Statistical Inference under Performativity
Authors: Xiang Li
Yunai Li
Huiying Zhong
Lihua Lei
Zhun Deng
Venue: NeurIPS 2025 poster ()
Keywords: statistical inference
performative prediction
Primary area: theory

Abstract: Performativity of predictions refers to the phenomenon where prediction-informed decisions influence the very targets they aim to predict—a dynamic commonly observed in policy-making, social sciences, and economics. In this paper, we initiate an end-to-end framework of statistical inference under performativity. Our contributions are twofold. First, we establish a central limit theorem for estimation and inference in the performative setting, enabling standard inferential tasks such as constructing confidence intervals and conducting hypothesis tests in policy-making contexts. Second, we leverage this central limit theorem to study prediction-powered inference (PPI) under performativity. This approach yields more precise estimates and tighter confidence regions for the model parameters (i.e., policies) of interest in performative prediction. We validate the effectiveness of our framework through numerical experiments. To the best of our knowledge, this is the first work to establish a complete statistical inference under performativity, introducing new challenges and inference settings that we believe will provide substantial value to policy-making, statistics, and machine learning.

Statistical Inference under Performativity
Xiang Li" Independent Researcher riangli0033@@maill.. com
Yunai Li" Northwestern University
Huiying Zhong" MIT zhong826@mit.edh edu
Lihua Lei Stanford University 1 @eteleee@staaorodee edu
Zhun Deng UNC at Chapel Hill
zhundeng@cs unc. edu
Abstract
Performativity of predictions refers to the phenomenon where reediction-ifformer decisions influence the very targets they aim to predict-a dynamic commonly observed in policy-making, social sciences, and economics. In this paper, we initiate an end-to-end framework of statistical inference under performativity. Our contributions twofold. First, we establish central limit theorem for estimation and inference in the performative setting, enabling standard inferential tasks such as constructing confidence intervals and conducting hypothesis tests in policy-making contexts. Second, we leverage this central limit theorem to study prediction- powered inference (PPI) under performativity. This approach yields more precise estimates and tighter confidence regions for the model parameters (i.e., policies) interest in performative prediction. We validate the effectiveness of our framework through numerical experiments. To the best of our knowledge, this is the first work establish a complete statistical inference under performativity, introducing new challenges and inference settings that we believe will provide substantial value to policy-making statistics, and machine learning.
Introduction
mrcdiction-informnn decisions ubiquitous in nearly all areas and play important roles in daily lives. An important and commonly observed phenomenon is that mrddiction-infommn decisions can impact the targets they aim to predict, which is called performativity of predictions. For instance, policies about loans based on default risk prediction can alter consumption habits of the population that will further have impact their ability to pay off their loans.
To characterize performativity of predictions, a rich line of work on performative prediction have been formalizing and investigating this idea that predictive models used to support decisions can impact the data-generating process. Mathematically, given a parameterized loss function the aim of performative prediction is to optimize the performative risk:
PR(0) := E
where = (x, X' x y is the input and output pair drawn from distribution D(0) that is dependent on the loss parameter 0. Typically, D(0) is unknown and the optimization objective PR(#) can be non-convex even if E(2; 0) is convex in f. Thus, finding performative optimal point 8po E arg min, PR(@) (there might exist multiple optimizers due to non-convexity) can be theoretically intractable unless we impose very strong distributional assumptions 20) As an
for equal contribution, listed in alphabetical order.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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alternative, (25] mainly study how to obtain performative stable point, which satisfies the following relationship:
Bps = arg min E=-)())))
The performative stable point could be proved unique under some regularity conditions, and it could be shown close to a performative optimal point when the distribution shift between different f's is not too dramatic, which makes it good proxy Opo. In partieular, Ops could be considered good proxy to the Stackelberg equilibrium in the strategic classification setting, Moreover, it could be calculated in distribution-agnosiie settings.
Previous work mainly focuses on prediction performance and convergence rate analysis for performa- tive prediction. On the contrary, another important aspect, inference under performativity, cludes the literature. Although a central limit theorem was established in [7] for the stochastic gradient update algorithm with single sample under performativity (a setting that is often impractical in applications such as any data-driven methods for estimating the covariance of the various quantities appearing in their central limit theorem. Therefore, 7 did not provide a complete statistical inference framework under performativity.
the authors assumed all structural knowledge to be given and did not provide
However; inference extremely important performative prediction because parameter in many scenarios represents concrete policy, such tax rate credit score cutoff. Thus, when it comes to policy-making, the aim of tackling (R(0) is not just for prediction, but more for obtaining good policy, As a result, knowing convergence to Ops is not enough, we need to build statistical inference for Ops so as to enable people to report additional critical information like confidence or conduct necessary hypothesis testing.
Our contributions. In light of the importance of building statistical inference under performativity, in this work, we build a framework including the following elements.
(I). As our first contribution, we investigate widely applied iterative algorithm to calculate Ops, i.e., repeated risk minimization (RRM) (see details in Section and establish central limit theorems for the 0, 's obtained in the RRM process towards Ops. Based on that, we are able to obtain the confidence region for Ops. Our results could be viewed as generalizing standard statistical inference from a static setting to a dynamic setting.
(2). As our second contribution, we further leverage the derived central limit theorems to investigate prediction-powee powered inference (PPI), another recently popular topic in modern statistical learning, under performativity. Our results generalize previous work [T] to a dynamic performative setting. This enables us to obtain better estimation and inference for the RRM process and Gps- More importantly, our results could also help mitigate data scarcity issues in getting feedback about policy implementation that often conducted by doing surveys that frequently encounter non-responses [13]. Thus, we also contribute to generalizing the line of work on performative prediction by introducing a more data efficient algorithm.
To sum up, our work establishes the first end-to-end framework for statistical inference under per- formativity for the celebrated repeated risk minimization algorithm. Mcanwhile, we introduce inference under performativiyy to enable a more efficient inference. We believe our work would inspire new interesting topics and bring up new challenges to both areas of perfora- mative prediction and prediction -oweeed inference, as well as add significant value to policy-making in broad range of areas such as social science and economics.
ceddcction-pweett
1.1 Related Work
Performative prediction. Performativity describes the phenomenon whereby predictions influ- ence the outcomes they aim to predict. (25] were the first to formalize performative predic- tion in the supervised-learning
setting; their work, along with the majority of subsequent papers 24 26 29) have been focused on performative stability and proposed algorithms for learning performative stable parameters. On the other hand, performative optimality requires much stricter conditions (e.g. distributional assumptions to ensure the convexity of PR()) than performative stability, a few papers address the problem of finding performative optimal parameters. 20 introduce two-stage method that learns a distribution map to locate the performative optimal parameter.
9 19 21
[17] study performative optimality in outcome-only performative settings. Finally, (I]

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provide a comprehensive overview of learning algorithms, optimization methods. and applications for performative prediction. Unlike these prior works on performative prediction, which focus on prediction accuracy, our work is dedicated to constructing powerful and statistically valid inference procedures under the performative framework.
Prediction-power powered inference. [2] first introduced the prediction- ppweeed inference (PPI) frame- work, which leverages black-box machine learning models to construct valid confidence intervals (CIs) for statistical quantities, Since then, PPI has been extended and applied in various settings. Closely related our strategies, ( propose PPI++, a more computationally efficient procedure that enhances predictability by accommodating a wider range of models on unlabeled data, while guaranteeing performance (e.g. CI width) no worse than that of classical inference methods. Other ex- tensions include Stratified PPI [10), which incorporates simple data stratification strategies into basic PPI estimates; Cross PPI [33 which obtains confidence intervals with significantly lower variability by including model training: Bayesian PPI [12 and FAB-PPI [ which propose frameworks for PP] based on Bayesian inference. PPI also connected topics such as cmi-paaraettiic inference and missing-data imputation
28 30 Our work is the first one to study PPI under performativity, and we validate the PPI framework in the performative setting both theoretically and empirically.
8
27
Inference in performativity. [5) studies identifiability and estimation error under a specific micro- foundation model with performativity. Yet it doesn't address confidence interval construction. A closely related work is 77 which also establishes the asymptotic normality and minimax optimality for performative settings. focuses on stochastic gradient update with one sample per iteration, whereas our work analyzes the empirical risk minimizer on batch updates. Moreover, 7 does not provide a data-driven approach for covariance estimation and thus lacks an end-to-end inference framework. In contrast, our work explicitly handles density estimation and provides an end-to-end inference method for constructing confidence intervals, which is missing in existing literature, to the best of our knowledge.
2 Background
In this section, we recap more detailed background knowledge about performative prediction and prediction-powered inference.
Repeated risk minimization. Recall that the main objective of interest is the performative stable point, which satisfies the following relationship:
Ops - arg min E,wD( ((;
Repeated risk minimization (RRM) simple algorithm that can efficiently find Ops. Specifically, one starts with arbitrary Og and repeat the following procedure:
O++1 arg min E,-:(e,)((=;0))
0
for E N. Under some regularity conditions, the above update is well-defined and provably converges to unique Ops at linear rate.
Theorem 2.1 (Informal, adopted from (25).. If the loss is smooth, strongly corvex, and the mapping D(-) satisfies certain Lipchitz conditions, then Ops is uniquely defined and repeated risk minimization converges to Ops linear rate.
We will further explicitly state those conditions in Section 3
Throughout the paper, we will mainly focus on building an inference framework under the repeated risk minimization algorithm.
condicttoo-powered inference. A rich line of work on prediction-pon predectionppowered inference (PPI) 1 considers how to combine limited gold-standard labeled data with abundant unlabeled data to obtain more efficient estimation and construct tighter confidence regions for some unknown parameters. Specifically, a general predictive setting considered in which each instance has an input x E X' and an associated observation {zr, ) that are i.i.d. drawn from a distribution D. Meanwhile, we have abundant unlabeled data {x"") Ri=1 that are i.i.d. drawn from the same marginal distribution as standaddd labeled data, i.e. Dx. where N > In addition, annotating model f X ++ (possibly off-the-shelf and black-
Y People have access limited set of cold-standard labeled data
3

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box machine learning models) is used to label data In M. the authors show that for a convex loss with unique solution, compared with standard M-estimator @ arg ming Di=1 ( yi: 0)/n,
@ (A) = arg min À
1
Σ
N
(x,,, (xx): + Σ (o )6:0)
_ a
Σ (x.. f(z)::)))
can be a better estimator of 0 = arg ming (2; 0) via appropriately chosen based on data.
Notation. For K E N+, we use [K] to denote 21, KK. We use 2 and 2 to denote convergence in probability and in distribution, respectively. For two set S and S", we use S + S' denote the set {s + s s E S, E S'). Wu, Σ) denotes Gaussian distribution with mean μ and covariance matrix Σ. Lastly, we denote k-dimensional identity matrix as Ig. We use ((c, r) to denote ball with center e and radius r. Lastly, we use || for fo-norm and denote a column vector with all coordinates 1.
3 Inference under Performativity
In this section, we initiate the inference framework for repeated risk minimization in the batch setting under performativity. We mainly consider the repeated risk minimization setting, Unlike standard inference problems, where estimators are built for fixed underlying data distribution, in our dynamic setting specified below, building asymptotic results such as CLT imposes extra challenges and this has not been covered by any existing literature so far.
Specifically, at time t = O, we have access to set of labeled data = + that is i.i.d. drawn from a distribution D(O)), where =0,; = ( Vo.s) and Og chosen by us. Then, we use the empirical repeated risk minimization to output
0, = arg min
no Σ (zzo,;
as an estimator of 0₃ = arg ming (0, (z; 0). Then, for t > 1, at time t, we further have access to set of labeled data { that are i.i.d. drawn from the distribution induced by last iteration, i... D(O)). Let us further define G((0) arg ming E EEND( (2; (). Then, we can obtain the output
0 = arg min Σ ((2)1)))
as an estimator of 02+1 G(Oz). This iterative process will incur two trajectories, i.e., (1) underlying trajectory: 0, + 01 ... ...;; (2) trajectory in practice: O, 02 ... -
Our aim is to provide inference on O, for any > 1. For simplicity, we let for all t.
3.1 Central Limit Theorem of 9
In order to build CLT for 0z, we first establish the consistency of 0, which is relatively straightforward given that (25) has built the non-asymptotic convergence results. Then, introduce main result on building CLT for 0g. Lastly, we provide novel method to estimate the variance of O,.
Consistency of 0.. We start with proving the consistency of 0.. Recall that have trajectory induced by the samples fn + 02 + ... - by the iterative algorithm deployed. Without consistency, CLT is not expected to hold. Our results are based on the following assumptions.
Assumption 3.1. Assume the loss function l satisfies:
(a). (Local Lipschitzness) Loss function ()=;)) is locally Lipschitz: for each 0, there exist neighbor- hood (0) of such that {(;; 0) is L(2) Lipschitz w.r.t for all 0 (()) and Eg I(=) oc.
'The annotating function f could either be a stochastic or deterministic function. It could even take other inputs besides but for simplicity, only consider the annotation with the form fx).

