Double Machine Learning for Causal Inference under Shared-State Interference
Authors: Chris Hays
Manish Raghavan
Venue: ICML 2025 poster (2025)
Keywords: double machine learning; interference; causal inference
Primary area: general_machine_learning->causality
TL;DR: We formalize the problem of shared-state interference in causal inference, prove a debiased machine learning theorem for inference in this setting, and instantiate the theorem in several settings of interest.

Abstract: Researchers and practitioners often wish to measure treatment effects in settings where units interact via markets and recommendation systems.  In these settings, units are affected by certain *shared states*, like prices, algorithmic recommendations or social signals. We formalize this  structure, calling it shared-state interference, and argue that our formulation captures many relevant applied settings. Our key modeling assumption is that individuals' potential outcomes are independent conditional on the shared state. We then prove an extension of a double machine learning (DML) theorem providing conditions for achieving efficient inference under shared-state interference. We also instantiate our general theorem in several models of interest where it is possible to efficiently estimate the average direct effect (ADE) or global average treatment effect (GATE).

Double Machine Learning for Causal Inference under Shared-State Interference
Chris Hays'
Manish Raghavan
Abstract
shared state may be prices, wait times, availability of in- ventory. In each of these settings, individuals both influence and are influenced by the shared state.
Researchers and practitioners often wish mea- treatment effects in settings where units inter- act via markets and recommendation systems. In these settings, units are affected by certain shared states, like prices, algorithmic recommendations or social signals. We formalize this structure, call- ing shared-state interference, and argue that our formulation captures many relevant applied settings. Our key modeling assumption that individuals' potential outcomes are independent conditional on the shared state. We then prove an extension of a double machine learning (DML) theorem providing conditions for achieving ef- ficient inference under shared-state interference. We also instantiate our general theorem in sev- eral models of interest where it possible to effi- ciently estimate the average direct effect (ADE) global average treatment effeet (GATE).
Assumptions are required to perform causal inference in the presence of interference. Prior work has studied this
kind of interference in particular applications (Johari et al.
2021
Basse et al., 2016
Wager 2024 Munro 2024 Simchi
Levi and Wang 2023
Dhaouadi al. 2023 Farias et al.
Li al. 2024). In some cases, they propose either experimental designs or analysis methods that account for interference and allow for valid inference. However, these approaches typically assume a parametric model of the system or domain-speciffs assumptions.
2023
2022
Our contributions. We provide a general framework to reason about causal inference under shared-state interfer- ence. We define formal model in Section [ where individ- uals arrive sequentially. Their covariates (i.e., unit character- istics), treatment assignments, and outcomes may influence the outcomes of future individuals through shared state.
1. Introduction
In Section S we provide conditions under which efficient inference is possible in the framework. We extend meth- ods from double machine learning (DML) (Chernozhukov et al. 2018), which has shown that it is possible to achieve efficient inference using expressive machine learning meth- ods without parametric functional form assumptions. A key assumption we rely on is that the shared state progresses according to a Markov chain. We also provide a consistent variance estimator, which is necessary for constructing valid confidence intervals and running hypothesis tests.
In causal inference, interference - where individuals' treat- ment assignments, outcomes or other characteristics impact others' outcomes - everywhere: On short-form video platforms, the consumption of one individual is used as an input to recommendation algorithms that are subsequently used to serve content to others. In ride-sharing or hous- ing rental services, providing discount to some people can cause them to increase their usage of the service and increase wait times or prices. In such settings, failure to account for interference may lead to biased estimates of causal effects, even in randomized controlled trials.
Next, we instantiate our framework with two applications: first, we show how our framework can be used to estimate average direct effects (ADE) in observational settings; then, we provide a variance reduction strategy for estimation of the global average treatment effects (GATE) in switchback experiments. Intuitively, the ADE measures the expected difference between treatment and control for a unit drawn uniformly random, keeping the treatment assignment dis- tributions of other units fixed. The GATE measures the mean difference between outcomes when all units are as- signed to treatment versus when all are assigned to control. In each instantiation, we provide simulations validating that our method produces estimators that concentrate around
Across many settings, like those of recommender systems and markets, interference between units often follow com- mon pattern: the outcomes of individuals depend on others through some shared state. In recommender systems, the shared state might be the outputs of recommender systems that are used to generate users' feeds. In marketplaces, the
MIT. Correspondence to: Chris Hays sjhays@mit.ddu
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Double Machine Learning for Causal Inferenee under Shared-State Interference
the true treatment effects in finite samples. We also show consistent variance estimators can be used to construct confidence intervals with the desired coverage probability. We refer the reader to the full version of the paper (Hays and Raghavan 2025), which we recommend.
2. Modeling Shared-State Interference
We first give a high-level description of our setting. Our model begins with a set of sequentially arriving units. Each unit has covariates and a binary treatment assignment drawn iid from joint distribution, as in canonical causal inference settings. There is also a (possibly vector-valued) observed shared state through which all interference is channeled: each arriving unit has an outcome of interest that may de- pend on the shared state, their covariates and their treatment assignment; then the shared state at the next time step may depend on the previous shared state, as well as the previous unit's 's outcome, covariates and treatment assignment. The key assumption that makes our setup tractable is that the shared state has a Markov property: that is, conditional on the shared state at the previous time step and the data of the unit that arrived in the previous time step, the shared state is independent of the previous data. We next introduce the model and notation formally.
1.1. Related work.
Shared-state interference. Several prior works have ex- plored causal inference in settings captured by or similar to our formulation of shared-state state interference (Johari ct al.
2021
Basse et al.
2016 Wager| 2024| Munro 2024| Holtz
et al.
2024| Simchi- Levi and Wang 2023; Dhaouadi et al.
2023
Farias et al.
2023
2022
Li et
2024 Brennan et al.
2024
Bajari et al. 2023 Bright al.) 2023). Some of these
papers have analyzed bias resulting from naive estimators or experimental designs and proposed less biased alterna-
tives (Johari et al. 2021 Farias et al. 2022 Brennan et al. 2024). Other works have proposed parametric models of settings that allow for inference in the presence of interfer- ence (Wager and Xu 2021 Li et al., 2024). Several works consider a Markov model of interference similar to ours in an experimental setting and explore less biased estimation
2.1. Setting
Model. We will consider a sequence of units indexed = 1, and observed up to time T. Each unit t will have features X, drawn iid from a sample space R2x for finite dimension px and binary treatment assignment D,. We will let H. on R25 for finite dimension ph, denote the shared state. The observed outcome for each unit will be realevalued and denoted Y,
methods (Farias et al. 2023 2022
Glynn et al. 20222).
Double machine learning. The double machine learning framework was introduced in Chernozhukov et al. (2018), drawing on a rich literature in semiparametric statistics (see, e.g., ge.eeyy (2023) for an overview). DML methods have been extended to several other settings, such those with continuous treatments (Kennedy et al.) 2017), estimation of quantile effects (Kallus et al., 2022) and settings with limited unobserved confounding (Rambachan et al. (224)). A few works, like ours, have developed semiparametric methods for settings with interference. Several works have explored interference channeled through individuals" social networks (Emmenegger et al. 2023 Ogburn et al. (2244). Munro 2024) develops DML theorem for settings like auc- tions where interference is channeled through centralized allocation mechanism. Zhan et al. 22044) develops DML theorem for a discrete choice model of content consump- tion under a neural network- based recommender system. Ballinari and Wehrli (2024) develops a DML theorem for a time-series model where single unit is observed over time, but where observations across time obey a mixing condition and where the target estimand is an impulse re- sponse function, which measure the effect of an intervention at a particular time on the outcome at a future time. We supply an extended comparison to some key related works in Appendix B
We will adopt notation so that potential outcomes are a (stochastic) function of treatment and the shared-state (i.e., Y H₂)) similar to the exposure mapping approach of Aronow and Samii (0117). The dependence of outcomes on the shared state is the primary difference between this model and canonical iid causal inference settings, and the avenue through which interference is assumed to occur. We will assume D, is independent of all data from prior units but that it may depend on X,. The observed data associated with time step will be denoted W. (X,, Dr, He, Yi) on the sample space denoted W. We will denote the full observed data WiT (W.)7=1 E W and the (unknown) data distribution of W1:0c {W as P E P for some set of distributions P. The set P specifies the set of probability distributions in which the true distribution may fall, and we will impose assumptions that must hold for all P P P.
We will assume that H; depends the data from the previ- ous time step (i.e., W,-1) but is independent of X,, and is conditionally independent of the data at time steps before time t - 1, i.c., that H,, has the Markov property: For all P E P and P- P-measurable A C RPH,, and == 1,2....
P(H, E A WI((-))> = P(H, E A W
This is assumption satisfied in settings where Hç is an update to H,-1, like when inventory at time function of
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Double Machine Learning for Causal Inferenee under Shared-State Interference
inventory at time - 1 and whether the customer at time t made purchase.
(b) m-deppendence i.e.,
W1. W1.... We IL Wemm+h--.. WT
(2.3)
Our setting allows for the shared state at time t to depend on the shared state at time t - 1, allowing for dependencies over time, as long as these dependencies are exclusively mediated by the shared state. For example, the purchasing decisions of two individuals may be correlated if inventory low during the time interval in which they arrive. The shared-state at time can also depend on the outcome at time t t-1 1. This is different from many other models for causal inference under interference, which allow for dependencies only on other units' treatment assignments. (See, e.g., (Eckles et al. (0144) for further discussion of this distinction.) The dependency structure assumed by a shared-state interference setting is summarized in Figure
for all [T].
Geometric ergodicity is a generalization (to the general state space setting) of aperiodicity and irreducibility in finite-state Markov chains; we define it formally in Appendix [A] As- sumption 2. a implies that the chain has a unique steady state transition probability distribution and that the chain converges to that distribution from any starting state (Meyn and Tweedie 2009 Proposition 17.1.6). Thus, writing K in Equation (.2) is well-defined. Intuitively, detailed bal- ance says that, for continuous distributions, the steady state probability density around a point times the probability density of transitioning from w to z is equal to the same expression with the roles of w, z reversed.
We will also assume that the distribution of H, and Dr conditional on W is invariant of t. Thus, we can specify a transition probability kernel Ки where, for w E W and measurable event A on W and all t E [T].
The m-dependence condition states that data observed more than m steps apart are independent. Observe that any m- dependent sequence can be written Markov chain, using the last m observations as the state. We will assume through- out when invoking Assumption 2. that (a finite upper bound on) m is known. We refer the reader to the literature on switchback experiments for methods of estimating m from data (Bojinov et al. 2022).
K (w. A) def. P PHI E A W W).
(2.1)
There will be an initial (known) shared-state Ho generated according to an arbitrary, unknown and possibly determinis- tic distribution Po. The t-step kernel will be denoted Ki,, where KJ H (w, A) = P(W,- E A W = w).
The fact that H, has the Markov property, X, is drawn iid, and D, is independent of the prior data trivially implies that [W,) itself is a Markov chain. Indeed, since all transi- tion probabilities are t-invariant, 1 is homogencous. W may be uncountable, so the chain may be defined on a general (not necessarily finite) state space. We will denote the (unknown) transition probability kernel of W1:00 as K where K(w, A) = P PWW E A W w). Note that K is distinct from KH; the former denotes the transition prob- abilities for W; and the latter only for H₁. As with KH, the t-step kernel will be denoted K*.
Estimand. We will assume there is a scalar-valued func- tional of interest ve" which depends on the data distribution P: e.g., in Section as* will be the average direct effect and in Section [5] it will be the global average treatment effect. We will assume there exists a function 4² : W - R so that & can be written as an asymptotic average over expected values of 4²" L.e. I...
T 2
10* lim T- T
We will assume that sp" is continuous.
When the Markov chain has a unique stationary distribu- tion, we will denote it K since we will only be con- sidering settings where, for all w, A it holds K (A) lim KTT (w, A). (We drop the argument w in K since K does not depend on the starting state.) When 00 exists, we will require 8 E P. We will also require that {Weje, satisfies natural Markov chain conditions.
Examples of shared-state interference. Our model is suited to describing marketplaces where interference is in- duced by limited supply or demand. For example, [ohari ct al. (20II) proposes Markov chain model of a rental plat- form consisting of sequence of arriving customers and a set of listing types, each with a fixed number of available list- ings. If a customer books listing, it becomes unavailable for time, temporarily decreasing the number of listings of the type by 1. Several other models of interference in specific markets can be well-captured by shared-state in- terference: for example, models of freelance labor markets (Wager and Xu (021) and ride-sharing markets (Li et al.) 2024]. Moreover, m-dependence is assumed in the large literature on switchback experiments (Bojinov et al. 2022
Assumption 2.1. Wion satisfies at least one of the follow- ing conditions:
(a) Geometric ergodicity and detailed balance: i.e., WI:00 is Harris ergodic Markov chain satisfying geometric mixing, and for w. E W. it holds
K (dw) K K(w, dz) K (dz) ((,))
(2.2)
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Double Machine Learning for Causal Inferenee under Shared-State Interference
H1-1
H
is trained on all but one fold and then the estimator is con- structed using the held-out fold. In our setting, in the vein of Angelopoulos et al. (2023). Ballinari and Wehrli (024) and many similar methods in statistics and machine learning, we instead assume that the nuisance estimators are gener- ated via an auxiliary sample of data that is independent of the data to be used for inference. We make this simple alternate assumption because sample splitting would not guarantee independence between folds of the data in our setting, so predictors trained on some folds may still not be independent of the others.
Yi-1
Y
X1-1 D
X₁
Dr
Figure 1: Dependency structure of shared-state interfer- ence.
3. DML for SSI
In this section, our main result is a double machine learning (DML) theorem for causal inference under shared state inter- ference (SSI). Double machine learning a meta-algorithm which allows for efficient inference with the use of expres- sive machine learning methods. It was proposed in Cher- nozhukov et al.] (2018), building on a large literature in semiparametric statistics. We provide background on the method in Appendix and give an informal overview here.
For our theorem in this section, we will assume that there is a known estimator & that depends on WIT and a set of data-dependent nuisance parameters/functions 7j E for some convex set S. We can imagine 1) to be the param- eters in neural network, the coefficients on possibly high-dimensional linear regression, or the feature splits and thresholds for trees in random forest. We will need that e identifies vi;; i.e., that P [wi(W: "]] = (P) for unknown true nuissances ny E S. We will also require that s is an empirical mean of time-t estimator yp, i.e., that
