Unsupervised Domain Adaptation for Binary Classification with an Unobservable Source Subpopulation
Authors: chao ying
Jun Jin
Haotian Zhang
Qinglong Tian
Yanyuan Ma
Sharon Li
Jiwei Zhao
Venue: Accepted by TMLR ()

Abstract: We study an unsupervised domain adaptation problem where the source domain consists of subpopulations defined by the binary label $Y$ and a binary background (or environment) $A$. We focus on a challenging setting in which one such subpopulation in the source domain is unobservable. Naively ignoring this unobserved group can result in biased estimates and degraded predictive performance. Despite this structured missingness, we show that the prediction in the target domain can still be recovered. Specifically, we rigorously derive both background-specific and overall predictive probabilities for the target domain. For practical implementation, we propose the distribution matching method to estimate the subpopulation proportions. We provide theoretical guarantees for the asymptotic behavior of our estimator, and establish an upper bound on the prediction error. Experiments on both synthetic and real-world datasets show that our method outperforms the naive benchmarks that do not account for this unobservable source subpopulation properly.

Published in Transactions on Machine Learning Research (04/2026)
Unsupervised Domain Adaptation for Binary Classification with an Unobservable Source Subpopulation
Chao Ying
Chooyying@wiseh
Deportments of Statistics ond of Bíostatistics & Medical Informatics
University of Baaoonsin-Madison
Jun Jin
jjin2@hfts.orr
Public Health Sciences Henry Ford Health
Haotian Zhang
haotiangeekd@gaiik.com
School of Computing University of Connecticut
Qinglong Tian
ginglong, tian @wwaterloo.cr ca
Department of Statistics and Actuarial Science Universily of Waterloo
Yanyuan Ma Department of Statistics Pennsylvania State University
yam63@psuu.c
Sharon Li
sharonli wisesssw..sssc..o edu
Department of Computer Sciences University of Madiisn
Jiwei Zhao
edu
Departments of Statistics and of Biostatistics & Medical Informatics
University of Aisconsin-Madison
Reviewed on OpenReview: https: // openreview. net/ pertfformm id= a0KculttBar 8mm
Abstract
We study an unsupervised domain adaptation problem where the source domain consists of subpopulations defined by the binary label Y and binary background (or environment) A. We focus on a challenging setting in which one such subpopulation the source domain mobbservbll.. Naively ignoring this unobserved group can result in biased estimates and show that the
degraded predictive performance. Despite this structured missingness, prediction in the target domain can still be recovered. Specifically, we rigorously derive both background-specdffc and overall predictive probabilities for the target domain. For practical implementation, we propose the distribution matching method to estimate the subpopulation proportions. We provide theoretical guarantees for the asymptotic behavior of our estimator, and establish an upper bound on the prediction error. Experiments on both synthetic and real-world datasets show that our method outperforms the naive benchmarks that do not account for this unobservable source subpopulation properly.
Key Words: Unsupervised domain adaptation, Structured missingness, Distribution matching

