coverage close to above the nominal confidence level (95%), with biases of around one order of magnitude lower than the true effect. As for the ATE, the direct method has low bias but the standard errors underestimate the true variance of the estimator, and so coverage based on asymptotic confidence intervals is poor?
As can be seen in the Appendix, ForestRiesz seems to have larger bias and low coverage when there are no confounders compared RieszNet (both using the TPS or the DR moments).
8

---

Under review as a conference paper at ICLR 2022
We can consider plug-in estimator benchmark. Using the knowledge that T is normally dis- tributed conditional on covariates X, the plug-in in Riesz representer can be constructed using Stein's s identity (Lehmann & Casella 2006).
(T, T (X) (x)
where (x()) and @² are random forest estimates of the conditional mean and variance of T, respec- tively. The results for the plug-in estimator are on Table A3 Surprisingly, we find that our method, which is fully generic and non-parametrie, slightly outperforms the plug-in that uses knowledge of the Gaussian conditional distribution.
Table 2: RieszNet and ForestRiesz: bias, RMSE and coverage over 1000 semi-synthetic datasets
based on the BHP gasoline price data (10 different random seeds). The DGP is based on a complex regression function with linear and non-linear confounders.
(a) RieszNet
(b) ForestRiesz (with simple cross-fitting and multitasking)
Bias
RMSE
Cov.
Bias
RMSE
Cov.
DR
0.062
0.504
0.877
Direct
0.053
0.562
0.056
DR
0.131
0.377
0.909
IPS
0.061
0.496
0.916
Direct
0.094
0.304
0.046
IPS
0.161
0.443
0.847
TMLE-DE -RL
-0.082
0.327
0.953
Benchmark:
RF Plug-in
0.055
0.327
0.912
Figure 2 shows the distribution of estimates under the most complex design for RieszNet and Forest- Net (simple cross-fitting and multitasking). The distribution is approximately normal and properly centered around the true value, with very small bias for the doubly-robust estimators.
reg RMSE: 0.379,
RMSE: 0.046
reg RMSE:
RNSE:
0.094, AMSE Coverage: 0 0.847
direct, Bies: 0.00 0.053, RMSE: 562, Coverage: 0,056
los. Bies: 0 0.161 RMSE: 0 644
ip. Bies: 0.11. RMSS: 0.696, Coverage: 0.916
dr. Bles: 0.131 R.SE: 0.377,
Coverage: 0.909
dr. Blss: 06062 RMSE: 0.44, Cover age: 0.677
Coverage
trum
mnh
moquett3
350
150
300
100
50-
55-
-25
-20
-0.5
0.0
3.6
-2.0
Extmates
2.5
(a) RieszNet
(b) ForestRiesz (simple xfit, multitasking)
Figure 2: RieszNet and ForestRiesz: Regression and Riesz representer R2 and RMSF, bias, RMSE, coverage and distribution of estimates over 1000 semi-synthetic datasets based on the BHP gasoline price data (10 different random seeds). The DGP based on a complex regression function with linear and non-linear confounders.
REFERENCES
Susan Athey and Stefan Wager. Policy learning with observational data. Econometrica, 89(1): 133-161, 2021.
Susan Athey, Julie Tibshirani, and Stefan Wager. Generalized random forests. The Annals of Statis- tics, (712):1448-118 2019.
Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, (44:962297 2005.
9

---

Under review as a conference paper at ICLR 2022
Richard Blundell, Joel L Horowitz, and Matthias Parey. Measuring the price responsiveness of gaso- line demand: Economic shape restrictions and nonparametric demand estimation. Quantitative Economics, 3(1):29 51, 2012.
Richard Blundell, Joel Horowitz, and Matthias Parey. Nonparametric estimation of a nonseparable demand function under the slutsky inequality restriction. Review of Economics and Statistics, 99 (2)24913304, 2017.
Marco Carone, Alexander R. Luedtke, and Mark J. van der Laan. Toward computerized efficient estimation in anfnite-dimensiona models. Journal of the American Statistical Association, 114 (127):174-1100 2019. doi: 10. 10.1000011224499 2018. 1482752. URL https:tpsss /tttps://do..ee org/
10.1080/ 621 459. 2018 1822442274 752
PMID: 32405108.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural pa- rameters. The Econometrics Journal, 21(1):C1-C68, 2018a.
Victor Chernozhukov, Whitney K Newey, and Rahul Singh. Automatic debiased machine learning of causal and structural effects. arXiv preprint 180.02289, 2018b.
