Efficient Adaptive Experimentation with Noncompliance
Authors: Miruna Oprescu
Brian M Cho
Nathan Kallus
Venue: NeurIPS 2025 poster ()
Keywords: adaptive experimentation
instrumental variables
causal inference
semiparametric efficiency
multiply robust estimation
sequential design
Primary area: probabilistic_methods
TL;DR: Adaptive instrumental variable design plus a multiply-robust estimator yields efficient, unbiased ATE estimates in sequential settings.

Abstract: We study the problem of estimating the average treatment effect (ATE) in adaptive experiments where treatment can only be encouraged—rather than directly assigned—via a binary instrumental variable. Building on semiparametric efficiency theory, we derive the efficiency bound for ATE estimation under arbitrary, history-dependent instrument-assignment policies, and show it is minimized by a variance-aware allocation rule that balances outcome noise and compliance variability. Leveraging this insight, we introduce AMRIV—an **A**daptive, **M**ultiply-**R**obust estimator for **I**nstrumental-**V**ariable settings with variance-optimal assignment. AMRIV pairs (i) an online policy that adaptively approximates the optimal allocation with (ii) a sequential, influence-function–based estimator that attains the semiparametric efficiency bound while retaining multiply-robust consistency. We establish asymptotic normality, explicit convergence rates, and anytime-valid asymptotic confidence sequences that enable sequential inference. Finally, we demonstrate the practical effectiveness of our approach through empirical studies, showing that adaptive instrument assignment, when combined with the AMRIV estimator, yields improved efficiency and robustness compared to existing baselines.

Efficient Adaptive Experimentation with
Noncompliance
Miruna Oprescu Cornell University
Brian M Cho Cornell University bmc2330cornell.. edu
Nathan Kallus Cornell University & Netflix kallus@cornell. edu
am0780corneel.edu
Abstract
We study the problem of estimating the average treatment effect (ATE) in adap- tive experiments where treatment can only be encouraged- than directly assigned-via binary instrumental variable. Building on semiparametric effi- ciency theory, we derive the efficiency bound for ATE estimation under arbitrary, history-dependent instruument-assinmmnn policies, and show it is minimized by a variance-aware allocation rule that balances outcome noise and compliance variabil- ity. Leveraging this insight, we introduce AMRIV an Adaptive, multtppy-Robuu estimator for Instrumental- Variable settings with suriance-optima assignment. AMRIV pairs (i) an online policy that adaptively approximates the optimal al- location with (ii) sequential, influence-function functinn--bseeddddd estimator that attains the semiparametric efficiency bound while retaining multiply-robust consistency. We establish asymptotic normality, explicit convergence rates, and anytime-valid asymptotic confidence sequences that enable sequential inference. Finally, we demonstrate the practical effectiveness of our approach through empirical studies, showing that adaptive instrument assignment, when combined with the AMRIV es- timator, yields improved efficiency and robustness compared to existing baselines.
Introduction
Adaptive experimentation enables efficient estimation of treatment effects in sequential settings by adjusting assigmment strategies based on accumulating data. Compared to traditional randomized controlled trials (RCTs), adaptive designs reduce estimation variance, thus accelerating discovery and limiting exposure to ineffective or harmful interventions. These methods are now widely used across domains-from from medicine online platforms-and have been formally endorsed by the U.S. Food and Drug Administration
21 driving both practical adoption and theoretical advances.
In many such settings, however, direct treatment assignment is not feasible ethical. The treatment must instead be encouraged through randomized recommendation or design choice- referred to as instrumental variable (IV)-leaving the final decision to the participant. This gives rise to noncompliance, where the assigned encouragement and the actual treatment may differ, and where self-selection based unobserved factors introduces confounding that biases standard estimators. For instance, in a real TripAdviso experiment, users were exposed to different premium sign-up interfaces that encouraged membership enrollment [522] The actual treatment -whtthe a user subscribed- could not be enforced, but the interface (the instrument) could randomized adaptively adjusted. Similarly, in clinical trials, physician may recommend but cannot compel adherence: the recommendation can be assigned, yet treatment uptake remains endogenous. Related challenges arise recommender systems and public health interventions, where engagement or behavioral uptake is voluntary.
medication
In such applications, reducing estimator variance is not merely statistical preference; it determines how quickly and confidently a study can reach conclusions. Because high variance delays both
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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the detection of harmful effects and the confirmation of beneficial ones, adaptive designs that learn to allocate instruments in curaacce-miimmizing ways can mitigate these risks, yielding tighter confidence sequences and enabling earlier, statistically valid stopping decisions. Despite extensive work on adaptive designs for settings where treatment can directly enforced 13 23 27 adaptive experimentation under noncompliance- whhre treatment is voluntary but encouragement can be adaptively controlled, remains largely unexplored, even though it describes many real-world scenarios.
This paper addresses this gap. We study the problem of estimating the average treatment effect (ATE) in a sequential experiment where the experimenter can assign only binary instrument, while the treatment itself is determined endogenously. Building on the semiparametric framework of Wang and Tchetgen Tchetgen [54], which identifies the ATE under an unconfounded compliance assumption and provides a multiply robust, efficient sircencc-function
estimator, we extend
this framework to the adaptive setting. Specifically:
We derive the semiparametric efficiency bound and characterize the variance-optimal adaptive policy that minimizes this bound through a covariate- -rrridtt-deeenneett instrument assignment.
We introduce AMRIV, an Adaptive, Multiply Robust estimator for IV settings, which applies a sequential. plug-in in version of the efficient influence function evaluated under the adaptive policy.
We establish strong theoretical guarantees, including asymptotic normality, explicit conver- gence rates, multiply robust consistency, and time-uniform asymptotic confidence sequences for valid inference at arbitrary stopping times.
We demonstrate practical effectiveness in both synthetic and semi-synthetic studies, showing improved efficiency and robustness over non-adaptive baselines and alternatives.
In contrast to prior work on adaptive design with instruments on point estimation of the ATE, achieves semiparametric efficiency, and supports multiply robust inference under adaptive assignment. To our knowledge, this the first method to bring the full suite of modern semiparametric tools efficient influence functions, adaptive policy learning, robust plug-in estimation, and anytime- -aalid inference to the adaptive IV setting with noncompliance.
61 our method focuses
2 Related Work
We provide a brief overview of related work here, with a more detailed discussion in Appendix A
IV-Based ATE Estimation. ATE identification in IV settings has traditionally relied on structural equation models (SEMs) that impose parametric assumptions on the outcome and treatment as- signment mechanisms. More recent work has proposed flexible alternatives such as DeepIV (24), kernel IV (50), and orthogonal moment methods 66 52 that enable conditional effect estimation in high-dimensional or nonlinear settings. However, these approaches do not directly target robustness or semiparametric efficiency for the ATE. We instead build on the framework of Wang and Tchet- gen Tchetgen [54], which establishes point identification of the ATE via an unconfounded compliance assumption without requiring SEMs. Their
that
estimator achieves semipara- metric efficiency and is multiply robust, remaining consistent under partial nuisance misspecification. We extend this framework to the adaptive setting and use as the foundation for our estimator.
Adaptive Experimentation for ATE Estimation. A large and growing literature studies adaptive designs where the treatment itself can be directly assigned, with the goal of minimizing estimator variance or its regret analogue. Early two-stage designs asymptotically achieve the semiparametric efficiency bound (33, and fully sequential approaches such as A2IPW attain variance-optimal
Neyman allocation [27] Subsequent extensions learn the allocation policy online (28) and add
principled policy truncation with the first anytime-valid confidence sequences [T3]. Parallel work from an online-learning perspective achieves sublinear logarithmic "Neyman regret" via clipped or optimistic algorithms 14 38 39], matches finite-sample lower bounds under low-switching policies [35), and extends to covariate- adaptive and off-policy settings (29 33 Together, these methods form a mature toolkit- inference but all assume the experimenter can randomize the treatment directly. Our work generalizes these efficiency guarantees to the harder and less explored regime where only an instrument can be assigned and treatment uptake is endogenous.
and toolkit-adaptive nuisance learning, cross- fittinn, policy truncation, regret-style allocation,
Adaptive Experimentation with Instruments. A small but growing literature explores adaptive design in settings where only instrument, rather than the treatment, can be assigned. Closest to our
2

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Adaptive Experimentation Noncompliance
Early Stopping
New
Unobserved confounder
Confidence Sequences
X
Adaptive policy
Non-adaptive
Noncompliance when (244) = 4
Estimatted
0
(X I (Ht-))
A
Y
Learned adaptive policy
Assignment (x,)
Endagenaus treatment
Outcome
Vatop
Tatop
Time
Figure 1: Left: Adaptive experimentation with noncompliance. Blue elements denote learned assigned quantities (Tt, Zt), orange elements represent observed variables (X,, A,, YD,, and the dashed red arrow indicates noncompliance (A₂(2,) # Z,). Right: The adaptive policy yields faster contidence-sequence contraction, enabling earlier stopping.
work is Chandak et al. Ω who propose practical inffuence-function--eeee prediction error nonparametrie regression. However, they focus on prediction accuracy and do not address semiparametric efficiency or robustness to nuisances. Other approaches focus on partial identification [3]. adaptive data acquisition [22], or regret minimization in bandit-style settings with endogeneity [16 61]. In contrast, goal is to enable efficient and robust adaptive ATE estimation under noncompliance, providing the first efficient and multiply robust estimator for this setting.
procedure to reduce
Additional related work on semiparametric inference, multiply robust estimation, and confidence sequences is discussed detail in Appendix
3 Background and Setup
We consider the problem of estimating the average treatment effect (ATE) of a binary treatment A E 10, on a real-valued outcome E R. the presence unobserved confounding, within an adaptive experimental setting. We adopt the potential outcomes framework, where each unit is associated with two potential outcomes, Y(0) and Y(),, corresponding to the outcomes under control and treatment, respectively. However, only the realized outcome Y Y()), corresponding to the treatment actually received, is observed.
Each unit is also associated with covariates X E R" and we assume that the random variables (X, Y (0), Y (1)) are jointly distributed according to some unknown distribution P. Our goal is to estimate the ATE given by:
= -XY
Because direct treatment assignment may be infeasible, we rely on a binary instrumental variable Z E {0, }} that influences treatment uptake. The instrument can be interpreted recommendation or
someethnn--smetinngg the experimenter can assign, unlike the treatment itself. We denote by Y (a, 2) the potential outcome under treatment level a and instrument value z, and by A(z) the potential treatment that would be taken under instrument assignment 2. Noncompliance arises when the potential treatment does not follow the instrument, i.e., A(z) * 2, reflecting endogenous treatment selection. Equivalently, unit complies if its treatment adheres to the assigned instrument (A(2) = 2).
The experiment proceeds over T N rounds. At each round f, from P. The experimenter observes X, and selects an instrument value Ze ~ Tt(- X,, H,-1), where n. is adaptive policy that depends the current covariates X, and past observations
unit with covariates X, is drawn
H4-1 == ((X1, A1, A1), 1X
At-1, Y-1))
This allows the instrument-assignneet policy to evolve over time based accumulated data. Fol- lowing the instrument assignment Z,, the treatment A, = Ax2),) is realized, and the outcome Y, (A,, Z,) is observed. The full observation time is thus (X,, Ze, A,, Y)). After T rounds, the experimenter estimates the ATE from accumulated data Hy = {(X; Z;, A,, Yo)
To identify the ATE under endogenous treatment selection (that may be influenced by unobserved confounders), we adopt standard instrumental variable assumptions, as well the unconfounded compliance condition introduced in Wang and Tchetgen Tchetgen (54). We summarize these below.