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Published in Transactions on Machine Learning Research (04/2026)
1 Introduction
Unsupervised domain adaptation (UDA) (Kouw & Loog 2019) addresses the challenge of transferring pre- dictive models from a labeled source domain to an unlabeled target domain under distributional shifts (Koh)
et al. 2021 Sagawa et al.) 2022). In this area, research methods aim to reduce domain discrepancy by align- ing feature distributions, using statistical measures such as maximum mean discrepancy (MMD) (Tzeng
ett al. 2014) and higher-order moment matching (HoMM) (Chen et al., (020). Deep adaplation frame- works, such as deep adaptation network (DAN) (Long et al. 2015) and aovman-adverssaial neural network (DANN) (Ganin et al., 2016], ; are also popularly used due to their strong empirical performance. There are also other approaches that integrate reconstruction objectives to disentangle fomain-invariant and domain- specific components (Chifary et al. 2016). These approaches often assume access to a representative and diverse set of source examples. However, real-world datasets may violate this assumption in systematic and non-random ways.
In many applications, the goal to build a target-domain prediction model to predict a binary label Y that identifies a specific object in an image (e.g., classifying a waterbird versus a landbird in the Waterbirds dataset). Additionally, the image typically contains contextual attributes represented by binary background environment variable A, such as whether the background consists of water land. In this work, we focus on more challenging and practically relevant UDA setting where structured subpopulation is entirely missing from the source domain. Specifically, we study the case where one subpopulation, defined by a particular combination of Y and A, is unobserved in the source. This structured missingness is not merely a sampling artifact, but often reflects real-world constraints in data, collection. For instance, in the widely studied Waterbirds dataset (Sagawa et al., 2019), waterbirds (Y = 1) photographed in water environments (A=1) can be rare or entirely absent due to the difficulty of capturing such images in the wild. This issue arises in many other disciplines well. In healthcare, certain patient subgroups, defined jointly by disease status and demographics, may be underrepresented or absent in historical datascts due to restrictive inclusion criteria or changes in clinical practice over time. When such models are applied to broader populations. unobserved subgroups can suffer from systematic mispredictions. This structured missingness (Mitra et al. 2023) fundamentally changes some statistical properties when comparing the source and target domains, and, if unaddressed, can lead to severely biased estimation and unreliable prediction in the target domain. These structured gaps pose new challenges that are not adequately addressed by conventional UDA techniques, which motivates our work.
To tackle this challenge, we develop a theoretical framework that accounts for the structured absence of a subpopulation, such as (Y = A, = 1), in the source domain. Our key idea is to model how prediction in the target domain can still be recovered by relating it to the observable parts of the source and target data. Under a mild assumption that the distribution of features X given (Y,A) stays the same across domains, we derive closed-form expressions for making accurate predictions in the target domain. These expressions depend on the proportions of different subgroups in the target, which are unknown. To estimate them, we propose practical method based on distribution matching that avoids modeling complex feature distribu- tions directly. Specifically, we frame the problem as estimating finiteddimensional mixture proportions under structured conditional invariance, and propose a (L-ddvergenneebasee objective that can be optimized using only observable quantitics. We also provide theoretical guarantecs, showing that our approach yields statis- tically consistent estimates and deriving upper bounds on the prediction error of the resulting target-domain classifiers. Overall, our framework provides the first rigorous characterization of model adaptation under structured subpopulation absence, and enables robust domain adaptation in such a challenging scenario.
We validate approach through experiments on both synthetic and real-world datasets. We simulate domain adaptation scenarios where one subpopulation is systematically excluded from the source data and evaluate our method against baseline approaches that do not account for this missing group properly. Across a range of settings, our method consistently achieves higher accuracy and F1 scores. These results highlight the practical value of explicitly modeling structured missingness and demonstrate that our approach leads to more reliable predictions in the target domain. To summarize, this paper makes the following novel contributions:
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We consider new unsupervised domain adaptation setting where an entire tabel-background sub- population is missing from the source domain, a scenario motivated by real-world data collection constraints.
We develop a theoretical framework that enables accurate prediction in the target domain by estimat- ing subpopulation proportions through distribution matching, and we provide rigorous guarantees and error bounds for our method.
We demonstrate the effectiveness of our approach on both synthetic and real-world datasets. Our method outperforms standard baselines that ignore structured missingness, particularly in recovering performance the unobserved subpopulation.
2 Related Work
-uttof-dist (OOD) generalization OOD generalization refers to the ability of a prediction model to perform well on test data drawn from a distribution that differs from the training data. In our context, the subpopulation (Y= = 1) the target can be regarded as the OOD data while the other three subpopulations are in-distribution data. For a comprehensive overview of OOD generalization, we refer the readers to the excellent survey (Liu et al., 2021), which reviewed real-world datasets, evaluation protocols, and key challenges in this area. In the OOD generalization literalure, different methods were proposed with different emphases: AArjovsyy et al. (2019) emphasized the need to minimize invariant risk across different environments to ensure consistent model performance, whereas Sagawa et al. (2019) underscored the importance of distributionally robust optimization (DRO) and various regularization techniques in reducing performance disparitics across subgroups. In addition, Bahnng et al. (2020) introduced adversarial training as a method for learning de- biase representations, which is critical for promoting fairness in machine learning models, and [Sohoni al. (2020) examined the issue of robustness in classification lasks involving coarse classes that contain finer subclasses, enhancing model performance across all subclasses.
OOD detection OOD detection is the task of identifying inputs at test time that do not come from the same distribution as the training data. Its goal is to prevent a model from making confident but incorrect predictions on unfamiliar or anomalous inputs by flagging them as OOD. There are a variety of techniques developed for OOD detection in the literature. For example, Hendrycks & Gimpel (017)) introduced a simple yet effective method for detecting both misclassified and 0OD inputs in neural networks. Liang et al. (2018) (ODIN) proposed an improved method for detecting OOD imputs by applying temperature scaling to the softmax outputs and adding small input perturbations during inference. ODIN significantly outperformed previous baseline methods, including the maximum softmax probability approach, and set a new standard for OOD detection in classification tasks. Other techniques include but not limited to, outlier exposure
(Hendrycks el al.] 2018 Papadopoulos et al. 2021 CoufGAN (Sricharan & Srivaslava 2018) and OodGAN
MMarek et al.) 2021). In addition, Fort et al. 2021) provided an extensive empirical study of OOD detection
methods across a wide range of datascts, architectures, and training rogimes.
Spurious correlation Spurious correlation is a major obstacle to OOD generalization, where models often rely on non-causal features that can degrade performance, particularly when these correlations do not generalize across domains. For example, a model trained to classify cows might rely on green pastures (background) instead of the cow itself. On desert background, it fails. This is also the case in the Waterbirds dataset where the spurious correlation exists between label Y and background A. Different learning strategies were proposed to discover and mitigate the impact of spurious correlation on model performance, as well to improve model robustness. For example, Wu et al. (2023) introduced attention- approach to automatically identify spurious concepts and apply adversarial training to reduce reliance on them. Another approach proposed by Kuunrr et al. (2023) used causal regularization to detect. and discourage spurious dependencies, allowing for scalable robustness across shifts. In addition, [Sagawa et al. (2020) investigated why overparameterization exacerbates spurious corrclations, and Kirichenko al. (2022) found that retraining only the final layer on a small, balanced dataset can restore robustness against spurious correlations. Also,
Wang & Wang (2024) developed theoretical model to analyze the influence of spurious correlation strength,
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sample size, and feature noise on learning. Spurious correlations were also investigated in feature learning
[Tzmailov et al. 2022 Qiu et al., 2024). reinforcement learning (Ding al. 2023 OOD detection (Ming
et al. 2022). and text classification (Wang & Culotta 2020). One can also resort to a comprehensive survey
paper Ye et al. 2024 on this topic.