At a high level, the method consists of two steps. First, an expressive machine learning algorithm is used to approxi- mate any nuisance parameters. Nuisance parameters often take the form of estimated conditional expectation func- tions (the expected outcome conditional on covariates and treatment) or propensity scores (probability of treatment conditional on covariates) which are useful for estimation of treatment effects but not of interest on their own. predic- tor for the expected outcome conditional on covariates could be used to create plug-in estimators by simply taking the average of predicted outcomes under treatment versus under control. However, such plug-in estimators may converge at slower-than-paramerio rates and may be very biased in finite samples. Thus, second, the estimator is constructed so as to satisfy a first-order insensitivity property, in a dis- tributional sense that we will precisely specify later. This property can be used to show that the convergence rates of the estimator depend only on products of convergence rates of the ML estimators, which allows for weaker conditions on the convergence rates for any one estimator.
* (Wirr:)) = T (WW:
In Sections and 5 & will be an estimator for the average direct effect (ADE) and global average treatment effect (GATE), respectively, which we define formally under the structural models in each of those sections. The estimator will be specially constructed so as to satisfy the double- robustness property characteristic of DML methods. Much of the challenge in applying this section's main theorem involves specifying structural model that fits a context of interest and identifying an estimator with the double- robustness properties.
We will assume access to a nuisance function estimator 7 (usually imagined to have been trained by a flexible machine learning method like a neural network or random forest). We will also require that 7 belongs to convex set containing 1), with high probability. This is for technical reasons that will become clear later and are common to all DML methods: the conditions required for our main theorem necessitate reasoning over convex combinations of r and ry" so it is useful to define a convex set within which ń and 17 fall. Finally, we will require that the convex set is shrinking around 17" asymptotically in T. This ensures that r converging to 17* (in an appropriate sense) as T scales. Formally, for a constant I 7 1, will require that the nuisance function estimator í) belongs to a convex set ST C S with probability 1 y- where it is also assumed 17" E Sr. The DML procedure for our context is summarized in Algorithm
We note that a naive approach to inference in our setting would be to treat the stared-state as a covariate and apply DML methods as if the data were iid. However, depending on the estimand, this may yield inconsistent treatment effect or variance estimators, as we will see in our simulations in Dections[aand33
An important part of any DML procedure is that the nui- sance parameter estimators must be independent of the data used to construct the target estimator. (Otherwise, typical L convergence rates are not sufficient to ensure that the nuisance estimators yield consistent target estimators.) In iid settings, this independence is usually achieved via sam- ple splitting: the data is split into k folds, the MI. estimator
We will need several assumptions about the data generat-
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Double Machine Learning for Causal Inferenee under Shared-Star Interference
Algorithm 1 DML for shared-state interference
Applying the theorem. At a high level, instantiating The-
Train a nuisance function estimator 1) from auxiliary data Waux.
orem 3.1 in specific settings is a matter of verifying Assump- tions 2. and C Once these are satisfied, Theorem 331 tells us that the DML procedure in Algorithm can be used to generate asymptotically normal treatment effect estima- tors using the provided variance estimators.
Construct the estimator &b def. (Wirr,
Return s
In the next two sections, we apply Theorem B. two struc- tural models of independent interest. In each section, we examine specific semiparametric structural model of out- comes given covariates, treatment and the shared state and specific causal estimand. In each case, verifying Assump- tions 2.1 and amounts to ensuring all interference is channeled through a shared state, making appropriate reg- ularity assumptions and proving the following three lemmas: first, that the estimator is Neyman orthogonal with respect to the nuisance estimators; second, that the average over t of the 1,2 norm of yo v)) < 1f") must converge to zero as T + oo; third, that the second Gateaux derivative of the estimator has order smaller than vT. The regularity assumptions amount to support conditions requiring a lower bound on the probabilities of each treatment assignment and requirements that nuisance estimates and parameters are bounded in probability.
ing process and nuisance function estimation. These as- sumptions are analogous to those necessary for the results in Chernozhukov et al. (2018). The first assumption, As- sumption C.2 requires that & is smooth function of 1j, is Neyman orthogonal and satisfies regularity conditions. We state it formally in Assumption C.2, The second assump- tion, Assumption C.3 requires that the nuisance estimators obey regularity conditions and converge at appropriate rates, for all 1) E ST. We state formally in Assumption C.3 We defer discussion and interpretation of the assumptions to the appendix. We are now ready to state our theorem, which says that the assumptions above are sufficient for efficient inference and consistent variance estimation under shared-state interference.
Theorem 3.1. Under Assumptions 2.1 C.2 and C.3 with probability no less than - y, Algorithm returns an esti- mator such that
VTo-ToW(Wnn (WWW:::)i )² S 1)
4. Inference on Average Direct Effects
(3.1)
In this section, we instantiate Theorem 3.1 to estimate aver- age direct effects in observational settings with shared-state interference. The average direct effect (ADE), informally, is the mean difference between treatment and control out- comes for each individual, keeping all other individuals' treatment assignments the same. It is also sometimes called the expected average treatment effect (Sävje et al.) 2019). We first define a structural model of outcomes under in- terference in Section 4.1 apply our DML theorem to this setting in Section 4.2 and then validate our method for finite samples via simulations in Section 4.3
where we define o² = Tim-- T Var (Wr((W .)).
Moreover, when the variance of the estimator o² is replaced with the variance estimator 0² given in Equation C.II. Equation (3.1) still holds.
The proof of our Theorem follows the pattern in that of Theorem 3.1 of Chhrnozhuukvv et al. (2018). We have simplified some of their setting for the sake of space and clarity, but our analysis in this paper could be extended to accommodate the additional generality in their theorems.
To prove the result, we must account for covariance between different terms (W,, in our analysis: Chernozhukov et al. (2018) do not have to handle any covariances since each observation is independent. To control covariance, we ap- peal to the combination of geometric ergodicity and detailed balance or m-dependence. In either case, correlation be- tween terms goes to zero sufficiently fast that these terms do not dominate. After these covariances are handled, we may replace the iid central limit theorem used in Chernozhukov al. (018)) with an appropriate Markov chain central limit theorem complete the result. To prove consistency of the variance estimates, under Assumption 2. we apply a consistent variance estimation theorem from the Markov chain literature. Under Assumption 2. b) we show that the plug-in variance estimator is consistent by showing the plug-in estimates of each variance and covariance term are themselves consistent.
4.1. Model, estimand and estimator
Informally, our structural model will consist of data gener- ated according to standard observational setup for units indexed t = 1, 1,.... T with outcome Y,, covariates X, and treatment assignment D, except that we will allow for each outcome to depend on shared state H,, which is dependent on data up to time t - 1. The shared state, which may be vector- valued, will be the channel through which spillovers may occur in the data: outcomes are dependent on shared states and shared states may be dependent on pre- vious outcomes, covariates, treatments and shared states. In particular cases, the shared state might represent available inventory in a market, like the number of drivers available on a ride-sharing platform, or public information, such as the popularity of songs on a music streaming platform.
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Double Machine Learning for Causal Inferenee under Shared-State Interference
We will have, for each individual i, potential outcomes, shared-state and treatment given according to
section as
cpli n)
def.
[(1, 5(0, Hz),
(i,D,, H1) f (D., (,, H1) + Yo, H1( = =h (W-1)) + He.
(4.1)
db (W,,
def.
Dr.
D- 1 1- m
(4.5)
(4.2)
(m()))
= mXi) D',
(4.3)
(Y: - S(Dr, Xx, H.)):
where the nuisance estimators are 17 = m)). The func- tion yp mirrors the augmented inverse probability weighting (AIPW) estimator (Robins et al.) 1994) that is standard in the semiparametric inference literature (see, e.g., WWager (2024) for background and history), with the exception that our out- come nuisance estimator f takes the shared state H2 as an argument. One way to interpret our estimator as a covari- ate adjustment for the shared state: omitting the shared-state at given time may result in form of confounding between units' outcomes. However, it is not sufficient to treat He as a covariate and apply DML methods as if the data were iid, since there is dependence between units which needs to be accounted for. We will see the contrast between our method and an approach that treats the shared state as if they were iid covariates in our simulations, and we note that, in Section [ treating the shared state as if they are covariates will not in general yield consistent treatment effect estimator.
where .). is potential outcome depending unit t's outcome and the shared state, H.(-) is the potential shared state depending on the data from time t - 1, Dr is the treat- ment assignment and X, are covariates. We use Y, and Ht to denote the random variables realized from the data generating process (as opposed to --)-.) and H.).) which describe the potential random variables under different treat- ment assignments and shared state prior data). The struc- tural model posits that Y, is determined by a function f" plus stochastic residual term Y, Similarly. H₂ is a func- tion of h- and residual H, and D, is function of m² and residual Dr. We will assume the residual terms Ye He, D. obey Ep [Y, Dr, X,, H.] = O, E Wr-1] = O, Thus, m is a propensity score func- x,) = Ep(D, X,) m" m=mmm""X (X,).
Ep[D, Xx]
tion since P PDD,-
By assumption, H, depends on data from prior time steps only through the observations at time t - 1. Thus, H, sat- isfies the Markov property. We also note that this model allows for H, to be stochastic, even conditional on Wi-1, through the residual term Hr Thus, the shared state may fluctuate due factors exogenous the previously arriving units' behavior. We also assume that H, satisfies either As- sumption 2. 1 a) or Assumption [bb This implies that the whole chain, consisting of observations (WiJT satisfies Assumption 2.1
4.2. Estimation and inference
Our main result in this section requires two assumptions beyond the Markov chain assumption. We defer formal definitions and further discussion of these assumptions to the appendix and discuss them briefly next.
Assumption D.T imposes regularity conditions which ensure that both the true data generating process and estimated nui- sance functions satisfy support conditions on the probability of treatment and that nuissance functions and error terms are bounded in L norm. Assumption D.2 requires that the nuisance estimators mn and f converge in L to their true values m² and f and that the product of their convergence rates vanishes at (T rates.
Our estimand in this section will be the average direct effect (ADE). It defined:
T Σ
0 def.
1
EP (1,1, H.) Yi(0, HO)
(4.4)
where the expectation is taken over WI.T. In other words, this is the average effect on each individual of changing their treatment from control to treatment, without altering the distribution of others' treatments. (Note that the expectation Ep marginalizes over values of H,, so the estimand is not conditional on realizations of Hz.)
The required rates for Assumption D.2 can be achieved if, for example, when averaging over time steps, I1(m - mx)( [f--f"((D, X,, H.)ll = o(T-1/4). On the other hand, if the propensity score function m is known (if, e.g., it is determined via a randomized experiment controlled by the researcher), then |[(f- f ) (D,, Xx, He) Ho)l can converge an arbitrarily slow rate and still yield an efficient estimator.
= o(7-1/4) and
Recall from Section 3 that, in our setting the estimator (Wirrs is the sample average of functions seWW;; m)) In this section (and the next), the functional form of the time-t estimator & can be written as the sum of two com- ponents: a plug-in term and a debiasing term. For conve- nience, we will denote these ypi and ddhh; i.., (WW,; = pp (W;; n) + (W::n). These will be defined in this
There is a rich literature giving learning rates for machine learning estimators trained on dependent data. A key im- plication of our Markov chain assumptions is that the data are p-mixing, which intuitively means that observations
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Double Machine Learning for Causal Inferenee under Shared-State Interference
that are far apart have diminishing correlations (see, e-g-, (Bradley 2007) for formal definitions of mixing conditions). Learning rates for mixing sequences have been proved for neural networks (Ma and Safikhami 2022), random forests (Goehry] 2020) and several other machine learning methods
(Wars'')
Danity
[Irle] 1997 Steinwart et al. 2009 Lozano et al. 2014 Wong
(et al. 2020). See (Ballinari and Wehrli 2024) for further
discussion.
With these two assumption in hand, our efficient inference result for this section.
ready to state
0
2
Bias
Theorem 4.1. For the model defined in Section 4.1 and
Figure 2: Estimates of the average direct effect with our double machine learning estimator versus naive Horvitz- Thompson, naive plug-in and naive DML estimators.
the estimand s and estimator 1/" defined in Equations (4.4)
and (45) respectively, under Assumptions D. and D.2 then,
with probability no less than 1 - 7 it holds
VTO-1(WWV fi) )² S 10,
ure 2 Full details of the simulations are available in Ap- pendix F and the code used to generate them is available gi hub.
where ā is defined in Equation
Theorem instantiates Theorem for the model de- fined in Section It replaces the general assumptions required for Theorem 3.1 with specific regularity and rates
Results for treatment effect estimator comparisons. In Figure 2 we observe that all the distributions of each of the naive estimators are substantially biased. By contrast, the DML estimator is centered around the true treatment effect. The bias in the plug-in estimator occurs despite the smoothness of f and m and the fact that the data is low dimensional. The bias in the HT and DML-N estimators comes from the fact that they do not account for the effect of the shared state and thus will in general be inconsistent. The direction and magnitude of the bias for each naive estimator is idiosyncratic to our synthetic data generating process and machine learners; in other settings the sign of the biases may be reversed, and the relative performance of different naive estimators may change.
assumptions in Assumptions D.I and D.2
4.3. Simulations and estimator comparison
Next, we validate our results through simulations demon- strating the performance of the double machine learning estimator versus naive alternate estimators.
Setup for treatment effect estimator comparisons. We compare our estimator, which we will sometimes denote by DML4SSI, to several benchmark estimators: The naive plug- in estimator viPi, defined formally in Equation (F.1). simply imputes the conditional expectation for each treatment as- signment using the estimated conditional expectation func- tion f. Plug-in estimators constructed using machine learn- ing models will converge at slower-than-v'T rates, and in finite samples may be very biased. The naive differrncce- in- in-means Morvitz-Thomppe
5. Regression adjustments in switchback experiments
In this section, we turn our attention to estimating the global average treatment effect (GATE) using switchback experi- ments. The GATE, as we will define formally shortly, is the difference in an the mean outcome of interest when all units are assigned to treatment versus when all units are assigned to control. Switchback experiments, which were formalized in Bojinov et al. (202)) allow for valid inference in settings where interference is mn-dependent.
(HT) estimator HT, defined formally in Equation F22). takes difference in means be- tween treatment and control units. The naive HT estimator does not account for the shared state and thus may be in- consistent and very biased in finite samples. A naive DML estimator (DML-N) treats the data as if it were iid using the AIPW estimator and is defined in Equation (.3).
Switchback experiments are implemented by assigning treat- ments in sequential blocks so that é > m units in row receive the same (randomized) treatment. For example, for a block of l sequential units, the experimenter may use a Bernoulli random draw to assign all units in the block to treatment or control. Intuitively, switchback experiments account for interference by arranging for the observation of some units where all m previous units received the same
In our simulations, we generated data according to smooth function of both De, Xx, and of He For all machine learning predictors f. m and f', we used random forests with de- fault parameters trained on auxiliary data of size T sampled independently of the data used for inference. Then, we com- puted each of the estimators (WWi:; 7), (Wirr; 7), sp (W1r1; f)) and (PMI.-N(1..T: (7') and created a den- sity plot for the bias of the resulting estimates in Fig-
7