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(b). (Joint Smoothness) Loss function {(2; 0) is 8-jointly smooth in both 2 and 8:
[V)((2;0)- - Vol(2; ll2 < 3 ||0 12 (Vl((z; - vol(z';0)1. & 8112 - 1 for any z, E Z and 0, 0' 6.
(c). (Strong Convexity) Loss function {(2; 0) is y-strongly convex w.r.t 0: 0) > ((z;0') + Fol(2;)') - 0¹) + 7 100 01(
for any = E and 0,0'.
(d). (e-Sensitivitt) The distribution map D(0) is s-sensitive, i.e.:
W1 (D(0), D())) < el10 - 8'1l,
for any 0, 8', where W1 is the Wasserstein- I distance.
Remark 3.2. The assumptions (b), (c), (d) follow the standard ones in 125), which are proved to be the minimal requirements for trajectory convergence. We additionally require (a) to build consistency
for 0e beyond convergence of 0g to Ops-
Proposition 3.3. Under Assumption 3.1 t [T]
if < f. then for any given T > O, we have that for all
0 P O2.
Building CLT for dz. In order to build the central limit theorem for 0. we need to introduce a few extra assumptions. Due to limited space, going forward, we defer the required assumptions in later theorems to Appendix.
Let us denote (0) = HV(0))-V((00H,(000) where H₂(0) 200) (z;; and Vi((0)
Cov sw) (2) And recall that G(Ø) arg ming E ,DD0 ((;;0)
Theorem 3.4 (Central Limit Theorem of 0t). Under Assumption 3.1 and A.1 ife < 3. then for any
given T > 0, we have that for all E [T].
v vn 02) 3 N (0, V2)
with
V1 = Σ (G(+)) (0,)
II VO(ON)
In particular, VG(02) - -Ho (0x+1) (Vo (VolE--DOo)) V Vol(z; 0k+1)), where V is taking gradient
for the parameter in D(0), Vo is taking gradient the parameter in E(2; 0) and II= 1-1 VG(O,) = La-
Estimation of (O.),, V, Given the established CLT for 0, in order to construct confidence regions for 0, in practice, the only thing left is to provide an estimation of (G.))) and V with samples. In previous results, with a more detailed calculation, we obtain
G(())) - He, (0x+1) (000+1)-'B.00 "EE-Do [ Vel(z, 0k+1) Ve log J.
where p(: 8) is the density function of distribution D()), and the score function Vo log p(2, 0) is thus d-dimensional vector (recall that 0 is of dimension d). In order to estimate it for any 8, we propose novel score matching method. Specifically, we use model M(z, 0; 1b) parameterized by sa to approximate p(2, 0). Inspired by the objective in [14], for any given 0 (e.g., 0,), we aim to minimize the following objective parameterized by va for all 0:
J(0; / p(z,0)]l p: - s(z, 0; 22
/ pl( f 0) (1Vo
log 0)| + || 1) 2V logp(z,0) ((,, 0())0:))))
5

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where s(2, 0; () Ve log (2, 2)) If we can learn e so that s(2,0; 0) Vo log p(2, 0) for all 8, then we reach the minimum ](0; (D) O.
Notice the first term is unrelated to yi; the second term involves the model M that is chosen by us, so we have the analytical expression of it. Thus, our key task will be estimating the third term, which involves (C; v) : p(z, Vo log p(z,(z,0 ((2,0)) 0) s(2, 0; v))2..
We remark that in our setting, instead of taking the gradient at 2, we have new challenges in taking the gradient at 8. So, we derive the following key lemma.
Lemma 3.5. Under Assumption
A.2 we have
a 2010)
/
Olog M(z, 0:
0olog 02 log M (z, 0; v)) de pl2,0)
(((0:b) Σ
p(z, 0)
dz
000
00
where g(3) is the i-th coordinate of
Based on the lemma, we propose novel gradient-free score matching method with policy pertur- bation to estimate K(0r; v) for any t E [7]. Policy perturbation is a commonly used technique in estimating the policy effect under general equilibrium shift 22 or interference [32). Instead of just getting samples for Og for each we additionally sample for all perturbed policies in (0: + ne,, 82 + "/e,,..... ,, + mea), where rj > is small scalar our choice and {ey hj are standard basis for R2d. Typically, for a policy 0, its dimension d low. One concrete example in practice is to use slightly different price strategies in different local markets.
Specifically, the term f'p(z, 0,) 68 log ((al dz could be easily estimated by using empirical mean, 8616)2
e.g.- 1 Ej=1
log (5,, ..:11 And for the derivative a
a063 , [(2 0.) 8 log ((,,0sa)) dz, if we draw (0(3)
additional & samples ( for each perturbed policy 0, mes, can use the following estimator:
k 2 log ( O, mes; v)
a log M(r, 0.s
0061
00(0)
Combining the above, we have a straightforward way to estimate G(O)) and Vv2 for any t E [2] by plugging in the empirical estimate. Let us denote the estimator of V. by V. Then, we would have
1/1 Vn(O. 0,) 2 (0,
the model M(2, A; th) is expressive enough.
Theoretical validity of estimation. Now we prove that the policy perturbation method provides valid estimator of VG((g). Recall that the gradient of G is given by
VG((2) = - Ha, (0x+1)~ -E Vo 0 V Vo Loggop 0k) J]
and the estimator defined by
gh = - HHg (6e+
V
Vol(2; O [s( oe: (0x))) J.
Theorem 3.6. Under Assumption
3.7 A.
A.9
and
A.l7
we have
Ig - VG(0,) Op(
+
+ an).
Vn
TV mimninn, k)
where a, o(1) is vanishing optimization error term defined in Assumption A.9 d).
The proof is by decomposing the error between the empirical and true gradient into several com- ponents, where each varies in whether it uses empirical or population quantities. We bound the deviations between the true parameters 0s. 0.+1 and their empirical counterparts 0z. 0x+1 us- ing Theorem 3.4 We bound the true score function and its estimation via a uniform bound of ||V log p(2 0) 8(2,0, (²))) over the perturbed policies using Theorem A.13 Taken together, the above result shows the consistency of 9A obtained by policy perturbation.

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3.2 Bias-Aware Inference for Performative Stable Point
Finally, we further provide way construct the confidence region for 8ps. This is directly followed from our previous results on building CLT for Oz. By the convergence results derived for the we have
underlying trajectory by [25], under Assumption 3.1
EB
||. - Opsll
110o Opsll-
Thus, we can immediately obtain the following corollary by using bias-aware inference a commonly
seen technique in econometrics 4 15 23)
Corollary 3.7 (Confidence region construction for Ops). Under Assumption 3.7
A.7 A.2
and A.3 if
< for any 8 (0, 1). obtain confidence region R,,,,, 8) for O₂ by using Eq.
such that
lim P 02 E Rn( = 11- n
Moreover, if Ops E {0: |00| < B}.
lim P P(Ops Re(n, 6) B) (2)) 1-8.
Corollary 3.7 provides way to construct the confidence region for the performative stable point based on the confidence region for 8t Notice that the derived new confidence region quite close to (n, 6) and the difference vanishes exponentially fast as grows. Thus, we expect the derived region be quite tight for moderately large meaning:
lim PP (POps E R.(n, 8) + B 0, 2B 1-000
(B
2 )')) = 1
For the condition G, Ops ||01 < B). it will be natural to satisfy and we can get explicit and feasible upper bound B under mild conditions. is because that 0g our choice and we can further derive explicit and feasible upper bound for ||Ops] by using the strong convexity. Specifically, by y-strong convexity of the loss function with respect to 0, we have
BB Vol(2; 0o) - ((0s)) V (p)(2;0ps) (0o - (ps)
|10 0p
Since (Ex)D(0))) V (z 0, this leads to [|EDD((0)) Vol(=; 00)1 > 110, Opsil - Thus, further have supzez || p 0,)1 B for B>0, which could be achieved and calculated by assuming Z is compact and use the continuity of Vol(-,0)). Then, it will immediately give us an upper bound for ||ps||| that |[0p||| B/r + |0ol].
Prediction-Powered Inference under Performativity
In this section, we further investigate prediction- cndliction-poeered inference (PPI) under performativity to enhance estimation and obtain improved confidence regions for the model parameter (i.c., policy) under performativity. This can also address the data scarcity issue human responses when doing survey to get feedback on policy implementation.
Specifically, at time t 0, besides the limited set of gold-standard labeled data f Vo.a == that are i.i.d. drawn from distribution D(0)), we have abundant unlabeled data ( i..d. drawn from the same marginal distribution as gold-standard labeled data, Dx (+0()), where No > no. In addition, an annotating model f X + y is used to label data D which leads to (2o.1, f200 and {28, ((x Then, we use the following mechanism to output
)= that are
X1 Σ
a (l(zu,s. No.:: X Xi f(a,)):
(PPI(A)) arg min
n
*Our theory can easily be extended to allow using different annotating function for each iteration, but for simplicity in presentation, we f for all iterations.
7

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for scalar λ1 an estimator of 01. After that, for t > 1, at time t, besides having access to the set of gold-standard labeled data ( Vt,s J that are ii.d. drawn from a distribution ((O,). Meanwhile, have abundant unlabeled data ( ) that are i.i.d. drawn from the same marginal distribution as gold-standard labeled data, i.e. (x)) where N₂ > ne. Similar before, we can estimate 0u+1 via
8.f1 (Ac+1) arg min
A411 N
Σ ( f(x)))))
f(
for scalar 1 This incurs trajectory in practice: 0, (A1) .... ......00(P(A
Notice
that if we choose A = 0 for all L E (T].th) this will degenerate to the case Section 3 Later on, we will demonstrate how to choose {>+) via data to enhance inference. Our mechanism could be adaptive to the data quality with carcfully chosen {}+}T= and could be viewed as an extension of the
classical PPI++ mechanism [25] to the setting under performativity.
Building CLT for (Pr) We start with building the central limit theorem for (Pel(Xe) with fixed
constant scalars {Ar)==1 for any E [7]. The proof similar to that of Theorem 3.4 Specifically,
we denote
Ex,0(0:)) r) H3(0)-1 (rV' 2,0
(0) + v2,)(0) H₂(0)-1
with V (0) = X² Cov (x))
and (0) = Cov (a.))))((()) (Vol(z,
oVll(z (z)) Then, we have the following theorem.
Theorem 4.1 (Central Limit Theorem of Ofr Under Assumption 3.1
A.4
and
A.5.
ife <
and N + r for some r 3 0, then for any given T z O, we have that for all 1 E
17,,
(np")))(>)) - 01) 2 N N
V. (0.V..
(1,,
with
(1A,,
II VG(&s) ((
II (G()))
Selection of parameters {Ae) Now, the only thing left is to select (Ax TT so as to enhance estimation and inference. As choosing A; = for all & E [T] will degenerate to Theorem 3.4 we expect that we appropriately choose {>e]²= to make F(V)) F(V.T) ((Ag, f (;))=);)) where F is a user-specified scalarization operator depending on different aims. For instance, if we are interested in optimizing the sum of asymptotic variance of coordinates of 6,, then, F(V) Tr(V)). Or if we interested in the inference of the sum of all coordinates 0,, i.e., 0,, then F(V)) 1 V.1.
We consider a greedy sequential selection mechanism as follows. Imagine that we have selected (A via the data, and our aim is to select a Až so as to make (V.P²) ((2), 0, 11 6r))) > X,
F (VPPl ())r))))) for any X. Thus, we choose
= arrg - F (V. (V (17,0015;110003))
However, in practice, there are still several issues that need to be addressed. First, we need to choose [Ar)T= via observations, and this could be handled by using our results in Section 3 to estimate (O)( and we obtain sample version X₂ by plugging in the estimation. Second, when obtaining X in Eq. 3 we actually need 0g Exx00:-1 (0;;r) But when using samples to estimate, we need to
get X, first before we obtain 0.rl. Thus, we propose similar optimizing strategy as inspired by M:
time > 1, given the obtained {Am) and ( Jim] we choose an arbitrary X. obtain 0.
as temporary surrogate Then, we further obtain
X arg min F ( ((,, 05, AA (A); 1)
Notice that (A) is still consistent estimator of 0,
8