Imbalanced classification and few/zero-shot learning In imbalanced classificaton, all classes are observed in the training data but appear with highly unequal frequencies, and many methods focus on reweighting or resampling strategies improve performance, particularly on minority classes. Few-shot learning (Wang et al., 2020), such one-shot learning {Li et al. 2006), refers to a learning paradigm in which a model learns from very small number of labeled examples and then generalizes to novel classes. Unlike traditional supervised learning that requires large amounts of data, few-shot learning leverages a limited number of examples together with task-specific prior knowledge or structural information. Zero- shot learning (Wang et al. 2019) addresses tasks in which no labeled examples for the target classes are available during training, typically relying on auxiliary information such as semantic attributes or textual descriptions to relate seen and unseen classes. In contrast to these settings, our work considers different challenge: a structured subgroup defined by specific combination of (Y, A) is completely absent from the source domain. This missing-subgroon setting introduces distinct type of distribution shift that cannot be addressed directly by existing methods designed for imbalanced classification or few/zero-shot learning.
Adversatial domain adaptation Adversarial domain adaptation methods, such as DANN Ganin et al. 2016) and ADDA Tzeng et al., 2017), aim to align the source and target feature distributions using ad- versarial training. However, in the setting with structured missingness that we consider, these methods can fail because the unobserved target subgroup may be incorrectly aligned to a visible source subgroup, a phenomenon termed as "collapse". This occurs because adversarial alignment enforces marginal distribution matching without modeling hidden subpopulation structure, which can lead to biased predictions. Differ- ently, our framework explicitly accounts for the unobservable source subpopulation, avoiding the collapse issue and providing more reliable adaptation in such scenarios.
3 Problem Setup and Notation
In our UDA setting, Y E {0,1] denotes the binary label, which is observed in the source domain but not
in the target. Let A E {0, 1} be binary background or environment variable and X E RR a vector of all other attributes. Let R E (0,1) be a domain indicator, with R = 1 corresponding to the souree and R = 0 to the larget. In our notation, we cousistently use the order of (R, Y, A) for indicator function [[,, sample size n(-): and population probability P(-}
We define
pr(R 1). For y 1,0, 8ye pr(Y = y, A == R = 0). For clarity, the total source sample size is n2 = 1101 + 71100 + 71000 and the target sample size is no = 10-1 no-o, so that the total sample size is n nn + no. Table 1 summarizes the observed data structure and key notation.
1,0, we define Gya = pr(Y y,
R 1), and
Table 1: Data structure and key notation used throughout the paper.
R
Y
A
X
Sample Size
Proportion
Probability
/ 1i101
P101 001
(1() pr = X x,
Source
0
v
R110
P110 0
{o(x) x
0
/
18:00
P100 = Ogoa
= x
0
1
0
1
v
Pip-1
Po-1
(811 + (o1)(1 - xx)
m (x)) x
Target
- nxx) ==\\=||X=x
0
V
0
0
no.
(810 900)
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In our context, we have 010 + duu + auo 1, 01 0, and 0 < a10, a01. Cuo 1. be consistently estimated by
The parameters
ã = m10/m1
0ot = /101/m1, ooo = m $ = n1/n.
(1)
More formally, on the following structured missingness condition:
pr(Y A - R 1)= 0.
(2)
Note that this assumption is made without loss of generality, alternative combinations, such (Y 0, A = 1), (Y = 1, = 0), or (Y = 0, A = 0), can be similarly assumed to have zero probability.
To characterize the distributional connection between the two domains, we impose a structured conditional invariance assumption:
p Y. R= 1) p Y, A, 0) p(X Y. = (X),
that is, the conditional distribution of features X given AY remains the same across domains. This be regarded conditional version, or, more nuanced version, of label shift where the marginal distribution of labels (now, the combination of both label and backgroun)) varies across domains (e.g.. Du Plessis
& Sugiyana 2014; Garg et al.] 2020 Lyer et al. 2014 Lipton et al.) 2018 Nguyen et al.] 2016 Tasche
2017 Tian et al. 2023 Zhang et al. 2013 Lee et al. 2025a This type of invariance assumptions was
also postulated in other contexts such as causal inference; see, e.g., Peters et al. (2016). It indicates,
conditional on background A, the label shift assumption holds. It is equivalent p(R X, Y, A, - p(R),, A), the independence between R and X, conditional on (Y, A). In practice, this assumption may be suitable in many applications. Below we give two examples to illustrate the rationality of this assumption. For instance, we aim to predict user clicks on advertisements for a new batch of users (target domain, R R=0 0) using historical data (source domain, R = 1). Conditional on the advertisement type A and whether the user clicks Y, the distribution of browsing behavior features X is assumed to remain stable across time periods. This is because user clicks are fundamentally determined by ad content and user interests, not by the time period in which data are collected. As another example, suppose we have datasets from two hospitals (R indicates the source hospital and R = O indicates the target. hospital). Here, X represents imaging features, Y is the disease type, and A denotes patient attributes such gender age group. Then, conditional on the discase type Y and demographic attributes A, the distribution of imaging features X is expected to remain the same across hospitals. This is because imaging characteristies for a given disease and demographic group are not systematically altered by the hospital. The main difference between hospitals lies in sampling proportions rather than conditional distributions.
This framework captures real-world scenarios in which a certain label-background subpopulation is absent from the source domain. For example, in the Waterbirds dataset, waterbirds water backgrounds (label == 1, background A A 1) are rarely observed, or completely absent, in the training set, making the adaptation to target domains particularly challenging. For illustration purposes, Table 2 below shows the three observed subpopulations in the source as well as the four subpopulations in the target in two real-world datasots.
4 Proposed Methodology
Our goal in this work is to correctly identify and successfully implement, under our UDA setting, the two backkround-speeiffcc predictive probabilities in (x) and no(x) and the overall predictive probability n(x), the target domain. All of the three probabilities were precisely defined in Table [
4.1 The naive benchmarks
We consider two types of naive benchmarks. The first, termed Naivel throughout, to blindly apply the three source domain predictive probabilities (x),, fo(x), and {(x) the target, ignoring the structured missingness issue in this problem. More precisely, the first naive benchmark incorrectly treats (x(x,, (x)).
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Table 2: Illustrations in Waterbirds and CelebA datasets. Note that the (Y = -, = 1) combination does not exist in the source domain but does in the target domain.
Dattwett
Souree Data
Target Dara
(F:4)
(0.4) Landaaddd Amidee backgnoud
11.
(0,4)
(4.3)
(0,3)
(4,0)
(0.0)
Amoit beckaworad
Y Wartehard
YwoLandaad Auiil her beriground
Y-aeWaaeciidd Water backgroard
AWaeW backgrourd
YoitLandlied
A-Diead bodetional
Vetelandiio A Amitlaan lackkpoodd
Y-olooa Inir A-ll
Y=1
Y-L A-1 Male
Y-ddilood
Y-l:
Swh.Boo
A-1.bale
and [(x) as V (x), (x), and 7xx,, respectively. Note that, even though (i = 0 as indicated in Table 1
one can still use the observable data to implement the overall source domain predictive probability 5(x) and the other background-specifi. predictive probability fo(x) as
{xx pr(Y = x, R = 1), and fo(x) pr(Y = x, R A 0).
(4)
The second lype, termed Naive2, is ignore the environment A and simply impose label shift assumption between the source and target domains. That is, one incorrectly specifies the following 7(x) the overall target domain prodictive probability n7(x). After some simple algebra, one can compute
Bu+Ba <(x) 011+010
7(x)
#100000 [(x) + 011+a10
Po +80 {1 E(x)}
001+
4.2 Model adaptation from source to target
The most challenging aspect of this work is adapt the model for the A = background since the component (Y = A 1) is entirely absent in the source. Nevertheless, we still correctly derive the three predictive probabilities for the target domain, as shown below.
Proposition 1. Define conditional probabilities To(x) = pr(A = 1 x = x, R= and
<(x) = pr( = x, A 1),
(6)
both of which can be implemented using the observed data in our UDA setting. Then the three predictive probabilities in the target domain are given by:
f10 {x()) 0.0
(xx) Por 1 x(x)
no(x)
and
π
1 - xx(xxx
Bin [x +
[1 -o(x))
(7)
ano
atea
7xx) = 7 (x))) + (x) (x)))
The proof of this result is provided in Appendix A.1 Proposition illustrated that, in general, the naive benchmarks presented in Section 4.1 fail. There are no explicit relations between mx(x) and (1(x) = 0 or between 7(x) and {(x). For the relation between (x) and fo(x), they coincide only the special case that f10/110 = S00/ooo which corresponds to a proportionality condition between the class-conditional densities across domains. Outside of this narrow scenario, the naive approach systematically misestinates the target posterior, leading to biased predictions.
This result also implics that model adaptation fundamentally relics on estimating the proportions of key subgroups in the larget population. In particular, for individuals with A = 1, one only needs to estimate Bor, while for those with = 0, suffices to estimate the ratio 310/300. Denote & = (810. 8oo)TT can be seen that, accurate estimation of the parameter B in the target domain enables valid model adaptation across domains. Before developing methods for estimating in Section 4.4 we first present some model identification considerations.