---

Double Machine Learning for Causal Inferenee under Shared-State Interference
treatment assignment the current unit. Our framework in- corporates doubly robust regression adjustments that allow for lower variance estimates in settings where covariates and shared-state variable exert significant influence on outcomes (or, equivalently, in cases where treatment effect sizes are small). The structure of the remainder of this section is the same is in Section 4 we define our setting in Section [5. state an efficient inference result in Section 5.2 and validate our results via simulations in Section 5..
1 [ m)st = =1]
1 ( = =0}
db
(We. def.
(- m) :t,
x+ m) :t, :)
(V2 -V- VV,- Xx, ((,
(5.5)
Note that the estimator has form similar to the AIPW esti- mator of the previous section with a few differences. First, in the plug-in component of the estimator, we plug in the all treatment or all control vector for treatment assignments. Second, in the debiasing component of the estimator, we substitute in indicator functions for the past treatment assignments being equal to the current units' treatment as- signment. These ensure that the estimator identifies the effect of all treatment versus all control.
5.1. Model, estimand and estimator
Our model in this section is similar to that of Section 4.1 for t 1, ,., T, we have an outcome Y, that depends on shared state Hd, covariates X, and treatment assignment D, This section is different from the previous in that we require He to consist of the last m observations of treat- ment assignments and covariates. This allows for satisfying the mn-dependence condition required for switchback ex- periments to yield unbiased estimates of treatment effects (Bojinov et 2022). Also, since we considering switch- back experiments, assume D1:r' to come from known distribution determined by the experimenter.
We will assume ("t((- m) : t, 1,, x ((t- :t, 0) > Ç for all I. This is not always the way that switchback experi- ments are defined: in some contexts, the switching points may be deterministic, which would mean that any t less than m time steps after (deterministic) switch point would have T"(( - m) 4, 1) = x ((1 m) O) = 0. One way to achieve the constraint is to pick a switching length >> m and then pick a first switch uniformly at random from the first l time steps. Then for each block of constant treatment assignments, Bernoulli randomize. This will give P(D 1) = P(D(-m)a = 00 2 (e mm)/2.
Formally, we will analyze the following potential outcomes model on a population indexed t = 2....... For a con- stant m E N and for each individual there is some binary treatment of interest Dr E [0, iid covariates X, E Rdd shared state He E Rd and outcome Y, E R given by
5.2. Estimation and inference
The assumptions necessary for efficient inference in switch- back experiments under m-dependence are analogous to those in Section are deferred to Appendix E The only crucial difference is that, since treatment assignments are randomized in an experiment, propensity scores are known. This means we do not need to impose rate conditions on the convergence of f to f*. We now state our estimation result for the GATE under
Y((D,, H)) (',, H1) +Y, (()(())))(--))
(5.1)
1); X (t))m)((t1))
(5.2)
Di. D
(5.3)
where D is distribution determined by the experimenter. We will drop the arguments to H, when D(t-m):(t-1) are the observed treatment assignments. The residual terms Y, will be assumed to be mean zero and obey Ep X,, H = O. Notice that the above implies that shared states are m-dependent as defined in Equation (2.3 For an index set t = (s : t), 8 1 E [T] and boolean vector b = {0, 1311 define n² (t, b) IP [De = b] to be the prob- ability that the subvector Dt is equal to b. The estimand will be the GATE, defined as
model.
Theorem 5.1. For the model defined Section 5.1 and the estimand and estimator defined in Equations (5.4) and (.5). under Assumptions E.T andd E.2 then with probability no less than 1 7/ it holds
VTo (+6 - --) - (N 1),
where ē is as defined in Equation C.1
T Ep (1,1,, Y, (O)). (5.4)
5.3. Simulations and estimator comparison
2²- = T
We next validate the DML method for estimation of the GATE in switchback experiments.
As in the previous section, our estimator ve will be defined by the sample average of the sum of a plug-in term yPi (W,, 77) and debiasing term e ddb(We, n). We define these terms for this section as follows:
To guarantee m-depennence, the data generating process used for the plots in this section is different from that in Section 4.3 In particular, the data generating process for H, in Section allowed for dependence on prior shared
Setting.
po (W,, def. ((()) (,0,X,,
8

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Double Machine Learning for Causal Inferenee under Shared-Stale Interference
g R (Warte)
(1
y (Waz))
Danity
Denetr
0.3
0.2
0.1
0.0
0.1
0.2
0.2
0.1
0.0
0.1
0.2
Bias
Figure 3: Estimates of the global average treatment effect in switchback experiments with our double machine learning estimator versus naive estimators.
Figure 4: Estimates of the GATE in switchback experi- ments with the double machine learning estimator versus the switchback Horvitz-Thompson estimator. The DML estimator has substantially lower variance.
states, which allows for long-range dependencies between observations over time. In this section, H₃ is function only of the last observations of treatments D Dt-m:t-1 and covariates X x-mm:t1-- The setup is otherwise similar to that in Section 4.3 f is a smooth function of D,, Xx, H, and we generate the predictor f using random forest with de- fault parameters. The switching length is chosen arbitrarily 2m. Full details of the simulations are deferred to Appendix F We compare the performance of our estimator against naive estimators in Figure 3 and against unbiased estimator for treatment effects in switchback experiments in Figure
covariates (and is learned by expressive machine learning models). and the fact that the DML estimator uses all of the data while the switchback HT estimator drops the first m observations after any switch. We leave more detailed exploration of the performance of the estimand (compared to (1) the Korvitz-Thompson and (2) across different choices E) for future work.
switchback estimator
6. Discussion
In this paper, we formally define shared-state interference and develop methods for efficient estimation of treatment effects using double machine learning. Shared-state inter- ference occurs when spillovers in a system are channeled through a low-dimensional statistic like prices, inventory information. Our motivating examples of shared-state interference are marketplaces and recommender systems, but there are many contexts where researchers and practi- tioners wish to measure treatment effects in the presence of shared-state interference. Our method allows for the use of expressive machine learning estimation of nuisance parame- ters while achieving VT We also provide consistent variance estimator.
Treatment effect estimator comparisons. In Figure 3 we compare our DML estimator to naive estimators. As in Section 4.3 we compare our estimator to a naive Horvitz- Thompson estimator, a plug-in estimator and a naive DML. estimator. Each estimator is defined the same way as in Section 4.3 except that we plug in the known true propensity score m. instead of the estimated propensity score mn since m is known in an experiment. Additionally, we show the shared state as covariates estimator defined in Equation (F.5). JSSAC is the estimator obtained by treating the shared state as iid covariates.
Note that the naive estimators are biased away from the true GATE, while the DML estimator is approximately centered at the true effect. The SSAC estimator in this section is bi- ased, intuitively, because it controls for the spillover effects between units, rather than incorporating them into the effect estimate: the global average treatment effect includes both the direct and indirect effects of treatment.
In future work, it would be valuable to explore a number of theoretical and empirical questions around shared-state interference. Theoretically, it would be interesting to ex- plore what other estimands (e.g., the average indirect effect as in Munro et al. (2023) or time-discounting as in Farias et al. (2023)) It would also be valuable characterize alter- nate forms of interference that have similar structure: for example, in some situations, a shared state may influence treatment assignments (which we assume does not occur in our structural models). Empirically, it would be interesting to apply our methods to markets or information systems that exhibit shared-state interference. We hope that this work inspires further investigations of tractable interference structures in semiparametric inference.
compare our estimator with unbiased switchback estimator proposed in Bojinov et al.] (2022), which is defined in Equation (F.6). Note that both the DML and switchback HT estimator distributions are centered at the true effect, but the DML estimator has substantially lower variance. This is due to the fact that some of the variation in outcomes is explained by the shared state and
In Figure 4

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Double Machine Learning for Causal Inferenee under Shared-State Interference
Impact Statement
h http://arxiv.orgg http:///..rr/a///00....o 00000000010000 48 ar arXiv:2009.00148 [stat].
This paper presents work whose goal is to advance the meth- ods for measuring treatment effects in settings like recom- mender systems and markets where units interact via shared social signals or information. There are many potential so- cietal consequences of our work, none which we feel must be specifically highlighted here.
R. C. Bradley. Chapter 3: Five Classic Strong Mixing Conditions. In Introduction to Strong Mixing Conditions, volume 1. Kendrick Press, 2007.
J. Brennan, Y. Cong, Y. Yu, L. Lin, Y. Peng, C. Meng, N. Han, J. Pouget-Abadie, and D. Holtz. Reducing Sym- biosis Bias Through Better A/B Tests of Recommenda- tion Algorithms, Oct. 2024. URL httt::///axiv... g abs 2209 07107 arXiv:2309.07107 [stat].
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Double Machine Learning for Causal Inference under Shared-State Interference
A. Background
Additional notation. For a vector-valued random variable V, when we write V Op(-), we mean that the relation holds element-wise. For sample space 92, let ((2) be the corresponding Borel set.
Geometrically ergodic Markov chains. geometric mixing rate property. A Harris ergodic chain is aperiodie, j--irreuuiblee where y is "maximal" irreuuciiiiity measure (see, e.g., (Meyn and Tweedie 2009 Theorem 4.0.1)) and positive Harris recurrent. We define each of these properties next.
geometrically ergodic Markov chain is a Harris ergodic chain that satisfies a
First, we review aperiodicity for our setting. The period of the chain l is defined in our context as the largest sequence of disjoint measurable sets Ag..... Ak-1 such that the chain cycles between the sets with probability one: i.e., that P(W, E A(c+1)%k W E A.) = for all i E {O...... ,, k - 1}. An aperiodic chain is one where the period is 1. chain p-irreducible for some measure p. if, for all measurable S W such that μ(A) O, it holds for all w and e [T]
e Wi+ E A W; = w) > 0.
This definition of irreducibility is a generalization of irreducibility in finite-state Markov chains. Harris recurrent means that for all p-measurable S W with p(A) and w E A, it holds,
P([W) E A i.. Wi = =w) =
where i.o. stands for infinitely often:
(WW)) A..)) def. nuaw A}.
Harris recurrence is generalization of recurrence in finite-state Markov chains. A chain is positive if there exists measure 7 such that, for all 7-measurable A,
((A) I, K (w,
ie., that there exists an invariant measure.
Detailed balance is defined Equation 2.2 The property means, for continuous state spaces, that the steady state probability density of each state times the probability density the transition to another state equal the steady state probability density times the density of transition to the first state.
A Harris ergodic Markov chain is geometrically ergodic if there exists function M W - R50 and constant 0 E (0, such that, for all w W.
|| (w,-) Pso(:)ll S Mwwoo"
where I1-IlTy is the total variation distance norm defined for signed measure μ as
||l|TV
sup p(A) inf AEB(W)
p(A).
AEB(W)
Double machine learning (DML). Double machine learning is meta-algorithm for semiparametric inference, introduced and analyzed in Chernozhukov al. (2018). It allows for the use of expressive machine learning models even when those models may not converge to a true data generating process at parametric rates. The algorithm consists of two conceptual components: sample splitting and bias correction in the estimator. The first component is the construction of nuisance function approximators from data that is independent of the observed data it will be applied to, typically achieved by sample splitting. The nuisance estimators can be constructed using expressive machine learning methods that may produce biased predictions and converge at dower-than-- than-parametric rates. The second component is the use of a bias correction term added to plug-in in estimator which ensures that errors in the nuisance function approximators depend only on second-orde -odder terms. We use the following definitions of the Gateaux derivative and Neyman orthogonality our results.
13