Victor Chernozhukov, Whitney Newey, Rahul Singh, and Vasilis Syrgkanis. Adversarial estimation of riesz representers. arXiv preprint arXiv:2101 10,0000,,,000,,,,, 2020.
Victor Chernozhukov, Whitney K Newey, Victor Quintas-Martinez, Martinez, and Vasilis Syrgkanis. Auto- matic debiased machine learning via neural nets for generalized linear regression. arXiv preprint arXiv:2104.14737, 2021.
Vincent Dorie. Non-parametrics for Causal Inference. ppci 2016.
https::/gittub.com//
Miroslav Dudík, John Langford, and Lihong Li. Doubly robust policy evaluation and learning, arXiv preprint arXiv: 103. 4601, 2011.
Jerome H. Friedman. Greedy function approximation: A gradient boosting machine. The Annals of Statistiics, 29(5):1189 1232, 2001. doi: /021///o/00012033551 URL https://doi. org 20g/110.1 201224////000111335555
Will Grathwohl, Dami Choi, Yuhuai Wu, Geoffrey Roeder, and David Duvenaud. Backpropagation through the void: Optimizing control variates for black-box gradient estimation. arXiv preprint 1rXiv:1711.00111 2017.
Ruth Gross. Infant Health and Development Program (IHDP): Enhancing the Outcomes of Low Birth Weight, Premature Infants the United States, 1985-1988. Inter-university Consortium for Political and Social Research, 1993.
Erich L Lehmann and George Casella. Theory of Point Estimation. Springer Science & Business Media, 2006.
Christoph Molnar. Interpretable Machine Learning (Section 8.1). Lulu. com, 2020.
Whitney K Newey and James R Robins. Cross Cross-fitting and fast remainder rates for semiparametric estimation. arXiv preprint 1eX.v:1801.090 2018.
Paul R Rosenbaum and Donald B Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):41-55, 1983.
Claudia Shi, David M Blei, and Victor Veitch. Adapting neural networks for the estimation of treatment effects. arXiv:/906 02 120, 2019.
Linh Tran, Constantin Yiannoutsos, Kara and Maya Petersen. Double robust efficient estimators of longitudinal treatment effects: Compar- ative performance in simulations and case study. The international journal of biostatistics, 15 (2), 2019.
Abraham Siika, Mark Van Der Laan,
10

---

Under review as a conference paper at ICLR 2022
Martin Wainwright. High-dimensional Staristics: A Non-asymptotic Viewpoint, volume 48, Cam- bridge University Press, 2019.
Qingyuan Zhao and Trevor Hastie. Causal interpretations of black-box models. Journal of Business & Economic Statistics, (911:272-288 2021.
A APPENDIX
A.1 RIESZNET ARCHITECTURE AND TRAINING DETAILS
As described in Section 3 the architecture of RieszNet consists of l common hidden layers that are used to learn both the RR and the regression function, and d - k additional hidden layers to learn the regression function. In our simulations, we choose = with a width of 200 and ELU activation function for the common hidden layers, and d - k 2 with width of 100 and also ELU activation function for the regression hidden layers.
We split dataset in training fold and test fold (20% and 80% of the sample respectively). Following [Shi et al. (2019), we train our network in two steps: (i) fast training step, (ii) a fine- tuning step. In the fast training step, epochs if the test error is smaller than 10-1, and with a maximum of 100 training epochs. In the fine tuning step,
learning
of 10-4 with early stopping after
learning rate of 10-5, with the same early stopping rule after 40 epochs, and with maximum of 600 training epochs.
We use L2 regularization throughout, with penalty 10-3, weight Å 0.1 the RRLoss and X2 = 1 on the targeted regularization loss (as defined in Equation (7)), and the Adam optimizer.
A.2 DESIGNS FOR THE BHP EXPERIMENT
For the average derivative experiment based on BHP data, we generate the outcome variable = X() + with six different choices of
1. Simple f:
f(T, 00.6T
2. Simple f with linear confounders:
f(T,X) = 0.66 + X1:21 b
3. Simple f with linear and non-linear confounders:
f(T,X) -0.67 + X1:21 b + NI(X
4. Complex f:
X?
f(T,X) 6
+ 0.5 3
10
5. Complex of with linear confounders:
x2 10
J(T,X) =
X19
c
78
X121 .