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4.3 Model identification considerations
The identifiability structure of our problem closely resembles that of the open set label shift (OSLS) framework [Garg et al. (022). Note that our target distribution consists of a mixture over four joint distributions: pr(Y 1,1y=1,A=0 11,A=11 1), pr(Y = 1, A = 0), pr(Y = 0, A = 0), and pr(Y = 0, A = 1). By treating the joint label (Y as the response, this setting can be viewed as a special case of the OSLS framework. However, our setup considerably simpler due the availability of the auxiliary variable in the target domain. a result, we can restrict attention the subset A 1, thereby discarding the A 0 portion of the distribution. This reduction simplifies the problem to recovering pr(Y = A = 1) from mixture of pr(Y unlabeled
1,A
and pr(Y = 0, A 1): given direct access to pr(Y = 0, A 1). This is a canonical posilive- posiline-unlabeled (PU) learning problem. Identifiability in this sctting is governed by the standard anchor set condition (sce Definition 8 of Ramaswany et al. (2016)): there exists a measurable subset Xanchor E X such that
0 and p(X E Xanchor Y = 1, = =0
p(X E Xanchor Y = 0
p(X E Xanchor Y 0 =
subset of the feature space that is occupied by the negative class (Y=0, A= 1), which is necessary for identifiability. Under the assumption 3 the primary difficulty arises from the fact that the component Pi1(x), corresponding to the subgroup (Y 1, = 1), is not directly observable in either the source or target domain.
This condition ensures that the positive class (Y = 1,A 1) has support
To elucidate this observation, we denote Pn(x)= (ooo(x)) and then the observed data log-likelihood of one generic observation in our UDA setting is proportional to:
(1ologp1o(x) + /oologpe1 (x) +
+ 1 { (8)P + (1 f11 - B 1 1) (x)) + lo.ulog Pp(()))
In this formulation, the paramcter with finite dimension is B. The model involves four nonparametric muisance components: (x)(x), (xo(x), Poi (x), and (x).
Lemma 1. Assume $n1 and P1o(x) + (x), then all components except Pı1(x) are identifiable. Assume B1 <1 and is known, and pio(x) # (x), then all components in the model identifiahle.
The identification conditions in Lemma D are intuitive and reasonable. If 311 = O, it degencrates to the situation that the source and target domains have the same support on both label Y and background A, then the component pm (x) is no longer relevant. Also, if Pio(x) = (x)()), the subpopulations of (Y = 1, = 0) and (Y = 0, A = 0) become indistinguishable, and hence the individual probabilities f10 and foo are not separately identifiable. Overall, these conditions are natural to ensure the problem is well-posed.
The proof of Lemma D is provided in Appendix A.1
4.4 Estimating parameters of interest
To estimate the parameter B, we consider the distribution of attributes x in the subpopulation defined by (R 0, 0). By the law of total probability, we have
p(x R 0, A O)p 0, = 0) = pio(x) 310(1 - л) + poo Boo - -),,
(8)
subject to the constraint
pr(R 0, A = 0) = 310 -)) + Boo x).
(9)
Note that the distribution p(x R 0, 0,A= 0) identifiable from the target population. The distributions (x) and Poo (x) can be consistently estimated from the source population subgroups (R 1, 1, 0)
and (R = Y 0, 0), respectively. Thus, the parameters B (Buo, (Io)TT can be estimated by minimizing a suitable discrepancy measure between the two sides of (5), such
T.2 norm divergence-
based criterion (e.g., Kullback Leibler divergence), subject to the constraint in (9). Therefore, we reformulate the cstimation / as a constrained distribution matching problem:
f = argmin D (p(x R = 0, A = (P)o())0)) + (oool))
= 0 = 0)}
(10)
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subject to (r(A = O 0) B10 + Soo: where D denotes discrepancy measure between probability distributions over the covariate space X' Among various choices for D, we adopt the Kullback-Teibler (KT.) divergence due to its favorable analytical and computational properties. To facilitate optimization, we relax the constraint in 10 and reformulate the objective under KL divergence, as summarized in the following lemma.
Lemma 2. Let D be the Kullback Leibler divergence. Then the solution 310 to the minimization problem 10 is given by
arg max P. (1og(Go(X)0)) B10 + (1-6o(X)) -1) -10)] 0, 0,4=0). 910
(11)
where, for simplicity, bj = pr(p = 1|R = 1, A 0), pr(A A = o|R = 0) and E represents the empirical average.
The proof of Lemma 2 is provided in Appendix A.1 A key advantage of minimizing the KL divergence is that it circumvents the need to explicitly estimate the generative probabilities P1o(x) and Poo(x), which are often difficult to model accurately in high dimensions. Instead, sullices to estimate one background- specific prediction probability (x(x) using standard classification techniques on the source domain restricted to A = O.
Finally, based on all of the above discussions, we summarize the implementation details of our proposed method in Algorithm 1
Algorithm Implementation details of our proposed method.
Input: Observed source domain data {(X,, Y,, A,, R, = 1) and target domain data ((X;, R )-
Output: Estimated benchmark predictive probabilities f(x) and (x(x), proposed predictive probabilities for the target (xx), V (x) and (x);;); and subpopulation proportions Bya: Bya.
1: Estimate f(x) (defined in (4) using data {(X,, Y,, R, = 1): ii= 1,... ,n1). as E(x);
2: Estimate (x( (defined in (4)) using data {X,, Y,, A; = 0, = 1 1, .....nnh m as (x(::
3: Estimale r,( (defined in Proposition 1 using data {(X;, A, R4 :.,
= 0, as F,
4: Estimate x(x) (defined in (6)) using data {(X,, R4, = =1) 1): =i: as R(x);
5: Estimate B and O'g.a following 11 and (1, as B and aya for (v,a) E 10,
6: Estimate 7n (x), (x) and ()) lollonigg
7 as क (x), (x)) and 7xx).
The above method adopts the idea of distribution matching. Alternatively, one may consider matching only certain moments rather than the full distribution. Due to space constraints, we defer the details to Appendix
5 Theoretical Results
For the interest of space, we only present the results for the background-speeifie predictive probability with A = O. The results for the other two predictive probabilities are parallel and can be similarly developed. To facilitate the analysis, we begin by formally defining the population-lewe (expected) objective function:
E(Eo, by Pro. e) E (log(G((X)6, 310 + (1 - (x)))) - bi) (1o) R O, A 0)
with its empirical version Ū(Eo, bI, Bl0, e)-
Assumption Define fxx {fo (x), fi (T) where fo(x) log(Eo(x)) - [log(Eo(x)) + logg1 {x and fx(x) = 1o
- [x)) [ - 1 (x)) + log(1 {x()) and the corresponding estimate is {fr {x)
There exist constant e > 0 and a sequence Tmio - 0 such that, for almost every x, we have
pr (1f(x)- >1) S exp (-p ( V1>0.
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Remark 1. Note that the toil bound described in Assumption is intended to hold uniformly for every RI-0 when estimating fK for k = 0,1. In other words, for each subsample size n1-0 we have a Tn such that the corresponding eslimators J& for k = 0,1 are required to salisfy the stated concentration inequakity. This inequality is analogous to Hoeffding's inequality and provides a con-asymptotic concentration bound on the estimation error. Similar assumptions have also been adopted in recent work (e.g., Maity et al. (0022) Tsybakov & Audiberl 2007)).
Theorem 1. Suppose Assumption holds. Define Xv. = Tn v (oog(no.)) + 11.
// + no-0 Then, there exists constant cın such that for any ó O, with probability at least 1 66, we have
11/2
|| B11 S c10Xn Vlog(188)
The proof of Theorem 1 is provided in Appendix A.2 Theorem Testablishes the consistency of the estimator
B, provided that a v log(noo) + as 11.-0 Ro.0
00.
Theorem
I establishes the upper bound of the parameter estimate. Beyond parameter estimation, an im-
portant question is how the resulting estimator performs in downstream prediction tasks.
With any loss function E(.) and some function le(-), for the background-sspeeifee prediction model with A = 0, the conditional risk is
Y) R = 0 0 = Y) w(Y)
1, 0],
(12)
where, for simplicity, write w(y) prig 1r One can derive that w(1) fio +aro)
+10) and
an ( (800
(0) Deo(a00-a10)
+191)) t.960 To evaluate the performance of the predictive probability, it can be approximatod
as (X), } ((Y) R= 1, A= 0]. Furthermore, the model can be fine-tuned specifically for the target subgroup by minimizing the reweighted empirical risk:
To E arg min min
Ele(h(X), (Y) R= 1, = 0).
(13)
KEF
where F is suitable function class. For the other two predictive probabilities, the relations analogous to 12 can be derived similarly. For the background-specffe predictive probability with A = 1, one can derive
E (()( Y 0, A = 1] as
S01
E[((((X) = =
(((ET(X)) = -)) 03] 0,A
Boo + B11
For the overall predictive probability, the conditional risk (]) 0] is
E[e(M(X))Y) Y) wxY R = 1, -, (] + 300) + E(E(M(X), Y 1|| 0, = 1](301 + B11) -E((E(((X), = 1} - (M(X), Y = 0Y] = 0, A = 1) 301.
The above relations show that the target-domain risk can be expressed in terms of appropriately reweighted risks computed from the observed source data. This motivates the use of weighted empirical risk minimization for learning classifiers tailored to the target domain.
We now establish a generalization bound for the fitted model 13 which is obtained via weighted empirical risk minimization over the source subgroup. Let F denote the hypothesis class of classifiers. For any h E F and a weight. function w(y): y + R, we define the population-level weighted loss and its empirical counterpart based on the source subgroup data as follows:
w))
E
= =
(1, w) = [((e)()), ()oY)) 0, R=
We also define the population loss on the target subgroup as: Lo(h) = E = Clearly, Li (h, w) = (h).
To establish our generalization bound, we utilize the concept of Rademacher complexity (Bartlott Mendel- son 2002), denoted as (9) (see Appendix A.2 for details), and impose the following assumption on the loss function:
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Assumption 2. The loss function & is uniformly bounded; that is, there erists a constant B 0 such that [(((()), vHl - B for ony h E F,x C R2, and y {0, 1}.
We now present the generalization bound for the learned model, with its proof provided in Appendix A.2 Proposition 2. Under Assumptions and 2 let h = ergmingerCiit w) be the classifier obtained by minimizzng the reweighted empirioal rish on the source subgroup. Then, there erist constants c, such that, with probability least 1 75, the following generalization bound holds:
log(1/0)
log 1/6)
Co(h
min Lo(h) S 2R dBB -811 + heT
c
{
+
V
111.0
110
where G = (w())e(((x) y) hh F}, and Ru (9) denotes its Rademacher complerity as defined in Ap- pendir A.2
Remark 2. Proposition indicates that the generalization bound depends on the estimation ervor ||@ I which can be directly controlled based on the conditions listed in Assumption [ implying that different estimation procedures for B will yield different upper bounds. In Theorem [ established an upper bound for the estimation error of B, which directly leads to refined generalization bound for the learned classifier ha. Specifically, for any 5 > 0, with probability at least 1 - 136, the following inequality holds:
min (h) S 2R (9) dBeivXn (1/() + J BEF
log( (Pg(1))) )
1og(1/8)
(h w
+1
111.0
n0
where Cin is the conslant appearing in Theorem and on characlerizes the converyence rale of B.
6 Synthetic Data Results
We consider a structured lata-generating process in which the covariate X E R is drawn from a distribution conditioned a pair (Y, A), where Y E {0, 1} denotes the class label and A E (0, 1} denotes the background. The generation begins by sampling (Y, according to a predefined distribution.
In the source domain, we consider (Y, E {(0.0), (0, (), (1, each occurring with probability 1/3. The covariate X E R$ is generated as X ~ N (PYAA, where μya denotes the mean vector for each combination and I is the 4 x 4 identity matrix. The stratum (1, is excluded from the source. In the target domain, all four combinations (Y, A) E {0, 132 appear with equal probability 1/4, and X is drawn from the same distribution N (HYA Ig) with distinct means:
1o0 01, Poi 10,,,,,0)*, Pi0 (0,1,0,0) 111 = (0,0,0,1)
We follow Algorithm D to compute the conditional probabilities required by the proposed approach as well the two naive benchmarks. Specifically, we calculate the five key conditional probabilities needed for implementation: (x)),, {(x), 70(x), 7 (x), and x(x), which, together with the estimators of the parameters {Bya y 0, a = 0} and {aya = = 0, 1; = 0, 1}, determine the predictive probabilities no(x), (x) and v)(x) for the proposed method. In addition, we compute the naive benchmark 7(x) following ( which corresponds to the standard UDA method relying solely on the label shift assumption without accounting for the subgroup information.