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Double Machine Learning for Causal Inferenee under Shared-State Interference
Definition A.1 (Gateaux derivative). For a convex set of functions S and function v) S, the Gateaux derivative of map IR with respect to n7:1" 1-= E S is defined as the quantity
Or f(n + +r(n ))),, r [0.
Definition A.2 (Neyman orthogonality). For a distribution P on data W, the function vé is said to be Neyman orthogonal with respect to S for 17" E S and all 1y E S the Gateaux derivative of + Er(s(W;;)) m)) with respect to 1); 17" vanishes at r=
Ōr [(1 + +r(1- 11")) = 0.
B. Additional Related Work
Here we provide detailed comparison of several prior works that are particularly closely related our paper.
Comparison to Chernozhukov et al. (2018). We prove theorem analogous to their Theorem 3.1 for the case of dependent data. We make two changes to their setup. The first is that we will not assume each individual's treatment assignment, covariates and outcomes (D, X, Y) are drawn iid from joint distribution. Instead, we will assume that covariates and treatment assignments are drawn iid, that units arrive sequentially, and that outcomes are a function of covariates and a shared state possibly depending on the treatment assignments, covariates and outcomes of previous arrivals. Second, we assume, in the style ongenoooosssss et al. (2023), there is access to an auxilliary sample on which to train the nuisance parameters. We will assume that the size of the auxilliary sample scales appropriately so that the nuisance functions converge in (P) appropriate rates and are independent of the data. In Chernozhukov al. (2018), independence is achieved through sample splitting of the iid data. In our context, where outcomes not drawn iid, the auxiliary training data serves the role that sample splitting does in the double ML for iid data. Notationally, our score function v is defined differently than theirs: ours must be a function of all of the data W whereas theirs may be a function of single draw W from the iid data distribution.
Comparison to Emmenegger et al. (2023). As in their main results, we derive efficient inference results for the average direct effect. In their appendix, they also provide results that are analogous ours for estimation of the global average treatment effect. Our model can be written as network interference model by constructing network where each node affects the potential outcomes of future nodes. However, they prove results about network interference under sparsity assumption, and our network interference model is not sparse (since there would be a directed edge from each node t to each future mode > 1). Instead, our framework makes the interference structure tractable by mediating interference through shared states. Thus, our work allows for potentially long-range dependencies between observations that are disallowed by the sparsity assumptions in the network interference literature. Additionally, our model allows for dependencies between units" outcomes and future units' outcomes, which is disallowed in their model (and the vast majority of work in the network interference literature).
Comparison to Ballinari and Wehrli (2024). Their DML results, like ours are about time series model where units are observed in a sequence over time. They instantiate structural model similar to our approach in Sections 4 Jand and their efficient influence results, like ours, go through mixing conditions. The estimand in their structural model is a impulse response function, which measures the effect of an intervention at time t on outcomes at time + h for some known h, whereas ours is the average direct effect and global average treatment effect. Our structural models are also different, since we require all interference across time to occur through shared-state variables, whereas they allow for generic mixing across time.
Comparison to Munro (024). This paper considers semiparametric estimation of the global average treatment effect in settings where outcomes are determined by a centralized mechanism like an auction, rather than a context- and mechanism- agnostic setting like outs. Informally, their key assumptions require that the mechanism allocates goods according to a known function of individuals' bids and market-clearing cut-offs. They also require individuals" "bids" to the mechanism obey the stable unit treatment value assumption (SUTVA). Their work establishes estimation of the global average treatment effect in observational settings, whereas we provide methods for estimating the average direct effect in observational settings and the global average treatment effect in experimental settings. Their model considers simultaneous arrivals of units to the market, whereas ours considers sequential arrivals.
14

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Double Machine Learning for Causal Inferenee under Shared-State Interference
Comparison to Zhan et al. (2024). We prove a theorem analogous their main theorem, except that results go through Markov chains and theirs go through martingales. We also do not make functional form assumptions about the relationship between outcomes and the shared state, which in their case is the set of recommendations chosen by the platform (they use discrete choice model). We also do not require that the shared state be computed by neural network, as they do.
Connections to the network interference literature. Our model can be interpreted as a network with a particular structure (where each unit affects the outcomes of future units), but our model allows for dependencies between outcomes of each unit and the outcomes of subsequent units. Typically, in the network interference literature, all dependencies are channeled through treatment assignments. rather than outcomes, since outcomes in their models are typically not sequentially determined (see, e.g. AArooww et al. 2020) for overview). Additionally, methods in network interference often make sparsity assumption on the interference graph (like those in Emmenegger et al. (2023), ), whereas our interference network can be dense. However, since correlation between units dissipates over time, it would be interesting to explore whether models of approximate neighborhood interference (Leung 2022) or approximate local interference (Chin) 2019).
C. Further details and proofs for Section 3
In this section, we prove our main theorem in Section 3
First, we state our consistent variance estimator.
-1-1
((+1)/22
-
(Wir::
under Assumption 2.
T2( -1
==
+1
2² def.
(C.1)
Σ
(p) (Wi.T: 2 Σ (WW,)) v (Wi.:; (Wir::)))<(Ww-::>) (Wi::11)
T
under Assumption 2. b)
where, for some 0 (1 (1+00 (+8)) we define T2 [ and Ti = [T/Tp].
Remark C.I. We briefly comment on the fact that our variance estimators under Assumption 2. a and under Assump-
tion [b) are different. Under Assumption [a] our consistent variance estimator not plug-in estimator. In fact, plug-in variance estimators in Markov chains are not in general consistent (Jones al. 2006). On the other hand, for
Assumption2..I|0 the plug-in variance estimator is consistent.
We next state the formal assumptions sufficient for the theorem. Assumption of eh with respect 17 E S, exist and are continuous. The Gateaux derivative is defined in Definition A.IJandd informally, is equivalent to the usual derivative with respect to E (0, 1) for (WW1: rr)" + (1 (1-r)n for 17 Sr. Assumption C.2 requires Neyman orthogonality, which is defined in Definition [A.2 and, informally, is the requirement that the Gateaux derivative above, evaluated at = 0, is zero. In applications, using our theorem in this section will require carefully designing estimators so that these conditions hold.
C.2 (a) requires that second Gateaux derivatives
Assumption C.2 (Smoothness and Neyman orthogonality conditions). For all T 1, the following conditions hold for all P =
(a) The map 17 + Ep [+(,.r:: m)] is twice continuously diffeet-di on Sr around 17"
(b) The estimator v is Neyman orthogonal with respect to the nuisance realization set Sr around 17"
We next state regularity and convergence rate assumptions for our main theorem. Equation
requires that the L norm of ((W,, 7) must be bounded for all E [T] for > 4. Equation (C.3 requires the average over I of the I,2 norm of <PWW; v) - (W,, (, 77") must converge to zero T oc. Finally, Equation (C.4) requires the second Gateaux derivative of 2 with respect to 17 must go zero at faster-than- /T rates. These are analogous standard regularity and convergence rate conditions for DMI. theorems (Chernozhukov et al. 2018).
Assumption C.3 (Regularity and convergence rate conditions). There exists constants 5, C 0 and To 2 1 such that for all T > To. the nuisance function estimator i(Wanx.) belongs to a realization set Sr with probability at least 1 - y where
ST contains 17" and satisfies the following conditions for all P:
sup ll:( C, Vt T
(C.2)
yESr
15

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Double Machine Learning for Causal Inferenee under Shared-Stale Interference
sup T
(ls,(W;; <(Warr")l = d(1), -")]|| = =((T-112)
(C.3)
7EST
sup (E.
Ep [WWiinn + +r(n - -
(C.4)
We next provide overview of the proof and prove the main theorem, At high level, the two parts of the theorem are VTT-asymptote normality of the treatment effect estimator and consistency of the variance estimator. Here we overview the proofs of each of these parts.
For -assmpototii normality of the treatment effect estimator, we make the simple observation that the difference between the treatment effect estimator <>(Wi: fi) and the estimand sh* can be broken down into the sum of three differences. (This an observation made in Kennedy (2023, page 19, equation 10) and is standard in proof's of DML theorems.) The three components in the sum are: (1) The difference between an estimator where the true nuisances are known (sometimes called an oracle estimator) and the estimand (Equation (C.5) ), (2) the deviation from its mean of the difference between the oracle estimator and our DML estimator where the nuisances are not known (Equation (C.6)), and (3) the expected difference between the oracle and plugin estimators. Then, the first term is shown to be vT /T-asymptotically normal in Lemma C.6, and the second two terms can be shown to go to zero in probability at faster than -T-tes in Lemmas C.7 and C.9 respectively. Together, this implies (Wi:T; fi) approaches its mean at vT rates and is asymptotically normal. The consistency of the variance estimator given by Lemma C.10 We describe the high-level intuition for each of the lemmas before their proofs.
Proof Theorem 3.1
Observe:
(Wir: - 4²= S (Wi:r;"") x0- + +W) - (Wi:::
S (W1:r;;")) -
(C.5)
+ (Wi::: n) - <W(1irn"" - Ep(e (Wia::"") Waix]]]
(C.6)
+ Ep (W (WW::;;)) Warx
(C.7)
from applying the triangle inequality twice. Then applying Lemma
C.7 and Lemma C.9 implies
VTo (W(Wi.:: w² = -To ( - x²) + (1))
Applying Lemma
C.6
we observe
(T .³) $ N( 1).
which, with, e.g., (van der Vaart 2000) Theorem 2.7(iv), implies the first statement in the result. Lastly, Lemma C.10 proves 03- P, 0², and applying Slutsky's 's theorem completes the proof.
In the proof of Lemma C.6 we will use the following two central limit theorems. The first is for geometrically ergodic Markov chains on general state spaces, and the second is for m-dependent sequences. Theorem C.4 generallis standard central limit theorems for Markov chains in finite state spaces, where the analogous requirements are that the chain is irreducible and aperiodic. See Appendix for further description of geometrically ergodic Markov chains. Theorem C.5 states a central limit theorem for m-deppenent sequences.
Theorem C.4 (Theorem 2, (Chan and Geyer 1994)). Suppose that a sequence of random variables { is a geomet- rically ergodic Markov chain with stationary distribution Also, suppose that there exists constant 8 > 0 so that measurable function f satisfies [| ( < cc. Then it holds
1
Σ
f(A)) -E+1)) $ 1),,
VTo
where we define
0² def. Var (S()) +2 Σ Cov=(((A),,JAA))
16

---

Double Machine Learning for Causal Inference under Shared-Star -tacc Interference
Theorem C.5 (Theorem (Hoeffding and Robbins 1948)
For sequence of m-dependent random variables A1, Ag,
if there exists constant C 0, satisfying, for all 1,2, .... Er|A, 3 C. Then,
T
T
Σ
(Ae B(A))) 4 (),
t=1
where we define
o²
def.
lim e- E-000
Varp(Acts) + Σ CovplActor Coup(Arts. Anori))
Lemma C.6 says that the oracle estimator approaches the estimand at V rates and is asymptotically normal. The proof of
Lemma C.6 applies Theorem C.4 To apply the CLT, we just need to verify that we can apply of the central limit C.4 for Assumption
theorems above: Theorem
2. (.T and and Theorem C.5 for (b]
Lemma C.6 (Corollary to Theorem C.4). For estimator vp satisfying Assumption 2.7 and Assumption
C33 it holds
VTo (WbWWWWWW +*) $ ().
Proof of Lemma C.6 From Assumption 2.1 under part (a) we have that {W,} is a geometrically ergodic Markov chain.
Also, m"") ) is bounded in L (P) by Assumption
C.3 Equation
CC2) So (WWirss"" -i² is bounded in probability
by the triangle inequality:
T
(W - 30" (P) I T (W (ll=+or) + |00 < 00
Thus, the result under Assumption 2. follows directly from the application of Theorem C.4
For Assumption 2. b) since { { is m-dependent, is ()) By Assumption C.3 that (WW"""") is bounded in T Thus, we can apply Theorem C.5
Equation
C.2
we have
Lemma C.7 says that the deviation from its mean of the difference between the oracle estimator and our estimator approaches zero at faster-than-v33 rates. Our proof pattern is similar to that Chernozuuuu et al. (0188): our strategy is to prove that the variance of the expression goes to zero at faster-than-T rates, so that we can apply Chebyshev's inequality, which implies that the expression itself goes to zero faster-than- rates. The core difficulty of proving the lemma, compared to the analogous result in CChrnnzhuuov et al. (2018), is that must account for the covariance between observations at different times. Chernozhukov et al. (2018) assumes independence between observations and thus does not have to account for such covariances. We handle the covariances separately for Assumption 2. {a) and for Assumption 2. [b) Under Assumption [.) we can invoke a theorem saying that geometrically ergodic Markov chains satisfying detailed balance have correlations between observations that decrease to zero at an exponential rate. This implies that the sum of
correlations is bounded by a constant, so they do not dominate. Under Assumption 2. [b) there are only finitely many non-zero correlations, so their sum is also bounded by constant.
Lemma C.7. Under Assumption 2.1 for an estimator dp and nuisance parameter estimators i) satisfying Assumption and Assumption with probability 7,, holds
(WWur;))) - (Wi Eple(Wi:r;)) (Wiir: (2
Proof of Lemma C.7 First, will introduce new notation. For all E [7]. let
S def.
p(Wa:))
def. ((We)"))
17

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Double Machine Learning for Causal Inference under Shared-State Interference
Let ET be the event that r ST (which happens with probability 7 by Assumption C.3 deterministic function of Waux. On the event Er,
Also, ij is assumed to be
Ep (Wb(WW) (WWiTss") peppp (i:::
w)ux.))2 [
=Ep
T-1 Σ
Gr 22 -Ellse sp8 W
Warx.
(Definition of vb)
T
T-2
Er - x/- -Errie 25 2 W
6-1
+ 2
Epl (5) - :0 - -Eplot - -97 waux))
(C.8)
8m11
(Bi- ii-- Eplei-e - li- (anx.)) ]
(Rearranging.)
< sup T
Σ
E Ep((p((W(;)) - -Eplo(We:n) to waaxx)) (ramu
SESr
1=1
+ 2
Ep (<o(W):)) - P8 -Eplo(W2: n) - 2o waux)))
(((WW -pie- - -EEppp(Wiee)) - - wanx))) [aax
(f E ST on ET)
T
sup 2 SE
Er Er((p(W(::) - -Erlp((W))))) 6;1)
(C.9)
+ 2
Epl (vo(W)))) - 20 -Fplp((s: -;1))
(v((W -ii-e- (Splp(W-ess)) -li
where the last equality comes from the independence of W from WI..- Now, for all 1 N, let F7" be the o-algebra generated by Wi.0 and let F be the o-algebra generated by WL... Recall the definition of p-mixing (see, e.g., (Bradley 2007 )
p(t) def.
sup (E
[Corr(f.g).
where the correlation, covariance and variance for functions has the usual definition:
Corr(f.g) (oo(f.))/(Va((/)aa(()))))
Cov(J,g) Ep [((W)..<) - Ep ()[Wae)))(((nnan
- Ep [9(Wa--o)))]
and Var(f) Cov(f, f). Next, state a lemma that will allow to bound the covariance between terms in the above sum under Assumption [a)
Lemma C.8 (Theorem 2, Jones 2005 If Markov chain A1:s is geometrically ergodic and satisfies detailed balance, for some e >> 0.
then it is p-mixing with p(T) O
On the other hand, under Assumption 2.1 b)
Wi:00 is mixing with p(l) = 0 for all f m. Note that these facts for some c > 0. Also, for all & E [7], E Sr and & < f, it holds 17 o - EEppp(W;; - ot] (FF00 and ((W,-e;)) -ir Eplp((,,eiit) - of E L2 (where expectations are taken with respect to P). To see this, note that for all t E [T] and E Sz, it holds (w, (.(P) < oo
2
imply Wi.0 is p-mixing with p(t) (
by Assumption C Equation (C.2). Also, since the expressions just depend on W. and W,- respectively, this implies (lseWW
1)) Also, is assumed ( (P(P)) Iwy: (A())))) and similarly (l, mll = ll
n" Sr, so by the same reasoning || I| ( < and || || oc.
18