6. Complex with linear and non-linear confounders:
1
/X? 10
f(T,X)=
+ +1.9 e+
c+ 0.5 1 + X1:21 b+ NL(X)
where NL(X) 1.5o (10xs)) for the sigmoid function o(I) = 1/(1 + e-)), and where the coefficients b ~ 0/..5 0.5) and ~ iid 0|-0.2, 0.2] are drawn once per design at the beginning of the simulations (we try 10 different random seeds).
11

---

Under review as a conference paper at ICLR 2022
A.3
FULL SET OF RESULTS FOR THE BHP EXPERIMENT
Table AI: RieszNet: Regression and Riesz representer R², bias. RMSE and coverage over 1000 semi-synthetic datasets based on the BHP gasoline price data (10 different random seeds).
Direct
IPS
DR
neg zt"
122
Bias
RMSE
Cov.
Bias
RMSE
Cov.
Bias
RMSE
Cov.
Simple f
0,890
0.871
0.008
0.047
0.063
0.022
0.044
0.939
0.009
0.041
0.926
Simple f with linear confound.
0.825
0.865
0.046
U.354
0.083
0,494
0.918
0.065
0.500
Simple f with linear and non-lincar confound. U.789
0.866
0,044
U.363
0.047
0,070
0.499
0.914
0.058
0.506
0.878
Complex f 0.873
0.852
0.021
U.064
0.107
0.020
0,053
0.957
0.021
0.056
0.920
5. Complex f with lincar confound.
0.826
0.864
0.057
U.348
0.051
0.074
0.491
0.924
0.072
0.500
0.878
Complex f with linear and non-linear confound.
0.790 0.867
U.362
0.056
0.061
0.916
0.504
Table A2: ForestRiesz: Regression and Riesz representer R², bias, RMSE and coverage over 1000 semi-synthetic datasets based on the BHP gasoline price data (10 different random seeds).
Direct
IPS
DR
TMLE-DR
mullit.
R
Bias
RMSE
Cov.
Bias
RMSE
Cov.
Bias
RMSE
Cov.
Bias
RMSE
Cov.
Simple f
Yes
01952
0.148
0.844
No
0.991
0.491
0.112
0.113
0.000
0.148
0.149
0.000
0.039
0.701
-0.005
0.844
Yes 0.981
0.290
0.136
0.138
0.000
-0.996
0.102
0.222
-0.025
0.037
0.892
0.009
0.025
0.962
No 0.945
0.250
0.114
0.116
0:000
-0.096
0.102
0.222
-0.022
0.039
0.848
0.002
0.024
0.962
No
0.147
0.000
-0.091
0.099
-0.022
0.038
0.861
Simple f with linear confonnd.
0
Yes
0.342
0.450
0.170
0,289
0.051
0,182
0.292
0.876
0.078
0.297
0.856
0.037
0.317
0.835
No
0.194
0.183
0.881
0.848
0.355
0.157
0.040
0.044
0.400
0.883
0.082
0.353
0.320
0.952
No 0.665
0.250
0.188
0.306
0.047
0.044
0.409
0.882
0.074
0.340
0.932
0.009
0.324
0.942
No 0.530
0.231
0.129
0.326
0.042
0.024
0.407
0.895
0.065
0.435
0.856
0.082
0.331
0.948
Simple with linear
confound.
Yes
0.330
0.150
0.154
0.283
0.051
0.163
0.284
0.588
0.057
0.296
0.862
0.014
0.320
0.842
0
No 0.684
0.491
0.168
0.295
0.055
0,164
0.264
0.888
0.048
0,274
0.584
-0.005
0.302
0.830
Yes
0.342
0.289
0.136
0.317
0.041
-0.001
0.415
0.878
0.040
0.353
0.931
0.061
0.317
0.955
No
0.639
0.159
0.113
0.312
0.954
No 0.488
0.231
0.105
0.322
0.059
0.021
0.415
0.898
0.016
0.453
0.961
0.041
0.327
0.961
Complex f
Yes
0.866
0.076
0.081
0.010
0.082
0.086
0.023
0.042
00072
0,250
No 0.866
0.19]