To assess the performance of the proposed approach as well compare with the two naive benchmarks, conduct 100 simulations for each configuration and summarize the results using boxplots that compare f(x), ((x), across varying sample sizes. The left panel of Figure displays displays the performance of {(x) and 7x) for 7i = 1000 and 6000, with ny ranging from 1000 to 8000, while the right panel shows the corresponding results for n1 = 1000 and 6000, with no ranging from 1000 to 8000. Performance evaluated using two standard metrics: accuracy and F, score. In both metrics, the proposed approach consistently outperforms the benchmark estimators. Notably, the accuracy of πxx) steadily improves with larger no (or mn) further demonstrating the robustness and effectiveness of the proposed method.
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no 1000
nO= 6000
1000
6000
08
Accuracy
F1 Score
Fi G001e
D1
Proposed
Nesvett
3po
Nakwet
Figure 1: Left panel: Accuracy and F1 score of (xx),, E(x), and 7(x) with fixed = 1000/ /6000 but n Accuracy and Fi score of fxx), É(x), and (x) with fixed 1000/6000 but no varies.
varies. Right panel:
7 Experiments
In this section, analyze the Waterbirds dataset (Sagawa et al. 2019 which consists of 11,788 images. This public dataset is widely used to investigale spurious correlations in image classification. It is also well aligned with our problem setting of unsupervised domain adaptation under structured missingness, where specific combinations of labels and backgrounds are systematically absent in the labelod source domain, while labels are entirely unobserved the larget. domain. The label Y denotes waterbird and Y landbird. The background A corresponds water background and 0 land background. It yields four label background subpopulations, as summarized in Table
Table 3: Empirical joint distribution of (Y in the Waterbirds dataset, with varied values of a, and c, 0 b, 1.
Y
Description
Count
Total Proportion
Proportion in Source
Proportion in Target
1
Waterbird on water
1832
0.155
0.155
0
Landbird on water
2905
0.246
0.246a
0,246(1 - a)
1
Waterbird on land
831
0.071
0.071b
0.071 (1 b)
0
Landbird land
6220
0.528
0.528c
0.528 c)
To construct structured domain adaptation problem, partition the full datasct into source domain
(R 1) and target domain (R 0). Specifically, allocate samples from three subgroups, (Y 0, A 1), (Y 1,4 0) and (Y 0, A = 0), into the source domain, with allocation rates denoted by parameters a, 6, and respectively. The remaining subgroup, (Y = 1, = 1), is deliberately ercluded from the source domain and appears only in the target domain. This setting reflects real-world scenarios in which a specific combination of label and background is structurally missing from labeled datasets due to systematic data collection biases or constraints. In the target domain, all four subgroups are retained, but the label variable Y treated as unobserved.
To implement the proposed method, we apply the distribution matching approach to estimate the subclass proportions in the target domain. For fcature extraction, we embed cach image into 512-dimensional feature vector using a ResNet-18 model (He et al.) 2016) and a ViT-16 model (Heo et al. 2021), both pre-trained on ImageNet (Deng et al. 2009), without additional fine-tuning. These embeddings serve as covariate X R512 in our downstream analysis. Based on these feature vectors, we fit logistic regression models with Leeegaalaioontonn to estimate five key conditional probabilities required by both our proposed method and benchmark procedures: Ex(x), [(x), To(x), (x) and s(x).
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ResNet18
VIT15
ResNet18
VIT16
0.90
Accuracy
0.65
Accuracy
0.85
0.85
0.00
0.00
0.80
6.75
0:25
0.75
0.3
0.2
0.4
03
66
0.7
0.4
0.
0.
02
0.4
05
0.4
0.5
6.
611
02
03
04
93
0.
0.8
0.0
0.1
0.
04
65
0.6
0.7
0.5
0.9
0120
0.80
T1-score
Fl-score
0.8
0.6
0.75
0.70
0.00
0.3
0.2
5.1
84
05
66
0.7
0.1
033 82
0.1
0.1
055 04
20
0.0
0.9
0.1
02
0.1
0/ 055 06
0.0 0.8
6.1 0.7
0.3
0.1
65
06 0.7 A.
b
Proposed
Naivel
Figure 2: Performance comparison of our proposed estimator 77(x), and the naive methods {(x) and 7(x) under the setting a = 0.7 with either and varying b or 0.5 and varying c.
0.950
0.977
0.975
0.900
Accuracy
0,55
Accuracy
2.05
0.800
0.000
0.80
0.60
0.955
0.85
0.15
2.15
2.70
0.3
0.4
2/
0J
08
0.9
0.1 0.2 0.
0.
04
da
S6
08
0.2
0.1
0.2 03
5.85
3.80
0.8
0.7
0.5
08
6.1 53
0.5
07 0.8
6.1 0.
Proposed
Nalvel F0XI
Nolve2 vir)
Figure 3: Performance comparison of our proposed estimator 17xx, and the naive methods {(x) and 7(x)
under the setting 0.5 with either 0.5 and varying b or 0.5 and varying
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0.955
0.925
0.900
0.900
Accuracy
0.85
Accuracy
0.65
c.10
0.030
0.50
0.00
cat2
0:825
0.75
2.75
0.800
8:0
0.70
0.1
0.4
05
5/
b-7
0.4
0.1
0.7
0.3
0.1
04
05
58
DT
0.0
0.4
0.1
0.7
0.3
0.0
0.85
0.85
F1-score
0.8
0.0
2.20
05
34
0.
08
0.9
0.1
0.2
03
0.5
0.3
0.6
0-6 0.7
0.1 0.2
033
0.5
Proposed
Naivel Elx
Naive2 vixl
Figure Performance comparison of our proposed estimator r7(x), and the naive methods {(x) and y(x) under the setting a 0.3 with either 0.5 and varying b or b 0.5 and varying c.
For empirical evaluation, we fix the subclass sampling rate 0.7 in the source domain and systematically vary the remaining subclass inclusion rates by setting either b = 0.5 with with b (0.1,0.2,.... 0.9}. For each configuration, the data generation process is repeated 50 times to account for sampling variability. We assess performance using two widely adopted classification metrics: accuracy and F1 score. Figure 2 presents boxplots summarizing the distributions of the metrics across repeated experiments for our proposed estimator f(() and the two naive estimators {(x) and 7(x). The left panel shows the results based on features extracted by ResNet-18, while the right panel shows the results based on features extracted by ViT-16. We also reduce the proportion from 0.7 to 0.5 and 0.3, and similarly demonstrate the results in Figure and Figure 4 respectively.
0.9), 0.2........
0.5
The observations and conclusions from this experiment can be summarized as follows. First, the proposed correct predictive probability πソ() consistently achieves the best performance across most scenarios. This is particularly evident when the subpopulation (Y= 0, the source domain (Figure 2 This behavior logical; because (Y = 0, A = 1) is the only class containing information about A 1, higher representation of this subgroup helps compensate for the absence of the (Y = A, 1) subgroup. Second, the performance of the proposed nxx) the two incorrect naive benchmarks, if the proportion of either (Y 1, A 0) or (Y 0, A = 0) becomes exceedingly small either in the source domain or in the larget. In such cases, the source domain may effec- tively suffer from multiple missing subgroups (beyond just (Y = A 1)), elevating the problem's 's difficulty well beyond the scope of this paper. Finally, choosing ViT-16 generally brings much better performance than using ResNet-18, especially for the two naive benchmarks.
constitutes a larger proportion (a = 0.7) in
degrade, even falling behind
8 Discussions and Conclusions
In this paper, we introduce a novel unsupervised domain adaptation setting where an entire babel-background subpopulation is absent from the source domain, scenario motivated by real-world data collection con- straints. Despite this structured missingness, show that accurate prediction in the target domain is still achievable. We develop theoretical framework that enables such prediction by estimating subpopulation proportions in the target through distribution matching. We provide rigorous guarantecs, including statis- tical consistency as well as upper bounds on the larget-domain prediction error. Empirically, our hod outperforms standard baselines that overlook structured missingness, especially in prediction performance for the unobserved subpopulation. Overall, our framework provides a rigorous characterization of model adaptation under subpopulation structured missingness, and enables robust domain adaptation in such a challenging scenario.
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Our theoretical framework is built upon structured conditional invariance and mixture proportion estima- tion. These tools naturally generalize to multi-class labels for By species and multi-level (or even continuous) environment variables for species. In fact, the identification strategy and distribution-matten tion carry over to larger joint label-onviromment spaces, though at the cost of heavier notation and more complex optimization. Technically, at this general multi-label and multi-background situalion, the model identification considerations (see discussion in Section 4.3) becomes more complex. At this situation, one can identify both pr(X, A all 0) as well pr(A alR 0), which in total 2n, 1 quantities, while one has in total ny unknown quantities, including pr(Y y. A a 0) and the unobservable subpopulation distribution pr(X 1 = 1). To make sure this model is identifiable, one needs to make (n, - 2mm + anchor set assumptions. For example, when ny 3 and na = 2, 3 anchor set assumptions are needed. Interestingly, as long as the label is binary Ny = 2, one anchor set assumption is sulficient if only one subpopulation is missing in the source. In the setting we consider in the paper, ny = na = 2, so we only need to make one anchor set assumption.
estima-
The invariance assumpliin imposed in ( can be viewed a conditional, or more nuanced, extension of the standard label shift assumption. While standard label shift assumess that the distribution of X has no change across domains conditional on the label Y, equation (3) postulates that its distribution remains the same conditional on both the label Y and the environment A. As discussed in Section B this assumption is justifiable in a variety of practical contexts. In practice, however, the environment. A itself may undergo shifts. For instance, the source domain might contain only two backgrounds (e.g., water and land), whereas the target domain introduces a third (e.g., sky). This scenario effectively transitions into an out-of-distribution (OOD) generalization problem, where the shift occurs with respect to the environment. A rather than the label Y. Addressing missing subgroup in the source domain within this OOD context extends significantly beyond the scope of this paper, the model identification challenges discussed in Section 1.3 become substantially more intractable. Consequently, this problem remains an important area for future work.
Broader Impact Statement
This work addresses practical limitation of unsupervised domain adaptation by enabling reliable prediction when entire subpopulation is absent. from the source data, situation that commonly arises in healthcare, ecological monitoring, and other domains where certain subgroups are systematically underrepresented due to data collection constraints. By explicitly modeling structured missingness, our approach can help reduce biased predictions on groups that conventional methods would otherwise misclassify, contributing to fairer and more robust machine learning systems.
Acknowledgments
Sharon Li is supported in part by the AFOSR Young Investigator Program under award mumber FA9550- 23-1-0184, National Science Foundation (NSF) under awards IIS-2237037 and ITS-2331669. 1669, Office of Naval Research under grant number N00014-23-1-2643, Schmidt Sciences Foundation, Open Philanthropy, Alfred P. Sloan Fellowship, and gifts from Google and Amazon. Jiwei Zhao is supported in part by NSF under awards 195-5993322, DMS-2122074 and 2MS-2310942, and National Institutes of Health under award R01DC021431. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the author(s) and do not necessarily reflect the views of the funding agencies.
References
Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez- Paz. Invariant risk minimization. arXiv preprint XrXiv:1907.0055 02800, 2019.
Hyojin Bahng, Sanghyuk Chun, Sangdoo Yun, Jaegul Choo, and Seong Joon Oh. Learning de-biased rep- resentations with biased representations. In International conference on machine learning, pp. 528-539.
PMLR, 2020.
Peter L Bartlett and Shahar Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3(Nov):400 (4NN-)4633-482 0863--424444 2002.
14