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Assumption 1 (Standard IV Assumptions). The following properties hold: (Exclusion) (aa 2) ((a) the instrument affects the outcome only through the treatment: (Independence) Z IL U X- the instrument is independent of unobserved confounders U given covariates; and (Relevance) Cov(Z, A X) + 0-the instrument has an effect on treatment uptake for almost every X.
Assumption 2 (Unconfounded Compliance, from (54) The treatment effect independent of compliance status given covariates: Y(1) - (()) 1 A(1) -(0) X.
Assumption is standard in the IV literature and ensures instrument validity. The independence assumption can often be satisfied by design, c.g., via randomization. While sufficient for identifying the local average treatment effect (LATE), these assumptions do not identify the ATE under effect heterogeneity or treatment endogeneity. To enable ATE identification, we invoke Assumption from Wang and Tchetgen Tchetgen (54), which assumes the treatment effect is mean-independent of compliance type given covariates, ruling out interactions with unobserved confounding.
Remark (Interpretation under violations of Assumption 2 If Assumption 2 does not hold, the ATE is no longer point- ciittideetified. The estimand then shifts to the average conditional local average treatment effect (ACLATE), which averages treatment effects among instrument reseeett-reposivvee individuals (compliers) across covariate strata. The ACLATE remains identified and interpretable, capturing how causal effects vary among compliers when compliance is confounded and cannot be fully controlled.
With Assumption 1 and 2 the ATE can be point-identified. For notational convenience, instrument-induced outcome and treatment models 1 (2, X) == E[Y Z = 2, X] and μ ( X) =
define the
E[A| Z 2, X] for {0,1}. The ATE then be expressed (Theorem 1 from [54] ):
(1) (0. T = Ex (1, X) = Ex Tx)(X) [A(X)
(I)
where 8x (X) and (X) denote the instrument-induced shifts in outcome and treatment, respectively, We refer to 6x(X) as the compliance score, representing the instrument's effect treatment uptake.
In the non-adaptive setting. where the instrument assignment policy is fixed over time-i.e,
Te XxH x(X) Wang and Tchetgen Tchetgen [54] (Theorem 5) derive the ef- ficient influence function (EIF) for the ATE estimator in Equation Let T()), n(x) =
{n² (0, (0, x)) (x), ((x)) denote the nuisances, where 0(X) (X))8A(X). The (Re- centered) EIF is then given by
(,,2,AA7 ;
(2)
7x(X) + 2)(1 (X)) (A(X) Y A8(X) +, + 5(X).
2Z
The corresponding estimator- as the multiply robust IV estimator (MRIV) 18 uses plug-in estimates of nuisance functions within the recentered efficient influence function. It attains the semiparametric efficiency bound when all nuisances are correctly specified and remains consistent under partial misspecification.
We extend this framework the adaptive setting, where the instrument assignment policy स. evolves with accumulating data. We characterize the optimal adaptive policy that minimizes asymptotic vari- ance and develop AMRIV, an Adaptive Multiply Robust IV estimator that combines adaptive policy learning with sequential influence- www.ffunnttt-ssteeeeeeeee estimation. Our method achieves semiparametric efficiency, ensures multiply robust consistency, and enables valid time-uniform inference.
Notation: We write TX,, H4-1) == (e (1) X,, H4-1) for the probability of assigning Z, = 1
given covariates and history. The Ta norm of a function f is ||/ == [r (x))112 and ft denotes an estimate of f based on samples. We use É to denote empirical expectations computed from data.
Notation used throughout the paper is summarized in Appendix B (Table
4 Efficiency Bounds and Optimal Instrument Assignment
To guide optimal experiment design under the IV setting, we derive the semiparametric efficiency bound for ATE estimation under a fixed instrument policy π (X). This characterizes the variance- minimizing allocation strategy and motivates our adaptive estimator.

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Theorem 1 (Semiparametric Efficiency Bound), Under Assumption efficiency bound for estimating the ATE T is given by
and the semiparametric
1
(1, T(X)
+ x² (0, + (8(X) 7)2
Va(aa) E
(3)
(A(X)2
1 - a(x)
where (2) = - Z = 2, X).
Corollary 2 (Optimal Instrument Assignment). The assignment policy Ht (X) that minimizes the efficiency bound Theorem 1 given by
(1
nt (X)
(4)
Vo + 00
Proofs of Theorem and Corollary 2 are included in Appendix E
Drivers of Efficient Allocation. From Corollary the optimal assignment policy x (X) allocates
more weight to the arm 2 with higher residual variance Var(Y - A8(X) Z = 2, X). where
Var(Y - A5(X) Z z, X) = Var(Y Z = 2, X) xxx) Var(A 7 = z,
26((X Cov(Y, Z = 2, X).
This expression reveals that, unlike in standard adaptive ATE estimation, the residual variance depends
jointly on both outcome noise (Var(Y / Z = 2, X)) and compliance noise (Var(A) | Z = z, X)) When compliance is more uncertain in arm, the estimator becomes noisier in that region, leading the optimal policy to allocate more probability mass to that arm to compensate.
Connection to Neyman Allocation. In the special case of perfect compliance (when A(2) 2), the treatment is fully determined by the (conditionally) randomized instrument and our setting becomes the classical adaptive ATE estimation scenario. In this setting, Var(Y - AA(() Z 2, X) = Var(Y - A5(X) A 2, X) = Var(Y = 2, X) and thus the optimal allocation
Var(Y A=L,X)
reduces to Oa (Vr((Y) which exactly matches the classical Neyman allocation
for minimizing the variance of difference-in-mest -n-mnnns estimator 27 40 This highlights that our policy generalizes Neyman allocation to settings with noncompliance and endogenous treatment, adjusting for both outcome and compliance-driven noise.
Motivating Illustration. Consider an example with one- is only accessible to those who receive the instrument, so y (0) = 0. This cap- tures scenarios such vaccine access, product rollouts, or behavioral nudges. Let compliance vary with X E R via (x) (1, (1.x) x) a(-2z), and let outcomes follow == ((, + uA+ EA, where u is a fixed unobserved con- founder and сд (O0 + 4(1 1---A A)), with higher variance in the control arm. As shown in Figure 2, the optimal policy T" (X) approaches Neyman allocation when (X) 1, but shifts toward uniform allocation when (x) 0. This re- flects key design intuition: under low compliance, assigning more units to Z compensates for scarce treatment uptake, helping preserve estimator efficiency.
Adaptive policy vs. compliance
sided noncompliance-
0.50
n (x)
0.45
t
0.40
0.35
Neyman Allocation
0.0
0.5
1.0
8
Figure 2: Optimal policy n' (X) as
function of compliance 8 (X).
5 Adaptive Estimation of the Average Treatment Effect
We propose an adaptive framework for estimating the ATE in sequential experiments with binary instrument. Our goal is to minimize the semiparametric efficiency bound from Theorem 1 by combining: (1) instrument assignment via a data-driven policy Ty (x, H.-1) that approximates the optimal allocation x (X): and (2) treatment effect estimation using adaptive plug-in version of the multiply robust estimator from Eqquatio 2, Although the theory assumes per-round updates, the method also applies in batch settings with fewer updates. We detail both components below.
5

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5.1 Adaptive Instrument Assignment
To stabilize nuisance estimation, we begin with burn-in phase of T rounds using a fixed policy Timi(X) such as uniform randomization. From round To + onward, instruments are assigned via data-driven policy Te (x( H4-1) that approximates the optimal allocation in CCroollary 2 Specifically, we compute plug-in estimate ( (H4-1)aa
(1,
Tx( Hc-1)
(5)
(0, + o1,
where 07- (2, is an estimate of the conditional residual variance Var(Y- 2 = based on data in H1-1- We then apply truncation step to Tx( (4-1) (described below) to obtain the final assignment policy (x H4-1) used at time t.
Residual-variance estimation. One option for estimating /2 X) is using the decomposition
Var(Y- - A8(X) Z 2, X) E AU(X))² Z x, (p* (0. x) (0,
We proceed in two stages: (i) fit (0, Nz (0, X) and 8 (X) using H4-1: (ii) form residuals
R Y - (X) and regress R on (2) to obtain 3 (2, X): = E[ Z = X).
Unbiased two-stage estimation via cross-fitting, To mitigate sample bias in estimating (2, X). we apply the sequential cross-fittingg scheme of Waudby-Smith et al. [55] Thus, we split H4-1 into two temporal folds H,!"), {(X,, Z., A,, Yi) : E jt 1], mod j}. j E {0, fit 6 on one and compute residuals R²_1 on the other, and vice-versa. The combined residuals are used to regress R on (Z, X) to estimate (z, X). Since 7o/1 and D4-1 not depend on other nuisances, they can be learned on the full history H4-1-
Nuisance learners. Any sequentially consistent nonparametric regressor can be used for 1 P
and s; e.g. k-NN [58, kernel smoothers [45] random forests [53] or neural nets 48]. /2-21 may
also be estimated via these methods or classifier such as logistic regression. We compute ), (X) 8Y (X) (X), where 02= is estimated via either difference of regressions P (1, X)- (0, X) or a direct IPW-style regression È YZ is obtained analogously by replacing Y with A.
Y(1-2)
X
An estimate (X)
-x- 1-x
To guarantee non-negativity of the estimated variances, we define 12-1 X)
,= (2, (0, (0 (0,
if (....>>0
(6)
otherwise
for small constant 0.
Remark (Choice nuisance and variance estimators). Sequential cross-fitting removes the need for restrictive conditions (e.g.
nonparametric convergence rates suffice for the guarantees in our main theorems. In practice, any consistent learner (e.g., k--N, random forests, or neural networks) can be used, depending data structure and sample size (see Appendix (C). The variance estimator in Eq. (6) ensures nonnegativity via a small floor E, though fully nonnegative alternatives such as self-normalized kernel estimators may also be used. Appendix Aopeedixx(furthe further discusses these options and outlines online or streaming implementations that avoid full data storage.
Truncation for Finite-Sample Stability. Following recent work on adaptive ATE estimation without endogenous treatment assignment (e.g., . [33 14 38)), we apply truncation scheme to stabilize the assignment policy (x (H)-)). After computing the raw plug-in policy म, (X H4-1) via Eq. 5 we define the truncated policy Ty (X) H4-1) as
(X Hz-1) == min { b.
1 |
H H1-1)
max
[
where h; E [2, oc) is truncation parameter satisfying h; + as / - xo. Truncation ensures that the instrument assignment probabilities remain bounded away from and 1, thereby improving finite-sample stability and leading to better theoretical guarantees.

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Algorithm AMRIV: Adaptive Multiply Robust IV Estimation
Require: Burn-in period To: initial policy Timi (X); regression/chassffcatiin learners for 1 (2, (X), X2,
(z, X).
for t 1 to I do
2: Observe covariates X,.
3: if - To then
4:
Assign Z, ~ (ern(Tini (X,)). else
5:
6:
Estimate nuisance functions , X), P/- (0, 8, and 12,1(2, X) from H,
using cross-fiting. Compute (2, using Eq. (6)
(1.x)
7:
Compute plug-in assignment probability: (x(X H4-11 = (0,x)++ 10
Apply truncation to obtain xx(X (H4-1) == min { ki max k #e(( X H4-1) }
8:
9:
Assign Ze
( Xx H4-1))-
10:
Observe instrumented treatment A, = A(Z;) and outcome Y, = -((A)-
11:
Construct ih ( X), /2_ (0, X), (X), ( (x))
by estimating (or reusing)
nuisance functions via cross -oots-fitting...
12:
Compute o, = (XX,, Z,, A,, Yú Te,
Te, m)) using Eq. (9)
AAMRIV return T/
Σr 9z.
5.2 AMRIV: Adaptive Multiply Robust Estimation of the ATE
We now introduce our estimator, AMRIV, which adaptively estimates the ATE using the recentered efficient influence function in Eq. ( evaluated on sequentially updated plug-in estimates of nuisance functions. The estimator is defined as
1 T
AMRIV T
Σ
6Xx, Ze, Ar, Yiime
(8)
denotes plug-in estimates of the nuisance functions at time t. constructed solely from the past data 74-1 (Note: the instrument assignment policy xe (X M4-1). defined by the experimenter based on the estimated optimal rule from Sec [ion 5.T is treated as known and does not require further estimation from data). This construction confers the estimator 7 7AMRIV near-martingale structure, that is,
where 9/.
(), (X)
be written as the sum of true martingale difference sequence and remainder term of order on (T'-1/2, enabling, we will show in Section 6, valid asymptotic inference under sequential dependence.
The nuisance functions (0, X). p (0, X). (x) (X), can be estimated using any flexible nonparametric regression method applied to the historical data H4-1 To reduce computational overhead, we can reuse the estimates of pi (0, X) and (0, previously obtained for instrument assignment in Section 5.1 Similarly, the estimate of (X) can be formed by averaging the cross-fitted estimates 0,21 and (1) trained on the two data folds 71 and H(
Nuisance Estimation.
respectively. The only remaining component is (X), which must be estimated separately if was not already computed as part of the 8 (X) estimation pipeline. For completeness, the final estimate of the (R)EIF at time t is given by
22:
o(Xi, Zi, Ar, Y1, (() ZA Xx Hi (1 (1) T. Hi 1)) 84 (X.)
Yy A.8 (x.) - (., + (0. x.) + (x.) (9)
where all quantities are constructed from H4-1- The full procedure is summarized in Algorithm
Unlike prior adaptive ATE methods without IVs, the estimator 7 + + (Xx, Ze, Ar. ) cannot be written as martingale difference sequence. Hence, standard MDS central limit theorems not apply directly, and we must instead decompose the estimator to recover suitable martingale structure.
7