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where 1.PPI is obtained via replacing VoG((t)) with their estimation in V, In particular, if F
satisfies F(U + +V- = F(U) + F(V) as our examples of F(V)) = Tr(V)) and F(V,) = 1 VV1 then
we only need to optimize F [xx0m, ( (c((A); $))
To sum up, by the above process, we have the following corollary.
Corollary 4.2, Under Assumption 3./ 1.2 A.3 A.4 andA.S
if 3 and N + r for some >
then for any given T 3 O, we have that for all [
m.r ("))) x)
-
vn
(0
(2x) 0.) 2 (o)
1.0
1.0
10-
8
0.9
0.9
0.8
1
0.7
0.7
0.7
1
(dee)
0.6
0.5
0.6
0.30
0.10
0.10
0.08
0.08
0.08
0.04
0.04
0.04
0.02
0.02
400
800
1000
200
500
1000
200
600
1030
Figure I: Confidence-region coverage (top row) and width (bottom row) with different choices of A. The left, middle, and right columns correspond to inference steps 2, = 3, and respectively. The solid and dashed curves correspond to the confidence-region coverage for 0, and 0rs, respectively.
Experiment
In this section, we further provide numerical experimental results to support our previous theory. A case study on semi-synthetic dataset is provided Section B.3
Experimental setting. We follow 20 to construct simulation studies on a performative linear regression problem. Given a parameter 0 E Rd, data are sampled from D(0) as
= a x + - e + v, x ~ N( Ez). V ~ (0,
The distribution map D(0) is e-sensitive with E = ||a||22 For unlabeled x;. the annotating model We use the ridge squared loss to
is defined as f(xx²) a x. + Ju 0 + vi, v NN(0022, measure the performance and update 0: ((xx,));0) - (y 0 0Tx'²² + / 11022 For easier calculation for smoothness parameter, we truncate the the distribution of (æ, y) in our experiment to deal with truncated normal distributions, but this
necessary in many other choices of updating rules. In
the following experiments, we set d = 2, z 0.02, 7 = 2, and oy = 0.2. We set N 2000 and vary the labeled sample size n.
Simulation results for PPI under performativity. To quantify the results of PPI under performativity, evaluate the confidence-region coverage and width for three strategies: À (only labeled data), 2₂ as defined Eq We vary the labeled sample size n and perform t E {2. 4} repeated risk minimization steps, averaging results over 1000 independent trials. In Figure [ we can find that all three methods approach 0.9 coverage as n grows, while our optimized X₂ (orange) achieves the narrowest interval width, supporting its effectiveness enhance the performative inference. The dashed curves denote the bias-adjusted confidence regions for the performative stable point 0ps. It can be observed that fps coverages upper-bound that of 6g (solid curves) across steps t, and the gap between them vanishes as I grows. This observation verifies the validity of Corollary 3.7
1 (full unlabeled data weight), and our optimization method
Verifying central limit theorem. To validate the central limit theorem, we sample different 0 (here t 4 and 1000) and visualize the distribution of 11 Vn(O, 0t). We plot the density map

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Dbserved Theoretical
05
0.
10.4
Observes quamtles
derally
5.05
0.52
-2
Theonetical quantiee
(a) Density map of observed dis- tribution.
(b) Multivariate Q-Q Plot for nor-
mality test.
Figure 2: Visualizations to verify the Central Limit Theorem. (a) plots the density map of sampled V
1/2 Vn(O, 0z). while (b) compares the observed distribution with theoretical one .N (0, Ia).
in Figure 2a and find it is close to a normal distribution. In Figure 2b we further do a normality test with the multivariate Q-Q plot of observed squared Mahalanobis distance over theoretical Chisg quantiles. The tight alignment of points along the identity line (red dashed) verifies that the observed distribution is well-approximated by asymptotic CLT.
Estimation results of score matching. We consider two implementations of the gradient-free score matching estimator (z, 0; v):
(a). Gaussian parametric: we assume p(2, 0) = (Npp Exp) and parameterize v = {Mpp Epp:
(b). DNN-based: a small deep neural network with two hidden layers of width 128.
collect 01. trajectory and corresponding data {=1=,)) to train both models via the SGD optimizer with a learning rate of 0.1 to minimize the empirical score-matching objective ((b).
In Figure we evaluate the estimation quality of models by their final training loss (()) and the estimated variance error VIV Vell over varying n and t. We sample 1000 independent trajectories of On. and report (1b) as averaged J(0; v)) over all 0 in the collection of trajectories. In all settings, both estimators achieve J((b) < 0.05, indicating the perfect approximation of our learned model M(2, 0;; 3)) to the true p(z, 0). Correspondingly, the pariance-estimation error remains negligible and decreases as grows, verifying the feasibility of using our score matching models to Vo log p(=, (z,0) for estimating VG(0s) and V.
12 VI
0.0073
0.03
t-2
t=1
0.0070
t=2
0.02
t-5
0.0069
t=5
0.005
0.03
500
1000
2000
500
1000
1500
2000
1000
2000
1000
2000
(a) Gaussian parametric model.
(b) DNN-hased model.
Figure 3: Evaluating the estimation quality of two designed score matching models.
6
Conclusion
In this paper, we introduce an important topic: statistical inference under performativity. We derive results asymptotic distributions for widely used iterative process for updating parameters in performative prediction. We further leverage and extend medietion-pooeeeed inference to the dynamic setting under performativity. Currently, our framework uses bias-awared inference for Ops. Obtaining direct inference methods will be of future interest. Our work can serve as an important tool and guideline for policy-making wide range of areas such as social science and economics.
7
Acknowledgements
We would like acknowledge the helpful discussion and suggestions from Tijana Zrine, Juan Perdomo, Anastasios Angelopoulos, and Steven Wu.
10

---

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12

---

A Theoretical Details
We will include omitted technical details in the main context. We first summarize all the required additional assumptions in Section A.I Then, we provide omitted proof's for Section 3 in Section A.2
and omitted proofs for Section Section A.4
A.1 Assumptions
In this subsection, we summarize all the additional assumptions we will use to build various theoretical results in this paper. Before state the assumptions, need some further notations.
Let us denote (0) as the collection of minimizers of J p(2, V)llV log p(z, 0) s(2, 0: 4/) [[2 de for the given O. We further denote the empirical estimation function for the two terms, i.e.,
/ p(z, 8) (ls((2, 8; -2V (og.(2. O)T s22,0; ())) dz.
as & (10; 8) following the methods in Section 3.2 for any 0 in the trajectory (0,)T- where n and are the number of samples we get at each iteration for O, and perturbed policies.
Meaning of each assumption. Assumption A. is used to establish the asymptotic normality of estimator under performativity. Additionally, use the fact that the population loss Hessians (0) = E (2; 8) are positive definite, which is guaranteed by the strong convexity of the loss function (Assumption 3.1). Assumption A.2 and A.3 are used in the analysis of score matching. Assumption A.2 based the differentiation lemma 18], ensures the interchangeability of integration and differentiation. Assumption A.3 guarantees the consistency of the score matching estimator. Assumption A.4 and A.5 parallel Assumption (.1 aa) A and used establish the consistency and asymptotic normality of our PPI estimator under performativity. Conditions
such local Lipschitz continuity and positive definiteness standard for establishing asymptotic normality. Similar assumptions are also imposed in [l]
Assumption A.1 (Positive Definiteness & Regularity Conditions for the Estimator). We assume the following.
(a). The loss function satisfies the gradient covariance matrix is uniformly bounded below:
(0) = Cov (01)) (ol(2;0)) = cl,
for any where e >> 0 is a constant.
(b). For any sample size n. assume the M-estimator 0, has a density function with respect to the Lebesgue measure, and its characteristic function is absolutely integrable.
Assumption A.2 (Regularity Condition for M). Assume that for Vi:
i log M (z,0;
(a). The function = + p(z,0 0)
Lebesgue integrable.
a0()
(b). For almost every E Z (with respect to Lebesgue measure), the partial derivative
a
2 log M (2, ()
p(z, 0)
3053
O0(0
exists.
(c). There exists a Lebesgue babbsguu-iintggrbble function (2) such that for almost every & Z,
θ log M(,, 0: db) #t] H()).
p(z,0)
a
2000
Assumption A.3 (Consistency of Optimizer). We let k grows along with such that - oo leads to k + 30. We assume that the class M (z, 0: vb) is rich enough that for all 0 θ there exists si (0) such that M 0; o0 (0)) = p(z, 0). Moreover, for the underlying trajectory {0, [0))
lim arg min J. (1b; 0.) (0.))
20+30
13

---

Assumption A.4 (Local Lipschitzness with f). Loss function {(z, f());;) is locally Lipschitz: for each 0 E O there exist neighborhood T()) of 0 such that ((z, f(x); is L (xx) Lipschitz w.r.t for all 0 T()) and (z) L' (x) oc.
Assumption A.5 (Positive Definiteness with f & Regularity Conditions for the PPI Estimator). We assume the following.
(a). Assume the loss function satisfies the the gradient covariance matrices are uniformly bounded below:
(0) = Cov Cov==D(ū) (vo((z))) = ct, (0) Cov (D) (Voll,, (o = ct,
for any 0, 0, where > 0 is a constant.
(b). For any sample size assume 0 has density function with respect to the Lebesgue measure, and its characteristic function is absolutely integrable.
A.2 Details of Section 3
Theory of Inference under Performativity
We provide the omitted details in Section 3
A.2.1
Consistency and Central Limit Theorem of o
Let us denote:
Lg(0) = F 20) (2;0)) (0) _ Σ ((z; 0):
where the samples 2; (xi, 11k) D(0) are drawn from the distribution under 0.
Proposition A.6 (Consistency of be. Restatement of Proposition 3.3). Under Assumption 3.1 if
E < 3. then for any given T O, we have that for all e [T].
0, P O2.
Proof. Let us denote G())= argmingeee 1. - (zz; 0') where the samples z; D(0) are drawn for some parameter 9 along the dynamic trajectory 0g - 02 +
...0. 0g +
|10 0.1 |[G(O:-1) ((,,-)))
||G(0,-)) -(0,- [[G(O,-)) G(O,-1)ll
||G(0,-1) -G(00-)) + _ 11:0111 0-11l
where the last inequality follows from the results derived by 25 under Assumption 3.1 we have
||G(0) - G & #0 - 8'|l-
Notice that [ E(L E( (0)) = L (0). By local Lipschitz condition, there exists 6p > 0 such that
1Cg (0) Lū, (0) 5.
sup
8: 0:10-GG66.111 LeD
Since is strongly convex for any 0, G(Ø,- 1) is unique. Then we know that there exists such that C6.1..10 (0)
(G(O-_1) for all in 10 1|0 G(θ,-1))l = so}. Then it follows that:
inf fa. (0) Lz. mm (G(O,-))
||0 | 1)11=50 (LCr Lū (8)) + (L (0) Lū, (G(O,_)))))))
inf 110- -0-0
+ (Lz__, (G(0)_z)) - L (G(0,_1))))
14

---

>6- 00-o
Then we consider any fixed such that 10 - G(0(-1)ll > Eo it follows that
G(0,-1)
(0) (G)
(Ca. (w) - Co (G(Or-1))
w G(0,-1)
|0 G(Oc-1))l (6 op(()) (8-ooll) > op(1),
E0
where the first inequality holds for any by the convexity condition of L. in (0), and the second inequality holds as we take w = 0-G(0,-1)
| [e-G(O._..) co G(O,-1) and using the above result. Thus no f such
that ||0 G(0,-1))l = can be the minimizer of Lū 1. (0). Then [[G(0,-1) G(0,-1)) 4 0.
We then have, for given T > O, we have that for all t [T].
100 Bell
p ( - - 4
Thus, we conclude that 02 Eg Or.
Theorem A.7 (Central Limit Theorem of 0z, Restatement of Theorem 3.4 Under Assumption 3.1 A.7 i A.
3. then for any given T > O, we have that for all t [T],
Vn(O. 0.) N (1)
with
h-1
Σ VG(Ok) I0,,
II
(G(OO))
In particular, VG()))) - -Ho (0x+1)- (VoE.--o Vol(z; 0x+1)). where V is taking gradient
for the parameter in D(0), V is taking gradient for the parameter in ((260) and I (G(()) = Ia-
Proof. Let U2 == vn 0z) and denote 0, = G(O)-1). We make the following decomposition:
0.-0, 0, (0.
0.)
(0,
0.)
(1) (2)
Step 1: Conditional distribution of U.!U,-1-
For term (1), we have
Vā 0,) = - G(((-))))
For term (2), the empirical process analysis in 1 establishes that
θ, 2 N, [0,, (0.)) Va - 0,)
where the variance is given by
(8.) Eg (8)) H (8,) -V, (0.) (0.)-1
Conditioning 0,- and considering the distribution D(0,1)) for any function h, we use the following shorthand notations:
E,h: :- Mar, v:), Gnhh == (Ti (L
7
15