0.049
0.058
0.020
0.082
0.086
0.289
-0.039
0.051
0.691
-0.078
0.086
0.246
Yes 0.734
0.250
0.997
0.101
0.000
00.25
0.264
0.000
-0.201
0.206
0.000
-01137
0.141
0.038
No 0.735
0.250
0.072
0,081
0.000
-0.259
0.261
0.000
-0.208
0.212
0,000
-0.147
0,151
0.018
No
0.118
0.000
-0.272
0.278
-0,200
0.205
-0,112
0.206
5. Complex f with linear confound.
0
Yes
01343
0.150
0.111
0,202
0.060
0,114
0.258
0.927
0.008
0.289
0.875
-0.037
0.319
0.835
No 0.710
0.491
0.096
0.263
0.060
0.114
0,258
0.000
0.267
0.895
0.008
0.304
0.838
0.115
0.113
0.426
0.856
-0.044
0.959
No 0.663
0.250
0.098
0.268
0.064
-0.113
0.426
0.866
-0.106
0.351
0.931
-0.01
0.320
0.951
2
No
0.500
0.231
0.094
0.315
0.056
0.150
0.446
0.860
0.100
0.441
0.845
0.052
0.327
0.952
Complex f with
non-linear confound.
Yes
0.332
0.450
0.095
0.259
0.064
0.095
0.253
0.936
-0.012
0.293
0.884
-0.000
0.326
0.835
0
No 01.883
0.191
0.070
0.260
0.053
0.095
0.253
0.934
-0.033
0.274
0.594
-0.079
0.314
0.627
Yes 0.343
0.289
0.094
0.301
0.016
-0.161
0.415
0.847
-0.131
0.377
0.909
-0.082
0.327
0.953
No 0.637
0.009
0.056
0.100
0.445
-0.135
0.365
-0.991
No
0.188
0.231
0.070
0.313
0.061
-0.196
0.168
0.809
-0.158
0.454
0.831
-0.094
0.358
0.950
12

---

Under review as a conference paper at ICLR 2022
Table A3: RF Plug-in Benchmark: Regression and Riesz representer R², bias, RMSE and coverage over 1000 semi-synthetic datasets based on the BHP gasoline price data (10 different random seeds).
RF Plug-in
RP Plug-in
R
R
Bias
RMSE
Cov.
R
R
Blas
RMSE
Cov.
1. Simple f
4. Complex f 0.491
0.960
0.491
0.043
0.051
0.535
0.806
0.030
0.791
Simple f with linear confound.
5. Complex f with linear confound.
0.711
0.491
0.090
0.328
0.912
0.710
0.491
0.065
0.323
0.914
Simple linear and non-linear confound.
Complex with and non non-linear confound. 0.683
0.684
0.491
0.080
0.332
0.915
0.491
0.055
0.327
0.912
13


Reviews:
[{"review_id": "hXZpEdFDGDb", "reviewer": "Reviewer_47Pi", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "**Strengths:** The method presented in the paper is applicable to a broad class of problems. The main idea presented in Lemma 1 is easy to understand and is applicable to other machine learning methods as well. The presented methods perform well in experiments, exceeding the performance of two alternative methods, although this comparison is only performed on synthetic variations of one real-world data set. Code is provided.\n\n**Weaknesses:** As someone who is not familiar with the type of problem discussed in the problem, I found some parts of the paper confusing (see below). Some explanations might be missing altogether unless I missed them. I think the paper could benefit from investing more time in the writing, including the notation. Additionally, in my opinion it could be stated more clearly what is and what is not novel about the presented methods (see below). Moreover, it appears to me that the paper hints at some theoretical guarantees in the beginning but then does not really discuss what theoretical guarantees can be given for the presented methods.\n\n**Specific comments / questions:**\n1. In the first eq. on page 1: The notation is confusing to me. It is not clear which variables the expectation is taken over. I assume the expectation is over $Z$. Also, it is not clear that $m$ even depends on $Z$. Moreover, in the following examples it is not clear whether $W = w$ or $W = w(Z)$ or something else. It would probably more clear to write something like $\\theta_0 = \\mathbb{E}_Z [m(Z; w, g_0)]$, or is it even $\\mathbb{E}_Z [m(Z; w(Z), g_0)]$?\n2. It seems to me that $w$ is always fixed throughout the paper except for the second function space definition in Eq. (3). Hence, it is not clear to me why it would not just be absorbed into the definition of $m$, such that we can write $m(g_0)$. Is the explicit dependence on $w$ in Eq. (3) needed to show that the resulting constant $C$ is uniform over different choices of $w$?\n3. The semicolon notation $m(W; g_0)$ is not used consistently throughout the paper.\n4. Sec 2: \"This framework gives rise to a score\": Which framework? Is it new or from the literature? In general, the steps here are not cited and the formulation does not make it clear whether they are novel. \n5. Eq (1): What does the index i mean in this context? Is it fixed or summed over? And why does $W$ even have an index? Is it because $W_i = w(X_i)$ ($W$ is not explicitly defined anywhere)? Explaining this might also make the paragraph below more understandable.\n6. Theorem 1: Why does $\\delta_n$ depend on $n$ even though the function spaces do not depend on $n$? Why is the assumption on all $f$ in the spaces of Equation (3) not formulated directly after the definition of the function spaces? Why would it be reasonable to assume that all functions satisfy $\\|f\\|_\\infty \\leq 1$ in the considered settings?\n7. Theorem 1: In the mean-squared continuity property equation, is the square root intentionally only taken on the left-hand side and not on the right-hand side? In this case, I would not call it \"mean-squared continuity\" because then it is not the same as in the beginning of Section 2.