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Theoretical Guarantees
This section provides theoretical guarantees for the AMRIV estimator. We establish its asymptotic normality, characterize its convergence rates, and demonstrate its multiply-robust consistency. Fur- thermore, in i/Appendix DJ we consider the sequential inference setting and derive asymptotically-vaadd time-uniform confidence sequences for the AMRIV estimator.
6.1 Efficiency and Asymptotic Normality of the AMRIV Estimator
We start by establishing the asymptotic properties of the AMRIV estimator 7/ FAMRIV We first introduce the following assumption:
Assumption (Bounded Outcomes and Nuisances). The potential outcomes and nuisance function estimates are uniformly bounded. That is, there exists constant C > O such that, for all t. and x:
[Y([)), [Y)(1] S C, 1P (0,z)), [2 (0,z)), (8(()), (()(=) SC
This boundedness assumption is standard stability of the estimator. With this assumption in place, we now state main result the asymptotic efficiency of the AMRIV estimator.
Cesennccffnnttion--e -based ATE estimation and ensures
Theorem 3 (Asymptotic Normality the AMRIV Estimator). Suppose Assumptions 7 to hold and there exists non- adaativee policy π (X) E [c, for some e. >> O such that the nuisances estimates îe and the adaptive assignment policy me (X Hz-1) 1) are Lz-consistent relative to the truncation sched- ule, i.e. kellft-1 fll2 (1) and ke del[Te T 1|2 = (p( for f Furthermore, assume 11ō,-1 0| 8A op Then, the AMRIV estimator is asymp- totically normal:
{e (0, -), 8(-), 8 (.))-
TT
AMRIV
-T) 4 N (0, Vag(T)),
(10)
where Ver is defined in Theorem 1. In particular, if we have T(X) = (X), then TAMRIV is semiparametrically efficient.
The key insight behind Theorem 3 is that the AMRIV estimator admits the following near- martingale decomposition: vT (7AMMIIV T) vT VT (+E VT
( (+Ei=) m.). where is martingale difference sequence (MDS), and m. ne, .7) is an asymptotically vanishing term that captures the impact of estimating the nuisance functions. The first term satisfies central limit theorem for MDS under standard Lindeberg-type conditions [60), while the second is controlled by the I---oonsistency of the nuisance estimates. We formalize this in Appendix F Importantly, this result holds under mild assumptions: we only require T.2 convergence (no pointwise convergence or Donsker conditions 77), bounded outcomes and nuisance estimates, and Lg-consistency of the nuisance components w.r.t. the truncation schedule. This allows AMRIV to accommodate flexible, data-dependent policies and sequential nuisance estimation.
o(X,, Ze, Ae, Ye: NT. m) 6(x,, Ar. YG ne, Tt) o(x,, A,, Ye:
The truncation schedule k plays a central role by ensuring positivity of (x()) (F(XXcrruca for variance control- while still allowing Te to approach an optimal policy л² k2 we must ensure lim;-+00 kt > supx ("X) so the truncation threshold does not constrain the optimal allocation in the limit and semiparametric efficiency can be achieved (see last line in Theorem 3). This mirrors tradeoffs efficient ATE estimation [33] where adaptive truncation stabilizes estimation without distorting the estimator asymptotically. In practice, when the plug- in policy #, is uniformly bounded away from 0 and 1, truncation becomes unnecessary: setting k; 1/ minx (#x()), (X)) ensures N, म. for all t.
For this to hold,
6.2 Consistency Guarantees under Partial Nuisance Misspecification
As shown in Theorem 3. the convergence rate of AMRIV is primarily governed by the estimation error of 8(X) and (4()). This reflects its multiply robust property: AMRIV remains consistent even when other nuisance components are misspecified. This robustness goes beyond prior work on IV methods in adaptive settings, where such guarantees were not established. The next two results formalize AMRIV's convergence rate and multiply robust consistency.

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Normalized MSE horizon
MSE horizon
95% CI coverage vs. horizon
MSCrit MSSEOOacct)
Empinccal coverage
3.0
AVRI
1.0
1.6
06
AMRIVINS
DM
1.2
1.4
MSE
0.2
0.4
62
AZEFW
DM-NA
DH MS
Grocce-N
2.0
Oracle
07
0.6 0.6
0.7
T/1000
7/1000
T/3000
(a) Efficiency
(b) Consistency
(c) Coverage
Figure 3: Performance of different estimators across increasing sample size T. (a) Efficiency:
Normalized MSE versus oracle benchmark. (b) Consistency: MSE standard error. (c) Coverage: Empirical coverage of 95% confidence intervals.
Theorem 4 (Convergence Rate of the AMRIV Estimator). Suppose Assumptions 1 hold, hold, and that there exists a non- non-adaptive policy (XX E [e, 1 - foror some e > 0 such that the adaptive policies ae
satisfy & I л la (1). Let îj = (i² (0, (i (0, )) 8()), (A)) denote possibly misspecified
limit of the nuisance functions, and suppose that kg fllf - fll = op(1) each f 17. Then the AMRIV estimator satisfies
AMRIV - -T| 0 + O.p (184-8 - 12211 8112)
(11)
Corollary 5 (Multiply Robust Consistency Guarantees). Under the conditions of Theorem 4 8, or 84h is Lz-consistent, then the AMRIV estimator TT 7AMRUV consistent for T.
if either
We provide proof of Theorem 4 and Corollary 5 in Appendix G
An immediate consequence of Theorem 4 is that if both 8(X) and (x) converge rate (T--4/), AMRIV achieves the parametric (T rate. This usually attainable under mild regularity conditions, even with flexible nonparametric models. Furthermore, Corollary 5) shows that AMRIV inherits the multiply robust property from its static counterpart [18]54]. the static setting, the MRIV converges either (i) p (0,-). -). (0,00.)) +), 8 are correctly specified, (ii) both and 8 are, or (iii âे and are. However, in the adaptive setting,
establish stronger result: even if the outcome-related nuisance functions 7.! (0, and f2, (0, :) are misspecified, AMRIV is still consistent as long as one of 0 or ô, converges. This is due to the adaptive setting design where we control the instrument assignment (X, H4-1) which confers robustness to misspecification in fo) (0.-) and 2 (0, :). Thus, our adaptive generalization preserves the multiple robustness property, making it particularly well-suited for practice where some nuisance components may be difficult to estimate reliably.
Experimental Results
We demonstrate the practical effectiveness of our approach in both synthetic and semi-synthetic studies. In each setting, we compare our estimator (AMRIV) to its con-adaptive counterpart (AMRIVV. NA), which assigns the instrument uniformly at random; the plug-in direct method from Eq. 1 (DM) and its non-adaptive version (DM-NA); the A2IPW estimator from [27]: and two oracle baselines: fully oracle-efficient estimator that the true nuisance functions (Oracle) and non-adaptive version (Oracle-NA), To assess robustness, we also evaluate misspecified variants of AMRIV and DM-denoted AMRIV-SS and DM-MS-in which the 8(X) estimator deliberately misspecified.
Across both experiments, we evaluate three desiderata: (i) efficiency, measured by normalized MSE relative to the Oracle estimator, (ii) consistency, assessed via MSE decay with sample size T: and (iii) coverage, computed from empirical 95% confidence intervals. We implement all estimators using Random Forests (RF, [8] and update the nuisance estimates in bini-batchess for efficiency. Further implementation details, including model hyperparameters and ablations, are provided Appendix H The replication code is available at httpps: /i thub. com CausalML/ CausalMI/Adaptiiee-
7.1 Simulation Studies with Synthetic Data
We construct synthetic environment with one-sided noncompliance, where the treatment A is only accessible to those who receive the instrument Z 1. At each time t, we sample covariates X,, assign

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C, where C3 is a latent compliance indica- tor sampled from Bern (8 (X,)). The outcome Y, depends on A, X,, an unobserved confounder U,. and heteroskedastic noise. The full data-generating process is detailed in Appendix H.1
the instrument Zi ~ TT(x, H)-1), and realize A,
We set T- 2000, To 200, and run 1000 trajectories. All estimators updated in batches of size b = 200 and implemented using Random Forests (RFs) when applicable. For the adaptive estimators, we use the truncated optimal policy in Eq. 7 with truncation schedule he- = 2/0 2/0.999'. AMRIV uses RF classifiers for (X) (clipped at 0.01) and RF regressors for je (z, X), while ((X) is computed via the plug-in in ratio. A2IPW follows Kato et al. [27] with Neyman allocation and RF regressors. To
induce misspecification, we replace TT X1 with the constant Rly (1, X)], flattening outcome heterogeneity. Figure 3 summarizes the experimental results.
Adaptivity. As shown in panel (a), adaptive design consistently improves the efficiency of all estimators. AMRIV approaches the oracle benchmark despite using estimated nuisances, while AMRIV-NA exhibits a constant efficiency gap due suboptimal allocation. This illustrates how adaptivity enables more effective data collection: by dynamically allocating instruments to regions of high uncertainty. AMRIV concentrates sampling effort where it contributes most to precision. The effect is particularly evident under one-sided noncompliance, where asymmetries in both outcome and compliance variance make uniform allocation especially inefficient (Theorem T). Panel (a) also confirms that AMRIV and AMRIV-NA converge the expected (T rate (Thcorem 4). whereas DM and DM-NA converge more slowly, as their normalized MSE increases with T.
Consistency. Panel (b) confirms that AMRIV, AMRIV-NA, and DM converge to the true т, with AMRIV variants achieving lower error due to variance-awan allocation. In contrast, A2IPW is biased and fails converge, as expected, since it does not correct for unobserved confounding in treatment selection. The comparison further highlights the importance of robust nuisance estimation: DM-MS diverges when 5( (X) is misspecified, while AMRIV-MS remains consistent. This robustness reflects the multiply-robus property formalized in Theorem 4) which ensures consistency as long as either the compliance model or the outcome model is estimated correctly- especially desirable feature in practice when some nuisance components are difficult to learn reliably.
Coverage. In panel (c), evaluate the empirical coverage of 95% asymptotic confidence intervals. Only AMRIV and AMRIV-NA achieve nominal coverage, consistent with our theoretical guarantees plu--u methods under-cover, with DM and A2IPW performing particularly poorly as T grows, owing to finite-sample bias and unaddressed confounding bias, respectively. AMRIV-MS provides partial correction but still falls short of nominal coverage, consistent with the requirement that 8(X) be estimated consistently for asymptotic validity.
(Theorem 3). The misspecified and plug-in
7.2 Simulation Studies with Semi-Synthetic Data
We also evaluate AMRIV on a semi-synthetic dataset based on the TripAdvisor customer simulator from Syrgkanis et al. 52 where we use customer features as covariates X, a simulated signup prompt as the instrument Z, and subscription revenue as the outcome Y. The DGP and oracle nuisances are described in Appendix H.2 Results are consistent with the synthetic setting: adaptive instrument assignment improves efficiency, AMRIV achieves superior coverage and consistency, and robustness holds under partial misspecification.
Overall, these findings confirm that pairing adaptive design with the AMRIV estimator improves efficiency, enhances robustness, and yields more reliable inference than non-adaptive baselines.
8 Conclusion
We develop AMRIV, an adaptive, multiply robust estimator for ATE estimation in experiments where treatment can only be encouraged via binary instrument. Our approach (i) derives the semiparametric efficiency bound and optimal assignment policy, (ii) constructs a sequential estimator that attains the bound under adaptive allocation, (iii) provides asymptotic normality, convergence rates, and multiply robust consistency, and (iv) supports valid inference through time-uniform confidence sequences. Empirical results on synthetic and semi-synthetic data confirm that adaptive instrument assignment improves both efficiency and robustness over non-adaptive baselines. This work represents step toward principled, data-efficient experimentation real-world settings where compliance is optional and uncertainty is unavoidable. We discuss limitations and broader impacts in in/Appenixx
10