---

Note that 8 = G() Recall that
n (lxs, Yei 0), where (ri, D(0).
Cg( =
(x.)))~D(0) faz, 0: Lgn
I
Under the assumptions, Lemma 19.31 in [11 implies that for every sequence hn = Op(1), we have
Gn vn (1
hn kn Vn
l(x, (ar.y:00)) 01) ((..
11,2 Vol(x, - 0.
Applying second-order Taylor expansion, obtain that
ha ) lxx, ((x,V;
eon((o) (C. (0) +
hn v Lūz (81)
nEn [1z, y; 01
vnl
vn
+ h Gn Vol (xi yr 0c) Op
2 n Ho. (0e hn + h Gr Vol(x, +
Set 11, - Vn(0,-00) and hn -HH (0.) Vol(a, yi 0.), Corollary 5.53 in 31 implies they are Op(1).
Since 0, is the minimizer of n. the first term is smaller than the second term. We can rearrange the terms and obtain:
2 hh - hn)) 2 (0) - hh,) op()).
which leads to hr - An (p()). Then the above asymptotic normality result follows directly by applying the central limit theorem (CLT) to the following terms, conditioning 0,-1:
vn 8.) 0 = - -Ho (0,)) - Op()),
S
VollIe, VollI1e,s 0c)
E (I.))~D(:: 1) Vol( y: v.1)).
Note that, conditioning on 0.-1. (1) is a constant. Therefore, (1) and (2) follow a joint Gaussian distribution. Consequently, given U the conditional distribution of U, is given by:
Up U--1 = vn(0, - 0() 0-1
Vv((. 0,) + Vn(0, 0.) 0:-1
= Vn(G(0-_1) G(0,-1)) + vn(0. - 0.) 0--1
SN N (Vn(C(()))) - G((,- Eú. (0.)
+ 0 - G(0,- Dee =N (vviia U Vn
UL-1 +
0 01-1))
(G
v +tll
Vn
For later references, we denote U, U,-1 Dg (,
Step 2: Marginal distribution of U.. We calculate the characteristic function of U, by induction. To begin with, we directly have
X, D A(O,V))) V Eo,
Now, assume that U D, (O, V2-1), we derive the joint distribution of (Ut, U,-1) and marginal distribution of U.. Then we have, the characteristics functions & and the probability density function p of distributions U.-1 and U, U follow:
(s) + (v(o)...)(s)) exp( exp(-1s -V (w) = (2m)d /
e it --t"" "du,,
16

---

(U,(U,_1(6) - (N(U(U (1)( = exp(is" p(U(-1) exp(is=((U<_1) ({{(U_))s)
2
Then have
oU, (s) Ee =B =E((((' ((-1)) 0u, (s(1)-1)
0U.UU,, (slu)pp pes (w) du
(81
/ (ouile (s| uu) (2T)d / c 1o.
1
ourn, (2) d2 du
1
// Quilv. (s u) (1) e dz du
(2
1
// explis 1(U--1)- - -3=""((U--1ss) exp(-2² =T=T1 V1-12)
(V-12) e -iz"
dz du
(2
1
exp(is) ((U(-1) - (((((-))s) (f exp(( = 2 iz u) dz) du
(2T)a
1
exp(is ((U(_1) 1 {(U)-)ss)
(2T)
(ʃ /exp(- - Vi=le)) exp( (2- vi (iniu)/ViVa (1-- dz dz)du
1
/ exp(is ((U--1) - ((U1-1)))((2))
det [V-11 exp( (xp(--5" --- Vv
1
du
(2x)d
1
/ exxpiis m(U,-1) (xUp(s""(U,--1> 8 E(U, " Vi-1u)du.
(2T) det [V-1l
2
Apply dominant convergence theorem li (s), we have:
/ exp(is" p(U(-1) 1 2/² E(U--1))
1 =u"Vi-uudda
lim ou, (s) n-000
lim n00 (2T) 2 det 1/-1l
2
n-000 (2T) 9 det Vi-1 / expliss VTlGG
UL-1
= lim
+ + G((--1))
Vn
1 T Ev, + (G U1-1 + 0c-1))s
1
VI-iu)du
2
Vn
2
U--1
lim exp(is" (n( n-000
+ +.-11) (1-1)-G(O--1))
(2r) det V/-21
Vn
1 T "EE +0,- (G(
U11 + ((-1))s 1 u
Vi- Vi-uu
2
Vn
2
1
(2T)* 4 det [V_11| explis VV
//2
Ee. (G((0-1))s /" TuTvo Vi-uuddu
= =exp(-> 2. x ([G(0-))+++VG(0,
5 -E,,, (0t)s),
1
2
which is the characteristic function of (0, Vi), where V2 VC(0--)VV-IV((0---)) +Lo., (0.).
Here the fact that lim,- vnl GG( y + 02-1) G(B1-1)) VG((,-1)9. and the domi-
nant convergence theorem holds as we have
UL1 + +0-1) - G(B,-1))
eex((is Exp(isTVVa(G(
Vā(G(
Vn
1 5 Ee 10,
UL-1 (0 Vn Vn
(al
+ +(-1)))
1 u" V--1u) & <lexp(>>
1 -T VV))
2"
Thus we conclude by induction that
U, 2 N Vi).
17

---

V4 Σ
vo((s)
Ea:
(0,)) k=
II
And by the theorem of implicit function, we can calculate the gradient of G as follows
VG(0)
2
E (..))~D(0)
((x, y: +)) 3))
(V
V (D.))~D())
E
((x, v: 2i)) a())
VG((0) = (an))
E
(V
E ((())
Vol(x,
Ho, (0x+1)
(V
R
(z.0))D((.)
Hep "eop""""_LL,,,,V/((001111
A.2.2 Score Matching
In this part, we provide details about score matching mechanism.
Given that
G(())) He, (0s+1) " Vo loggo,,, 0,)
once we have good estimation of Vo logp(z, for all 2, (G(+,) could be easily estimated by samples.
Recall that we use a model M (z, 0; (b) parameterized by & to approximate p(z, 0). Inspired by the objective in [1], for any given 0 (e.g.. 02), we aim to optimize the following objective parameterized by v
J(0: w) / p(z, (22 log 0) - s(2, 0: 2d
/ p(z, 0) Vo log ((2 12 + [|s(2, A: +) 2V logp(z,0) s(,,)))) dz
where s(2, () = log M 0; v²
As mentioned in the main context, the first term is unrelated s/; the second term involves model M that is chosen by us, so we have the analytical expression of s(2, 0; vb). Thus, our key task will be estimating the third term, which involves K0(0;) v) = p(z 0) V 9 log p(z, 0) s(2, 0; v)22.
Lemma A.8 (Restatement of Lemma 3.5
Under Assumption A.2 we have
i 200) p(2,6) 0) log M(, 0,v))
/
02 log M ( 0; 0; de p(2,0)
K((:: +b) Σ
dz 00(1)2
o0(1)
where ((8) the i-th coordinate of
Proof. Recall that 0 is of d-dimension.
I o(2,0)Vo logpo,, s(2,,, 0(2,0:0))d =
Σ
p(2,0) 0) Olog Ologp(z,0) 0)
i log M(z, 0; vt)
dz
ag(()
a0(1)
2 Olggp(z, 0) d log ( 0; 0;4)
p(z, 0)
dz
O0
a0(3)
Öp(z, 0)
olog M (z, 0; 0;4) dz.
20(1)
00(1)
18

---

Then, we study f of the following equation is exchangeable, i.e.,
Opl=.0)
log (2.00e) dz. Under Assumption A.2 the integral and differentiation (00e
/
p(z, 0) (M 02 dz / 00()
p(z, 0) aM 0: 1p)
dz.
ag(i)
According to integral by parts, have
/ a0(0)
Öp(z,0) 0) log (z, 0: dz = 0061) / p(2,0)
(2 0:
dz f pl2.0)
0² log (2,
de.
a0(i)
a0(i)
00(6)2
Thus, our proof is completed.
The rest of the estimation process via policy perturbation is provided in the main context in Section
The other part omitted in Section 3 is the details about Eq. 2 that
V. 1/2 V=(0, - 0,) (0,
Here Vi denotes the sample-based estimator of the variance, obtained by plugging in the empirical Hessian and empirical covariance matrices:
i2 (0.) E EDDD(A-))) Vol(=:0.), Cov "VL
well the estimator for (O((;-)):
Ma_ (0.) E DDD Vol(z, log (2, JJ
where &b is obtained by minimizing Jn...
Eq. is direct result following Slutsky's theorem. Assumption A.3 makes sure the empirical optimizer set can converge to the population optimizer set. Then other parts such as estimation of the Hessian matrix etc. could all be directly obtained by standard law of large numbers. Thus, we can directly use Slutsky's theorem to obtain Eq.
A.3 Policy Perturbation
In this part, we prove the validity of Tr as an estimator of (G(2)). Recall that we have s(2, 0; v)) Va log (z, 0; v) and we further denote s; @M(x,::)) and use Eg, Eg, for the expectation
agt0
EznD(D) and empirical expectation (., (o) respectively.
Firstly, we define the following function families:
Fi.: : {s(,
v) (+))
E E)
(() i
2.0 (000) 91(:0;0) & w).
F0 {s:(;;; v).
Further, we set
Eo,nl a
In,((::)) = Eam]]1s(2, 0; +)] 1o 0)12 + Σ
2010 su(2, s((,0,66)) []]]
i=1
d
- 2
>
1
- Eo,n [s((z, 0: [s((2, 0.))
77
)) Eollls(z, (1
Σ
[Bo Dair sc(2, s 0: +t]]
Botneto [o,(2,0 me (0; : - E. (z, 0:
17
19

---

Assumption A.9. We assume that the score function distribution map D and the corresponding function families (Fi,,, 2.0 F(l) .,..., df has the following properties:
(a). There is positive constant C > 0, such that for v0 E 0; E v.
0000 E Bo(st(2, 8 +) & C co.
(b). (Enveloping function) For 40 E O there is function Ho(z), such that Eg (2)(2])) < co, and for any function f E F1.o Ul 2 U( (F.)), we have
[f(z)] < (z(())
(c). (9-uniform Donsker) Let NN F. -1 II) denote the covering number, that the minimal number of || |l-balls of radius needed to cover the set F, there exists p(=) >> 0, such that
L. Vple)de 0o,
and
sup log Ho (o, O Q
Fo, L2(1)) < p(e),
where Fe E (F1.0.
(1) 3,0
I,..., d} and Q is any distribution on 2.
2,01
2,0
(d). (Vanishing optimization error) There exists an = o(1), such that
(0; (()) min Jn (0; v/) + an.
NEW
(e). (Richness of class) There exists 40" (0) for each 0, s.t.
s(2, 0: 30 (f)) = Vo lologp(s p(,, 0).
Remark A.10. Assumption A.9
c) holds with p(e) = C log + when Fo is VC-subgraph class for I.....d}. d}.
all Fg (Fi,,, F(1) F T(()
Assumption A.11. Assume that the following conditions hold:
(a). (Smoothness) (0) is L-joint smooth in (0,0) for some < cc.
(b). There exists C > 0, such that
sup Eg [el(z;(ps)] <C<00
Lemma A.12, Under Assumption A.9 A.9(a), we have
sup [00; - (0( < ndCn. 0:a
Proof. By mean- value theorem, there exists 77()) E [0, < i < d such that
d
d
J(0:b) - (n( 2
Bo [si(2,0 0 me Eo [st [5]]
2
Σ 2o1 Eo(s((2,0,$)]
{=1
LE E (s0,
[l] 2 Σ
[0(3)
Eg (2, 01,0)1111]
20()
1=1
1=1
20
()
2dCn.
the first inequality follows from direct application of mean-value theorem.
20

---

Theorem A.13. S., = (0, nel i- :-
j= =011, i......,d ......,,TT Under Assumption A.9
for v0 dg the following inequality holds
1
1
Ee [[VVlogp(2 -.,
0p( Vn mmmnn(n,k)
1 + an).
Proof. Fix 0 E θ by Dudley's 's uniform entropy bound, c.f. Corollary 19.35 in 31, we have
E₂
sup 4
Eg.n n
[1, 0; +)]l Eo
Eg [11((2, 0 }
0
1
Eg
sup
9 (2,0;
o0 (2 0; +))
00(1)
vn'
1
Eg
sup
Eg
[3/(2, 0; +) - Eg [1, (2 0: )]]
Vk
Since = (1), from the above results, the inequality below holds
(0: Jn w-) Op( Vn my min(n,k)
sup sup NEW #еa.,
by Lemma A.12 we know
1
1
sup sup NEW BEBAT
Op(
+
+ +1)).
Vn
TI T min(n,n) k)
From Assumption A.9 d), we know that for any E OA,T,
(0; ((0) < min Jn,, (b) + On
NEW
Using Assumption A))re), we know that mingev J(0; v) 0, and assume there exists xi (0) E arg mingev (0; v)
Take s = 1/" (0) and (0) respectively in inequality we have
(n. "0 (0)) Opl
+
+ t),
Vn 17 mi/miinnn k)
((() - (0: (0)) Opl
+ +n). vn n (wmin(nkk) kn
+
Since (0,,(0: & Jn (0: (0)) + an we have
J(0:((0)) (0; (0) - (0; (()) In,, ((0) (0; (0)))(()) + (0:
1 +
1
=0, Vn " mmmin(n,
+ 1 + ax))
By definition J(0; v). we have proved
1
1
[o[[lVoogp(z, ((2,,00000))))) Op(
n + an).
Vn mvmin(n.k))
Recall that
VG(Ok) Ho, (0k+1) -EEe Vo
and the estimator defined by
g+ = - -H (0x+1)
21