---

Published in Transactions on Machine Learning Research (04/2026)
Peter J Bickel, Chris AJ Klaassen, Peter J Bickel, Ya'acov Ritov, J Klaassen, Jon A Wellner, and YA'Acov Ritov. Efficient and adaplive eslimation for semiparumetric models, volume 4. Springer, 1993.
Chao Chen, Zhihang Fu, Zhihong Chen, Sheng Jin, Zhaowei Cheng, Xinyu Jin, and Xian-Sheng Hua. Homm: Higher-order moment matching for unsupervised domain adaptation. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pp. 3422 3429, 2020.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei, Imagenet: A large-scale hierarchical image database. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognàtion (CVPR), pp. 248 255, 2009.
Wenhao Ding, Laixi Shi, Yuejic Chi, and Ding Zhao. Seeing is not believing: Robust reinforcement learning against spurious correlation. Advances in Newral Informalion Processing Systems, 36:66328 66363, 2023.
Marthimus Christoffel Du Plessis and Masashi Sugiyama. Semi-supervised learning of class balance under class-prion change by distribution matching. Neural Networks, 50:110 119, 2014.
Stanislav Fort, Jie Ren, and Balaji Lakshminarayanan. Exploring the limits of out-of-distributio detection.
Advances in neural informalion processing systems, 34:7068 7081, 2021.
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario March, and Victor Lempitsky. Domain-adversarial training of neural networks. Journal of machine learning research, 17(59):1 35, 2016.
Saurabh Garg, Yifan Wu, Sivaraman Balakrishnan, and Zachary Lipton. A unified view of label shift estimation. Advances in Neural Information Processing Systems, 33:3290 3300, 2020.
Saurabh Garg, Sivaraman Balakrishnan, and Zachary Lipton. Domain adaptation under open set label shift. Advances in Newral Information Processing Systems, 35:22531-225444 2022.
Muhammad Ghifary, W Bastiaan Kleijn, Mengjie Zhang, David Balduzzi, and Wen Li. Deep reconstruction- classification networks for unsupervised domain adaptation. In Computer Vision-ECCV 2016: 14th Eu- ropean Conference, Amslendam, The Netherlands, Oclober 11-14, 2016, Proceedings, Part IV 14, pp- 597-613. Springer, 2016.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. Proceedings of the JEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770-778, 2016.
Dan Hendryeks and Kevin Gimpel. A baseline for detecting misclassified and out-of-distribution examples in neural networks. In International Conference on Learning Representations, 2017.
Dan Hendryeks, Mantas Mazeika, and Thomas Dietterich. Deep anomaly detection with outlier exposure. arXiv preprint Xrxi::1812 04606, 2018.
Byeongho Heo, Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Junsuk Choe, and Seong Joon Oh. Rethinking spatial dimensions of vision transformers. In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pp. 11916-11925. IEFE, 2021.
Arun Iyer, Saketha Nath, and Sunita Sarawagi. Maximum mean discrepancy for class ratio estination: Convergence bounds and kernel selection. In International conference machine learning, pp. 530-538 PMLR, 2014.
of spurious correlations. Advances in Neural Information Processing Systems, 35:38516 38532, 2022. Pavel Tzmailov, Polina Kirichenko, Nate Gruver, and Andrew G Wilson. On feature learning in the presence
Polina Kirichenko, Pavel Izmailov, and Andrew Gordon Wilson. Last layer re-training sufficient for robustness to spurious correlations. Xixiv preprint or Xiw:2224.00099 2022.
15