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Acknowledgements
We thank the anonymous reviewers for their thoughtful feedback and insightful suggestions, which have helped improve and strengthen this work. Miruna Oprescu was supported by the U.S. Department of Energy. Office of Science, Office of Advanced Scientific Computing Research, under Award DE-SC00231 12. Brian M. Cho was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program. Nathan Kallus was supported by the U.S. National Science Foundation under Grant No. 1846210. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U.S. Department of Energy, the U.S. Department of Defense, or the U.S. National Science Foundation.
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A Extended Literature Review
We contextualize our work by surveying six strands of prior research. We group these into two categories: (1) core threads that directly motivate and inform our methodology, and (2) auxiliary threads that provide important theoretical and practical foundations but are not specific to our design.
A.1 Core Related Work
Identification and Estimation with Instrumental Variables. Instrumental variable (IV) methods are widely used to estimate causal effects in the presence of endogenous treatment selection due to unmeasured confounding. Under classical IV cetmmpptonn-xxclussiii independence, and relevance-
these methods typically identify the local average treatment effect (LATE) 2 4 TO 42 which pertains compliers: units whose treatment responds the instrument. However, compliers represent an unknown and potentially unrepresentative subpopulation, limiting the policy relevance of LATE.
To target the population average treatment effect (ATE), IV methods have traditionally relied on linear structural equation models (SEMs), which the ATE corresponds to a regression coefficient under correct model specification [9)]. Two-stage least squares (2SLS) is the canonical estimator in this setting, but its consistency and interpretability depend on strong linearity assumptions [2] 57]. More recent SEM-based IV approaches focus estimating conditional effects, including kernel IV [50], DeepIV [24], and other moment-basee based estimators [6 52].
54 who introduce an al- ternative identification strategy based on an unconfounded compliance assumption. This allows point identification of the ATE while separating identification assumptions from estimation model assumptions. Their approach avoids reliance on parametric SEMs for ATE estimation and instead yields an efficient semiparametric estimator with an efficient influence function (EIF) that confers the estimator a multiple-robust property, i.e. the estimator remains consistent if one or several nuisance components are misspecified. This structure makes the estimators well-suited to nonparametric plug-in in estimation using modern machine learning tools. This insight has been extended to develop multiply-robus CATE estimators that leverage binary instruments to adjust for unobserved confound- ing [8], and to debias confounded observational data by incorporating (potentially weak or imperfect) instruments 43 Other recent work extends the framework of Wang and Tchetgen Tchetgen [54] to nonparametric identification of ATES under related assumptions 34
We instead build on the framework of Wang and Tchetgen Tchetgen
We adopt the framework of Wang and Tchetgen Tchetgen [54] as the foundation for our estimator, with the goal of efficiently and robustly estimating the ATE under adaptive, sequential data collection. Specifically, we derive the semiparametric efficiency bound for ATE estimation under arbitrary, aatrummnnt-assigamen
covariate-dependent
policies and identify the optimal adaptive policy that minimizes this bound. We then introduce AMRIV, adaptive, multiply robust estimator that attains the bound and enables valid inference in scquential experiments.
Adaptive Experimental Design for Treatment Effect Estimation. A substantial literature studies adaptive algorithms for estimating the average treatment effect (ATE) efficiently and with minimal variance when the treatment itself can be directly assigned. This line of work was initiated by Hahn al. 23 who proposed two-stage explore-thee sesoore-the--cce wwwetteennooeee -heee-oommttttttttteeeeeeee design that asymptotically achieves the semiparametric efficiency bound, echoing early bandit-style adaptive allocation schemes. Fully sequential designs soon followed: Kato et al. (27) introduced the Adaptive Augmented Inverse Propensity Weighting (A2IPW) estimator, which achieves variance-optimal Neyman allocation; Kato et al. (28) extended this to settings with estimated policies and nuisance functions and demonstrated multiply robust consistency; and Cook et al. [13] added principled policy truncation and developed the first anlli confidence sequences for adaptive ATE estimation.
Parallel progress has come from an online-learning perspective. Recent methods such as Clip- OGD and its optimistic variants attain sublinear or logarithmic "Neyman regret" for the ATE [4 38 39
low-switching policies achieve finite-sample optimality bounds [35]; and newer designs jointly optimize over covariate and treatment dimensions [29]. Off-policy estimators with adaptively collected data have also obtained sharp error rates and regret guarantees [3 Together, these advances form a mature toolkit for adaptive experimentation that combines adaptive nuisance learning, cross-fitting, policy truncation, regret-minimizing allocation, and time-uniform inference.
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Our work builds directly on this literature but extends it to the far less explored setting where only an instrument can be assigned and treatment uptake is endogenous. We integrate core components from the direct-treatment literature-in confllence-functoon--ie
estimation, adaptive policy learning, cross-fitting, policy truncation, and sequential inference- to construct a unified estimator that retains semiparametric efficiency, multiply robust consistency, and time-uniform inference under noncom- pliance. Specifically, we are the first to: (i) derive the semiparametric efficiency bound for ATE estimation when only an instrument can be adaptively assigned; (ii) identify the variance-optimal allocation rule that balances outcome noise and compliance variability; and (iii) develop AMRIV, a multiply robust estimator that attains this bound and supports anytime- valid inference.
Adaptive Experimentation with Instrumental Variables. Recent work has begun to explore adaptive experimentation in settings where treatments cannot be directly assigned, requiring the use of instrumental variables (IVs) to estimate causal effects under unobserved confounding. Broadly, these efforts fall into two categories: methods aimed at improving predictive accuracy in the presence of confounding, and approaches focused on adaptive design and data collection or regret minimization using bandit-style feedback.
The first group focuses on improving estimation efficiency in indirect experiments. Gupta et al. P2 propose an adaptive framework for selecting among multiple data sources to efficiently estimate causal functionals such as the ATE. Their method, Online Moment Selection (OMS), chooses which source to query at each step based on moment conditions implied by causal graph. While they address efficient data acquisition under structural constraints, their setting assumes passive data collection and differs from our focus on adaptive experimental design with noncompliance and endogenous treatment. Ailer et al. 3 study sequential indirect experiment design in instrumental variable settings, focusing on partial identification of nonlinear treatment effect queries. Rather than aiming for point estimation, their method adaptively tightens upper and lower bounds on a functional Q[f] of the treatment effect by selecting experiments that reduce the gap between these bounds. In contrast, our work targets point identification and estimation of the ATE, and provides semiparametric efficiency and robustness guarantees under adaptive instrument assignment. Most closely related to our setting, Chandak et al. [9| study adaptive instrument selection to improve sample efficiency in indirect experiments. They propose a general influence-functe based optimization procedure for selecting instruments that minimize the mean squared error of nonparametric IV estimators, such as DeepIV (24). However, their objective is variance reduction for prediction, not inference for causal estimands like the ATE. Their analysis is estimator-specifit and does not characterize semiparametric efficiency bounds or multiply robust inference, which are central to our approach.
The second line of work focuses on regret minimization in settings with instrumental feedback. Zhao et al. [61] use randomized instruments within linear structural equation model to enable pure exploration for policy learning under unobserved confounding. Their focus is on identifying the best treatment arm using bandit-style algorithms with finite-sample confidence intervals and opar-optimal sample complexity guarantees. Unlike our work, which targets semiparametric inference for the ATE, their goal is policy optimization rather than estimation. Della Vecchia and Basu [16] study online instrumental variable regression with bandit feedback and propose regret bounds under endogeneity. Their focus is on prediction in stochastic settings with instrumental bandit structure, rather than causal effect estimation or statistical inference.
Our Contribution. To our knowledge, we present the first estimator that is both semiparametrically efficient and multiply robust for ATB estimation under a binary instrument with adaptive assignment. We characterize the efficiency bound, derive the optimal allocation policy that balances outcome and compliance variance, and develop the AMRIV estimator that asymptotically attains this bound. Our results generalize prior work ATE estimation to settings with endogenous treatments, establish asymptotic normality, and construct time-uniform asymptotic confidence sequences. Empirical studies confirm our method's efficiency, robustness, and practical viability.
A.2 Auxiliary Context
Semiparametrie Efficiency and Inffuence-FunetionnBin Methods.
Our estimator builds on a long line semiparametric inference techniques, particularly those using influence functions to achieve efficiency and robustness in the presence of nuisance components 47 Recent work has adapted these methods to flexible machine learning settings by incorporating sample-spliting and cross fots-fitting [1] [8 30). In adaptive experiments, such techniques have been
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shown to yield efficient estimators without requiring Donsker conditions 13 27 28 these tools to a setting with endogenous treatment and adaptive assignment via binary instrument.
We extend
Multiply Robust Estimation.
Multiply robust estimators remain consistent if any one of multiple nuisance components is correctly specified. In the IV context, this structure has been exploited to enhance robustness of ATE and CATE estimators [8 [4]. We extend these ideas to adaptive settings, showing that AMRIV retains consistency even when some nuisance functions are misspecified, as long as at least one of 8(X) or 8 (X) is consistently estimated.
Confidence Sequences and Anytime-Valle Valid Inference.
Confidence sequences (CSs) provide coverage guarantees that hold uniformly over time, making them well-suited adaptive experiments with interim monitoring early stopping. Recent work has developed CSs for iifuuencefuunciion-eee
estimators using martingale techniques and empirical Bernstein bounds [13 56). We build this construct asymptotically valid confidence sequences for our adaptive IV estimator, accounting for sequential dependence and cross-fitted nuisances.
B Notation
Table I: Notation
X
Observed covariates (feature vector) in R
Z
Binary instrument / encouragement, Z (0. 13 (experimenter-assineee Binary treatment, A E {0, 1). determined endogenously by the unit. Real-valued observed outcome.
A
Y
(a,
Potential outcome under treatment a (and instrument when shown). Potential treatment taken under instrument 2.
A(z)
U
Unobserved confounder(s).
True (ohservable) distribution of (X, Z, A,Y) Population average treatment effect (ATE): 7 = ([Y - E[)
n (2)
F|Y 7 2, X s/, instrument-comdditooaa outcome regression.
a (2,x)
F[4 2 =, X = x). instrument-condiiione treatment regression.
8Y (x), (x))
Instrument-induced shifts in Y and A, respectively.
8x (x) = n² (1, - (0, (x)x) = = (1,z)) (a:0))=A(000 (0, Conditional average treatment effect (CATE) a: 5(x) 6" (x)
6(x)
melar H(-1)
Instrument assignment policy at time Pr(Z,
X, = x, He-1).
t (x)
Efficiency-optimal instrument assignment (Eq. History before round t: {(X,, Zi, A,, Y:) 2-] Total number of rounds samples. Burn-in rounds (initial exploration). Truncation parameter time (ensures E [1/ka, 1/kka). Residual variance: Var(Y A6(X) 7 x=
Ho-1
T
To
ke
(2,
(aE(T)
Semiparametric efficiency bound under policy я (Eq. (3)
((=T,))
Recentered efficient influence function (EIF) used in AMRIV (Fq. 2
f.
f: estimate of f: f: plug-in in or candidate function. AMRIV estimator after 7' rounds (Alg.
P GAMRIV
E
Empirical expectation (sample average).
llallz
La norm: llall2 := x
(1),) O,(-)
Standard probabilistic asymptotic order notation.
Small positive constant (e.g., variance-floo fflorr in Eq. 6
Temporal cross fotsing fold (e.g.. j E 1)) used for sequential tross-fitting.
C Practical Implementation of Nuisance and Variance Estimators
This section complements Remark 2 by outlining practical choices for nuisance and variance estima- tors, including nonnegativity and online-update considerations.
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Nuisance estimators. Because AMRIV uses sequential cross coossniitting,, we require only standard nonparametric convergence rates to establish asymptotic normality (Theorems the analyst may flexibly choose estimators according to data structure and sample size. Representative options include:
3 Consequently,
Table 2: Representative nuisance estimators, convergence rates, and suitable applications.
Estimator
Convergence rate
Best suited for
k-NN / kemel smoother
for Hölder-B smooth
Low-dimensional. smooth problems
functions 41
Random forest
Ô n 9/(28+4) -=-0((54+)
for Hölder-B smooth
Moderatee-imennionn tabular
functions
49
Fully connected neural nets (ReLU)
WWWLoog
1/2) for width and High-dimensional or structured inputs
depth L 59
Neural nets (1 (I-Lipschitz, bounded weights)
11 M//n//4 where M; bounds the Frobenius norm of layer I's weights 20
High-dimensional or structured inputs
The best choice depends on sample size, covariate dimension, and smoothness. In adaptive experi- ments with directly assigned treatments, k-NN and random forests are standard baselines [3 27
Online or streaming implementations. Theoretical results assume that nuisance functions are re-estimated each update, but full data storage not required. Practical alternatives include:
1. Online learners: gradient-descen (SGD) updates, online random forests 32 or GLMs can update parameters incrementally using new data.
2. Block sample-splitting reuse recent batches to update nuisance fits and discard older data while maintaining cross cross-fitting validity (26).
3. Sufficient-statistie updates: for parametric or binned regressors, sufficient statistics such as running sums of (X, A, per cell suffice.
Non-negative variance estimation. Equation (6) enforces non-negativity via small floor pa- rameter e. As an alternative, one may use a self-normalized kernel variance estimator that is fully nonparametric and guarantees (2, x) 2 b construction.
For each (). define
0f(2,)) E.sr KA KA xl)) = 2] (R, (2,
E.sr Kn (1x,, 12, =
Ensr KA - x11) I(2, = R.
D: (2,
Dost Kn([Ix, T(7 2]
where R, Y, 1 and (.)() is kernel with bandwidth h.
This estimator satisfies o7 (2, x) 2 by construction and is consistent for Var(Y A5(() X xx) under standard kernel conditions. Caveat: updating kernel weights online can be ( in general; for large-scale data, the plug-in estimator Eq. 6 more computationally efficient.
Both the plug-in estimator with the floor and the kernel-based version above preserve all asymptotic guarantees; the choice between them is primarily trade-off between computational efficiency and strict non-negativity.
D Asymptotic Confidence Sequences
The fixed-time intervals Section guarantee (1 a) coverage solely at pre-specified sample size T. In practice, however, analysts often peek at interim results and may stop the study carly once
18

---

a decision rule is met [46], behavior that invalidates fixed-tine sime-time intervals. To remain valid under such data-dependent stopping one needs a confidence sequence (CS) collection of intervals
[L,, Uzl121
satisfying the time-uniform guarantee
P (VI E N+ : E [T,, U.)) 2 1
Constructing non-asymptotic, anytime- valid CSs can be difficult when the target estimand contains estimated nuisance functions. Fortunately, for AMRIV the nuisance-induced remainder is op(t (o/tt//) under Theorem 3 Jassumptions, so an asymptotic CS-valid after a finite, burn-in phase- phase-remaing remains both tractable and practically useful.
Def. 2.1 & 2.3]). A sequence of random intervals Č. = [L,, Uele21 is an asymptotic time uniform (1 a) confidence sequence (AsympCS) for a parameter if the following two conditions hold.
Definition (Asymptotic time-uniform coverage
15
(i) Asymptotic confidence sequence: there exists an exact (potentially unknown) (1-o) confidence sequence C; = [LL,,U D22 such that L./ Li + 1 and Uz/U; + 1 almost surely.
(ii) Asymptotic time-uniform coverage:
lim P Po To-xo
To:
a
Definition can be read follows: if one waits "peek" until the sample size is sufficiently large (T' 2 To for some burn-in To). the band then covers the true parameter at every later time with In practice, the rare coverage failures occur almost exclusively during this short initial window; once past it, the intervals tighten rapidly and deliver appreciable power gains over fully non-asymptotic sequences 13 55)
probability approaching
Building on the methodologies of Waudby-Smithe et al. (56] and Cook et al. 13 we now present the corresponding asymptotic confidence-sequee sequence (AsympCS) results for estimator:
Theorem 6 (AsympCS for AMRIV). Suppose Assumptions 7 to 3 hold and there exists a non- adaptive policy π (XX E [c, <) for some > 0 such that the nuisances estimates ii and the adaptive assignment policy ox(X He-1) are Lz-consistent relative to the truncation schedule,
i.e. fe11.r-11 - fll (1) and kg || - T|12 = -,, for f E {er (0,-),HA(0 (0,-),, (,,,(),88())) 8-)).
Furthermore, assume [18c-1 8|12 = Ou.s. (1) and 118,-1 - 6112110,1 - 1/1 = Oa.s.
log
Let
Vr = T
1 1X
Σ (o(X,, Z,, A., Yos Te. nd)
T7 +AMRRV)2
1=1
be the estimated variance of (Ø(X,, , Ze, Ar, Yr; Te, mr and fix user-specified > O. Then, for all T 0, the interval
VTVpp2 +1
+2
C OAAsmmCCS
AMRIV TT
log
T2p
forms an asymptotic (1 11-o0 a) confidence sequence (as in Definition 7 for Furthermore, width of C AssymmCSS is (approximately) minimized at
2 log a + log( (2g-22ogger +1
T
Remark 3 (Difference in convergence rates for fixed-time and anytime-valid inference.). The conditions under which the AsympCS in Theorem valid are largely the same those imposed for fixed time inference in Theorem
3 The main difference lies in the convergence conditions for the
nuisance functions. While 18,-1 S||2 and 1|θ,-1 0|,11 8^112 are assumed to converge in probability for fixed time inference, confidence sequences require convergence almost surely, as noted by Waudby-Smith et al. (53]. However, the conditions for valid inference are not necessarily stricter than fixed-rime time inference, as the product error term is allowed to converge slower vlog t/t rate as opposed to 1/2
19