---

Theorem A.14 (Restatement of Theorem 3.6.. Under Assumption 3.1 A.1 A.9
and A.1I
we have
gh VG(0=)] =0p( Vn +
+ 1 an).
"I mymin(n,k)
Proof. By Assumption 3.1
[Vl((2; 0k+1) - Fgl(2;0x+1) & 010 - 0x+1ll,
by Assumption 3.1 there exists C₁ > 0, such that
| &
The proof falls into five parts.
Let
Ik.1 -= 1 ((
Step 1: Convergence of 1 - Th.2 ll.
We have
19 Dk.1 ll S 110k+1 0k+1 1
[1] Ou: (0x))
|| Os: ((+)) = - 10 0k+11l Σ
1=1
From Assumption A.9 b), there exists C2 > 0, such that
Eg, [1]s(2, 0a: ((xx))11 & C2 00;
by the law of large numbers, we know
1 Tx.1 0p - 0k+1ll),
thus combine the above result with Theorem 3.4 we have
1
9x.1 | Opl
Let
2v.2 = - -1 (0+11))
Vol(2; 1)) (0)
Step 2: Convergence of | Tr.2ll-
By Schway-Sehwarz inequality, we know
Eg Vol(=; O.s ((02)) Eg. (1 [1]a(2, On: ((+)))
From Assumption A.9 b), have
Eg [2, 8e: ((x)) < Eg. [H2 { 00,
02
where H3. (z) is the enveloping function for 0..
By Assumption 3.1
[[ol(2 - < - Arsl,
< 8110
since
11 0psll < ( 1100 0psl < 110 - 0psll
thus by Assumption b.I b),
E5g (2Vo(220041)000 E3 [|]o[[ 110 0rsll O(1).
(5)
Hence we have
Eg_ 00 11) 0(1),
(6)
22

---

by Chebyshev inequality and Assumption 3.1 this lead to the following bound of ll Tk.21l
[ō 02.211 x
Opl
Let
9k.3 == - (0x+1) (0 (x 0us (0x)) "..
Step 3: Convergence of 1 Tr.3ll. By Assumption A.11 a) and (6), have
l0 Tn,all x |H (0++1) Hz. (0,+ 1|
Ho, (0++1) )) (Ho, (0++1) - Hz (0x+1)) Ho: (0u+1) |||
further, using Assumption 3.1 and A.11 a), we know
1 Ōk.all S 100- 0x1l 110+11 Ost2ll.
thus by Theorem 3.4
11 - Ixc.3l = p
Let
I == Ha, (0&+1)
(ol(z;;0+1) Voggl(z 0z)].
Step 4: Convergence of [ Tl.
By Assumption
3.1
and Assumption
A.I
we have
10,3 I2.al1 x
E V (e; 0x+1)) 1)]
x 1|:((,
Vo logp(2, 0+)1
B Vo ( Ox+ )|
Eg. []]s(2, Oni v5 (0x))
Vo logp(z, 0+)l (by Cauchy Schwarz inequality)
$ Eg. [|) 0z: (0%))
Vologp(2, 0+) (by formula 5
Opl Vn +
+ + min(nn,
am) (by Theorem A.13
Step 5: Convergence of ||gx VG(O,)]
Finally, recall that
VG((o) = -Ha (0x+1) (
0w]]
by Assumption 3.1 and A.I
| VG(0w)] S Eg, [o x 110 0x
x ~100 - 0z1l
O (By Theorem 3.4
Combining the above results, we thus have proved
1 VG(0,) Op( Vn +
+ 17 + an). n/mmin(n k)
23

---

A.4
Details of Section 4 Theory of Prediction-Powered Inference under Performativity
Without loss of generality, we let N, N and n, = n for all e [T].
Let us denote
(0) E (1) (0) (0) x (F)
8nn (0)) (0),
where
(0)
1 Σ (z., yi: 0), C/ (0) n
1 N 1 Σ l ((o, E2 (0)
E ((a))
0,n
Here the samples (zt, V1) ~ D()) and x. ~ (+) are drawn from the distribution under 6. Recall that we have defined E (0) = Hg(0)-1 (rV. A. (0) + vxo(0)) Hg before Theorem 4.1 (in
the following sometimes omit for simplicity).
Theorem A.15 (Consistency of ).rl). Under Assumption 3.1 and A.4 if < 7o. then for any given
T 3 0, we have that for all t E [T].
OPT1 (Ax) - Oz11-
Proof. Let us denote (0) arg min N Dim i {x;)); + (l(zi, ye 03;0 Ab(m,, f(x:); 0'), where the samples (Iri, ye) ((0) and x. (x) are drawn for some parame-
ter 0 along the dynamic trajectory 0o - 01 ...
110, 1 IIG(0--1) 0% (0
[[G( - G (0, + (0,(:=_))+++(((-(-1) - G(
3
|G(()))) - (L,0))) ++1===100--- 11
0
7
where the last inequality follows from the results derived by [25], under Assumption 3.1 we have [|G(0) - G(0')]
< =0 0 - 0'|]
Notice that R C(C..A (0)) Cam (0). By local Lipschitz condition, there exists Eg > 0 Ouch that
F-Ae (0) Lirn (0) F 0.
sup 0:110- G E
4-1
Since é is strongly convex for any 0, G(O:L)) is unique. Then we know that there exists å such that Cam (G(0PeL)) > 8 for all 0 in {0 ||0 G IlI = =o3. Then it follows that:
Cl.de (0) mmm
inf 10- G( |l|=
cl (0) 00m
(G((FL))
6mm
inf 10- -(0m) :11=co
(Cfooh (8) Carn (8)) +
(Cam,, (0) Lim (G( (((0)-1)))
+ (Cgen, (G(O,Ll))
L.Aee
(G (G(O-1)))
28 - (1()
Then we consider any fixed such that 00 G( > en it follows that
0 - G(OPT)) )
Ch.de (0)
c. 8m
(G(OPP1)
cf.de (w) Lide (G(O,²))
800
G(OPPl)
8:9
gmm
||0 G(0 )|
(6 - op(1)) > 5- or))
Eo
24

---

where the first inequality holds for any by the convexity condition of Llide (0). inequality holds we take
and the second
+
0 o-G( 1 1 _ DI
+ G(O and using the above result. Thus no 0 such
that ||0 G(
|I = En can be the minimizer of L. (0). Then [|G(0,"] 2 (0 )ll 4 0. A.m
We then have, for given 7 > O, we have that for all t E (T).
110,01 Oell & Σ [(=
( - (x,00)1))) (0,")) f
==0
Thus, conclude that 0. - O2.
Theorem A.16 (Central Limit Theorem of (PP(())) (Ae), Restatement of Theorem 4.1 iff & and & for some > O, then for any given T
Under Assump-
tion 3.7 A.4
a.5
0. we have that
for all t
(T(( - 0.) 24 N( (1A)
with
Veeº (1))
VG((a) Exao (0
II A=i
VG(())
Proof. Let denote the variance terms by V. for simplicity, while omitting explicit dependence on parameters in the notation. Let U, = (0 0,) and denote 0, G( We make the following decomposition:
0.h 0g (Ú. 0.) 0.)
(1)
Step 1: Conditional distribution of U.!U--1-
For term (1), have
Vī((O, 0₂) VT(GG( G(O:-1))-
For term (2), the empirical process analysis in [ establishes that
Vn 0. D, N(0, (10,E,,,P Am, (Bur)),
where the variance given by
Σ Epm (0:r)= H Ham
(0c)
rrl A, (0.) + Vx, 70m (0)) H Ham. (0x)-1
Conditioning on pper for any function h, we use the following shorthand notations:
h =
Ma. n.) G := Vn(EEnh ) M,
B&hh = N f())))
Gyhh= VNN E BaND MM (x))))
Bih = 1 Σ
A(a, f(x)),
G/6 = Vn(Enn - E (h
Note that 0, G( Recall that
Cz(0):=
E (D.))
((x, V),, c)2(0) Ci,n (0) + +
(0) (0) c22, (0)).
25

---

Under the assumptions, Lemma 19.31 in [3]] implies that for every sequence hn Op(1), we have
Gn
vn
((x, y:
An
- E(x, y: 81)
hn Vol(s, y: 0b)
E 0,
vn
Vn A(x, yi 02 +
hn
- f(x,, yr 0s)
11 Vol yi 0z)
- 0,
Vn
G
Vn ((r,V: + hn - f(x, y: Bi)
hn Vol(z,
Eg 0.
vn
Applying second-order Taylor expansion, we obtain that
nEn ((x.vv:0. + hn vn) E(x, (xx B1) = n (Lar,, (0. + vn
kn Larn (8:) + G, Vol(o, + 01)
In
vn
1 h Ham. (0.)lin + G, Volls, o1)
2
Based on similar calculation of the previous two terms, we can obtain that:
7
cf gre1
(0.
+
hn /n
L.f.A (0c) 1
11 H yeen (0t)len + - 2
Gn vN
T
GG
xo%)
+ 0,)
N
By considering it; = - 0.) and he = - (0.)-1
(N - -xOx) Vol(x,
Corollary 5.53 in 31 implies they are Or(1) and we obtain that
CL. pm
(PPl))
(8b) = 2 nn Hamn (B.) + hig Gn + / N
Te Gāk Vollz, + (1)
n
--1
cli
(0.
Chr 6pm
(0.)
- Ham (0.)hm + op()).
n
Oren
2
Since 0.P1 is the minimizer of C the first term is smaller than the second term. We can rearrange the terms and obtain:
(h* hnm
T Umm (0.)1)) h,)) (1),
2
which leads to h: - A, (L).). Then the above asymptotic normality result follows directly by applying the central limit theorem (CLT) to the following terms, conditioning 0,
v 0.)10 = - H1mm (0.) (Si + S2) ol(),,
S >o
n
ff
E
Volle, f()(),((=)(0,))
2~
S2 = Voll - Xe Vol(E (f 0z)
(D Vol(z.v:0) 0.) - (V))
Note that, conditioning on 6 (1) is constant. Therefore, (1) and (2) follow joint Gaussian distribution. Consequently, given U.-1, 1. the conditional distribution of U, is given by:
U.UU-11 (m 0,) 0.
- - 0.) + (, 0.
=Vn(G(0,11 G((,-1)) + vn( 0.)
26

---

NN (2 (Vi(G((.-)) G(Oc-1)), A,, (Biir).
(Vvaal UL-1 Vn
Up-1 + 00-1):)))
=N
+ 0c-1) - G(O2-1)); S de, aes +0-, vn
(G(
+00-1 For later references, we denote U. UL-1 D2 ((L(UU--),,
Step Marginal distribution of U.. We calculate the characteristic function of Ug by induction. To begin with, we directly have
X1 DS N/ V. VI (01;r))
Now, assume that - D, (O, v, we derive the joint distribution of (Uz, U--1 and marginal distribution of Uz. Then we have, the characteristics functions & and the probability density function of distributions UL-1 and Uz U follow:
(s) + (ot exp( st " vPTs). (2) (2
1
/
le-is" -i== ou. "o,'" (2)
2
(U.(U,_(8)) -
(s) = exp(is" ((U(-1) 2
- "'E((U-1;)))))
Then according the proof of vanilla CLT under performativity in Section A.2 we have:
f exp is" (T((UU))) 2 E(U u² ve"Vv-1 v--1" du.
ΦU, (8) (2m) & del |
Apply dominant convergence theorem 1im du, (8), we have:
2 (2r) 2 det | / exp ((U)-) " - V du
lim (s) lim #
lim n-100 (2r) del |
exp iis Vā(G(
-i- + 0(-1) -((,-1)) vn
I Exm"" Uh1 10-+
+oa-1 (G( Vn + (0(11);)s 2u / du V
exp(ii isT /n(GG Uي + 0(-1) - G((0-z1)
(2T) det 1P-l
o UnL1 +0.1 (G( U0 +0-1)):)s [ du
vn
- exp is V o Ex. prr (G((f-1):r)s (G
/u du
(2T) 3 dot |erl
= exp 4- 3
s- - - i', (0
which the characteristic function of (0, V.PPI,, where V VC(0--)VV Ex,0, r). Here we use the fact that lim,- /n GG in +0+-)) -(((--1) VG((:-1)V. and the dominant convergence theorem holds as we have
exp(iss Vn(G( UL-1 + +.-1) - G(Or-1)) 3 T de. Ja vn
+0.- (G( UL1 +0--1):r)s " T.
un1
vn
expl 1 u v2)
2
Thus conclude by induction that
U. D, N (0, V,PH),
:-1
PPP Σ G((+))
II vG(())
27