---

Published in Transactions on Machine Learning Research (04/2026)
Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Balsubramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Trena Gao, et al. Wilds: A benchmark of in- the-wild distribution shifts. In International Conference on Machine Learning, pp. 5637 5664. PMLR, 2021.
Wouter M Kouw and Marco Loog. review of domain adaptation without target labels. IEEE transactions on pallern analysis and machine intelligence, 43(3):766 775,, 2019.
Abhinav Kumar, Amit Deshpande, and Amit Sharma. Causal effect regularization: Automated detection and removal of spurious correlations. Advances in Neural Information Processing Systems, 36:20942-2088 2023.
Seong-ho Lee, Yanyuan Ma, and Jiwei Zhao. Doubly flexible estimation under label shift. Journal of the American Statistical Association, 120(549):278-299) 2025a.
Seong-ho Lee, Yanyuan Ma, and Jiwei Zhao. Efficient inference under label shift in unsupervised domain adaptation. arXiv preprint arxi:2208.. 1 17780, 2025b.
Fei-Fei Li, Rob Fergus, and Pietro Perona. One-shot learning of object categories. IEEE transactions on pattern analysis and machine intelligence, 28(4)5594-111 2006.
Shiyu Liang, Yixuan Li, and R Srikant. Enhancing the reliability of out-of-distribution image detection in neural networks. In International Conference on Learning Represenlations, 2018.
Zachary Lipton, Yu-Xiang Wang, and Alexander Smola. Detecting and correcting for label shift with black box predictors. In International conference on machine learning, pp. 3122-3130. PMIRR 2018
Jiashuo Liu, Zheyan Shen, Yue He, Xingxuan Zhang, Renzhe Xu, Han Yu, and Peng Cui. Towards out-of- distribution generalization: A survey, arXiv preprint XrXi::2188. 2021.
Mingshong Long, Yue Cao, Jianmin Wang, and Michael Jordan. Learning transferable features with deep adaptation networks. In International conference on machine learning, pp. 97-105. PMLR, 2015.
Subha Maity, Mikhail Yurochkin, Moulinath Banerjee, and Yuekai Sun. Understanding new tasks through the lens of training data via exponential tilting. arXiv preprint arXiv:2205 2022.
Petr Marek, Vishal Ishwar Naik, Vincent Auvray, and Amuj Goyal. Oodgan: Generative adversarial network for out-of-domain data generation. arXiv preprint 2rx:v22004 04.44484. 2021.
Yifei Ming, Hang Yin, and Yixuan Li. On the impact of spurious correlation for out-of-distribution detection. In Proceedings of the AAAI conference artificial intelligence, volume 36, pp. 10051-10059, 2022.
Robin Mitra, Sarah F McGough, Tapabrata Chakraborti, Chris Holmes, Ryan Copping, Niels Hagenbuch, Stefanie Biedermann, Jack Noonan, Brieuc Lehmann, Aditi Shenvi, et al. Learning from data with struc- tured missingness. Nature Machine Intelligence, 5(1):13 23, 2023.
Tuan Duong Nguyen, Marthinus Christoffel, and Masashi Sugiyama. Continuous target shift adaptation in supervised learning. In Asian Conference on Machine Learning, pp. 285-300. PMLR, 2016.
Aristotelis-Angele Papadopoulos, Mohammad Reza Rajati, Nazim Shaikh, and Jiamian Wang. Outlier exposure with confidence control for of-distriinttonn detection. Neurocomputing, 441:138-150 2021.
Jonas Peters, Peter Bühhlman, and Nicolai Meinshausen. Causal inference by using invariant prediction:
identification and confidence intervals. Journal of the Royal Statistionl Sociely Series B: Statisticnl Method- ology, 7844):947 1012, 2016.
GuanWen Qiu, Da Kuang, and Surbhi Goel. Complexity matters: feature learning in the presence of spurious correlations. In Proceedings of the 41st International Conference on Machine Learning, pp. 41658-4166) 2024.
16

---

Published in Transactions on Machine Learning Research (04/2026)
Harish Ramaswamy, Clayton Scott, and Ambuj Tewari. Mixture proportion estimation via kernel embeddings of distributions. In International conference on machine learning, pp. 2052-200 2060. PMTR, 2016.
Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Perey Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. arXiv preprint arXiv:191100879) 2019.
Shiori Sagawa, Aditi Raghunathan, Pang Wei Koh, and Percy Liang. An investigation of why overpa- ramcterization exacerbates spurious correlations. In International Conference on Machine Learning, pp. 8346 8356. PMLR, 2020.
Shiori Sagawa, Pang Wei Koh, Tony Lee, Irena Gao, Sang Michael Xie, Kendrick Shen, Ananya Kumar, Weihua Hu, Michihiro Yasunaga, Henrik Marklund, et al. Extending the wilds benchmark for unsupervised adaptation. In International Conference Learning Representations, 2022.
Nimit Sohoni, Jared Dunnmon, Geoffrey Angus, Albert Gu, and Christopher Ré. No subclass left behind: Fine-grained robustness in grasee classification problems. Advances in Neural Information Pro- cessing Systems, 33:19339 19352, 2020.
Kumar Sricharan and Ashok Srivastava. Building robust classifters through generation of confident out of distribution examples. arXiv preprint arXiv:1812.00239 2018.
Dirk Tasche. Fisher consistency for prior probability shift. Journal of Machine Learning Research, 18(95): 1-32, 2017.
Qinglong Tian, Xin Zhang, and Jiwei Zhao. Elsa: Efficient label shift adaptation through the lens of semiparametric models. In International Conference on Machine Learning, pp. 34120 34142. PMLR, 2023.
Anastasios A Tsiatis. Semiparamebric theory and missing dala. Springer, 2006.
AB Tsybakov and J-Y Audibert. Fast learning rates for plug-in classifiers. Annals of Statistics, 35(:): (08-633, 2007.
E Tzeng, J Hoffman, K Saenko, and T Darrell. Adversarial discriminative domain adaptation. In Proceedings- 30th IEEE Conference Computer Vision and Pottern Recognition, CVPR 2017, pp. 2962 2971-971, 2017.
Eric Tzeng, Judy Hoffman, Ning Zhang, Kate Saenko, and Trevor Darrell. Deep domain confusion: Maxi- mizing for domain invariance. Xiv preprint arXiv: 141.94444 12.3474, 2014.
Wei Wang, Vincent W Zheng, Han Yu, and Chunyan Miao. A survey of zero-shot learning: Settings, methods, and applications. ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):1 37, 2019.
Yaqing Wang, Quamming Yao, James Kwok, and Lionel M Ni. Generalizing from a few examples: survey on few-shot learning. ACM computing surveys (csur), 53(3):1-33 34, 2020.
Yipei Wang and Xiaoqian Wang. On the effect of key factors in spurious correlation: A theoretical per- spective. In International Conference on Artificial Intelligence and Statistics, pp. 3745-3753. PMLR, 2024.
Zhao Wang and Aron Culotta. Identifying spurious correlations for robust text classification. In Findings of the Associalion for Compulational Linguistics: EMNLP 2020, pp. 3431 3440, 2020.
Jon Wellner et al. Weak convergence and empirical processes: with applications to statistics. Springer Science & Business Media, 2013.
Shirley Wu, Mert Yuksekgomul, Linjum Zhang, and James Zou. Discover and cure: Concept-aware mitigation of spurious correlation. In International Conference on Machine Learning, pp. 37765 -77866.7776. PMLR, 2023.
Wenqian Ye, Guangtao Zheng, Xu Cao, Yunsheng Ma, and Aidong Zhang. Spurious correlations in machine learning: A survey. Xiv preprint ar arXiv:2402.11 12715, 2024.
17