---

Proof of Theorem 6 For the proof of Theorem 6 we rely on an existing result for asymptotic confidence sequences [55]. For completeness, we provide this result in Lemma bbllow.
Lemma 7 (Corollary 3.4 of Waudby-Son -mithh al. [55]
Suppose 0g is an asymptotically linear
estimator of with influence function & that satisfies
0. -
(XX, Zi, A,, YA 11 On...
(Vlogt/t)
(12)
Furthermore, suppose that Var(Ø) < cc. Then, (0.
2(tp2+11 p'p
log
Vip2+I
forms a valid
--a)--sympCSS (as in Desimition for 0.
we only need to show that (i) the residual error our estimator is a smaller order than Vlog 1/t almost surely and (ii) the variance the limiting influence function e is bounded.
Using Lemma 7
Verifying Residual Error. Our proof of the residual error bound follows similar steps to the proof of Theorem We first rewrite the difference between our estimate s/ +AMRIV and r as
AAMRIV
Σ
o A., Zi YOT. 11)
Σ m
(13)
where m. = (X,,, A, Z, Y; ( A, ,, Z, Y;: Te. .). We repeat the same arguments as the proof of Theorem 3] which shows that the cumulative residual error (i.e. the sum of m²) vanish
(1/v() rates. Replacing assumptions | 8112 (1) with 11;-1 8112 Oo.a. (1) and
1 511211,, 8 (t-3-//3 with [| 61|2|1 5 |l Oa.s.
logg
we obtain
Ei m = 0a.o (Vtlog Normalizing by we obtain the desired result. m,
Finite Variance of c. The finite variance of limiting influence function ø is immediate from Assumption 3 and the condition that x (X) [e. for some >> Under these assumptions, for any tuple [X, 7, A, Y), á is bounded almost surely function of some constant C,
2
1Ø(X, 2, A, Y; m. m) (3C² Ca) +C.
Using this bound, we now upper bound the variance as follows:
Var (Ø) = E[ E[ol < Bio"] $ (302 + +c)²
Because the constant C' is finite, the variance must also be finite, which completes our proof. By Lemma 7 the confidence sequence in Theorem 6 is a valid (1 - -syympCS for т. The proof for the approximate choice of p" that minimizes the relative width of the confidence sequence at time T provided in Waudby-Smin et al. [53 Appendix B.2].
E Proof of Theorem 1 and [Crolllary 2
In semiparametric theory, the efficiency bound is determined by the variance of the EIF characterized Eq. 2
VTn(T) = E[[((X 2, - - = E (, -) X
(Law of iterated expectations)
where
((X, 7, A, A,Y;n) -T
2Z
Y (--0(X)- , (O( =(000--000 +8(X)
ZT(X) +1 Z) (X)
20

---

First, we show that 1[A1 X] = 0:
E[Ar X| A - +
- T(X) 1 (1.
=(X) (x(X)
1 - (
+ 1- (X) (x)
(0, (0, + 1 (0,))
(Using the 2 N² definitions)
(x( (M'(L, (1; - (0,
0
where in the last line we used the fact that 5(X) = 11 (0.00 (0.X) " (1, - μ (1, (XX = μ² (0, X) / ())
-0 which implies the identity
Thus, we can expand Ven( (T) as:
Va((#) = [E (( [1 + - x]]
= E [ + (8(x)
- -} 2R - -
| X]]
= =E [
X] (6(x)-7)²
(Using B[A, | x]=0)
It now remains to expand [R[A]] x]:
B[A
X] = N (X) A Z X] + (1 - x (X)) 2 0,
(X)
+ 1 E [(Y A6(X)- -M² (0, +00 1-= [
( (x)2
1 (XX)
1
+ (1 - xX))² 8A(812
E [(Y A5(X)-
+ x0,
2=
(x() Var(Y - AO(X) 1==1,x)
8x(X)2
+ xxx Var(Y - A5(X) 2 200,7)
(1, x0)
8A(X)2
x(X)
lx)(X)
where used the fact that Y Ao(X) 7 X| (0. (0, X)) and
E|Y - A5(X) 7 z= 1, n² (1, - (1, x) = n. (0,
Putting
everything together, we obtain the result of Theorem 1
1
(1,
+ x²(0,X)
+ (8(X) +(8(X) -
m)²
(T = E
(A(X)²
T(X)
1-x(X)
Then, the optimal policy n² (X) is given by:
n"" = arg min (TT
1
+ 1 T()) ) [ (0,
(1,
o² (0,
= arg min E
8A(X)2
T(X)
1
+ 1- (X)) [x]
20,0,
= arg min E
(A(X)2 E
x(X)
0² (1,
o²(0, X)
(X) = arg min P
p
1 1-p p
21

---

(0 (I-p)*
x(11x) 0.
The minimum is obtained when the derivative of the argument w.r.t. p 0, i.e.
p
By solving for p. we obtain p 1 [
Thus, we obtain the result of Corollary
(1) x²) = (o²(1, + (0)
F
Proof Theorem 3
F1 Preliminaries
Our asymptotic argument relies on martingale central limit theorem under Lindeherg-tye condition. We use streamlined version of the MDS central limit theorem originally due to Dvoretzky 177, as presented in Zhang al. [60] Theorem 2].
Theorem 8 (Martingale CLT, adapted from
660 Thm. 2)). Let ((2,, Ho)) be real-valued
sequence where दт
+ such that:
1. (Martingale difference sequence) {2))== is martingale difference sequence; that is, E[z H4-1] = for every t E [1,T].
2. (Conditional variance convergence) There exists a constant o² > such that
E H4-1l 2 02
3. (Lindeberg condition) For every > O,
T
Σ
E
=VT) Hi-1 2 0.
Then,
VT 4 N(0,
To begin our proof, we define th == oxxx, , Zn Ar. Y:: n,, T T, where is given in Eq. 9 Letting 17 {nY (0, () X), 8 (X), (X), ((X) denote the true nuisance values, decompose t//
d/e = o( Z,, Ar, Yi Te 77) T (Xt, Ze Ao, To - o(x,, Z, A, MA
such that
VT AMRIV =
= VT
1 T
VT
T
Then, the proof of Theorem 3 proceeds in three steps. We first verify that f2, forms martingale
difference sequence. Then, we show that { satisfies conditions (2)-(3) of Theorem with
x² = (T). Finally, we will show that VT Σt m²) = (1).), thus concluding that
VT -T) d (0, Vor(T)).
F22
MDS structure of 2t
We now show that {2t} (X, Z,, Ar, YA - 7 forms an MDS, i.e. E[z Hz-1] 0:
E 24 H4-1)
27,
ZiTO 1H111) + +1 2,) [1]11 H4-1))
8A(),)
22

---

[Xc A,8(X;) - 1² (0, + (0, 8(x,) Hc-1 7
27:- -1
E E
(Xx (H1-1) + (1- - Z,)(1 (X, He-1)) (*()))
y2 A(8(X)) -
// (0, + (
Xx, He- 11-1 |
+ From Eq. ①
(x,) (L, - (1, - N² (0, + (0,
=
8a(x)) [u² (0, 10, (0, xx)((x)) n²" 2/ (0, + + (0, [4-1
-0
where we used the identity 1a (1, X2) 11 (1, xx) = μ.¹ -0. (X) which follows from the definition of (.)). Thus, {2t} an MDS, owing to the fact that n7 constructed from historical data only.
F.3 24 satisfies conditions (2)-(3 of Theorem 8
For condition (2), we first show that E[2? H4-11 VoH(TT) _ 0:
[| He-1] - (T)
(o)(1,
+ o²(0,Xi))
+ (6(()) 7)2
= -Var22 1H4-1) (A(X))²
(((1)
1 1- (Xii)
(55/2) I He-1l = 0)
Var(o(X,, Zr, Ar, Y(ite.)) (H(-1)
(1,
0² (1)
+ +8(()) - -)2]
E
+
(x)()))
x(1i)
1 1- (X,,)
X.)
0² (0,
+ +((X)) 7) H
= (x,))
TT(x H2-1)
+ 1 1- (X, (H4-1)
(n the oracle nuisance set)
(1, (0)
x(Xi) + 1- x(X,) ) + +(((X,)
7)² -7)²
E
(x)))
X.)
((X,) - Tx(,, (14-1)
Tx(X, 14-1) (X,) [
(A(X))²
o (0, X²)
(X, H4-1) x(x))
)
E
14-1
8- (x()))2
( (1- (X, (1 (Xxi))
36C
2guks
E (T(((x)) <x, HH- |}]| X ky || 1=
where in the last line we use the following boundedness conditions: (i) < C from Assumption 3
and thus 18(X) < 2C and 0² Xt) IE[(Y - A8(X,)² 2 = 2, X,] E A0(X.)
x, < 18C². (ii) 18 (X)
< εạд for some сĄ사 > 0 implicit in the (conditional) relevance in
Asssumption (ii) ग (x).
T (X,) > e from Theorem 3 staatement, (iv) (X,), =(X,) 2 1/k,
by construction, and (v) the L1 norm is bounded by the L2 norm. Thus, setting 0² := Van
have that each term converges in probability o², iee. E[2 H4-1] PS o², where o² is finite by Assumption Assumppttttooon 3
To complete condition (2), we now show that
T
Σ
E
He-1] - 02
P, 0.
T
Let a, := E[2? H4-1] and a := o We have just established that a 4 a, and under our bounded- ness assumptions, sup, E[ae] < oo, so the sequence {a,} is uniformly integrable. By the L' conver- gence theorem (c.g.. Loève (36)), this implies that a; - a in L1, i..., E [[ Hi-1] o2| + 0.
23

---

Therefore, by Cesàro averaging and Markov's inequality, we obtain ΣΣ= Lt=1 Elz? completing the verification of condition (2) in Theorem
Hi-1] _ o².
Now we verify that žn satisfies condition (3), the Lindeberg condition. We follow the same steps as in
Cook et al. [13. Let by := z 1[ > 8vT). Then by = z with probability Pr(Izal > 5vT), and
by 0 otherwise. By Chebyshev's inequality,
Var(=+)
Pr(Izd > 8VT)
8T
Since Var(z)) M[a]] < co, it follows that limy
Var(a)) which implies & 0 and hence 62T
S 0. Moreover, note that [b,| S =f and R[(2]] < co. By the dominated convergence theorem,
lim bbbb] T-
lim by
+00
Therefore,
T T
B
> SVT} H6-1]
" o,
t=1
verifying the Lindeberg-type condition required for the martingale CLT.
F4 VT mc) is (1)
We first decompose vT [ mi) as:
T
T
m
T
VT
1
Σ
(o(X,, Ze, Ar, Yo - -((,, Ze, Ar, Ys; Te
T
t=1
VT
1
H4-1] [E [E|pb X,, 7,, A,, (. 1 - EE( Xx, Zz, Ac, Yi (P6-111) (AA)
T
1=1
(o(X,, Zt, As, M)) o(Xi, Tx Ac, ();
T
[Xx, Zz, Ar, Yo ne, me) (X,, Ze, Ar, Yo Te, 1) H
(A
where A is an asymptotic bias term due to nuisance estimation and A B is the empirical pro- cess term. We bound these independently. Let A Elo(X,, Ze, Ae, Yo ie) He-1] E[o(X, Ze, Ae, Ye (t, 7) H.-1). Then:
A
Ei Ze, A., Yú Te 11) H4-1] - - Ze, A,, Yii He-1]
E[E[o(X,, Z,, A,, Yi m
X1, 7:-1] H, E(((X,) [H411]
[E A,, Y(i T 71
Z, = 1,,,, ,, H,-1 H,-1) Ha-1]
+ [E[EOO(X, Zr, Ar, Yo Te, mn) Z. = X, H. 7x(x, (H411)) Hr-1]
E(((X.) 14-11
E
- (X)
P4 (0.
842 (X,)
(4)(0, (0, - p (0, (0. (X,) +X - R._ (0, (0.)))))
8/1 (Xt)
24