---

Double Machine Learning for Causal Inference under Shared-State Interference
Thus,
Corr (p(We: ip
(Eplp(W)))))) vil. pi- - - - p()).
This implies we can rewrite Equation (C.9) as
T
sup nESr
T
2
Ep ((p((W:)))) ot - -Brl:((::))) -v:))
t-1
+
Er (p(WW))) vi - -rrlp(Wss)) - sill)
8=1
((((W-cin) -vi-e- Bplp(( (0)e)))
sup T neSr
( - -Brl(((Wss)) villia(p)
p()) (lp(We:)) - - Eppppp - gilll 1-
+
_
1=1
(|x((W Pi - -m)) - pieellea(P))
sup T nESr
(W(W)) *
t=1
p()) (lp(WW:)))))))) vill wll.eellllllwwwwww ri-el
1=1
Now, notice
T (W( - -illamp o(1)
T-1 1
Σ
by Assumption C.3
Equation
C.3
so
T
T
- t.. )
Next, since p(l) O(e-et) for some
0 (by Lemma C.8 and
T
T
Σ
(l pill = o(1)
for all E (T), holds
(())
- - - -iiellee
1=1
T
T
(())
Σ ((((W-ei)) -p -) pp (FllLa(P))
(Exchanging sums.)
1=1
1=++1
T
2/2
T
1/2
-
((E)
Σ
( -p-elleapp))
Σ
(ww(Wn)) - (illea())
1=1
1=t+1
(=+11
(Cauchy-Schwarz inequality.)
T
1/2
1/2
-T-1
Σ p())
T-1 Σ -(il-cm))--ellleap
T-1
Σ
Ilp(Wan) sp ((l()rr))
(=(+1
1=#11
(Distributing 1
19

---

Double Machine Learning for Causal Inference under Shared-State Interference
T
p(()((1)
(T Et lke(W;;) pill = o(1).
-
( (o())
(p(1) O for some >> 0.)
S {)
where the last line comes from the fact that E e c- e0 - e- Thus, applying Chebyshev's inequality, we have
(wi (Wir:n"" - Eplp (Wi.r:)"" w W =P((T-1/2).
Finally, the fact that conditional convergence implies unconditional convergence (Lemma 6. of (Chernozhukov al. 2018 implies the desired result.
Lemma C.9 says that the absolute value of the expectation of the difference between the estimator and oracle goes to zero at faster-than-v VT rates. The proof is nearly identical to the analogous argument in Chernozhukov et al. (201 18: we apply Taylor's 's theorem by differentiating on the path between f and 1f" to show that small deviations of nuisances r around 17" do not dramatically affect the deviation of the estimator from the oracle estimator. The first-order term is zero by the assumption of Neyman orthogonality. The second-order term small by assumption (Equation (C.4).
Lemma C.9. For an estimator and nuisance parameter estimators ij satisfying Assumption holds
C.2 and Assumption
C.3
Ep [#(Wi:;; (WWii: Waux (T
Proof of Lemma
C.9 We Neyman orthogonality, Define
f(r) def.
Ep (WWiiin" r(A- - ()) Waxx Ep (Wi.ssn"))
r E [0,
Then, notice
f(1) = Ep [ (Wi::: f))
Wanx - Ep (=((W.r:)")))
the quantity we want to bound probability. By Taylor's Theorem
02
f(1) f(0) + g.. 5(0) + 0, 10)/2
for some f E (0. 1). Notice that ʃ(0) 0 since <W(1::;3"") is independent of Wauxx Also, notice that 05(0) by
Assumption C.2 b) Finally, on the event εr,
82 0y
82
(()
< sup rE(0.1)
Ōr (r) = (pp(T1122
where the convergence comes from Equation
Lemma C.10 states the consistency of our variance estimator 8². We prove consistency for the estimators under Assump- tion 2. a and Assumption separately. For Assumption a we can directly apply a consistent variance estimation result from prior work. For Assumption [b] we prove consistency of the plugin variance estimator directly. We handle the variance terms and covariance terms in v separately. For the variance terms, we follow the argument in Chernozhukov et al. (2018). For the covariance terms, we follow a similar pattern to the argument for the variance terms: show that the average of the covariances converge in probability to covariance estimates computed using oracle nuisances, and then showing that the covariance estimates computed using oracle nuisances converge to their expectation.
Lemma C.10. For o7 as defined Equation C.I and o² lim Varp( and under Assumptions 2.1
C.2
and C.3 it holds o7 P o²
Proof. We split the analysis into two cases. The first case handles when Assumption 2.1 a holds and the second handles when Assumption 2. holds.
20

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Double Machine Learning for Causal Inference under Shared-State Interference
Case I: Under Assumption 2.1 a) We will apply the following result adapted from (Jones et al. 2006). For simplicity,
will use TI; T2 as defined in the statement of the result.
Lemma C.11 (Proposition 3, (Jones al 2006)). Suppose Markov chain {At)== is geometrically ergodic with stationary distribution л. Also suppose that there exists constant 6 such that measurable function f satisfies [(+s)))) C.
Define
2 def. Var (f(Ii) + (fv)
Σ
and
(1+1yTs
0² def.
Σ s
(
T
> s(Ai)
T2 1)
Then it holds δ² + o².
Notice that by Assumption C.3
Equation C.2
and the fact that Kx E,, we have assumed (( < C.
Thus, we can apply Lemma C.I1
and the result under Assumption 2.1 a) follows.
Case 2: Under Assumption
2. b)
Define
T-m
def.
+ 2 Σ
Cov(p(Wm+t:7)) ((Wm)>>-m)(mm
T
We will prove
07 of B 0
which will imply the result, since of o² trivially. Using the notation from Lemma
C.7
notice
T
07 - of = T- Σ (i:- (Wirr:;1)) E [(p] - )
(C.10)
min(t- 1mm]
27
(G. - (W (.W:r 1)) ($:- - (Wirr: - E [(p] (')( v²)
(C.11)
by definition. We will bound Equation
C.10 and Equation C.I1
separately. First, for Equation CC.I0).
T
(g. v (WIT; 1:;; i)) - E [(p] }/
T
Σ (.: (WiT:: n)) - (42 2/²²
(C.12)
+ (4p w0/² - E [0] - 16-)
(C.13)
by adding and subtracting terms. Next, bound Equation C.122 in probability, notice
(g. (Wi:r;;)) (1)) - -17)²
T
2
= -
(+) ((W1:: of + +)) (fe ((Wa.))) -
21

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Double Machine Learning for Causal Inference under Shared-State Interference
(g (Wi::; 87 + 12))
(C.14)
t=1
(6t (Wi::; + + 1//)2
(C.15)
The first (in)equality comes from rearranging: the second comes from the Cauchy-Schwarz inequality. Now, we can bound Equation (C.14) as
T
T
(g)
(WI:::
07 + 10)2
1=1
T
< 2T
Σ
+ +2(b(W1r:)) )2 (g: v0i)²
1=1
< 27
Σ (5) v2)²²
(C.16)
4 (W<WW::T 1) Ep [ob Waaxx)2²
(C.17)
4Ep (WW ?) x/² W
(C.18)
where the first inequality comes from the fact that (a b)2 2(a² B2) for all a, by the second comes from adding and subtracting the conditional expectation and then again applying (a b)² s 2(a² + b²)). Equation (C.16) is (1(1) by Markov's inequality and Equation CC33 since
T
T
Ep
(v0x vi)o]
7-
11 se (illi(())
tml
1=1
o(1)
Equation C.17 is (1( by the following sequence.
(wb ( ?) Er (Wl(::: ()) waux,))
T
2
-
c Er [ppr]
(Definition of vb.)
T
2
Σ
(9.: - Er (+x1)2 + Σ (+)
Ep (pr) - Er (3)
(Rearranging.)
1=1
T
1/2
2
(p: - Er (G.1)2 + 22/m((8 - Er (P,))
Σ (p.- - Er 11)
(Cauchy-Schward inequality.)
T'
1/2
= - Σ (()))22m00P))) Σ Op((L)
The last line comes by Markov's inequality and the fact that Ep (gp - Ep [+:))]] = 0(1) for all by Equation the fact that m is a constant independent of T. Equation (C.18 is o(1) by the following sequence.
C.2 and
Ep (Wi:;; ?) 3² Waax. 2 Ep [WW::: (WWr:;;)") | 3
T
2
Σ
Ep L++. s77l
22

---

Double Machine Learning for Causal Inference under Shared-State Interference
T
2
T
Σ
o(1)
= =
The first line comes by the fact that Ep [o( 77") W 1², the second is by definition of s/, and the last is Equation
C.3 Thus, Equation CC.14) is op). We can bound Equation (C.15) in probability as
T
T
(6.
(WW.:: + 20 -²)²
T
$ 21
Σ
(0r + pi)2
+ +2(x + +2²²²
1=1
T
T
2
Σ 47
p + pp²
+ 4
Σ
8t
+ 41ba2
1=1
S sup 47 yESr
Σ
T
() + 1t?
+ -2
Σ ((Wa:n)
2 + 466²2
(1(1)
(C.19)
where in each incquality, apply the fact that (a + b)² 2a² + 26 and in the last line apply Markov's inequality with Equation (C.2) Namely, from Equation (C.2) there exists a constant C' such that, for all s, E [T].
sup E [[((W)))]]
= lle S C,
nESr
E [x:]]
[1ppil C, and
sup E (P((s:)))(WW:)) (1s (<P(W.:)) C. n
(The last line applies the Schwarz inequality.) The fact that Equations (C. 14 and respectively imply that the expression in Equation C.122 is (1)).. Next, we bound Equation C. 13
are (p(1) and Op(1) C.15
in probability. Notice
T
E
T
Σ
(1)
---² - E [((0 - (1))9]]
1=1
((0) - 1b= - E [109 ())
7-7
t=1
+ Σ E ( (1p) - 2b/2² E [1pt - )))
(117): 2 - -E - - -
i=1
< 2
1:0 2b (
t=1
1/2
+ /m Vm - *'(lLA(e)
Σ pii 26² (1/2ppp9
-
1:0 36 (
1/2
m
+ /m 1lip 11(PP
Σ
ppj_ 3z
= or(1)
23

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Double Machine Learning for Causal Inferenee under Shared-State Interference
where in the last line we apply the triangle inequality llpt w/4 II (P(P) S I1 + |vb*| and Equation
Thus,
the fact that Equation (C.12 is (1(1) and that Equation (C.13) is op(1) imply that Equation (C.10) is op To bound Equation (C.II.
T
m
(ft
(Wi::;; R)) (Pi-: - (Wi::: f)) E [[p] - a (pp - [5))
1=1
T
T
(5t - (Wi::: ()) (Gi-1 - (Wi.: i)) - (9g - -1-)) (vi - -15")
(C.20)
+ (1t x))) (pt- x²)) - E [(p] ( 2)
(C.21)
To bound Equation CC20
m
(p: - (Wi::: f)) (fi-s - ((W(::;)) - (0] - -b²) (fi - 2b4))
T m
T 1
Σ
(g: (Wi1.: - pt + +5) (81-1 WW
1=1 1=1
+ (42 - v² (G.- - *W - si-- + 15*)
m
T
Σ (6t- - (W1::: - p/ + +5² Σ
(6i-i (W1r: f))
T
m
+ T
Σ
(P7 - w³)
Σ
(6L- 26 (W - :/i_ + x*)
(=1
i=1
T
Or (1)
Σ
(6i-) (WV:r:
1=1
1=1
m
+ T
Σ
(4c 4²)) Σ (1(
The first equality comes from the fact that for all a, b, a', V. it holds aa' - bb (a - b)a' + (a' bd)b; the second comes from rearranging; the third comes from Equation (C.3) Now, we just need to argue that yp -1² = =1p( and &e (Wi::;; = (1) for all to show that that Equation C.20 is or(1).
E [(p]] - " < 2 1| (r() + 21/²²² SC
(1p. - (W1::: 1)) < 2 1 +
( C
by Equation C.2
so Chebyshev's inequality completes the argument.
To bound Equation CC.21
T
27
E
T
Σ
(vp 3¹ (pi_.- - -1-)- 1E [(vt 2/ (p/ -wy]
t=1
t=1
T
2
Ep ((PT - 3²) (19=- x²- - E [((pt - (p) - -2-1)
i=1
m m+i
2
Ey ((1)) - x²²) (1pi - -b)) - E [(p] - (1p' - -}
1=1
(1p7-n - v²t) (pi - -²- - E [(vi_ ('Pi - ((]i
(Rearranging: m-dependence.)
T
< 2
Ep ((p) - 3²) (v07 (5)))²]
24