\n8. Sec 4: I don't understand the first paragraph. \"One could potentially minimize the Riesz loss ...\" - Why would an empty node change the Riesz loss? In general, the Riesz loss is formulated in terms of $Z = (X, T)$, making no distinction between $X$ and $T$. Why would there arise problems in $T$ that do not arise in $X$? Also, in the statement \"Such discontinuous in $T$ function spaces will typically not satisfy the mean-squared continuity property\", you could refer to the corresponding equation in Theorem 1, or even explain why this is the case.\n9. Eq. (9): What does conditioning on $X = x$ mean for the empirical expectation $\\mathbb{E}_n$? Is it only allowed for $x$ from the training set?\n10. Figure 1: It seems (at least by text search) that the abbreviation IPS is never explained. Ideally, the abbreviations would be explained in the Figure caption. Is there any reason why only MAE is used for evaluation even though the methods are trained on RMSE? If you want to make Figure 1 look more like a typical comparison table in the Deep Learning literature, you could consider the following suggestions:\n- Put both random forests and NNs in a single column, name them RieszNet-DR, RieszNet-Direct etc., and cite the sources (e.g. RieszNet-DR (Ours) or Dragonnet (Shi et al., 2019))\n- Mark the best-performing method(s) in bold (the differences seem to be at least partially significant)\n- Change the LaTeX environment from figure to table\n\nWhy is it not possible to have coverage estimates in this setting? What does it mean to \"redraw the treatment\" to get coverage estimates?\n\n11. Figure 2: I think you should label the axes, and maybe not put experimental results in text form above the plots.\n12. Why do you use the auxiliary loss in eq. (7) if you could also use the debiasing term from Eq. (1)? Do I understand it correctly that neither of the debiasing methods are novel?\n13. You could mention the experimental evaluation at the end of Section 1, to have a more complete exposition of the contributions of the paper. \n14. Sec 2, 3rd paragraph: Why does it debias the naive plug-in estimator? Is it explained anywhere, or in the literature?\n15. Could you discuss what theoretical guarantees can be given for (idealized versions of) the presented methods? What can be said about the infimum in Eq. (4)? The asymptotic statement $\\delta_n = O(d_n n^{-1/2})$ is not helpful if it is not clear how $d_n$ can behave. Moreover, the constraint $\\|f\\|_\\infty \\leq 1$ seems to be restrictive, and even in the best case, the conclusion of the theorem gives what I suppose is meant by parametric $\\sqrt{n}$-rates only for $\\alpha$ and not for $\\theta$.\n16. How are the confidence intervals computed?\n17. In my opinion Table 2 and Table 1 might be better placed in the Appendix, with perhaps some smaller summary table/figure in the main part of the paper. Also, making the best values bold might aid interpretation of the results. The gained space in the main paper could be used for explaining some other parts in more detail.\n18. I did not find any details on the training of the neural networks. Although they are of course contained in the supplementary code, maybe they should be briefly described in the appendix. (Number of layers, optimizer, activation function, etc.)\n19. Since the method is not that easy to summarize in the introduction, the paper might benefit from a conclusion section.\n\n\n**Minor comments:**\n1. Example 3: It should be $Z = (T, X)$ instead of $X = (T, X)$\n2. Sec 2: \"mean-square continuous functional\" -> \"mean-square integrable continuous linear functional\"?\n3. Sec 2, second paragraph: maybe it should be $f_0$ instead of $f$\n4. Sec 2: Could mention here that $\\alpha_0$ is in general unknown and will be estimated.\n5. Sec 2: \"This framework gives rise to a score\": One line uses $\\theta$ and the other one $\\theta_0$. Is this intentional?\n6. Eq (3): should this be a $\\mapsto$ instead of a $\\to$? Why $a \\in \\mathcal{A}$ and not $\\alpha \\in \\mathcal{A}$?\n7. Sec 2: \"focus on objects\" -> \"focus on problems\"?\n8. Figure 3: Orthogonality has never been discussed in the paper, or is this standard to report in the literature?\n9. Lemma 1: You could place the qed symbol in the same line as the last equation (using \\qedhere).\n10. The terms \"mean-independent\" and \"parametric rates\" were unclear to me. Also, for non-mathematicians, it might be helpful to explain a bit more about Riesz representers.", "summary_of_the_paper": "This paper considers the problem of estimating an expectation over covariates of some functional of an unknown regression function. The authors propose two estimators, one based on neural networks and one based on random forests. In contrast to many (all?) previous estimators which were derived for specific functionals, these estimators are applicable to general functionals. They employ a (new?) debiasing technique. Moreover, they use a novel multi-task architecture based on the observation that it suffices to estimate the regression function as a function of the Riesz representer. The paper demonstrates improved accuracies compared to prior work and good coverage on semi-synthetic tasks derived from two data sets.", "summary_of_the_review": "The paper contains interesting new methods which exhibit very good results in a limited experimental evaluation. However, the exposition is lacking in clarity to me. Therefore, I would currently classify the paper as borderline, but I am open to adjust my score based on the authors' response.