---

+ ( 6(X,) M1-1
1
E
(X,) 8 (X))) 8(Xx)
He-l
8 (X.)
(x()))
E
(8(X,) - -8, 8 (X,)) + +x,1 - 5(X,) H661
(x,)
1
E
(8^ (x,))(() - (xx1)
Hi-1
(X,)
& 0 - 8|12 18/21 - 81112
Assumption and Cauchy- Sennhy-Schwwrz)
(t
Using similar argument as in the previous section, we have that vT - 4/4 oz
We now focus the empirical process term A We will show that E| 0 and Var( (1) and then apply Chebyshev's inequality to reach the desired conclusion. We now turn to the empirical process term A Our goal to show that [[A B] and Var(A B) = (1), which together imply that A B (1) by Chebyshev's 's inequality.
Te, 1). We tackle the mean: Let de := (X,, Ze, Ar, TE, Te) and (1)) :- Φ(Xr. Ze, Ac. Yei
T
E
(che (x(n) - (Tt)) - <b(1)) He--il)
(fle)(fft) - (x(1 - - (1))) = 0.
(Iterated expectations)
Let us now bound Var( A B). Since the summands in A B are conditionally mean-zero and adapted to the filtration H4-1, the cross-terms vanish by martingale difference independence. Thus:
T
Var B) Var
Σ
(o.
(n) (() -v((1)))))))))))))))))))) Ho--ll)
T
Var (or (Tp) - (1) E( (e(r) | He-1l)
T
T
E Var (7x) d.(1) (H4-1)]
T
E B((()(TT)) -x))))² H1-i]
where we used the inequality Var(X E[ FF)= S [[ for any square-integrable X. We now stochastically bound E[(()((T))) - 2² 1. First, we note:
(Ti) - -1)(
2z
(Xx) +11 - 2) (X)))
8 (x())) - -8, (Xi) At
(8, 8(X.)
+ (0.)). (0, (x,)
Y,
84(X.)8211(0) (X.)
(xA)
54(Xi)
8A(X.)
(0, (0, 10,, Xt)
/ X,)6(0, Xt)
+ (X,) -8(x))) }
+
8,,
(A()))
25

---

Then:
[[(()((T))) - (e(1))) H4-1] R2 [ (sd)(Tt)) - )c(1))²
]]
X.. He
Ze = 1, x,] (61(X))
(x))2
S 2E
E|y?
M4-1
(a)
(X,)
(
2
(X,)
8(X,)
2E
E[A]] Z1 = 1,
H4-1
To (X,)
54(X,)
1
(0, X1)
Nº (0,
21E
Hi-1
(X,)
84
8*(X))
1
(0,100 (0, 11)8)88101.11
11_100 (0, (A(),)
2
21E
H1-1
Ne (X,)
(X,)
21E
(6, 1 (Xt)
8(xx))) ((xx)))((H(_1)
(x,) - 8/_ (x.))
21E
ETY??
E. Xxl
HP:11
Te. (X.)
(8^ (x2))2
1
(x,(
8(X))
21E
B[A
Z. Xt
Ha-1
To (X+)
(x,,
oA(X.)
1
(0, X1)
1 (0, X.)
2E
Hi-1
(c)
(X,)
0/2 (X,)
(A.,))
1
T (0, (0,X)) (0,
2EE
H4-1
(d)
(X.)
822 (X,)
(A(X,)
- -(xx)) ]
2E
(e)
Bounding all terms using
Assumption
and
Assumption 3
have:
B[( (x))) H,-1]
I 4kg 2.. 18,,1 - 1/12
(a)
+ 84, (11ō-- 611 + 40 -8113)
(b)
+ (3. - P 1- -8
(c)
))1 +1011182 - + 09118
6112)
+ (x,) 5(x,)
(e)
=1)
where the last line follows from the fact that (a-e) are (1) from the premise of Theorem Since each
term E (o. )²)) | H4-1 nonnegative, uniformly bounded, and satisfies op (1). follows
by Cesàro averaging that + Σ E (o.( (n))) H4-1] = (1). Thus, Var (A B) (1) and we can apply Chebyshev's 's inequality to obtain P(|A 2 E) S Var(AB)/e* Ve > 0. Therefore, A B is (1), as desired. Putting everything together, we conclude that vT ] ET m4) (1) and the conclusion offTherrem Joolds.
26

---

G Proof of Theorem and Corollary
Letting (r m) (X,,, Ze, Ac, (T for any 7, and we decompose AAMRIV follows:
T
FAMRIV
7
-
( n1)) -
T
1=1
T
T (B (Te, 7e) H4-11 7)
(oe(
- (T Re) H4-1) +
T
T
1=1
We will now show that A is (T222) via similar argument as in Appendix F and A B is
Op [107 1-112 - 8112)
Write A = (e( m (T ît) He-1] and note that 4 is an MDS by construction, i.e. EIAA [4-1] Let V (T) == (or ii)) where Var, indicates the variance over data where Z ~ Bern (T (X,)). Thus, it suffices to show that E{( H4-1] ", o², where σ² = V(T).
Then, the result follows by tracing the rest of the proof in Appendix
Write A, = (T.,, Tc) - Φr (T,, 77 and note
Var Var(or (Te To) H4-1) = Var(or (T,, 7) H4-1) + Var(A | H1-1) + 2 Cov(or(1)) A. H1-1)
and thus
Var(d 7h) He-1) Var(o 11)
H1-1)1
I Var(Ar H1-1)) + 2/ Var(Ar H(-1) Var (Te ?) Hc-1)
Since, (ar( H1-1) is bounded by Assumption 3 we just need show that Var(Ar He-1) = (1):
Var(dr(TT, ?) H.-1) < E [(Te(TT, Th) de(T., H1-1]
$ (k( - 8|2 + (0,-) - (2x(0.))1 + |1 - (20(0-) + 18 8
(Parallelogram law)
-1p(
(Theorem assumptions)
where Ò encompasses the constants e and C' from Assumption and the theorem's premise. Thus, He-1) _ Vn. can use Theorem 8 from
since Var Var( 7c) H.-1) _n, Var(or(TT, 7)
Appendix F and retrace the same arguments to obtain A Op 1/2) due to the Martingale CLT.
Now, we study AB;
AB
= (T 1e) 74- B[](X)]
E|E[o(Xx Z₁, A,, YI m X4, H4-11 E(8(X)) [4-11
EX Zz, A, Yo: TT, 7 Z 7 He-1) Ho-11
+ +E[EEo(X,, X,, Ze, A,, Ye: nn) 0. 7X (44-1) H4-1
-E(((X.) H1
E 1
8AL (x,)
(Xi, (0)(0, (0, Xt) - / ~ x, (0, (x,) + + (0, - M., (0, (x)
+ ( 66-1(X)) Ho-1
27

---

1
E
+ (6 (X,) 8 (Xx) 1 He-1
8 (X,)
0 (Xe) (8(X,) - -8. + ( oe- 6x.) Ha-1
E
8,2 (Xx)
E
1
(x))
-8e-1 (x) H4-1
(X,)
1 C 66-1 - 8|12 1/ - 81112
Assumption and
Thus, A is Op( 18r - 01 8A1 I|2), as desired. Corollary 5
B
follows immediately by noting that if
either - or 8 are consistent(i.e. (1)), then A 1 also consistent and (7 AAMRI = (1).
H Experimental Details
This appendix provides additional details for the simulation experiments described in Section 7 including exact hyperparameters, model components, and execution setup. All experiments were run on Perlmutter compute node with 256 CPU cores at the National Energy Research Scientific Computing Center (NERSC) 37 and required approximately 40-50 minutes per configuration. Random Forests were implemented using scikit-leary [44), and parallelization was handled via joblib. Full code for generating data, running experiments, and reproducing all figures is available at btttts:://githbb com wttpa//a...oo/aaaaaaaaaaaa/////. hatpsaaaaMM//aaaaieieeee
with instructions in the README md.
Each estimator was evaluated on 1000 independent synthetic trials. Simulations were run over T = 2000 rounds with a To = 200 burn-in period, and nuisance estimators were updated in mini- batches of 200. For all adaptive methods, we applied the truncated optimal allocation policy from Eq. (7). with a truncation schedule k2 2/ 2/0.999'. Oracle methods used ground-trun nuisance functions, while misspecified estimators were constructed by replacing p (1, X) with a constant regressor fit to the average oracle value.
Unless otherwise stated, outcome and residual variance functions were modeled via Random Forests with 100 trees, maximum depth 5, and minimum leaf size 5. The compliance model (1, X) was learned with shallower forest (depth 3, minimum leaf size 30), and (0, X) was zero by construction due to one-sided noncompliance. For the A2IPW estimator, we followed Kato et al. [27 and estimated outcome means and second moments using random forests (depth 5, leaf size 100) and used a Neyman-styy allocation based on observed outcomes. All figures report results averaged over replicates, with confidence intervals based on empirical standard errors.
H.1 Simulation Studies with Synthetic Data
We generate the data sequentially for each time E [1, + To] using the following one-sided noncompliance setup:
X₂ ~ Unif(0, 2)d 2, ~ Bern(T,(X) H(-1)) (0, X.) 0, (X,) = n1 (1, (2X,(11)
(Covariates & Instrument)
(Compliance Scores)
C. Bem(5
A, C. 7+
(Treatment Assignment)
U. u0==ul1-01 1 -C))
(Unobserved Confounder)
Y, f(A,, X.) Up CA.: CA. g(A,
(Outcome Function)
where To is the burn-in period, C, is the (unknown) compliance indicator, the logistic sigmoid, Unif and Bern are the uniform and the Bernoulli distributions, respectively. We utilize the following instantiations for d, u, J.g:
d d= 5
" 220
= 1+ ++ =
X 0.75a X(1]²
x(A,X) (v A (vo X[I| + +t) (1 A)), Up 4.0, V1 0.25
28

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Normalized MSE horizon
MSE horizon
95% CI coverage vs. horizon
MSE(E) / (SE(Oracle)
Empirical coverage
1.0
ACRIV
2.75
ACRRV NA
MSE SE
9.6
114-MA
0.4-
1.00
02
50
A21FW
Grocie
0.0
0.7 0.4 0.6 0. 15
1.0
7/1000
7/1000
(a) Efficiency
(b) Consistency
(c) Coverage
Figure 4: Performance of different estimators on TripAdvisor simulated data. (a) Efficiency: Nor- malized MSE versus an oracle benchmark. (b) Consistency: MSE + standard error. (c) Coverage: Empirical coverage of 95% confidence intervals.
where X[1] denotes the first coordinate of the covariate vector X Rd and 3 R Unif[-1. 1. 1] is parameter vector that is fixed over the 1000 simulations (we used seed of 1 for re- producibility purposes.). ((A, X) was chosen such that Var(ea XX) = 81 u (1])+ where Up, i1 constants.
H.2 Simulation Studies with Semi-Synthetic Data
To complement our synthetic evaluation, we conduct additional experiments using a semi-synthetic setting derived from a real-world dataset collected by TripAdvisor. The original data-generating process (DGP) was introduced by Syrgkanis al. [52] and is publicly available on GitHub) In the original A/B test, users were randomly assigned to one of two groups: group A (instrument Z 1) was offered a simplified membership sign-up experience, while group B (Z = 0) saw the default interface. This encouragement increased the likelihood of signing up for membership (treatment A), though actual uptake remained endogenous due to user-specific factors.
The covariates X E R.00 capture rich pre-treatment user behavior and demographics. These include: prior platform revenue, visit frequencies different TripAdvisor sections (hotels, restaurants, experi- ences, flights, and vacation rentals) over 28-day pre-experimental window, engagement through free channels (e.g., email), locale information, and operating system type. The binary treatment A indicates whether the user became member during the experiment, while the outcome Y records the total number of days the user visited TripAdvisor during the study period.
We preserve the original covariate structure and instrument assignment mechanism, but modify the outcome model to introduce heteroskedasticity by adding tog-normal noise whose variance depends on treatment status. This choice reflects the heavy-tailed nature of usage metrics in online platforms [S] 31, and results a more realistic and challenging estimation task. The full data- generating process is provided below ("*" indicates same as original).
TripAdvisor Data-Generating Process. We simulate tuples (X, A(0), A(1), (O), Y()) via:
X ~ TripAdvisor pre-treatment covariates,
Unif[-5,5], 5],
(latent user heterogeneity")
A(1) ~ Bernoulii (0.8 (0.44x000.X,,,, + v)), A(0) ~ Bemoolli(0.006), (compliance") (new: heavy-tailed errors)
61 () 6o LogNormal(0,
Y(1) f(X) - 2v + 5 [[[ [1] > 0] + E1,
(potential outcome for A = 1*)
(0) 2v + 5-
0] + Eo,
(potential outcome for A 0*)
where = 1.5 and σ = 0.25, and the structural CATE function is defined as:
f(X) 0.8 + 0.5 ((X1) 37).
with o(X,) = 5- [[ > 5] + 10 X 15] + 5 I[X; > 20).
We illustrate results on Figure 4 To maintain readability in the plots, define the misspecified outcome model /. (1, X) as the oracle ! (1, plus a constant shift.
Adaptivity. As shown in panel (a), adaptive allocation improves the efficiency of both the DM and AMRIV estimators (particularly at larger 7), with AMRIV again approaching the oracle benchmark. Interestingly, AMRIV, AMRIV-NA, and Oracle-NA slightly outperform the fully adaptive Oracle
29