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Double Machine Learning for Causal Inference under Shared-State Interference
m m++
Ep ((p) - 15*) (1i - 16*) - E [[p] - 3/ (pp= -()0
2
Ep [(pi__ - )²m) (fi-e - 15)) E [(pi_ (PE_ -5))) 1/2
(For I.V. X, B(X B(X))² & E(x?); Cauchy-Schward inequality.)
T
Ep [0p - -5)) Ep ( 1/2
< 2
m-+ mit
2
Σ
Ep ((p2 - 3/' (vi - +5))
t=1
Ep [((Pi_g 15)) (507 - -3 )0) 1/2
172
Ep (0] - -50") Ep [1pi-=- - )5))))))) 1/2
T
m+ m++
Ep (1pī 2b (- Ep -
2
t=1
Rp -1t))- v3))4)) 1/2
Ep ( Bp
T m 2
(88 (811971l(A4()) + 8o0-t) (11802 + 8)/+4) 1/2
m atimmi
(Bl1<elliaa0) + 80079) 8 pllzi-ilizeee + +8004)
1=1 t=1
(8l1et-sllzaep + twptt) (8110i- 8 1/2
(for constants a, b, it holds (a + by < 8a. + 8b4) (Equation (C.2): there are (TT) terms in the sum)
o().
Thus, with Chebyshev's inequality, Equation c,21) p is 0l(1). Then, the fact that Equation (C.20) is 0p(1) and Equa- tion (C.2T) is oP(1) imply that Equation (C.11) is (1). Finally, the fact that Equation (C.10) and Equation (C.1l are
each (1) completes the proof of the lemma.
D. Further details and proofs for Section 4
Before we provide an overview of the proofs and prove the results, we state the necessary formal assumptions. In Assumption D.I we require that each individual has true and predicted probability of assignment to treatment and to control bounded away from zero, almost surely. We also require that the outcome function be bounded in L norm, for q > and that the error terms Y, are bounded in L norm.
Assumption D.1 (Regularity conditions for ADE estimation). There exist constants C, 0 such that the following regularity conditions are met for all (S, m) ST and all / [T]
m(x, 1 <1-,, a.s.
(D.1)
[|( X,, ( <C,
(D.2)
( <
(D.3)
Equation DD44 says that, on average over units /= I,..., T, the expected squared difference between and m² must go
25

---

Double Machine Learning for Causal Inference under Shared-State Interference
zero. Equation D says the same for and f". Eqquatio (D.6) says that the average of the product in m m and over the data must go zero at faster-than-v77 rates.
f
Assumption D.2 (Rate conditions for ADE estimation). The following rate conditions are met for all (f, m) ST
T
11 - =2) =
(D.4)
II(f - X = o(1),
(D.5)
(m)(m)
( --
Hol = opy (T (T-1/2).
(D.6)
In this section, we prove our results for Section 4 The result follows directly from Theorem 3.1 once we have verified the assumptions necessary for the theorem to apply. Thus, our main task is verifying these assumptions.
There are three parts of the assumptions in Section D that are not trivially verified by analogous assumptions in Section [ These are: the Neyman orthogonality of v, L convergence of (WW, 17) to (W,, 17") on average over t, and a second-order condition stating that the second-order misestimation of 1) does not dominate. Each of these is stated in separate lemma, and we provide high-level descriptions of each of them before they are stated and proved.
Proof of Theorem If we can verify that Assumptions 2.1 and C.3 hold, then we can apply Theorem 3.1 Recall
that Assumption 2.1 is verified by assumption in the definition of the model in Section 4.1 Assumption C2 a is trivially
verified from the definition of <b. Assumption C.2 [b) is given by Lemma D.3 Assumption C.3
Equation
(C.2 is trivially
verified by the definition of e and Equations (D.1) and (D.2) Assumption
Equation CC3)
is given by Lemma D.4
Assumption Equation
is given by Lemma D.5
Lemma D.3 states that the estimator 3 must be Neyman orthogonal with respect to the nuisance realization set ST. The argument is standard to any proof of Neyman orthogonality of an augmented inverse probability weighted (AIPW) estimator.
Lemma D.3 (Neyman orthogonality). For the model defined in Section 4. and the estimand db and estimator up² defined in
Equations 4.4) and (45) respectively, and under Assumptions D.I and D.2 (WW1:; 17) Neyman orthogonal with respect
ST.
Proof. Notice, for all
0 Er (p(miinn + r(1
77')]
Ör
r==
E (f (1, Hi) -f- (0,
Dr 1 Dc
(f- (D,, X1, H.)
(Equation (.2) and the dominated convergence theorem)
(X,)
1- mX,e
) (1-1
Hi) (1- mmxx)))) (f Hi) 11 1 1-D;
Di
Ep
(x,=(X,) (x))
(Rearranging)
0.
[Dp[Di/m"(XX) X,] = -Ep [11 (1)/(1-m"(X)) mX,)) X1] = 1)
Lemma D.4 states that, on average over time, (WW;: must converge to 17"" in L2, uniformly over the nuisance realization set Sy. The proof consists of showing that convergence of y reduces to conditions on convergence of f to f
and to m² which assumed in Assumption D.2
Lemma D.4 (Consistency). For the model defined in Section 4.1 and the estimand v and estimator 3 defined in Equa-
26

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Double Machine Learning for Causal Inference under Shared-State Interference
lions 4.4 and
respectively, under Assumptions D.
and D.2
T
sup WEST
T
(((Wn)) - -oW o(1)
Proof. Notice, for 17 Sr,
(WW(;)) "(Wssn")
(f ()) H1) - (f (0, Hi) J(D,,
D. (M- IDD,, Nt, H.) Y - X1, H.)
Yt
m(x))
(Xi)
IY (D,, H1)
Y f Xc, He) 1-m (Xi)
-(1 -Di)
(S (*)(1, H.) - (S-5'(0, X,, H.)
(f" -1--)( (D(,, X1, H.) Y. Y,
+D.
m(x))
mx+)))
(f (DD, X1, + Y
Ye
-(1 -D))
Dd)
(Equation 4.1
1- 1-m (X,)
1 (X,)
Ss (1, X1, H.) -SS-5)(0. H³)
D ((Lf - (((--))) -f))(D, Hi)) m² (Xi) Po((m= m)) (X)))
+ - -D)) ((1S ([[[-fp -f')(D,, X1, H.)) (1 - -m²(xx)) fi((m-m'((x:)))
(Equation D.
S - (V--1)( (1) X,, H1) - -( H.)
+ +2 ((1S ((),,) H.)) - Ve((m- ((((m-m))(X))) -x)())))
(D,, 1 - De, (X,), 100mmm"(XX mx(X,) are all in [0, 1)
Thus, by the triangle inequality:
T
(W - ((W:r"))luu
T
Σ
ll(f - (i),
T 1
X,,
+ + (I0) X1, H.ll + +1(( [lm(X,)) (X.)ll
(Equation D.3
Next, the assumptions Equation (.II and Equation
D.5 combined imply
II( - f-) X, H,, o(1)
T
T +
|| -
= =(1).
And finally
T
2
Il( -
= o(1)
T
27

---

Double Machine Learning for Causal Inference under Shared-State Interference
1 T 2 E
( o
T t=1
by Equations (D4) and (D5).
Lemma D.5 states that the second-order term in Taylor's theorem converges to zero at faster-than-v rates. This ensures we can apply Taylor's theorem and establishes that slow rates of convergence of i) to ') do not lead sub-optimal rates on convergence of (WT.. fi) v². The proof relies on showing that the second Gateaux derivative of on the path from any 7 in the nuisance realization set Sr to 17" reduces conditions on the products of convergence rates of m m" and f to f* Thus, it is sufficient for each of m and f to converge to their true values at faster-than-777 rates, or for m. to be known priori while f converges at arbitrarily slow rates.
Lemma D.5 (Second-order condition). For the model defined Section 4.1 and the estimand ab and estimator 1/² defined in
Equations 4.4) and (4.5) respectively, under Assumptions D.1 and D.2
0²
Ep (W(Wirin + r((n 7"))]] o(2-1/2).
sup rE(0.1):NEST
Proof. Observe, for 7) E ST and r E (0,
0² Ep [p(wEn" + r(n - n]"]]
0,2
De
1- Dr
E.
(X,) + - mx)]m))) (1- - m (X,) r((m m (X)())
(m-m")(X)) (xar))((x(f-2) ff Xt, Ho)
where we first switch the derivative with the expectation using Equation (D22 and the dominated convergence theorem, and then we evaluate the derivatives. This implies
8²
Ep
(Wi:W: + -r(1 - 17 )7))]]
θr
D.
1-
T
Ep
(Xi) + +(m+mmmmmm-- (x)()) 11-m((x) r((m -m -X))(
(m (m-mm))XX) (0--()(Dr,)) X H+)
7-
Blc-2 (m-m")(X)) (T,) Xx, H.)
(Equation D.1
S OP 1/2).
(Equation
D.6
E. Further details and proofs for Section 5
We first state the formal assumptions for inference in Section In Assumption E.1 we require that the probability that any
m + 1 sequential treatment assignments are the same (either all treatment or all control) is bounded away from zero. We also require that the conditional expectation estimator f and the true conditional expectation function f be bounded in 7 norm for q > In Assumption E.2 we require that the time-average of the L22 norm of ʃ f- f is going to zero as T > cc. Assumption E.1 (Regularity conditions for GATE estimation in switchback experiments). There exist constants C, Ç > 0 such that the following regularity conditions are met for all /. E [T]
"t +(t - m) t, d) 2 C, a.s. [ He) L4400) <C
Vd E {0, 1}
(E.1)
(E.2)
|1 (D,, Xx, H.) 2416() <C
(E.3)
28

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Double Machine Learning for Causal Inference under Shared-State Interference
Assumption E.2 (Rate conditions for GATE estimation in switchback experiments). The following rate conditions met:
T Σ 1 - x,, Hille o(1)
(E.4)
In this section, provide proofs for results in Section 5 The proof of Theorem 5.1 mirrors that of Theorem 4.1 just need to verify the assumptions necessary for applying Theorem 3.1 This again reduces to verifying Neyman orthogonality condition, a consistency condition and second-orde Gateaux derivative condition.
Proof of Theorem 5.7
As in the proof of Theorem 4.1 we just need to verify Assumption C.2 and Assumption C.3 and apply
Theorem 3.1
Assumption
C.2 a is trivially verified by the definition tb. Assumption C.2 is verified by Lemma F.3
For Assumption C.3
Equation (C.2) is verified by Equations
E. to E.3 and the definition of y. Equation (C.3) is verified
by Lemma
E.4
Equation
C.4
is verified by Lemma E.5
Lemma E.3 states that our estimator is Neyman orthogonal with respect to the nuisance realization set. The proof resembles standard arguments AIPW estimators, except that in our case the inverse-propent is the replaced with the probabilities that the last observations where all assigned treatment or control.
Lemma E.3 (Neyman orthogonality). For the model defined in Section 5.1 and the estimand and estimator defined in
Equations 5.4 and 5.5). under Assumptions E. and E.2 then (W 1:T; 1) is Neyman orthogonal with respect to ST.
Proof of Lemma
Notice:
E [[p(Wnn'' +
0r
f" H, - -f H.()))
1 [1((e-m): 1] 1(Dii-m): =
=0] 0}
(Vr (f++(-- X,, H.))
([t m) :t,1)
('t m) : t,
1 1 (( = =1]
B[(f -[ ff H()))
((t m) 1)
(1- 1 (Dir-m):t 0}
(f,,
<-((1 m) : (t,
where the first equality follows from Equations equality follows by the fact that E
E.1 to E.3 and applying the dominated convergence theorem. The second Yg D m-]t and rearranging. Now, using the facts that
Ep [1 (D() 1} X [(=m)) Ep [1 (Dic-m) 1}}] =x
*=' ((1 m) (,11) and
Ep [1 (t-m:t 0} X(t-m)a Ep [1 (D(c-m): 0}] =x'((t= m) :t,0),
and iterated expectations, the Gateaux derivative is zero.
Lemma E.4 states the 2 convergence of mWW; uniformly over the nuisance realization set ST, to (W Similar to the previous section, verifying the result reduces to conditions on the convergence of the nuisances, which in this case is satisfied by the requirement that f converges to f and treatment probabilities are known.
Lemma E.4 (Consistency). For the model defined in Section 5.1 and the estimand and estimator defined in Equations 5.4
and (.5) under Assumptions E. and E.2
then
sup T
(eWW)) ((Winn o(1)
SE
29

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Double Machine Learning for Causal Inference under Shared-State Interference
Proof of Lemma E.4
Notice
- -1 H())
1 ((tmm)t = 1]
1 (c-mm =0}
(f - Xe, H.).
(-((1 - m) :,1)
π ((1 - m) (,
Now, since, by Equation and Equation E.4). D). it holds
|1( f (b, X,, H(b) LO(P) = o(1)
vb E {0.1)
and by the fact that
(t-mit = 1} 1 (t-)+= =0} ((t m.) 1) ((t m) :t 0) max ( m) :t, 11,1)",mm , ((t
m) +,0) 55-
almost surely, the triangle inequality proves the result.
Lemma states the condition that second-order terms on the path from yj in Sr to 17" do not dominate. this case, the condition easily verified by the fact that the setting experimental and therefore treatment probabilities are known.
Lemma E.5 (Second-order condition). For the model defined in Section 5.7 and the estimand and estimator defined in
Equations 5.4 and (5.5). under Assumptions E.1 and E.2
then
82 Ep [ub(War: + r(n r"))]] o(2-1/2).
sup (E(0,1).nEE5
0r
Proof of Lemmal E.5 The result holds trivially by Equations EE.1 to EE. and applying the dominated convergence theorem (to switch the derivative and expectation) and the fact that the second derivative
F. Additional simulation details
F.I. Simulations for Section 4
For each simulation, T 1,000 and dx 10. For all i d,. let X N(1,1) iid. D. Ber(min/max (,,,) {}) and ç 0.1. The outcome function is defined as
Y((D,, Hi) def sin(2mX,,1) + 2D, 2H,D, YY,
HWWW- def, 0.75 +1 Hr.
where Y, ~ 1/10), H, (O0 1) iid. For Figure 2, we run 50,000 simulations. In Figure 5 we run 1,000 simulations for each time horizon T and plot 10 values of T evenly divided on logarithmic scale from 100 10,000.
In Figure 5] for each time horizon T, we compute 1,000 simulations. We also compute the empirical standard error for each estimate in ribbons around the point estimates by computing the empirical standard deviation of the point estimates across simulations and dividing by the square root of the number of simulations. We next provide additional comparisons between confidence intervals constructed via our estimator and other estimators.
Estimators for treatment effect comparisons. The naive plug-in estimator defined formally as
pi (Wi.T:1)
def. T-1
Σ po (Wen)
(F.1)
where pp is defined as in Equation (4.5
The naive Horvitz-Thome Thompson is defined formally as
T Σ t=1
HT (
(W.T:)) def. T-1
Dr
Y.. 1 m(x))
1- 1-D;
(F22
m(.))
30