\n\n**Edit**: Due to the author's response, I raise my score from 6 to 8.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "gkqT58CLHcU", "reviewer": "Reviewer_S7Am", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "\nStrengths\n- The proposed methodology that the estimation problem of the moment function is converted to the problem of learning the Riesz representer seems to be novel. To learn the Riesz representer, the authors introduce a loss function according to a theoretical result of recent work.\n- The author proposed two methods to learn Riesz representer, including a Neural Network method (RieszNet) and a random forest method (ForestRiesz). The experimental evaluation seems to be very strong and can achieve state-of-the-art performance in estimating the average treatment effect.\n\nWeakness\n- The notations in this paper are very confusing. For example, $W$, $W_i$, $w_{1:k}$, $w_{1:d}$, $w_{k:d}$ are never defined in this paper. The notation $w$ is abused, referring to a function in Examples 2 and 4, but refering some certain values with a subscript such as $w_{1:k}$.\n- The contributions of this paper is unclear. It would be better to clarify which parts are the contributions of this paper and which parts are results from previous works. Some parts of this paper are the conclusions of previous works, while some parts of this paper are methodologies and discussions of comparing methods, but these parts are all mixed together.\n- The detail steps of the proposed RieszNet and ForestRiesz should be clarified. For RieszNet, if the network architecture and training details of RieszNet are similar to Shi et al. (2019)? What $w_{1:k}$, $w_{1:d}$, $w_{k:d}$ are referred to, and after the minimization of equation (7) what should we do? For ForestRiesz, if readers are not familiar with Athey et al. (2019), it would be very hard for readers to follow the idea of ForestRiesz.\n- Although the paper aims to solve a general problem of estimating the average value of a moment function, the focus of this paper is still causal inference (example 1 average treatment effect). In the experimental part, the paper only evaluates the average treatment effect and average marginal effects, which are both problems in causal inference.\n\n\n", "summary_of_the_paper": "The authors address the problem of estimating the average value of a moment function that depends on an unknown regression function, which is commonly used in causal inference. \nBy the Riesz representation theorem, the authors design a loss function where the Riesz representer turns out to be the minimizer of the proposed loss. \nThe authors propose both a Neural Network method (RieszNet) and a random forest method (ForestRiesz), i.e. a multi-tasking Neural Network method where the loss function is a combination of the Riesz representer and regression loss, and a random forest method that leans representation of both the regression function and the Riesz function. \nThe authors conduct experiments of estimating the average treatment effect and average marginal effects, and show that the proposed RieszNet and ForestRiesz beat the state-of-the-art methods.\n\n\n", "summary_of_the_review": "The idea of this paper is original, and both RieszNet and ForestRiesz show strong experimental evaluations. But the paper needs to be well-organized. Therefore, I would recommend acceptance of this paper if the paper was well-written.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "DTuvwaLNze4", "reviewer": "Reviewer_DMWH", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I like the premise of the paper. I think it is an important step towards making the estimation of policy-relevant causal estimands more accessible to practitioners working in data-rich environments in which nuisance functions are expected to be complicated. Although conceptually and theoretically most of the content of the paper is already known / described elsewhere, its novelty lies in combining multiple ingredients to propose practical methods and provide a concrete implementation.\n\nI have some suggestions for improving the paper:\n\n1) Compared to other papers in this line of work, it appears to me that the scope of the paper is relevant to a broader audience and especially so for applied data scientists. Yet, \nthe paper feels rushed (more on this below), e.g., with a lot of notation remaining undefined. The consequence is that the paper is not self-contained and would be very hard to understand, even for somebody with background knowledge in causality (who is not aware of the latest theoretical developments). Exposition is very important for a paper that could reach a broad audience! \n\n2) I felt the description of the methods is a bit too short and probably one would not be able to reproduce the results based on the text description. This is particularly the case for the ForestRiesz procedure which seems to me to be less \"automated\" than promised.