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in some regimes- likely due to extreme compliance scores in this DGP (8) + 0), which inflate the asymptotic variance when using oracle denominators. In contrast. estimators with learned denominators can perform better in finite samples 28 51
This also helps explain the narrower variance gap between AMRIV and AMRIV-NA relative to the synthetic setting, as the variance dominated by low-compliance regions rather than outcome variance between instrument arms.
Consistency. Panel (b) confirms that AMRIV, AMRIV-NA, and DM all converge to the true т, with AMRIV variants consistently achieving lower error. As expected, A2IPW fails to converge due to uncorrected confounding, while DM-MS diverges due to misspecification of 6(X). ). In contrast, AMRIV-MS remains consistent- further validating the multiply-robus guarantee from Theorem 4
Coverage. Panel (c) shows that AMRIV, AMRIV-NA, and AMRIV-MS maintain valid 95% con- fidence interval coverage, consistent with the asymptotic normality result in Theorem 3) All other estimators-iiclldinng DM-MS and under misspecification or confounding.
severely T increases, reflecting bias
I Limitations and Broader Impacts
Limitations
While AMRIV is grounded in semiparametric theory and achieves strong empirical performance, there are several limitations we highlight. First, our method relies on standard IV identification assumptions [Assumption 1) and the unconfounded compliance assumption (Assumption 2). which-while weaker than ignorability- are still untestable and may be violated in practice. particular, the exclusion restriction and the unconfounded compliance assumption may not hold even in observational settings where the instrument is randomized. Second, AMRIV assumes access to flexible, sequentially consistent nuisance estimators, which may be difficult to train or tune in low-data regimes or in the presence of heavy-tailed outcomes. Third, our analysis focuses on a binary instrument and binary treatment; extending the framework to multi-valued or continuous instruments remains open challenge.
Broader Impacts
This work contributes to the growing intersection of causal inference and adaptive experimentation, enabling more data-efficient and statistically principled estimation in settings with noncompliance. Potential applications include health interventions and online recommendation systems, where experimenters can encourage behavior but not enforce AMRIV allows experimenters to make better use of limited resources while supporting robust inference under endogenous treatment selection under unobserved confounding. However, we caution that the validity conclusions drawn from AMRIV hinges on the identification assumptions and data quality. In high-stakes settings, particularly those involving marginalized or vulnerable populations, improper use or misinterpretation could lead to harmful decisions. We strongly recommend pairing AMRIV with domain expertise, sensitivity analysis, and uncertainty quantification to ensure responsible deployment and interpretation.
30