---

Double Machine Learning for Causal Inference under Shared-Star Interference
The naive DML estimator is defined formally as
DMLL- (Wi:Ti def. T-1 Σ +
DMMLNNWWW:
(F.3)
where
DML- DOML-N(WWe)) (Wan)) def. (1) - (1)
Dr D- -Y, ('D,, X.))
+
m(x))
1- mx,()
The naive DML estimator 4DML--N is simply the canonical iid AIPW estimator, ignoring the shared state. The nuisances are v/ = (f", m), where we learn an outcome model f' by learning a predictor of Y, given X, and D, but omitting the shared state, and learn in as before by predicting Dr as a function of X,. (We use the notation f' to disambiguate from f. which takes He as an argument.) To learn f', we use an auxiliary sample as in Algorithm I
Setup for confidence interval comparisons. We next investigate the coverage of confidence intervals constructed from treatment effect and variance estimators. Confidence intervals for an arbitrary pair treatment effect, variance estimators (v, 6²) are constructed as
2:00--0010 =.0/vT. CT (5=
200/vT]
where 2a is the (1-a/2)-th quantile of a standard normal distribution. To compute coverage rates, we run many simulations, constructing confidence interval for each one and calculating the proportion of confidence intervals that contain the estimand
The DML4SSI confidence intervals are constructed using ( 1.1 i) and the consistent variance estimator 6² in Theo- The Horvizz-Thompson (HT) confidence intervals use the treatment effect estimator ei (Wi.T; defined in and a standard variance estimator Th defined as
rem 3.1
Equation F.2)
T-
T Σ t=1
(Y, - - m(x))
D.
HT
1 Dt
(Vr (V6- )²
- m(x))
where (1) for E {0,1} is the mean observed outcome conditional on D; = E. The plug-in confidence intervals use siPi (W1:1: as defined in Equation [.Il and the variance estimator defined Theorem 3.1, substituting yP for ip and aipi for sb The naive DML. (DML.-) estimator uses the treatment effect estimator DMLM defined as
-ML-N T
DML--N N (Wa:
(F.4)
DML (WiT::1))
In addition to the HT, plug-in and DML-N estimators compared above, we also compare our method to one where we treat the shared-state as covariates and construct treatment effect and variance estimates accordingly. We call these shared-state as covariates (SSAC) estimators. In this section, the SSAC treatment effect estimator is mechanically the same our treatment effect estimator, since both of them involve learning conditional expectation function estimate f. (In 5 this is not true, and our treatment effect estimator and the SSAC estimators will be different, leading to inconsistent SSAC treatment effect estimates.) However, the SSAC variance estimator ignores covariances between terms and treats the data as if there were no covariances between terms. This is different from our variance estimator, which does account for covariances between terms. The SSAC variance estimator 0 is defined as in Equation (F.4), substituting for yo DMI.-N and va for DMI.NN These are the variance estimates that would have been constructed had the shared states been treated like iid covariates.
31

---

Double Machine Learning for Causal Inference under Shared-State Interference
DML4SSI
HT SSAC
Plug-in
DML4SSI HT DML.-N
Plug-in
DML.-N
SSAC
0.8
2.0
Coverago
0.6
Width
1.5
0.4
1.0
0.2
0.5
0.0
10
10a5
10 1025
10
1020
10
10
10a
10
7
Figure 5: Coverage rates of 95% confidence intervals for the ADE constructed using our estimators (DML4SSD, shared state covariates (SSAC) estimators, naive forvitz-Thomppe estimators, plug-in estimators and naive DML estimators. The HT, plug-in and naive DML estimators coverage overlap around O. We plot standard errors of the simulations in lighter-colore ribbons around the coverage point estimates.
Results for confidence interval comparisons. In Figure 5 we run simulations for different values of T' and construct confidence intervals for each estimator. In the left plot, show coverage rates of the confidence intervals, and in the right plot, we show the confidence interval widths. We observe that the DML4SSI confidence intervals approach the target 95% coverage as T increases, while each of the other estimators fall substantially below the target coverage. In particular, each of the HT, plug-in and DML-N confidence are substantially biased as we saw in Figure 2 leading to confidence intervals with coverage close to zero. The SSAC confidence intervals use the same (consistent) treatment effect estimator as the DMI but have variance estimates that do not account for covariance between observations over time. This leads to underestimates of the variance, confidence intervals that are too narrow and seee--taa-desireee coverage. In this case, the SSAC 95% confidence intervals result in coverage between 0.6 and 0.7, and do not seem to be converging to the target coverage as T' increases.
F.2. Simulations for Section
We let dx 1 and X, Unif([O, 1)). Also, we let D, (max (C, X,,1,11 (}) where Ç 0.1. We let H, be the last m observations Xt-mat-i; Di-mat- where 5. We let
(eDD,, def. 22 2D, +
Σ
e HH.0 - -1+1 Y,
km|m/6]
where 9, ~ NO, 1/10) iid. We set the switching period to 2m.
In our simulations, we let T' 1,000. In Figures and we run each simulation 50,000 times. In Figure 6 for each time horizon T, we run 1,000 simulations.
Estimators for treatment effect comparisons.
The shared-state covariates (SSAC) estimator is defined formally as:
T f(1, X,, H.) - f(,,X,, H.)
MASSAC (WW::; def. T
1=1
(F.5)
D
1- D-
(Y - [(D,, X,,,,,,HH H.))
(X,)
1 - m(xx)
The unbiased switchback estimator is defined as
(e = 1}
((D = =0} ((1 m))
SB (W.T)
def. Σ
Y,-
(F.6)
a-((1 m) 1)
t=1
32

---

Double Machine Learning for Causal Inference under Shared-State Interference
DML4SSI HT
Plug-in SB
DML4SSI- HT
Plug-in
DML.NN
SSAC
DML.-N
SSAC
1.00
1.00
0.75
0.75
Coverago
Width
0.50
0.50
0.25
0.25
0.00
0.00
10 2.0
1025
1020
10a3
10
1020
1025
10 3.0
10ah
10
T
Figure 6: Coverage rates and widths of 95% confidence intervals for the GATE constructed using our estimators (DML4SSI). (HT) estimators, naive HT estimators, plug-in estimators and naive DML estimators. In the
switchback Horvitz-Thompe left plot, the coverage for confidence intervals constructed using the naive HT, plug-in and naive DML estimators overlap around O. We plot standard errors of the simulations in lighter-colored ribbons around the coverage point estimates.
Confidence interval comparisons. We next explore the coverage of confidence intervals constructed from treatment effect and variance estimators. We use the same procedure to generate confidence interval coverage Figure Oas we do for Figure 5 we run many simulations where we generate confidence intervals and compute the fraction of them that contain the true effect vh* The definitions of the treatment effect and variance estimators for the naive HT, plug-in, naive DML estimators are the same in Section 4.3 We use DML treatment effect estimator <<(1:: ft) specified by Equation (5.5) and the consistent variance estimator from Theorem 3.1 For the switchback HT estimators, we use the treatment effect estimator from Equation (F.6 and the conservative variance estimator provided in Bojinov et al. [2022 Corollary 1). For the SSAC estimators, we use the treatment effect estimator from Equation F.5 and the naive variance estimator as defined in
Section 4.3
We observe that the DMLASSI confidence intervals are close to the target 95% coverage, while the switchback HT estimator provides substantially greater than 95% coverage. This consistent with the fact that their variance estimator is conservative and will not be consistent in general. The mean confidence interval width constructed from the SB estimator (yellow) is between 3 and 7.5 times the width of that constructed from the DML4SSI estimator (red): put another way, in these simulations, the DML4SSI estimator requires many fewer samples to produce confidence intervals the same width as the SB estimator for given time horizon T. Finally, we note that the naive estimators and plugin estimators all provide coverage near zero. This occurs as a result of the biases of each of these estimators for estimation of the treatment effect.
33