\n\n3) There are many ingredients to the proposed procedures (beyond what I described above) and from the experiments it is not clear which ingredient is relevant (or if all ingredients are relevant). Ablation studies would be very helpful here for the reader.\n\nBelow I summarize these three points in more detail:\n\n# Exposition\n\n1) The introduction is very short and one immediately is faced with a formula, the components of which are never defined. It would be very helpful to at least define potential outcome notation early. $W$ is also never defined.\n\n2) I think it would be very helpful to a reader to discuss the assumptions of Theorem 1 in more detail. What is the take-away for a practitioner? What should they be careful about?\n\n3) After Lemma 1, please make sure to cite Rosenbaum and Rubin (1983). I think it would also be helpful to make a remark of how Lemma 1 is used. The \"naive\" (and perhaps traditional) approach would be to just nonparametrically regress on $\\hat{\\alpha_0}$. But this requires very small estimation errors for $\\alpha_0$. Instead the innovation is the use of e.g., the last layer of the neural network, which may still be capturing all relevant conditioning information!\n\n4) The first paragraph of page 4 talks about cross-fitting and double cross-fitting. What is actually being used in the implementation? For example if double cross-fitting is being used, then I think the reference to that could be removed (the audience for whom this point is relevant would likely already be aware of this point and practitioners would use the \"automatic\" software).\n\n5) In Section 3, I think that having a figure on the architecture (similar to Fig. 1 of Shi et al.) could be very helpful for a reader.\n\n6) Page 8, Riesz representer with Gaussian errors: It could be worthwhile to point out here that this is the case by Stein's identity (and point a reader e.g. to a supplement that explains this).\n\n\n# Details for procedures\n\n1) I think it is very difficult to follow what ForestRiesz is doing, e.g., it is not explained what is being computed in the leaves. \n\n2) Also the caveats and description in the first paragraph make the whole procedure seem less \"automatic\" and the implementation appears to need substantial additional modifications to account for treatments. This makes the procedure less appearling than RieszNet.\n\n3) Please provide more details for the simulations and implementation of procedures, e.g., how many folds are being used? Where is IPS defined? Could there be more details for architectures and tuning parameters chosen? What is the \"RF Plug-in\" benchmark? Is it the default recommended method in the GRF package?\n\n# Ablation studies\n\nFor example for the ATE simulations I would be interested in ablation studies showcasing the following aspects:\n\n1) What is the impact of the regularization term for RieszNet? \n\n2) What is the impact of cross-fitting? Shi et al. (surprisingly to me) claim that cross-fitting does not help for their DragonNet architecture.\n\n\n# Some potential Typos / minor points\n\n* Page 2, \"integration by parts gives\" -> Is a negative sign missing?\n* Page 2, Linearity after the first eqn. of Section 2?\n* Page 3, after eqn. (3), \"square\" on the RHS?\n* Page 4, before eqn. (6), $(k+1):d$ instead of $k:d$?\n* Check formatting of references, e.g., Grathwohl et al. after example 4 should presumably be in a parenthesis?\n\n\n\n\n\n", "summary_of_the_paper": "The authors study the nonparametric estimation of an important class of (causal) estimands that includes the average treatment effect (ATE) in experiments and observational studies under unconfoundedness. The main innovation is the \"automatic\" nature of the procedures, one based on deep learning and one on random forests. While previous work has developed hand-tailored constructions for the ATE (a traditional problem in statistics), the authors show that existing constructions can be generalized considerably based on recent advances in semi- and nonparametric statistics based on Riesz representers. The key challenge this work addresses is the fact that the Riesz representer is typically unknown.\n\nThe main proposed method based on deep learning, RieszNet, may be seen as a generalization of the DragonNet procedure by Shi et al. (2019) from the ATE to more general estimands. Even in the case of the ATE, DragonNet and RieszNet are not the same and RieszNet outperforms DragonNet in simulations: RieszNet directly targets the inverse propensities, while DragonNet estimates the propensities and then plugs in their inverse.", "summary_of_the_review": "This paper provides a solid contribution towards the automatic estimation of policy-relevant causal estimands. I think this work should be accepted. However the paper could be substantially improved by improving exposition and making the paper accessible to a wider audience.