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Reviews:
[{"review_id": "evQ88pyROV", "reviewer": "Reviewer_bwti", "summary": "This paper considers ATE estimation in the context of indirect treatment (instrumental variables) and adaptive experimentation. In this context, the authors derive a semi-parametric efficiency bound and use this to justify a specific adaptive experimentation strategy. Their approach is therefore an extension of MRIV to the adaptive setting. The instrument assignment policy used plugs data estimators for $\\\\sigma^2(0, X),\\\\sigma^2(1,X)$ into the formula of equation (4). These are computed by fitting several regression models as outlined on page 5. Theoretical guarantees are provided. Experimental validation on synthetic data shows that AMRIV performs better than several alternatives.", "questions": "- you say that $k$-NN, kernel smoother, random forest and neural net regressors can be used. How would one make this determination in practice?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n- non-compliance in experiments is a common set-up that warrants study\n- the paper provides an extremely detailed exposition\n- assumptions are clearly stated\n- a semiparametric efficiency bound is derived for adaptive instrument assignment\n- thorough theoretical treatment is provided giving asymptotic normality of the estimator, convergence rates and consistency, and confidence sets based on anytime-valid approaches\n\n# Weaknesses\n- the write-up is rather notation heavy. Perhaps consider including a glossary in the appendix so that we can easily find e.g. the definition of $\\\\delta$ on line 132 more quickly\n- little is said about which nuisance estimators are to be preferred, despite these being an absolutely crucial part of the algorithm. Perhaps some empirical investigation could have been presented here to help ground the practitioner.", "was_revised": true, "final_justification": "Thank you for providing a thorough and detailed response to my question. I maintain my score at \"Accept\"", "rebuttal": "**Strengths**\n\nThank you for your encouraging feedback and for highlighting both the importance of the problem and the theoretical contributions of our adaptive experimentation framework with instrumental variables. We respond to your suggestions and offer additional clarification below.\n\n**Re: Glossary of terms**\n\nThank you for the suggestion. We agree this would be helpful and will add a glossary table at the start of the Appendix to clarify notation and make the paper easier to navigate.\n\n**Re: Choice of nuisance estimators**\n\nDue to the sample-splitting approach, we do not impose any strong restrictions on the nuisance estimators (e.g. Donsker conditions). Instead, standard nonparametric convergence rates suffice to establish asymptotic normality of the semiparametric estimator and desirable convergence rates (Theorems 3 and 4). Thus, the choice of nuisance estimator in practice is driven by the structure of the data and the convergence rates needed to potentially attain parametric rates. For example:\n\n| Estimator           | Convergence Rate   | Best for…                  |\n|----------------------|-----------------------------------|-------------------------|\n| $k$-NN / kernel smoother | $O(n^{-1\\beta/(2+d/\\beta)})$ for Holder $\\beta$-smooth functions [1]             | Low-dimensional, smooth problems        |\n| Random forest  | $\\tilde{O}(n^{-1/(2+d/\\beta)})$ for Holder \\beta-smooth functions  [2, 3] | Moderate-dimension, tabular data        |\n| Fully connected neural nets with ReLU activations       | $O(\\sqrt{WL \\log W} \\, n^{-1/2})$ forwidth $W$, depth $L$ [4] | High-dimensional or structured inputs   |\n| Fully connected neural nets with 1-Lipschitz activations and bounded weights |   $\\sqrt{\\Pi_{l=1}^L M(l)}/{n^{1/4}}$ [5], where $M(l)$ bounds the Frobenius norm of layer $l$'s weight matrix| High-dimensional or structured inputs   |\n\nIn practice, the best choice depends on the amount of data and the complexity of the problem. In related work on adaptive experimentation with directly assigned treatments, $k$-NN and random forests are commonly used [6, 7]. We will add this discussion to the camera-ready version to further provide guidance for practitioners. \n\n[1] Nickl, R. and Potscher, B. M. Bracketing metric entropy ¨ rates and empirical central limit theorems for function classes of Besov-and Sobolev-type. Journal of Theoretical Probability, 2007.\n\n[2] E. Scornet, G. Biau, and JP Vert. Consistency of random forests. Ann. Stat., 2015.\n\n[3] S. Wager, and S. Athey. Estimation and inference of heterogeneous treatment effects using random forests\" Journal of the American Statistical Association, 2018.\n\n[4] D. Yarotsky. Error bounds for approximations with deep relu networks. Neural Networks, 2017.\n\n[5] N. Golowich, A. Rakhlin, and O. Shamir. Size-independent sample complexity of neural networks. In Conference On Learning Theory. PMLR, 2018.\n\n[6] M. Kato, T. Ishihara, J. Honda, and Y. Narita. Efficient adaptive experimental design for average treatment effect estimation. arXiv preprint arXiv:2002.05308, 2020.\n\n[7] T. Cook, A. Mishler, and A. Ramdas. Semiparametric efficient inference in adaptive experiments. In Causal Learning and Reasoning. PMLR, 2024."}, {"review_id": "xgNGkCfFFf", "reviewer": "Reviewer_zuRV", "summary": "This paper studies the problem of estimating the average treatment effect (ATE) in instrumental-variable (IV) settings when the researcher can adaptively allocate the instrument. Under a strong “no-confounding compliance” assumption that ensures point identification of the ATE, the authors derive an influence-function lower bound on the asymptotic variance and propose a sequential experimental design that attains this bound. Their strategy adaptively chooses the distribution of the instrument based on past data to minimize estimator variance. The paper provides both the theoretical derivations and a simulation study illustrating variance reduction compared to non-adaptive designs.", "questions": "1. Practical Motivation for Assumptions\nAs claimed in the weakness part, it’s not intuitive to think about scenario where the experimenter can actively control the instruments, willing to do that to reduce variance and also accept the strong assumption made.  Clarifying this will help justify the paper’s focus and demonstrate its applicability beyond toy settings.\n2. Partial Identification and Trade-Offs\n   If the “no-confounding compliance” assumption is violated or misspecified, can you characterize the resulting partial identification bounds for the ATE? Would your adaptive allocation ever widen these bounds (trading off identification tightness against variance)? This analysis would illuminate possible trade-offs when the key assumption fails.\n3. Finite-Sample Variance Bounds\n   Can you derive finite-sample (non-asymptotic) upper bounds on the mean-squared error of your estimator under the proposed adaptive design? Providing such guarantees would significantly strengthen the practical relevance for moderate sample sizes.\n4. Optimality under Adaptive Policies\n   Your influence-function lower bound applies under IID sampling. Does the same asymptotic variance lower bound hold when restricting to all history-dependent adaptive policies? In other words, is your algorithm provably optimal within that broader class?\nThe paper [1] addresses similar questions in the randomized-treatment setting. Can you explicitly map their finite-sample and asymptotic results to your IV framework? Such a comparison would clarify which aspects generalize and where new technical challenges arise.\nA clear answer to these questions will likely lead to higher score.\n\n[1] Li, J., Simchi-Levi, D., & Zhao, Y. (2024). Optimal adaptive experimental design for estimating treatment effect.", "limitations": "See above", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall the paper is well organized and clearly motivated. Definitions, assumptions, and the statement of the main theorem are clearly separated, and the adaptive algorithm is described in pseudocode. The asymptotic efficiency analysis is rigorous. The derivation of the influence-function bound follows established semiparametric methods and is presented clearly, with all regularity conditions stated. They introduce adaptive instrument allocation to minimize variance addresses an important practical challenge when instruments are costly or limited. It extends classical IV estimation by adding an experimental-design layer. Combining semiparametric efficiency theory with adaptive online experimental design in an IV context is novel.\n\nWeakness:  \nPractical Aspect: Under what real-world scenarios is it credible to assume both (a) the researcher can fully control the instrument distribution, and (b) there is genuine motivation to tune that distribution solely to reduce estimator variance?\nTheoretical aspect: 1) In the broader IV literature, identification typically dominates focus. Because the paper assumes point identification via a strong, untestable restriction, some may view the variance-optimization objective as a secondary concern. The work’s impact hinges on the prevalence of settings where such an assumption is plausible and the instrument is manipulable. 2) All variance guarantees are asymptotic. No finite-sample (non-asymptotic) variance bounds are provided, which limits practical guidance for moderate sample sizes. Also, influence function-based arguments only hold for iid sampling, so whether it’s the optimal variance that an adaptive policy can achieve remains unknown.", "was_revised": true, "final_justification": "I maintain my score. There is nothing in the rebuttal that change my opinion; the results are not surprising. Thank you.", "rebuttal": "**Strengths**\n\nThank you for your thoughtful feedback. We appreciate you highlighting the quality of our work as well as the novelty of combining semiparametric efficiency with adaptive design in the IV setting. Below, we address the main concerns and provide clarifications, which we hope will further strengthen your assessment.\n\n**Re: Practical motivations**\n\nThank you for raising this important point. To clarify, our framework is motivated by real-world scenarios where treatment cannot be enforced, but encouragement can be adaptively controlled. Consider the following illustrative scenarios:\n\n* **TripAdvisor (semi-synthetic) setting.** Our simulation is based on a real experiment where users were exposed to different sign-up flows for a premium membership [1]. The actual treatment (signing up for a TripAdvisor membership) cannot be enforced, but the assignment mechanism (e.g., interface shown) can be randomized and adaptively controlled.\n\n* **Clinical trials.** The U.S. Food and Drug Administration has explicitly endorsed adaptive designs for clinical trials [2]. Some of these clinical studied can involve non-enforceable treatments. For example, a novel drug may only be recommended (but not administered) resulting in non-compliance, yet allocation of the recommendation (instrument) is under the clinician’s/experimenter’s control.\n\nSimilar setups arise in recommendation systems (e.g. encouraging specific media content on streaming platforms), public health interventions (e.g. nudging certain actions on fitness trackers), or advertising campaigns with opt-in actions. \n\nIn such mission-critical settings, minimizing estimator variance is crucial for enabling early stopping: 1) If effects are harmful, high-variance estimators delay detection, potentially causing harm, and 2) If effects are beneficial, high variance delays rollout to the broader population. Our anytime-valid confidence sequences provide a principled framework for early stopping, but their usefulness depends critically on variance control: lower variance leads to tighter intervals and more reliable early decisions. Thus, reducing variance via adaptive design is not only theoretically justified but practically necessary in these high-stakes settings. We will add a more detailed description of these scenarios and the importance of adaptivity in the camera-ready version.\n\n**Re: Identification and trade-offs**\n\nThank you for this suggestion. Assumption 2 (unconfounded compliance) trades off with other common (but also untestable) IV assumptions. It rules out unobserved confounders that jointly affect compliance and the treatment effect (but not the outcome), enabling two key advantages: (i) we do not require monotonicity (so defiers may be present), and (ii) we identify conditional ATEs rather than conditional LATEs. Several classical IV models—such as those with additive outcome confounding—are nested within this framework, and recent work [3–5] has increasingly adopted it due to its generality.\n\nIf Assumption 2 is violated, the ATE is no longer point-identified, but the estimand shifts to the ACLATE, the average of conditional *local* average treatment effects $\\text{CLATE}(X)$. This object remains identified and interpretable: it captures how the instrument-responsive (complier) effect varies across $X$, and is often the relevant estimand when treatment cannot be enforced and compliance cannot be targeted. Unlike the LATE, which summarizes the effect for an aggregate (marginal) complier population, the ACLATE preserves covariate-level heterogeneity by averaging $\\text{CLATE}(X)$ over the covariate distribution. When CATEs or ATEs are not identifiable, the ACLATE provides the next best actionable quantity for policy decisions based on the given instrument. We will add a remark about this in our main text as it will serve as useful guidance for practitioners.\n\nIf the goal is to recover the full ATE under assumption violations, partial identification bounds (e.g., à la Manski) can be constructed (see fo e.g. [6]) though characterizing or tightening those bounds efficiently in an adaptive setting is an open and interesting direction for future work.\n\n**Re: Finite sample guarantees**\n\nTheorem 4 already gives a *finite-sample* error rate once the $O_p(\\cdot)$ notation is expanded. Concretely, the proof (Appendix B) is based on a self-normalized martingale Bernstein inequality, so for any confidence level $\\delta\\in(0,1)$ we obtain the explicit bound\n\n$$ \\big|\\widehat{\\tau}^{\\mathrm{AMRIV}}_T - \\tau\\big| \\leq  \\frac{C1\\sqrt{\\log(2/\\delta)}}{\\sqrt{T}} + C_2 \\|\\widehat{\\delta}^A_T - \\delta^A\\|_2 \\|\\widehat{\\delta}_T - \\delta\\|_2 $$\n\nwith probability $1-\\delta$, where the constants $C_1, C_2$ depend only on the boundedness parameters in Assumption 3 (i.e. they do not grow with $T$). We stated the result in $O_p$ form for brevity, but will add this high-probability corollary to the appendix so practitioners have an explicit non-asymptotic guarantee.\n\nSimilarly, Appendix B provides finite-sample variance control via our anytime-valid confidence sequences, which offer fully non-asymptotic guarantees after a brief burn-in phase: they bound the estimation error uniformly for all $t \\le T$ with $(1-\\alpha)$ coverage and width $\\tilde O\\big(\\widehat{\\Sigma}_t / \\sqrt{t}\\big)$. This yields practical, finite-sample variance control for sample sizes beyond a (small) threshold. \n\n**Re: Optimality under adaptive policies**\n\nAs per Theorem 3, our estimator does achieve the efficiency bound that was derived using a non-adaptive (i.i.d) policy for every history-dependent adaptive policy, subject to mild restrictions on the adaptive policy due to the truncation schedule: if $\\pi_t$ approaches 0 or 1 too quickly, the EIF, which contains a $1/(\\pi_t(X)\\cdot (1-\\pi_t(X)))$ term blows up; keeping $\\pi_t$ bounded away from the extremes (in line with the truncation schedule) is the only extra restriction we need.\n\nAs for the connection with Li eta al., their lower bound assumes enforceable treatments; in IV settings, only encouragement is controllable and compliance is stochastic, adding the compliance noise term that enlarges our bound. However, we will cite their work in the related-literature section and add a short paragraph explaining the connection. Your suggestion about bandit-style optimality (e.g., cumulative regret relative to the oracle $\\pi^\\star$) is well-taken. Extending Li et al.’s low-switching framework to the IV setting (and characterizing regret in presence of non-compliance) poses interesting technical challenges and could be a promising avenue for future work.\n\n**We hope our responses have addressed your key concerns and would be grateful if you would consider a higher score in light of these clarifications.**\n\n[1] V. Syrgkanis, et al. Machine learning estimation of heterogeneous treatment effects with instruments. Advances in Neural Information Processing Systems 32, 2019.\n\n[2] D. GUIDANCE. Adaptive designs for clinical trials of drugs and biologics. Center for Biologics Evaluation and Research (CBER), 2018.\n\n[3] L. Wang and E. Tchetgen Tchetgen. Bounded, efficient and multiply robust estimation of\naverage treatment effects using instrumental variables. Journal of the Royal Statistical Society\nSeries B: Statistical Methodology, 2018.\n\n[4] D. Frauen, S. Feuerriegel. Estimating individual treatment effects under unobserved confounding using binary instruments. International Conference on Learning Representations (ICLR), 2023.\n\n[5] M. Oprescu, N. Kallus. Estimating heterogeneous treatment effects by combining weak instruments and observational data. Advances in Neural Information Processing Systems 37, 2024.\n\n[6] A. W. Levis, M. Bonvini, Z. Zeng, L. Keele, and E. H. Kennedy. Covariate-assisted bounds on causal effects with instrumental variables. arXiv preprint arXiv:2301.12106, 2023."}, {"review_id": "zNIBtZRMK2", "reviewer": "Reviewer_Btdo", "summary": "A new estimator AMRIV is introduced for adaptive experiments in noncompliance scenarios. The paper derives the optimal assignment policy by minimizing estimator variance, and proposes an efficient estimator using influence functions. The method supports valid sequential inference and outperforms some baselines in simulations.", "questions": "There is only one minor question from me. The (R)EIF estimator in Equation 9 kind of comes out of nowhere and I cannot find any explanation or reference. Could the authors briefly describle the motivation of eq 9, or point to the reference if it was part of previos work on nuisance function estimation?", "limitations": "The authors have discussed their limitations and broader impacts in Appendix.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "This paper does a very good job in providing the motivation for adaptive experimentation and non-compliance. AE is a growing topic in causal inference and experimentation and non-compliance is very common in real life. The problem statement is very clear and the connection/contribution to related work is also articulated. A semiparametric efficiency bound and optimal policy is derived on optimal instrument assignment, and the toy example in Figure 1 illustrates how the optimal policy reacts under different compliance scores. Although the proposed AMRIV integrates recent advances of adaptive nuisance estimation, sample splitting, policy truncation and time-uniform inference and hence might be considered incremental, the paper provides a thorough theoretical analysis that is not in previous work. The experimental evaluation is also strong, two studies involving both synthetic and semi-synthetic data and many  non-adaptive and oracle baselines.\n\nFor (minor) weakness, see questions below.", "was_revised": true, "final_justification": "I recommend acceptance.", "rebuttal": "**Strengths**\n\nThank you for your positive and insightful review. We're glad you found the motivation, relevance, and theoretical contributions of our method compelling, and we appreciate your recognition of the strength of our empirical results. We offer further clarification below.\n\n**Re: EIF derivation**\n\nThank you for pointing this out. The (R)EIF expression in Equation 9 is the adaptive analogue of the static expression in Equation 2 (from [1]): it incorporates a history-dependent policy and data-dependent nuisance estimates, each fit on past data prior to the current policy update. This adaptive formulation is essential for enabling valid inference in the adaptive setting. We agree this connection should have been made more explicit and will clarify it in the final version.\n\n[1] L. Wang and E. Tchetgen Tchetgen. Bounded, efficient and multiply robust estimation of\naverage treatment effects using instrumental variables. Journal of the Royal Statistical Society\nSeries B: Statistical Methodology, 80(3):531–550, 2018.472"}, {"review_id": "LWHm29H82c", "reviewer": "Reviewer_KT2q", "summary": "This paper addresses the problem of estimating the Average Treatment Effect (ATE) in a sequential experiment where treatment assignment is endogenous. To tackle this, the authors propose a covariate-dependent instrument assignment strategy using historical data. The derived efficient influence function leads to a doubly robust ATE estimator, which enhances estimation robustness. Theoretical guarantees, including asymptotic normality and consistency rates, are established for the proposed method.", "questions": "- The parameter $\\epsilon$ in Equation (6) is used in both the simulation studies and real data analysis. How sensitive is the proposed method to the choice of $\\epsilon$? \n    \n- How does the method perform with different choices or dynamics of the variance term $k_t$ in Equation (7)? \n\n- For the variance estimation step, would it be possible to use a simple empirical variance estimator that avoids producing negative estimates?", "limitations": "See Weaknesses and Questions sections for limitations.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n- The paper provides a well-defined and motivated problem statement.\n- The authors conduct a thorough investigation, integrating estimation theory with experimental design considerations. \n- The proposed covariate-dependent instrument assignment and doubly robust estimator are rigorously derived, with theoretical guarantees.\n\nWeaknesses\n- The literature review in the main text is inadequate, particularly concerning adaptive experimental design in sequential settings. \n- The paper does not clearly articulate how it addresses or advances prior work. As currently presented, the main results appear to be derivable easily from established literature, raising questions about the incremental contribution. \n- The simulation study does not explore important practical scenarios, such as the presence of always-takers and never-takers. \n- The estimation procedure assumes access to the complete historical data up to time $t$, which may not be realistic in certain applications. It would be valuable to discuss whether and how the method could be adapted to scenarios where only summary statistics of the historical data are available", "was_revised": true, "final_justification": "My concerns have been addressed. I have raised my score.", "rebuttal": "**Strengths**\n\nThank you for your careful review.  We appreciate your recognition of the well-motivated problem, theoretical rigor, and integration of estimation with adaptive design. We address your concerns below and hope these clarifications will prompt a reconsideration of your score.\n\n**Re: Literature review**\n\nThank you for the feedback. Due to space constraints, we included an extended literature review in Appendix A (2+ pages) that covers core related works on adaptive experimental design in sequential settings, as well as additional related context. If there are additional papers you believe should be discussed, we would be happy to incorporate them; please feel free to suggest them.\n\n**Re: Contribution**\n\nWe appreciate this concern and clarify that, while our estimator builds on components from prior literature (e.g., influence functions, sample-splitting, adaptive policies, and IV treatment effect estimation), the integration of these elements in the adaptive IV setting is both non-trivial and novel. Specifically, we derive a new semiparametric efficiency bound tailored to adaptive IV designs, propose a variance-optimal instrument assignment strategy, develop a multiply robust estimator suited for adaptive data, and construct anytime-valid confidence sequences for sequential inference. To our knowledge, no prior work provides this unified framework or the associated theoretical guarantees in the IV context. We will revise the introduction to make this contribution more explicit.\n\n**Re: Simulation study**\n\nThank you for the suggestion. To clarify, the fully synthetic simulation adopts a one-sided non-compliance design (also used in our illustrative example) where units not encouraged (i.e., $Z=0$) do not have access to treatment. This setup, which rules out always-takers, reflects common scenarios like vaccine eligibility, product rollouts, or behavioral nudges. Importantly, never-takers are still present, as some units never comply even when encouraged. The semi-synthetic TripAdvisor study, based on real experimental data, includes the full range of compliance behaviors, including both always-takers and never-takers. Together, these settings cover a broad range of practical IV scenarios and demonstrate the utility of our variance-minimizing adaptive strategy. We will add clarifying details on the compliance structure of each data generating process in the final version.\n\n**Re: Historical data**\n\nThank you for raising this. Our theoretical development assumes the learner can *re-estimate* nuisance functions at each update, but it does not require storing the entire raw history. Thus, the method can be implemented using summary statistics or streaming updates such as: 1)  *Online learners* (e.g., SGD, online RFs, GLMs) can update nuisance functions incrementally without storing raw history; 2) *Block sample-splitting* allows reuse of recent batches to fit nuisances and then discard past data; 3) *Sufficient statistics* suffice for many parametric or binned regressors such that the needed quantities reduce to running counts and sums (e.g., of $Y$, $A$, and $X$), which are $O(1)$ per covariate cell. We will clarify in the final version that full data storage is not required, and discuss these practical alternatives.\n\n**Re: Formatting Concerns**\n\n* Definition of non-compliance: You're right that we overlooked defining “noncomplier” clearly in the main text and illustrative example. We will revise the example and include a more precise explanation of compliance behavior\n\n* Motivating scenarios: Agreed - adding real-world examples will strengthen the introduction. We will include scenarios such as: 1) TripAdvisor-style experiments [1], where randomized sign-up flows influence membership decisions but treatment uptake is voluntary; 2) Clinical trials with non-enforceable treatments [2], where encouragement is randomized but compliance is not guaranteed; 3) Other opt-in settings like streaming recommendations, fitness nudges, and advertising. These examples reflect the practical relevance of adaptive IV design in settings with noncompliance.\n\n* Title suggestion: Thanks for the suggestion, we’ll take it under consideration when finalizing the title.\n\n**We appreciate your thoughtful feedback and hope the clarifications provided show the value of our contributions and justify a higher score.**\n\n[1] V. Syrgkanis, et al. Machine learning estimation of heterogeneous treatment effects with instruments. Advances in Neural Information Processing Systems 32, 2019.\n\n[2] D. GUIDANCE. Adaptive designs for clinical trials of drugs and biologics. Center for Biologics Evaluation and Research (CBER), 2018."}]

Author rebuttal:

Meta-review: This paper introduces AMRIV, a multiply robust estimator and adaptive assignment strategy for estimating ATE in non-compliance settings, together with a semiparametric efficiency bound and anytime-valid inference guarantees. The work is technically solid, offering a rigorous theoretical contribution and empirical validation on synthetic and semi-synthetic data.

Reviewers were overall positive. One reviewer noted that the results are not surprising or groundbreaking. However, NeurIPS explicitly values rigor and quality contributions rather than only surprising results, and this submission exemplifies a careful, well-executed piece of work that advances adaptive experimentation in IV settings. 

Final Recommendation: Accept
Decision: Accept (poster)