Reviews:
[{"review_id": "thnGTOqEsJ", "reviewer": "Reviewer_YLJC", "summary": "This paper studies causal inference in the presence of *shared-state interference*, a common structure in real-world systems such as online marketplaces and recommender platforms, where outcomes of individuals are influenced by a low-dimensional global variable (e.g., price, inventory, recommendations). The authors formalize this structure and develop a general semiparametric framework under which treatment effects can still be estimated efficiently using **Double Machine Learning (DML)**. The core contribution is an extension of the DML theorem (Chernozhukov et al., 2018) to settings where units arrive sequentially and interfere through a shared Markovian state. The paper derives asymptotic normality and consistent variance estimators for two estimands: the **Average Direct Effect (ADE)** and the **Global Average Treatment Effect (GATE)**, using switchback experiments. Simulations confirm the theoretical properties and demonstrate that the proposed estimators outperform naive plug-in and Horvitz-Thompson alternatives.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your constructive feedback! We also thank you for noting the expressivity of our formality, relevance to practical settings and gap in research on causal inference in algorithmic systems and markets. We respond to each of your questions below:\n* **Estimating $m$ in Practice**: We note that for the procedure to be valid, we just need an upper bound on $m$, so that the switch period can be chosen to be greater than this upper bound. The literature on switchback experiments proposes procedures to estimating $m$ if it is not known. See, e.g., Bojinov et al 2022, Section 4.4. Informally, the procedure involves computing the estimator using different candidate values $m$ and running a series of hypothesis tests to see whether the results are the same for different values. If the results are different, then the smaller candidate for $m$ can be shown to be less than its true value, with high probability. The uncertainty introduced by unknown $m$ can be incorporated in our high-probability bound by choosing $\\gamma$ to account for the Type 2 error of estimating $m$ is smaller than it actually is. \n* **Auxiliary Data requirement**: We will add more discussion of this point to the paper. It would be a worthwhile direction for future research to explore how approximately independent data (such as blocks observed far away) may be used to train nuisance predictors that are approximately independent from the data they are evaluated on. \n* **Applicability to Recommender Systems**: Application of our framework to real-world empirical contexts would be a valuable direction for future work. Given the theoretical focus of the paper and the constraints of space, we don’t do this. We strongly believe that our framework is applicable to relevant empirical settings.\n\nI. Bojinov, D. Simchi-Levi, and J. Zhao. Design and Analysis of Switchback Experiments, Apr. 2022. URL http://arxiv.org/abs/2009.00148. arXiv:2009.00148 [stat].", "claims_and_evidence": "The paper claims that:\n- Causal inference under shared-state interference can be handled with an appropriately extended DML framework.\n- Under certain assumptions (Markovian dynamics, ergodicity, sample splitting or auxiliary data), valid and efficient estimation of ADE and GATE is possible.\n- The proposed DML estimators outperform naive baselines in both bias and variance.\n\nThese claims are well-supported:\n- The extension of the DML theorem is rigorous and builds on a solid foundation.\n- The authors provide precise conditions and detailed proofs for asymptotic normality and consistency.\n- Simulations effectively illustrate both downward and upward biases in naive estimators, and the superior performance of the DML estimators.", "methods_and_evaluation_criteria": "The methodology is carefully constructed:\n- The formalization of shared-state interference is novel and expressive, capturing dynamic systems where individuals influence and are influenced by a global state.\n- The use of DML with orthogonalization and bias correction is well-motivated.\n- The paper defines clear estimands (ADE and GATE), and designs plug-in and debiasing terms to achieve robustness.\n- Simulation settings reflect realistic assumptions (e.g., market congestion, algorithmic competition).\n\nEvaluation is primarily based on simulation, which is reasonable given the lack of real-world counterfactuals.", "theoretical_claims": "Theoretical contributions are central to this paper:\n- The paper extends DML to settings with shared-state interference by leveraging Markov chain properties.\n- Theorems 3.1, 4.1, and 5.1 establish conditions for asymptotic normality of the ADE and GATE estimators.\n- The variance expressions are derived under both geometric ergodicity and m-dependence.\n\nThe proofs are detailed, with attention to assumptions like Neyman orthogonality, Gateaux derivatives, and mixing conditions. The adaptation of CLT results for Markov chains is particularly well-handled.", "experimental_designs_or_analyses": "The simulations are thoughtfully designed:\n- Two estimands (ADE and GATE) are estimated across multiple conditions.\n- Competing estimators (plug-in, Horvitz-Thompson) are included for comparison.\n- The authors use random forests as flexible nuisance estimators, with auxiliary datasets to avoid sample leakage.\n\nFigures clearly show the bias and variance behavior of each estimator. Although limited to synthetic data, the experimental results strongly support the theoretical claims.", "relation_to_broader_scientific_literature": "The paper is well-situated in the literature:\n- Builds on Chernozhukov et al. (2018) for DML\n- Extends recent work on interference in experiments (e.g., Johari et al., Farias et al., Munro 2024)\n- Addresses limitations of social network-based interference models, offering a complementary approach via global state dynamics\n\nThe shared-state framework fills a notable gap in modeling real-world algorithmic and market-based interference.", "essential_references_not_discussed": "No critical omissions were found. The paper cites essential works in DML, interference modeling, ergodic Markov chains, and semiparametric inference. It also offers comparison with network-based approaches, positioning its contribution as orthogonal and novel.", "other_strengths_and_weaknesses": "**Strengths:**\n- Novel and realistic modeling of shared-state interference\n- Theoretical rigor and clear assumptions\n- Methodologically sound extension of DML\n- Strong simulation results\n- Clean writing and logical structure\n\n**Weaknesses:**\n- No real-world experiments, though the authors acknowledge this.\n- Some assumptions (e.g., availability of auxiliary data) may not always hold in practice.\n- The setup assumes known m in m-dependence, which could be hard to estimate in some applications.", "other_comments_or_suggestions": "NA", "questions_for_authors": "1. **Estimating m in Practice**: While the paper assumes known m for m-dependence, in real-world experiments m may be unknown. Can your method accommodate estimation of m, or how sensitive are results to misestimation?\n\n2. **Auxiliary Data Requirement**: The use of an auxiliary dataset for nuisance estimation avoids dependence issues, but may not always be feasible. Could you discuss alternative strategies, such as block-splitting or approximate independence?\n\n3. **Applicability to Recommender Systems**: Can your shared-state model be concretely instantiated in modern large-scale recommender systems, such as those used in e-commerce or social media? Have you explored any such datasets?\n\nClarifying these would enhance the paper’s applicability and help practitioners adopt your framework.", "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": "", "extra_scores": {}}, {"review_id": "FxP6Y6YQzw", "reviewer": "Reviewer_c4Lj", "summary": "This paper introduces a double machine learning (DML) estimator for sequentially collected samples where dependencies follow a Markovian structure through a shared-state variable $H_t$.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "We thank the reviewer for their feedback, as well as their comments about the soundness of theoretical results and strength of foundation for the work. We will incorporate your comments into our work. We clarify several points and answer questions below.\n* **Data generating process details.** We provide specific explanations of the data generating process in Appendix F, as we note in Line 300, column 2.\n* **Naive Horvitz-Thompson (HT) versus switchback HT estimators.** The HT estimator in figure 2 is a naive HT estimator which merely takes a difference in means across treatment and control observations. The SB estimator in figure 3 is a different estimator which averages only over observations where the last $m$ have been all treatment or all control. The naive HT estimator, in general, may be biased since it does not account for the shared state, but the SB estimator is unbiased. We will clarify this distinction, especially since the SB estimator is a HT-style estimator but is different from the naive HT estimator.\n* **$H_t$ only affects $Y_t$, not $X_t$ and $D_t$.** This problem formulation would indeed be more general and would be a valuable direction for future work. We made the choice to have $H_t$ affect $Y_t$ but not $X_t$ and $D_t$ for two reasons. First, we found it hard to imagine situations in which $H_t$ exerts significant influence on unit characteristics and treatment assignments: most often, unit characteristics should be innate to the unit (e.g. their price sensitivity or preferences for particular types of content) and treatment assignments should be exogenous (we imagine treatment conditions like new features on a platform or inducements like discounts). Second, our context presents the simplest possible change from canonical iid models and so our work can be directly instructive for comparison with models where no shared-state is present by simply removing the shared-state variable from the outcome structural equations. Assuming dependence of $X_t$ and $D_t$ on $H_t$ would be possible under Theorem 3.1 as long as the data $W_t$ still obeys geometric ergodicity and detailed balance.\n* **Notation.** We use standard potential outcome notation with exposure mapping where potential outcomes are typically not written as a function of covariates (even though outcomes depend on covariates). In other words, $Y_t$ *does* depend on $X_t$, but this dependence is omitted from notation. We note that we inadvertently included $X_t$ in the potential outcome notation of some equations, like 4.5, and we will fix this. Thanks for the question.\n* **Independence of $W_{aux}$.** $W_{aux}$ need not be iid as long as the nuisance parameter learners converge at appropriate rates. The tradeoff between quality (say, independence) and quantity (number of observations) of data is implicit in the nuisance learner rates assumptions, but exploration of this question would be a valuable direction for future work.\n* **Choices of $\\gamma$.** You can assume $\\gamma = 0$ as a special case of Theorem 3.1. We include the high-probability bounds because it follows the results in Chernozhoukov and because it allows for the possibility that eta falls outside of the nuisance realization set with some probability. $\\gamma$ might be useful if, as we note in our response to Reviewer YLJC, $m$ may be mis-specified with some probability.", "claims_and_evidence": "1. The paper has a strong theorical foundation. \n2. Empirical simulations are lacking of specific explanations (e.g., the data generating processes). No empirical evidence for the fast convergence (debiasedness) and doubly robustness are provided. \n3. Intuitive explanations on Assumptions C1 and C2 are required for assessing the claims, since they are key assumptions in the paper. \n4. Even if it's mentioned that HT estimator is unbiased for estimating GATE, it seems that the HT estimator is biased in Figure 2. Could you please explain the simulation detail?", "methods_and_evaluation_criteria": "1. Empirical simulations are lacking of specific explanations (e.g., the data generating processes), so it's hard to assess. \n2. It would be great if the experiments are done with real-world examples.", "theoretical_claims": "1. Theoretical results are sound.", "experimental_designs_or_analyses": "1. Empirical simulations are lacking of specific explanations (e.g., the data generating processes), so it's hard to assess. \n2. Even if it's mentioned that HT estimator is unbiased for estimating GATE, it seems that the HT estimator is biased in Figure 2. Could you please explain the simulation detail?", "relation_to_broader_scientific_literature": "This paper tackles an interesting problem, where samples are accumulated in Markovian sense. However, I think the paper makes pretty strong and unrealistic assumptions such that Ht only affects Yt, not Xt and Dt. If you assumed so, what are the most difficult challenges?", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "1. I think $Y_t(D_t, H_t) = f^*(D_t, X_t, H_t) + \\tilde{Y}_t,$ is a wrong formulation. $Y_t$ should be dependent on $X_t$. \n2. $D_t$ is not affected by $H_{t-1}$? This doesn’t make sense. \n3. Is W_{aux} is iid, while W1,…,WT are dependent in Markovian? \n4. Why Assumption C.2 has been made with high-probability? I meant, can we just assume with $\\gamma = 0$?", "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": "", "extra_scores": {}}, {"review_id": "18ROUxSLVN", "reviewer": "Reviewer_s6JS", "summary": "This paper addresses causal inference under unit interference in systems like markets and recommendation platforms. To model inter-individual interference without strong assumptions, the authors introduce shared-state variables, assuming individual outcomes depend on others only through these shared states. The authors apply double machine learning (DML) for efficient inference, using an auxiliary data sample for nuisance estimators instead of cross-fitting, to account for the sequential nature of sampling. The methodology is used to estimate the average direct effect (ADE) and the global average treatment effect (GATE), with simulations showing DML’s advantages in debiasing and variance reduction.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "We thank the reviewer for their comments. We will incorporate this feedback in our updated manuscript.\n\n* **Application-based simulations.** Exploration of more complex and real-world scenarios (using either simulated or real data) would be a valuable direction for future work. Our focus in this paper is on the novel theoretical methodological contribution. Our simulation settings were intended to be simple, to demonstrate how naive estimates can be biased even in simple, one-dimensional settings. Thank you for the suggestion to add coverage rates of variance estimates. We will include this in the appendix.\n* **Notation.** Thank you for pointing out the notational inconsistencies. We will fix these and adhere to potential outcome notation where the dependence on $X_t$ is implicit.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes, the proofs for the theorems on the asymptotic properties of DML appear to be largely correct.", "experimental_designs_or_analyses": "This paper presents simulation results for estimating the average direct effect (ADE) and the global average treatment effect (GATE). However, the simulation settings—particularly the one-dimensional covariance structure—seem somewhat simplistic. Exploring more complex scenarios would enhance the generalizability of the results. Additionally, while the evaluation focuses on the magnitude of bias and variance, it overlooks an examination of the consistency of variance estimation. Including the coverage rate of parameter estimates as an additional metric could provide valuable supporting evidence.", "relation_to_broader_scientific_literature": "This paper combines shared-state interference with the DML method, both of which have been extensively discussed in the causal inference literature. A related work, *Causal Inference under Interference through Designed Markets*, applies localized debiased machine learning (LDML) for causal inference under shared-state interference in a two-sided market. In this context, the current paper extends the application of DML to causal inference in interference scenarios.", "essential_references_not_discussed": "I believe this paper includes the vast majority of essential references.", "other_strengths_and_weaknesses": "Strengths: This paper clearly introduces the two core aspects of DML: Neyman orthogonality and cross-fitting. It also points out the difference between share-state interference and the traditional i.i.d. data generation mechanism, thus requiring additional modifications to the DML procedure.\n\nWeaknesses: The notation in some places of this paper needs to be consistent. For example, in Section 4.1, Equation 4.1 is written as $Y_t(D_t, H_t)$, while Equation 4.5 uses $Y_t(D_t, X_t, H_t)$.", "other_comments_or_suggestions": "No.", "questions_for_authors": "No.", "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": "", "extra_scores": {}}, {"review_id": "EKX65aJjpV", "reviewer": "Reviewer_wSH2", "summary": "This paper unifies the set of problems in causal inference with interference where the outcomes of individuals depend on others' treatment assignment only through an observed shared state. The paper assumes the units arrive sequentially and models this shared-state problem using the Markov chain. The paper then proposes a double robust machine learning meta-estimator to estimate causal quantities of interest and proves the estimator's asymptotic properties. The paper shows that both average direct effects and global average treatment effects can be estimated by this meta-estimator. The paper conducts numerical experiments to demonstrate the effectiveness of the proposed estimator.\n\n## update after rebuttal\nThe authors have addressed my concern. I will retain my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "We thank the reviewer for their thoughtful feedback. We will incorporate your suggestions, including providing coverage rates in Appendix F, in our updated manuscript. In the coverage rate plots, our consistent variance estimator approaches the target coverage rate as $T$ grows, while the coverage rates for naive estimators is near zero due to their bias.\n* **Dependence of $X_t$ and $D_t$ on $H_t$.** (Copying this from our response to reviewer c4Lj.)  This problem formulation would indeed be more general and would be a valuable direction for future work. We made the choice to have $H_t$ affect $Y_t$ but not $X_t$ and $D_t$ for two reasons. First, we found it hard to imagine situations in which $H_t$ exerts significant influence on unit characteristics and treatment assignments: most often, unit characteristics should be innate to the unit (e.g. their price sensitivity or preferences for particular types of content) and treatment assignments should be exogenous (we imagine treatment conditions like new features on a platform or inducements like discounts). Second, our context presents the simplest possible change from canonical iid models and so our work can be directly instructive for comparison with models where no shared-state is present by simply removing the shared-state variable from the outcome structural equations. Assuming dependence of $X_t$ and $D_t$ on $H_t$ would be possible under Theorem 3.1 as long as the data $W_t$ still obeys geometric ergodicity and detailed balance. \n* **Existence of auxiliary data.** Auxiliary data may exist when there are multiple similar systems or markets where unit behavior will be similar. For example, on a social media platform with multiple forums, the data from one forum might be used to construct nuisance estimates for measuring treatment effects in another forum. This is an assumption common to other methods using machine learning for inference or uncertainty quantification (see, e.g., Angelopoulos et al 2023), but of course, this is a limitation on how widely applicable any such method requiring auxiliary data can be. On the other hand, conceptually, the auxiliary sample assumption cleanly separates the machine learning from application of the learned predictors for inference. Relaxations of this requirement may be possible, but we wanted to preserve the conceptual clarity in our paper, so we leave these extensions for future work.\n* **Real-world validations.** Analysis of real-world settings would be a valuable contribution for future work. We wanted to include simulations for the sake of simplicity and brevity. \n* **Related work.** We are not aware of other methods that formalize causal inference through shared-state interference as we do. Several other papers formalize interference through markets or through recommender systems as we note in the related work, but these are context-specific. For example, Munro 2024 allows for causal inference under shared-state interference in settings like auctions where the shared-states are prices and allocations of goods exhibit a cutoff structure. Our work is complementary by offering a context-agnostic approach that does not require, e.g., knowledge of the allocation mechanism if the data generating process satisfies our Markov chain assumption.\n\n\nA. N. Angelopoulos, S. Bates, C. Fannjiang, M. I. Jordan, and T. Zrnic. Prediction-Powered Inference, Nov. 2023. URL http://arxiv.org/abs/2301.09633. arXiv:2301.09633 [cs, q-bio, stat].\n\nE. Munro. Causal Inference under Interference through Designed Markets. 2024.", "claims_and_evidence": "On line 83, could the authors explain why it is realistic to assume that the covariate X_t is independent of the hidden state H_t, perhaps using the example described in the introduction section? Is there a way to relax this? This also applies to invariance of the conditional distribution of H_t and D_t.\n\nOn line 182, the authors assume that the nuisance estimators can be generated via an auxiliary sample of data. In reality, when is this auxiliary sample of data available?", "methods_and_evaluation_criteria": "Yes, they make sense.", "theoretical_claims": "Yes, I checked the correctness of the proofs.", "experimental_designs_or_analyses": "Yes, I did. Since the authors have derived the asymptotic normality for the estimator, they shall also report the actual coverage of the estimator in the experiment.", "relation_to_broader_scientific_literature": "The problem studied in this paper appears to unify applications across various fields, as suggested by the authors in the introduction. However, the paper does not include an analysis of a real-world dataset. While the oracle causal effect is elusive in real data, it would still be beneficial for the authors to demonstrate their method on at least one such dataset. Additionally, I am curious to see how the authors justify their assumptions and the existence of auxiliary data used for estimating the nuisance parameter in a real-data setting. \n\nIf the authors do this, I will raise my score.", "essential_references_not_discussed": "Could the authors discuss more about how the prior works in various fields relate to the concept of \"shared-state interference\", and how the paper unifies this concept ( perhaps In the appendix)?", "other_strengths_and_weaknesses": "It is a theoretically rigorous paper with significant practical implications. The assumptions made are generally reasonable.", "other_comments_or_suggestions": "line 82: \"independent of the indentities of...\"\n\nline 85: \"...develops a DML theorem a discrete choice model...\"\n\nline 151: \"Hence, they are nuisances.\" is redundant\n\nline 307: \"Next, we our validate our results\"\n\nline 316: \"averate treatment effect\"\n\nThe word \"adjustment\" only appears once in the title of section 5. Consider to replace the title with \"Inference on Global Average Treatment Effect \"", "questions_for_authors": "Is this the first paper to propose the concept of shared-state interference? Which existing papers is the most closely related to this one?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper studies causal inference in settings where different units/individuals have shared state variables that affect them and are affected by them. It formalizes a modeling framework for this setting that determines the effect of the units/individuals on the shared state and vice versa over time, that is the units/individuals arrive sequentially and interfere through a shared Markovian state. It also provides extensions of double machine learning theorems for achieving efficient semi-parametric inference, which are then applied to estimating the average direct effect and the global average treatment effect.

The reviewers found the theoretical claims of the paper to be well-supported with clear arguments. The simulation results also support the theoretical claims and show better performance over the baselines; however, they are limited and only capture synthetic settings and not real examples or data sets. As the reviewers pointed out and the authors responded, I recommend that the authors add further discussion and remarks about the practicality and limitations of the assumptions such as the existence of auxiliary data or knowledge about ‘m’ in m-dependence.
Decision: Accept (poster)