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "4JucnrFtUsi", "reviewer": "Reviewer_WPay", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Main review:\n- The proposed method is very innovative and important in the field. This paper will be of great value to the field. \n- On the other hand, it is difficult to evaluate the novelty of the paper because many of the results rely on existing studies, such as Chernozhukov et al. (2021). \n- This paper may be more of a summary of the existing studies.\n- If the contribution of this paper is the proposal of the methods, then the experiments to show the validity of the methods will be important as the contribution of this paper. \n- However, the experiments are very simple, and it is not clear whether they are superior to other methods. \n- Even though the performance depends on the architecture and hyperparameters of the model, there is a little description of them.\n- Besides, although the proposed method is persuasive, it seems to me that the paper is not well organized. There were several points that I couldn't catch up on. For instance, \n   - What are the definitions of $Y$, $X$, $Z$, $W$, and $T$? We can guess the meanings, but the rigorous definitions are needed.\n   - In Example 3,  $X=(T, X)$ should be $Z=(T, X)$?\n   - Theorem 1 is cited from Chernozhukov et al. (2021), but there is no description about which theorem is used from Chernozhukov et al. (2021).\n   - In the first set in Equation (3), the authors might have forgotten to add $\\gamma \\in [0,1]$.\n   - In page 4, what are the definitions of $\\beta$, $w_{1:k}$, $w_{1:d}$, $d$, and $k$? I can guess about them, but the clear definitions are needed.\n   - Equation (5) and (6) are correct? The LHS of these equations are not functions of $w$ because they are minimized over $w$. I think that $min$ operator should be removed.\n   - In the RHS of Equation (5), $\\theta$ is $\\beta$?\n   - What are the definitions of $f_2$ and $f$?\n\nQuestions: \n1. Can we apply the proposed method to estimate the conditional (potentially heterogeneous) average treatment effect?\n2. Can we check that the conditions of Theorem 1 hold? In particular, the multi-task neural network architecture can satisfy them?\n3. The estimator has $\\sqrt{n}$ consistency? I think that we can derive the convergence rate from Theorem 1, which states the convergence rate of $\\|\\hat{\\alpha} - \\alpha_0\\|^2_2$, but it is not clear, at least for me.\n\nMinor points:\n1. Figures 2 and 3 are a little difficult to see because the histograms overlap. How about using boxplots?", "summary_of_the_paper": "This paper shows novel methods based on the recent findings on the Riesz representation in econometrics and machine learning, such as Chernozhukov et al. (2021). While the proposed applications make sense and are persuasive, the contributions of this paper are a bit limited. Although the authors support the soundness of the proposed algorithms in experiments, the experiments are a bit simple and may not be sufficient for the justification. ", "summary_of_the_review": "I find it difficult to evaluate the contribution of this paper because it is not clear what the contribution of this paper is. The method is novel, but the theory relies on existing different work, so I think the evaluation of the method highly relies on the novelty of experiments. However, the experiments are very simple and not enough to claim the usefulness of the method. If the methodological and theoretical contributions are significant, the paper should be highly evaluated even if the experiments are simple. However, in this case, because this paper has less novelties in  the theoretical part, the experiments are important. In summary, the method in this paper is very novel and interesting, but the experiments to support it are lacking. Besides, there are several issues in writing even though we can guess what the authors insist. This is the reason why I vote for weak rejection.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: In this paper, the problem of estimating the average of a moment function that depends on an unknown regression function.
It heavily relies on prior papers by e.g. Chernozhukov et al. and the actual novel material consists of making these theoretical 
results more practical. Experiments for two practical approaches based on neural networks respectively random forests are also reported.

Initially, the presentation of the paper was heavily criticized by the reviewers, but during the rebuttal phase at least some of the issues were removed. Together with some other improvements this lead to an increased average score. However, it seems fair to say that reading the other papers first, is still kind of necessary.

Despite the still unclear novelty the paper has some merits, which in principle make it acceptable. Compared to the other good papers in my batch, however, it is more incremental and the overall contribution is not as strong. For this reason I vote for rejection, but a comparison to other papers outside my batch is probably a good idea.
Decision: Reject