---

And we have:
VG((k)) (LN))<((0.)
E
Villo, y: 0--1)
(V E Volla,11(++1)
(1.))
Hyp (0x+1)-1(
Tol(x,1004+11
((0))
= =H (B2DD(0))))))
(V Vol(z, Vo 0k) I.
B Experimental Details
B.1 Additional Experimental Details
As described in Section 5 construct simulation studies performative linear regression problem, where data are sampled from ((0) as
y=a x = N(H, Ex). v o0)
At each time step t, the label is updated with 6,-1 via the above equation, and then 8, is obtained by empirical repeated risk minimization with the updated data 2; = (xe, v/. The objective this task is to provide inference on an unbiased Og with low variance, that is, the ground-truth 0, is covered by the confidence region of 0g with high probability, and the width of this confidence region is small.
Given set of labeled data, we can obtain the underlying O:
0₂ (E, + Pa + yTa) (Mar) 0,1 + (Ex + pz
To compute the coverage and width of confidence region Re, 8) for 0r, we run 1000 independent trials. For each trial j. we sample ôt,5 together with with its estimated variance Vi,j. and construct two-sided normal intervals for each coordinate
1,...,d: dd:
-
/n
41 5 = =
2d
where d is the parameter dimension, d 0.1 the significance level, the data size, and 4-1 the standard normal quantile. The interval width of each trial is averaged over d coordinate intervals, and we count this trial as covered if the ground-truth 6, lies inside all coordinate intervals simultaneously. Finally, we report the average width and coverage rate over all trials.
Similarly, compute the coverage for performative stable point Ops, we obtain the close-form Ops for this task as follows:
0ps
(Eg Maltz PzP + ylaa) (xPT))
(8)
As defined in Corollary 3.7 the confidence region for Ops is constructed with
Ri(n, 5) + B ( 83 )'
where ||l|22, B 1100
Ops|1a and
| (x - -1 + [|x11]]
B max
max { |1 + 7},
max
NEX
(X.).
Here we take (a : || < 20). Note that the closed form expressions for the update and the performatively stable point in Eq. 7 and Fq. holld for any distribution of z with mean パar and variance E.y, and with mean 0 and variance o.. For easier calculation for the smoothness parameter, we truncate the normal distribution of (x, y) such that ||æ||/ < 20. The mean and variance of the resulting truncated distribution be well approximated by those of the original normal distribution due to the concentration of Gaussian.
We experiments NVIDIA GPUs A100 in single-GPU setup.
28

---

B.2 Additional Experimental Results
Ablation study on effects of y In Figure Al we compare confidence-reginn performance under regularization strengths y = 1 and 3. Together with results of = 2 in Figure [ we can find that as y increases, the gap between the coverage for 0e (solid curve) and the bias-adj aasassdudted coverage for Ors (dashed curve) vanishes more quickly across iterations. For example, at ( = 3. the dashed and solid curves are tightly closed for y = 3, while substantial gap remains for ? = 1. This phenomenon derives that the larger Y yields more strongly convex loss, which both accelerates convergence of the estimate 8 to its stable point and reduces the performative bias ||0, 8ps|l- Consequently, the bias-awared intervals converge for Ops to the original ones for 0g in fewer iterations when 7y is larger.
Ablation study effects of c. In Figure A2 we compare confidence-region performance under sensitivity& & = 0.003 and e = 0.03. We can find that as E increases, the gap between the coverage for 0, (solid curve) and the bias-adjusted coverage for 0rs (dashed curve) vanishes more slowly across iterations. For example, for 2 0.003, the dashed curves tightly upper-bound the solid curves at I = 3. whereas for e x 0.03, a noticeable gap persists even e amplifies the performative shift (the dependence of the label distribution on 0), which increases the performative bias. That is, stronger sensitivity requires more iterations for 0g to approach its stable point, slowing down convergence of the two confidence regions.
5. This behavior is because a higher
Ablation study on effects of oy- In Figure A4 we compare confidence-region performance under noise level of 0.1 and o 0.4. We observe that across all settings, PPI with greedy-selected À is essentially never worse than either baseline À = 1 or X = 0. When the noise is low (o) = 0.1), greedy À behaves similarly to À = O. placing almost all weight on the true labels. Conversely, when the noise is high (o² = 0.4). greedy À behaves like À = 1, relying more heavily on pseudo-labels to reduce variance. For the intermediate noise level of = 0.2 in Figure | greedy significantly outperforms both baselines by hitting the optimal bvrinner balance.
B.3 Case Study on Semi-synthetic Dataset
In Section 5 we originally consider experiments on a synthetic dataset because the performative prediction is an on-policy setting, which means we need to collect the corresponding data every time we update the parameter (policy). In all the previous literature on performative prediction, no such dataset is provided. Alternatively, previous work always uses a semi-synthetic dataset, which one will need to specify how the data distribution will react and shift according to the new policy.
Following Perdomo et al. 25 we further conduct a case study in a semi-synthetic way on a realistic credit scoring task using a Kaggle dataset [ The dataset contains features of individuals and a binary label indicating whether loan should be granted or not. Consider the setting where a bank uses a logistic regression classifier trained on features of loan applicants predict their creditworthiness. while the individual applicants respond to this classifier by manipulating their features to induce a positive classification. Following [2]. we can formulate this task as performative prediction because applicants' feature distribution D(0) is strategically adapted in response to A. By applying repeated risk minimization, performative stable point fps can be achieved.
We treat the data points in the original dataset as the true distribution to compute Ops. We add Gaussian noise to the original data feature generate an unlabeled set of the same size. Then, sample varying n labeled points with N = 18000 unlabeled points and perform / = 5 repeated risk minimization steps to compute the estimated 0, and build the confidence region for it over 100 independent trials. From the experimental results, we find that a coverage of 0.9 is achieved with decreasing width as n increases. Notably, our optimized greedy X₂ (orange) achieves the highest coverage and narrowest confidence width compared with when À is fixed to 0 or 1. The results support our proposed theory and strengthen the practical significance of our methods. We hope this case study can inspire future work for practical settings of PPI under performativity.
https//wwwwiaagggcccmmc/oooeeeee
29

---

t=2
1.0
10
1.01
(0ss)
0.7-
0.7
0.7-
A=1 (Bu)
x=1
greedy (8..)
gready
0.10
0.10
0.10-
0.08
0.08
0.08
width
0.04
0.04
0.04
0.02
0.02
0.02
600
800
1000
800
1000
800
1000
(a) Confidence-region coverage and width with y 1.
2
t=4 4
1.0
coverage
0.8-
0.8
0.8-
A (8xc)
A=
A-1 (0gg)
(0bg)
greedy
0.10
0.10
0.10
0.08
0.08
0.08
width
0.06
0.06
0.06
0.02
0.02
0.02
0.00
0.00
0.00
200
400
600
800
1000
200
400
600
800
1000
200
400
600
800
1000
(b) Confidence-region coverage and width with 7
3.
Figure A1: Confidence-region coverage (top row) and width (bottom row) with different choices of X. The setup is the same Figure [ only change 7 1 or 7 =
30

---

t= 2
1.0
10
1.01
(0ts)
0.7
0.7
0.7-
A=1 (Bu)
x=1
greedy (8..)
gready
0.10
0.10
0.10-
0.08
0.08
0.08
width
0.04
0.04
0.04
0.02
0.02
0.02
600
800
1000
800
1000
800
1000
(a) Confidence-region coverage and width with = 0.003.
3
1.0
0.8
0.8
0.8-
Am 0 (8xc)
A=
1- (0gg)
greedy (0bg)
greedy
0.10
0.10
0.10
0.08
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Reviews:
[{"review_id": "riXIt768JI", "reviewer": "Reviewer_CPwq", "summary": "This paper initiates the study of statistical inference under performativity, where prediction-informed decisions influence the outcome variable, leading to a feedback loop between predictive models and the data-generating process. The authors focus on the performative stable point and propose a framework for valid inference in this dynamic setting. The main contributions are (1) the derivation of a central limit theorem for estimators obtained through repeated risk minimization (RRM), enabling confidence region construction for the performative stable parameter, and (2) an extension of prediction-powered inference (PPI) methods to performative settings, leveraging unlabeled data and machine learning predictions to obtain more efficient inference. Theoretical results are validated through synthetic experiments.", "questions": "1. Would the framework extend to non-convex loss functions or to settings where D(θ) exhibits more complex dependence on θ?\n2. Are there diagnostics that practitioners can use to check whether the model is correctly capturing performativity effects in practice?", "limitations": "See weakness.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\nThe paper tackles the statistical inference in performative prediction, which is relevant for domains where model-informed decisions recursively shape the data (e.g., economics, policy, and social sciences). The theory is solid.\n\nWeaknesses:\n\n1. Motivation: The motivation of this paper is unclear to me. A more realistic example where the methods could work will be helpful. Otherwise, it seems to be a simple combination of PPI and Performative prediction.\n\n2. Empirical Scope: The experimental validation is currently limited to synthetic linear models. Application to more complex or realistic settings (e.g., non-linear models or real-world feedback systems) would further strengthen the practical significance of the proposed methods. Real data application will also be important.\n\n3. Practical Implementation: The paper introduces novel estimation strategies (e.g., policy perturbation for score matching), but more details or discussion on computational aspects, tuning, or diagnostics for these methods would make the framework more accessible for practitioners.", "was_revised": true, "final_justification": "I recommend a borderline accept, as I find this to be a theoretically solid paper. However, I did not assign a higher score because I still have concerns about the method’s applicability. In the authors’ response, they indicated that no real data can be used and that all applicable datasets are semi-synthetic. As a result, I remain unsure about the intended target audience for the method.", "rebuttal": "Thank you for your comments. Below, we address your concerns point by point. We hope that our answers can further ease your concerns and raise our scores.\n\n**Q1:**\nMotivation: A more realistic example where the methods could work will be helpful.  \n\n**A1:** To clarify, there are two significant contributions motivated by real applications in this paper. \n\nFirst, as mentioned in our paper in Section 3, we initiate the study of the inference perspective of the celebrated repeated risk minimization (RRM) algorithm for performative prediction. This algorithm has already been widely used in applications regarding performativity, but previous work did not provide inference results like asymptotic normality and confidence region, etc., which makes the obtained parameter (policy) lack uncertainty quantification. When it comes to policy, only knowing that the algorithm will converge to a stable policy is not enough; we need to provide the concrete parameters as well as error bars incurred by randomness and optimization. So, that motivates the whole Section 3. To give a concrete, realistic example of the setting in Section 3, one can think of applications relevant to domains where model-informed decisions recursively shape the data (e.g., economics, policy, and social sciences). Indeed, in policy-making contexts, such as setting tax rates or credit score thresholds, policymakers often adjust policies dynamically in response to observed outcomes, which in turn influence the behavior of many agents. \n\nSecond, people often conduct surveys to collect information and feedback for policy development. As noted by [1], these surveys often suffer from non-response issues. This highlights the importance of addressing data scarcity in the performative setting, which motivates our Section 4 that considers how to derive a principled method to use unlabeled data to improve the estimation and inference under performativity. The principled method (PPI) is **based on the asymptotic normality derived in Section 3, which is a novel result itself**. We want to emphasize that the normality results are highly non-trivial to obtain. In Section 4, we extend the whole PPI framework in the performative setting, which is our other contribution.\n\n\n**Q2:**\nEmpirical Scope: Application to more complex or realistic settings would further strengthen the practical significance of the proposed methods. \n\n**A2:** Thanks for the suggestion. We originally only considered experiments on a synthetic dataset because the performative prediction is an on-policy setting, which means we need to collect the corresponding data every time we update the parameter (policy). In all the previous literature on performative prediction, no such dataset is provided. People always use a **semi-synthetic** dataset, which people will need to specify how the data distribution will react and shift according to the new policy.  \n\nIn order to ease your concern, we further conduct a case study in a semi-synthetic way following previous literature on a realistic credit scoring task using *Give me a credit* dataset (due to NeurIPS policy, we cannot include links, but this is a public dataset online). The dataset contains features of individuals and a binary label indicating whether a loan should be granted or not. Consider the setting that a bank uses a logistic regression classifier $\\theta$ trained on features of loan applicants to predict their creditworthiness, while the individual applicants respond to this classifier by manipulating their features to induce a positive classification. Following [2], we can formulate this task as performative prediction because applicants' feature distribution $\\mathcal{D}(\\theta)$ is strategically adapted in response to $\\theta$. By applying repeated risk minimization, a performative stable point $\\theta_{PS}$ can be achieved.\n\nWe treat the data points in the original dataset as the true distribution to compute $\\theta_{PS}$. We sample varying $n$ labeled points with $N$ unlabeled points and perform $t=5$ repeated risk minimization steps to compute the estimated $\\hat{\\theta}_t$ and build the confidence region for it over 100 independent trials. From the experimental results, we find that a coverage of 0.9 is achieved as the width decreases with increasing $n$. The results support our proposed theory and strengthen the practical significance of our methods. We will complete this case study and include it in our final version. We hope this case study can inspire future work for practical settings of PPI under performativity.\n\n| $n$   | 10k   | 12k   | 14k   | 16k   | 18k   |\n|:------|:------|:------|:------|:------|:------|\n| coverage | 0.85  | 0.88  | 0.90  | 0.87  | 0.90  |\n| width    | 0.110 | 0.102 | 0.094 | 0.088 | 0.084 |\n\n\n**Q3:**\nPractical Implementation: More details or discussion on computational aspects, tuning, or diagnostics for these methods would make the framework more accessible for practitioners. Are there diagnostics that practitioners can use to check whether the model is correctly capturing performativity effects in practice?\n\n**A3:** We indeed include more details in the appendix. We refer the reviewer to Appendix B.1 in the supplementary material for additional experimental details on computational aspects and tuning of the proposed method.\n\n\n**Q4:**\nWould the framework extend to non-convex loss functions or to settings where $\\mathcal{D}(\\theta)$ exhibits more complex dependence on $\\theta$?\n\n**A4:**\nAs we point out in Remark 3.2, strong convexity of the loss function $\\ell(z,\\theta)$ and sensitivity of the distribution map $\\mathcal{D}(\\theta)$ are necessary to guarantee the convergence of the trajectory $\\\\{\\theta_t\\\\}_{t\\geq 0}$. \n1) If the loss function is convex but not strongly convex, RRM may fail to converge. Additionally, if the loss function is non-convex, the optimizer may not even be unique, which makes doing statistical inference pointless.\n2) The conditions of $\\mathcal{D}(\\theta)$ are mild enough to contain a wide range of cases where the distribution admits a density function $p(z,\\theta)$. \n\n\n\n[1] Huang et al. \"Insufficient effort responding: examining an insidious confound in survey data.\"\n\n[2] Perdomo et al. \"Performative prediction.\""}, {"review_id": "CUdWP7G1qU", "reviewer": "Reviewer_BWhj", "summary": "This paper studies how to perform statistical inference when a predictive model influences the data-generating process — a phenomenon known as *performativity*.  \nAt each round the learner deploys a model parameter \\\\( \\\\theta_t \\\\), new data \\\\( z_{t,i} \\\\sim D(\\\\theta_t) \\\\) are generated in response, and the learner updates the model by empirical risk minimisation.  \nUnder standard smoothness, convexity, and a Lipschitz condition linking \\\\( D(\\\\theta) \\\\) to \\\\( \\\\theta \\\\), the authors show that the sequence \\\\( \\\\theta_t \\\\) converges linearly to a unique performatively stable point \\\\( \\\\theta_{\\\\mathrm{PS}} \\\\).\n\nThe main theoretical contribution is a central limit theorem: for fixed \\\\( t \\\\) and sample size \\\\( n \\\\), one has  \n\\\\[\n\\\\sqrt{n}\\\\,(\\\\hat\\\\theta_t - \\\\theta_t) \\\\; \\\\xrightarrow{d} \\\\; \\\\mathcal{N}(0, V_t),\n\\\\]\nwhere the asymptotic covariance \\\\( V_t \\\\) depends on a product of Jacobians \\\\( \\\\nabla G(\\\\theta_k) \\\\) and a sandwich matrix built from the score of the unknown density \\\\( p(z, \\\\theta) \\\\).  \nBecause \\\\( p \\\\) is not observed, the authors propose estimating its score by fitting a flexible surrogate density \\\\( M(z, \\\\theta; \\\\psi) \\\\) with a score-matching objective augmented by small “policy perturbations” in \\\\( \\\\theta \\\\).  \nCombining the CLT with the geometric convergence of \\\\( \\\\theta_t \\\\) yields asymptotically exact confidence regions for the stable point \\\\( \\\\theta_{\\\\mathrm{PS}} \\\\).  \nThe framework is further extended with *prediction-powered inference*: abundant pseudo-labels are blended with scarce ground-truth labels to reduce variance, and a second CLT quantifies the resulting efficiency gains.  \nSimulations in a performative linear-regression setting confirm nominal coverage of the confidence sets and illustrate how the pseudo-label weight improves precision.", "questions": "- **Meaning of \\\\( p(z, \\\\theta_k) \\\\).**  \n  I understand that \\\\( p(z, \\\\theta_k) \\\\) refers to the probability density (or mass) function of the distribution \\\\( D(\\\\theta_k) \\\\) from which the data at round \\\\( k \\\\) are drawn.  \n  A brief reminder of this definition in the main text might help readers.\n\n- **Missing of appendix.**  \n  I understand that not all assumptions and proofs can be included in the main text.  \n  However, the appendix containing these details is missing from the submission, which makes the paper difficult to understand on its own.\n\n- **Precision of the variance estimates.**  \n  Estimating \\\\( \\\\nabla G(\\\\theta_k) \\\\) and the covariance \\\\( V_t \\\\) requires approximating an intricate score function by the surrogate \\\\( M(z, \\\\theta, \\\\psi) \\\\).  \n  Because this involves sums and products over all past rounds, a discussion of finite-sample error — perhaps via non-asymptotic bounds or sensitivity analysis — would strengthen the practical relevance of the method.\n\n- **Dependence on the model class \\\\( \\\\mathcal{D}(\\\\tilde{\\\\theta}) \\\\).**  \n  The accuracy of the gradient estimates naturally depends on how well \\\\( M \\\\) can approximate the true family \\\\( \\\\{ \\\\mathcal{D}(\\\\theta) \\\\} \\\\).  \n  I would appreciate comments on how the complexity of this family affects approximation error, and whether model-selection guidance could be offered.\n\n- **Robustness to annotation error in experiments.**  \n  The numerical study assumes an annotator that is almost unbiased, with only a small mean shift.  \n  In many real applications, annotation models may be substantially misspecified.  \n  Exploring scenarios with larger mean or variance shifts, or other structural errors, would give readers a clearer sense of the method’s robustness.", "limitations": "This is the same the last three questions in the questions \n\n- **Precision of the variance estimates.**  \n  Estimating \\\\( \\\\nabla G(\\\\theta_k) \\\\) and the covariance \\\\( V_t \\\\) requires approximating an intricate score function by the surrogate \\\\( M(z, \\\\theta, \\\\psi) \\\\).  \n  Because this involves sums and products over all past rounds, a discussion of finite-sample error — perhaps via non-asymptotic bounds or sensitivity analysis — would strengthen the practical relevance of the method.\n\n- **Dependence on the model class \\\\( \\\\mathcal{D}(\\\\tilde{\\\\theta}) \\\\).**  \n  The accuracy of the gradient estimates naturally depends on how well \\\\( M \\\\) can approximate the true family \\\\( \\\\{ \\\\mathcal{D}(\\\\theta) \\\\} \\\\).  \n  I would appreciate comments on how the complexity of this family affects approximation error, and whether model-selection guidance could be offered.\n\n- **Robustness to annotation error in experiments.**  \n  The numerical study assumes an annotator that is almost unbiased, with only a small mean shift.  \n  In many real applications, annotation models may be substantially misspecified.  \n  Exploring scenarios with larger mean or variance shifts, or other structural errors, would give readers a clearer sense of the method’s robustness.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Very good results and answering important questions. Clearly presented. \nI commend the work and consider it a valuable contribution. However, I am giving a borderline accept due to concerns regarding the **precision of the variance estimates** (detailed below).  \nIn my view, this issue is central: if the estimates prove unreliable or highly sensitive in finite samples, the practical value of the results would be significantly diminished.  \nOn the other hand, if the variance estimation procedure is shown to perform well, either through theoretical guarantees or empirical validation, I would view the work as a very significant contribution to the literature and would be inclined to change my rating to **strong accept**.", "was_revised": true, "final_justification": "Dear AC and authors, \n\nI would like to recommend an acceptance to the paper. \n\nBest wishes,\nYour reviewer", "rebuttal": "Thank you for your valuable comments and for finding our paper meaningful. Below, we address your concerns point by point. Hope that our answers can further ease your concerns.\n\n**Q1:**\nA brief reminder of the definition of $p(z,\\theta)$ in the main text might help readers.\n\n**A1:** Thank you for your suggestions on the presentation. We will include reminders and more explanations for readers in our revision.\n\n\n**Q2:**\nMissing of appendix.\n\n**A2:**\nWe are so sorry to cause confusion. But **we indeed include our appendix in the submission**. We did not attach it directly following the paper, but our appendix is included in the supplementary material. Both ways are lawful and stated in the guidance of the NeurIPS submission.\n\n**Q3:** \nVariance estimates: finite-sample error, perhaps via non-asymptotic bounds or sensitivity analysis would strengthen the practical relevance of the method ... I would appreciate comments on how the complexity of this family affects approximation error, and whether model-selection guidance could be offered.\n\n**A3:**\nThank you for raising this point. We agree with your assessment of the estimation error. While important, addressing this issue is beyond the primary scope of our current work because **this will require case-by-case assumptions** (see below for an example).\nOur main result is the consistency guarantee of the estimator and an end-to-end method for confidence interval construction. Depending on the specific distribution map family (e.g., location-scale family) and the estimation method used (e.g., regression, score matching), different estimation error guarantees could apply.\n\n**A concrete example.**  Consider a specific parametric form of $\\mathcal{D}(\\theta)$: the location-scale family defined as $z_\\theta\\sim \\mathcal{D}(\\theta) \\iff z_\\theta = z_0+\\mu \\theta$, where $z_0$ is from a base distribution, $\\mu\\in \\mathbb R^{m\\times d}$ is an unknown parameter, and $m,d$ are the dimensionalities of $z,\\theta$, respectively. In the specific parametric model for $\\mathcal{D}(\\theta)$, the parameter $\\mu$ can be estimated by least square regression using all samples collected from $T$ iterations, where at the $i$-th iteration, samples are drawn for parameter $\\hat{\\theta}_i$. \n\nLet $\\hat{\\Theta}=(\\hat{\\theta}_1,\\cdots,\\hat{\\theta}_1,\\cdots, \\hat{\\theta}_T, \\cdots,\\hat{\\theta}_T)^\\top  \\in \\mathbb{R}^{nT\\times d}$ denote the augmented sample-wise design matrix, and $\\Theta=(\\theta_1,\\cdots,\\theta_T)^\\top \\in \\mathbb{R}^{T\\times d}$ denote the population-wise design matrix. Then standard concentration bounds imply that  \n\n$\\\\|\\mu - \\hat{\\mu}\\\\|_{\\text{op}} $\n\n$=\\\\| (\\hat{\\Theta}^\\top \\hat{\\Theta})^{-1} (\\hat{\\Theta}^\\top Z_0) \\\\|_{\\text{op}}$\n\n$\\leq \\frac{1}{\\lambda_{\\min}(\\hat{\\Theta}^\\top \\hat{\\Theta})} \\\\|\\hat{\\Theta}^\\top Z_0\\\\|_{\\text{op}}$\n\n$\\lesssim \\frac{1}{n \\lambda_{\\min}(\\Theta^\\top \\Theta)} \\sqrt{n T (d + m)},$\n\n\nwith high probability, given that $n$ is sufficiently large and $\\{\\theta_i\\}$ span the whole space. In the second inequality, the operator norm bound follows from the standard $\\epsilon$-net arguments and Bernstein-type inequality, and the eigenvalue perturbation is controlled using Weyl's inequality.\n\n\n**Q4:**\nRobustness to annotation error in experiments: Exploring scenarios with larger mean or variance shifts, or other structural errors, would give readers a clearer sense of the method’s robustness.\n\n**A4:** We refer the reviewer to Appendix B.2 in the supplementary material, where we conduct ablation studies on the effects of different choices of regularization strengths $\\gamma$, sensitivity (mean) $\\varepsilon$, and noise level (variance shift) $\\sigma_y^2$. The results shown in Figures A1-A3 suggest the robustness of our proposed methods."}, {"review_id": "D77KY8KYuo", "reviewer": "Reviewer_HBQs", "summary": "The authors establish a central limit theorem for estimates of the performatively stable solution ($\\theta_{\\text{PS}}$) which are constructed using repeated risk minimization (RRM). Using this central limit theorem, the authors construct confidence regions for $\\theta_{\\text{PS}}$. Finally, the authors also consider a problem setting where prediction-powered inference (PPI) is used in the presence of performativity. The authors prove a similar central limit theorem and a corresponding estimate for the optimal weights between ``gold-standard\" labeled data and the annotated data.", "questions": "* Can you provide non-synthetic experiments? How do the constructed confidence intervals perform on non-synthetic data?", "limitations": "* Consider adding a limitations section to the main text", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n* The results on selecting optimal $\\{\\lambda_t\\}$ for performative PPI are interesting and potentially useful.\n* The paper develops asymptotic results for a generic $t \\in [T]$ as $n$ grows; this is distinct from prior work on asymptotics of performative estimates.\n\n**Weaknesses**\n\n* The authors are missing a comparison with the closely related work [1]; this work provides asymptotic results for the repeated stochastic gradient descent estimator of $\\theta_{\\text{PS}}$. The asymptotic results here are in the regime $n=1, T \\rightarrow \\infty$ so the results in the submission are distinct, but discussion of this work is still needed. For example, how are the proof techniques in this work novel to those in [1]? Also, what are the motivations for this asymptotic setting? It seems somewhat less realistic then what is considered in [1]. \n\n* The claim that 'the work initiates the study of inference in performativity' is not true; see, for example, [1], [2]. This is repeated throughout the paper and re-writes are needed to eliminate this.\n\n* The theory does not account for any optimization error in the risk minimization step of RRM\n\n* Use of the constructed confidence intervals requires a density estimation step where model $M(z, \\theta; \\psi)$ is used to estimate the performative distribution $p(z, \\theta)$. Furthermore, the theory requires an assumption of perfect convergence on the optimization over $\\psi$.\n\n* All experiments are on simulated data; I think this is an issue for the PPI experiments in particular. I am not aware of practical settings which both require the use of PPI and have performative aspects. The authors adding a data set where this is plausibly the case would greatly strengthen the results.\n\n**Minor comments**\n\n* 'Lastly,' repeated twice in the notation section\n* Line 53 'perforamtive'\n\n\n[1] Cutler et al. Stochastic Approximation with Decision-Dependent Distributions: Asymptotic Normality and Optimality. https://arxiv.org/abs/2207.04173\n\n[2] Bracale et al. Micro-foundation inference in strategic prediction. https://arxiv.org/abs/2411.08998", "was_revised": true, "final_justification": "The authors agreed to include relevant work, and addressed my concerns over all synthetic experiments.", "rebuttal": "Thank you for your valuable comments. We will revise our paper according to your helpful suggestions. But we think there exists some misunderstanding. Below, we address your questions point by point and hope that our answers can ease your concerns.\n\n**Q1:** The authors are missing a comparison with the closely related work [1]. The claim that \"the work initiates the study of inference in performativity\" is not true.\n\n**A1:** Thank you for providing the related papers for us. We will discuss and cite them in our future revision.\n\nHowever, we acknowledge that [1] establishes the asymptotic normality and minimax optimality for performative settings. While both [1] and our work study the asymptotic normality of performative estimators, the problem setups and theoretical focus differ a lot. More importantly, **[1] does not provide an end-to-end inference framework** (please see further discussions below). Therefore, we respectfully disagree that the statement of \"the work initiates the study of inference in performativity\" is not true. But we can totally understand that our claim can cause confusion at the first glance, and we will modify in our future revision to clarify that we initiate the inference framework for repeated risk minimization in the batch setting under performativity.  \n\n1) Setup: [1] focuses on stochastic gradient update with one sample, whereas our work analyzes the empirical risk minimizer at each round. \n\n2) Theory: [1] establishes only the normality result. Their asymptotic covariance involves the gradient of the density function $p$ in their $\\nabla R(x^*)$, as in our $\\nabla G(\\theta_k)$. However, they only provide examples assuming a parametric form of $p$ and treat all the structural knowledge as given. Thus, they don't propose any covariance estimation method from data. In contrast, we further deal with the density function and provide an end-to-end inference method for constructing confidence intervals. All the theoretical results are validated by extensive experiments.\n\n3) Technical details: Our analysis draws on M-estimation techniques from empirical process theory, while [1] follows the literature on stochastic optimization. \n\nAs the reviewer noted, their setting is analogous to a stochastic approximation of our model with $n=1, T\\to \\infty$.  \nOur model adopts a “batch” formulation that **better reflects practical scenarios**. For instance, in policy-making, policies are often updated based on population-level feedback, which is often collected through surveys. These surveys often suffer from non-response issues, so we extend the PPI framework to help address the data scarcity issue.\n\nFor [2], it studies identifiability and estimation error under a specific microfoundation model with performativity. They did not discuss about confidence interval construction. In contrast, our main contribution is establishing an end-to-end inference method for constructing confidence intervals, which is missing in existing literature, to the best of our knowledge. We will clarify that in the related work section.\n\nTo sum up, we will revise the phrasing to more accurately reflect the existing literature, and will add citations [1][2] and comparisons with the works you mentioned.\n\n**Q2:**\nThe theory does not account for any optimization error in the risk minimization step of RRM.\n\n**A2:**\nWhile our theoretical analysis focuses on the empirical risk minimizer, the convexity of the loss function allows for incorporating optimization error with standard guarantees. \nFor instance, if the loss function $\\ell(z,\\theta)$ is $\\beta$-smooth and $\\gamma$ strongly convex w.r.t. $\\theta$, a gradient descent with $M$ iterations with an initialization $\\theta_{\\text{init}}$ and a learning rate $0<\\eta\\leq \\frac{1}{\\beta}$ guarantees that the distance between the gradient descent iterate $\\hat{\\theta}_1^{\\text{opt}}$ and and the empirical risk minimizer $\\hat{\\theta}_1$ as follows (we refer to [3] for more details)\n\n$$\n\\\\|\\hat{\\theta}_1^{\\text{opt}}-\\hat{\\theta}_1\\\\|_2^2\\leq \\frac{8\\eta\\beta^2}{\\gamma^2}(1-\\frac{\\gamma}{\\beta})^{M-1}\\mathcal{L},\n$$\n\nwhere $\\mathcal{L}=\\frac{1}{n}\\sum_{i=1}^{n}\\ell (z_{0,i};\\theta_{\\text{init}}).$\n\n\nIn practice, gradient-based methods with a sufficient number of iterations can be used to ensure the optimization error remains desirable. We will discuss this part more in our revision.\n\n**Q3:**\nUse of the constructed confidence intervals requires a density estimation step where model $M$ is used to estimate the performative distribution $p(z,\\theta)$. Furthermore, the theory requires an assumption of perfect convergence in the optimization over $\\psi$.\n\n**A3:** As our goal is to provide an end-to-end inference method, it's a necessary step to obtain an accurate estimation of the density (gradient) function and thus the variance term. The accuracy of this step depends on the structure of the distribution map and the estimation method used. \n\nRegarding the assumption of perfect convergence over $\\psi$, it's a simplifying assumption that the model class $M$ is rich enough to approximate the true density. In practice, our optimization procedure achieves high accuracy, as demonstrated in our experimental results.\n\n\n**Q4:**\nAll experiments are on simulated data ... The authors adding a data set where this is plausibly the case would greatly strengthen the results.\n\n**A4:**\nThanks for the suggestion. We originally only considered experiments on a synthetic dataset because the performative prediction is an on-policy setting, which means we need to collect the corresponding data every time we update the parameter (policy). In all the previous literature on performative prediction, no such dataset is provided. People always use a **semi-synthetic** dataset, which people will need to specify how the data distribution will react and shift according to the new policy.  \n\nIn order to ease your concern, we further conduct a case study in a semi-synthetic way following previous literature on a realistic credit scoring task using *Give me a credit* dataset (due to NeurIPS policy, we cannot include links, but this is a public dataset online). The dataset contains features of individuals and a binary label indicating whether a loan should be granted or not. Consider the setting that a bank uses a logistic regression classifier $\\theta$ trained on features of loan applicants to predict their creditworthiness, while the individual applicants respond to this classifier by manipulating their features to induce a positive classification. Following [4], we can formulate this task as performative prediction because applicants' feature distribution $\\mathcal{D}(\\theta)$ is strategically adapted in response to $\\theta$. By applying repeated risk minimization, a performative stable point $\\theta_{PS}$ can be achieved.\n\nWe treat the data points in the original dataset as the true distribution to compute $\\theta_{PS}$. We sample varying $n$ labeled points with $N$ unlabeled points and perform $t=5$ repeated risk minimization steps to compute the estimated $\\hat{\\theta}_t$ and build the confidence region for it over 100 independent trials. From the experimental results, we find that a coverage of 0.9 is achieved as the width decreases with increasing $n$. The results support our proposed theory and strengthen the practical significance of our methods. We will complete this case study and include it in our final version. We hope this case study can inspire future work for practical settings of PPI under performativity.\n\n| $n$      | 10k   | 12k   | 14k   | 16k   | 18k   |\n| :------- | :---- | :---- | :---- | :---- | :---- |\n| coverage | 0.85  | 0.88  | 0.90  | 0.87  | 0.90  |\n| width    | 0.110 | 0.102 | 0.094 | 0.088 | 0.084 |\n\n\n\n\n\n**Q5:**\nMinor comments.\n\n**A5:**\nThank you for your careful examination. We will change the fix them in our revision.\n\n\n[1] Cutler et al. \"Stochastic Approximation with Decision-Dependent Distributions: Asymptotic Normality and Optimality.\"\n\n[2] Bracale et al. \"Micro-foundation inference in strategic prediction.\"\n\n[3] Polyak. \"Gradient methods for the minimisation of functional.\"\n\n[4] Perdomo et al. \"Performative prediction.\""}]

Author rebuttal:

Meta-review: The authors propose a new approach to statistical inference in the setting of performative prediction, where the data distribution $D(\theta)$ depends on the model parameters $\theta$, and thus one must search for a performative stable point $\theta_{PS}$ for which the parameters are optimal given its corresponding data distribution.  A typical way to obtain $\theta_{PS}$ is by repeated risk minimization (RRM): for some large number of steps ($t = 1 \ldots T$), find the loss-minimizing parameters $\hat \theta_t$ with respect to an $n$-element sample drawn from the current data distribution, then update the data distribution to $D(\hat \theta_t)$, and repeat.  The authors provide a rigorous proof of a central limit theorem for the estimate $\hat \theta_t$ (after $t$ rounds of repeated risk minimization), for large sample size $n$, showing convergence to a Gaussian with mean $\theta_t$ and variance $V_t / n$.  This is important because, combined with prior work demonstrating that $\theta_t$ converges to $\theta_{PS}$ as $t\rightarrow\infty$, it means that the estimation error in $\hat \theta_t$ does not compound and cause the RRM to diverge from the true stable point $\theta_{PS}$, i.e., repeated risk minimization will work to find $\theta_{PS}$ in practice when the typical convergence conditions are met. Moreover, the amount of estimation error can be expressed in terms of $n$ and $t$, allowing for confidence intervals or hypothesis tests for the estimate of $\theta_{PS}$. The framework is then extended to the case of prediction-powered inference (PPI), where we have a small amount of data with true labels and a larger amount of data with classifier-predicted labels, and a similar RRM procedure is conducted: the authors prove another central limit theorem result for this case.  The approach is validated with a simulation experiment of PPI under performativity in a linear regression setting.

Strengths:
* Performative prediction is an important, emerging area of machine learning.
* The work fills a critical theoretical gap, showing the effectiveness of repeated risk minimization (in practice, with finite sample sizes) for identifying the performative stable point, and providing error/uncertainty estimates that can be used for statistical inference.
* The development of the main theoretical results (explanations, theorems, and proofs) are strong, rigorous, and well-written.
* Extensions to PPI are interesting and potentially useful for real-world problems with large amounts of noisily labeled data but limited ground truth labels.

Weaknesses:
* The claim that this work "initiates the study of statistical inference under performativity" is overstated, and needs to be qualified and clarified. (The authors have agreed to tone down this claim in the final version.)
* The empirical study in the submitted version is very limited, with only a single, synthetic, and relatively simple dataset.  (The authors conducted an additional semi-synthetic study on a credit dataset during the discussion period, and will incorporate this study into the final version.)
* The motivation for extending the work from the standard performative prediction setting to performative PPI is not entirely clear, and would benefit from a real-world example where we have both large amounts of noisily labeled data and performativity.
* It took me several readings of the paper to understand why we should care about the setting with fixed $t$ and large $n$.  I now understand that this is important, both from the standpoint of what happens when $n$ and $t$ are large (successful convergence of $\hat \theta_t$ to $\theta_{PS}$) and being able to quantify the estimation error of the RRM procedure in practice (since in practice one would have to choose finite $n$ and $t$).  But it would help to provide additional clarification up front.

Reasons for accept/reject decision: This is a "borderline accept" paper with strong and important theoretical results on performative prediction, but limited empirical validation.  I lean toward acceptance because I believe that the theoretical results are sufficiently important to stand on their own (even without the very nice, additional proofs for the finite-sample case that the authors developed during the discussion period), but am willing to be overruled given some of the weaknesses above and the concerns raised by the reviewers below.   While these concerns make me hesitant to say that this is a clear accept (or that it should be given a spotlight or oral), I do think that the strengths of the paper (specifically, novelty and importance of the theoretical contributions) outweigh its weaknesses, and again, lean toward acceptance of the paper.

Discussion: The authors and reviewers had a robust and productive discussion which helped to clarify the importance of the contribution, differences from related work, and specific points in the theoretical derivation.  The authors presented both an additional, semi-synthetic case study using real-world credit data, as well as an extension of the theory to incorporate error from finite sample size $n$.  These additions would greatly benefit the accessibility and impact of the work, so it is important that they are included in the final version. After discussion, reviewers were convinced to increase scores (2 x borderline accept; 1 x accept; note that we had only three reviews for this paper).  Some minor-to-moderate concerns remained about insufficient connection to real-world examples and datasets, insufficient motivation of the performativity + PPI combination, and whether density estimation (necessary for the variance estimates $V_t$) would be effective in practice.
Decision: Accept (poster)