---

Published in Transactions on Machine Learning Research (04/2026)
Kun Zhang, Bernhard Schölkopf, Krikamol Muandet, and Zhikun Wang. Domain adaptation under target and conditional shift. In International conference on machine learning, pp. 819 007-827. Pmlr, 2013.
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A Appendix
A.1 Proofs and More Details in Section 4
Proof of Proposition
For = case, note that
p(x R = 0, = (p A 1)
p(x R 0, = 1, 1011 + p(x R - 0, P00101
Thus,
Pn((x) p(x
R= A 0, poi (x)
Pul1
Then,
P11 Pii (x)
pr(Y = | R= 0, 1)
p(x, R = 0, 1)
p(x - 1)r 1 Po1 (x P001
p(x R = 1 -0.0--004-- =
p(x = 0, A 1)po.1 P01 (x) B01 x)
p(x
R 0, - Poo1
Note that
Poi (x)001
pr(R A
poi px)(x)0011 + p(x R 0, = 1)po.1
gives
Poi (x)
pr(R = x, A 1)
p(x R = 0, A = 1)P0-1
0o1 pr = x, A 1)}
Hence,
Bo1 (1 - σ)
pr(R 1 x, A = 1) pr (R = x, A=
x1 1 pr(Y -
=
0, - 11=
QU1
Note that
pxx R= = 1,A = 1) π
pr( p(x
R 1, A
1)001 p(x R 0, A 1)po.1
gives
p(x R = 1, 1)
- 1
1=
p(x R 0,A = 1)Po.1
x= 1)}
Hence,
Bon - T)) pr[R11|xAA]]
| = 0, = 1)
pr x, R 0
001
1 1
R=
For A = 0 case, note that
- 0
R=
Y = mrt.-=11.--114-000022224
pr(x, Y = R = A = 0 -=
- p 0, 1,:2=1A--1 1,
Bia (x
010
3I0 (x( + 1oo (1)
010
00g
19

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Published in Transactions on Machine Learning Research (04/2026)
By Bayes' rule, we obtain the following equation rxx) = (x(x) + (1xxx()-)(()))
Proof of Lemma 1 It is easy to see that, n, a10, a01, Pio(x), Po1 (x) and poo(x) are all identifiable Now suppose that there are two different sets Pı1(x); S10: 8ao and (x1(x), B1o, Poo such that
(1) + (1 - -B1 B10 - poo)pol (x) = (1)51(() + (1 311 310 Poo)po1(x) BIOPIO(x) BooPoo (x).
(10 + (x)
(14)
Now taking the integral with respect to x on both sides of the second equation above, it clear that $10 + 900 = 810 + Boo.
Plugging in back to the first equation above, obtain
B1 f1I - P11 = O.
Since 311- > O, we obtain Pı1 (x) = Pi1 (x). Finally, 14
leads to (510 - 310)P10(x) = (0oo - (oo(),, which
only hold if $10 310 and Boo Boo since Pio F (x) This completes the proof.
]
Proof of Lemma
1 Σ
1 π
p(x = 0,A
p(x =, A= 0k
pr R = 0, A = 0)
k=0
/ p(x|R= = 0,A =0llgg
p(x R = 0, A= 0)
dx
xxx(x k. A 0)3 pF(R=0,A=0)
| 0, A 0)log
p(x R 0, A 0) dx
( R 1, A 0)
( hi A 3)0 A dx
/ p(x|R= = 0, 0)log
p(x R = 1,A=0) 0)
/ p(x|R= -0 0)log
x 0, 0) dx
p(x R 1, = 0)
- -(xx|R=0,44 0:AR=00A00000 O)log
pr(Y k|x, = 1, - x) 1, =0) dx.
pr = = 1, k. 0)pr 0, 0 0)
Minimizing the above equation is equivalent to maximizing
Bko pr(Y = kR = 0)
argmna E log pr = k|x, R = A 0)
R= 0 0
k=
subject to pr(R = 0, A) 31 (1 T) 300 (1 m)
We enforce this restriction constraint in the distribution matching problem: where D is discrepancy between probability distributions on X.
Define
E(Eo, by 310. e) E' (log)6((X)67 810 + (1- (1)))(1-6))) So)][ -e 0, A
Its empirical version is
E(Eo b,, B10, e) E (log(G((X)6)) Bi0 + (1 (1 310) R = 0, =).
20

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Published in Transactions on Machine Learning Research (04/2026)
An Alternative Approach for Estimating 8
In the main lexi, we explore the use of distribution matching for estimating B. Alternatively, it is sufficient to only consider some moments instead the whole distribution. For any measurable function m(x), the law of total expectation yields the identity:
((m(x) R = 0, 0 = 0, =0
1, 1,03 (x)
-) + [mm(x) 0,0 - T).
(15)
Rewriting equation 15 obtain the following linear system:
(1 - x Poo [x(((x) 1,00, [xm(() 00,0]] 8 = [(m(x) R 0, A 0},
which leads to the expression
B = (1 - - Pu.o (m(x)) 1,00, E(m(x) 10,00 E(m((x R = 0, -}:
provided that the 2 x 2 matrix [E{m(x) 1, 1, (xm(x) 0, 0,0) is invertible. To use the idea of moment matching, one has the flexibility of choosing different moments mx). Certainly, a further research ques- tion of interest is to identify the optimal choice of this moment function, say, (x), by borrowing the semiparametric techniques (Bickel et al. 1993
Tsiatis 2006
A.2 Proofs and More Details in Section
We define the Rademacher complexity (Bartlett & Mendelson 2002) that has been frequently used in machine learning literature to establish a generalization bound. Instead of considering the Rademacher complexity FF we define the class of weighted losses G(C,F) [w(x, v)((g()) g Fl and N we define its Rademacher complexity measure
(g) Excryy
Egg sup hEF
Σ Ewlu,, o:)(( u1)
where ( are iid. Rademacher random variables, taking values #1 with equal probability 1/2.
Proof of Theorem
1 For probabilistic classifier: (fo(x), 1 - fo(x)) X - 2 and the parameter ,TT
(310, Boo) and b pr(Y = I = A 0), we define the centered logit function f X' + R as
fo(x) = logEo(x) (x) 2 [logSo(x) + logg1 - {x) and (x = log(1 {x()) [logEo(x) + log[1 (x)
We define the functions p(fo, b1) {x(x)b- - {1 (1 - b and w(fo. b,, 11,810.00 2) = (x)o 810 +
{1 - (x) - -b,) bi)-
B10), and notice that the objective is
i(fo, b,, B10. 2) = E flogw(fo. b, 910. 2) R 0, 4 4= = 0}.
whereas the true objective is
L(fo, bi, Bi0, 0) = E (logw(fo, b1, 810. R = 0,A = 0}.
We see that the first-order optimality conditions estimating Po are
0 0g L( by 3r0. @)
(16)
= Oy,o E { logw by 310. 2) R= 0, [}
B 1eem (w/ Fo. b, 3 0
R = 0, 0, 0
w by B Ø)
21

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Published in Transactions on Machine Learning Research (04/2026)
Similarly, the first order optimality condition at truth (for P10) are
0o T.Jo. bi, B10: !)
99o0 [E flogw (S: b1, 310: 2) R 0, A 03]
10 (w(/o, b1, B10, Ø)}
R 0, 0
w(fo 61, $10. Ø)
We decompose 16 using the Taylor expansion and obtain:
0 = 030 F. 81, $10: 2) + (Po fo. 0p T by, P10. 2))
where fo is function in the bracket [fo, fal, i.. for every x, fo(x) is number between fo(x) and fo(x).
Bound on (fo- for 1r.00.. [(Fo b, 310. 2)):
To bound the term, we define So = Fo - fo and notice that (So. 23 0s. [ by B),,
m( bi)
R R=0.A-00
Co. 0%
w(Fo, b1, B10.
25o - Eo)
[ +(1 b11
So
w(o, b,, 310 2)
b1)
1 310 (1-
R 0, 0,A=00 -0)
"Fo, by, 310, 2)
The derivative third equality in the above display calculated in Lemma
3 Assume @ Q E E> 310 > >E>0 0
and 1
1
bj
61 0, i.e., there exist a 01 > 0 such that
1 1 310 (1 - -bi--'(0 Bnn3]]
2Ea(1-Ea) 7 (1 - -b,--1 31000)
((fo.d))
< <.. This implies the fol-
wi.o.bi wi
lowings: we have
Go 25 - Eo)
1 + (1-b,)-1
w(o. 8,, B10, 2)
m.Fo. B1)
17 310 - - (1-bb) -
(0 310)
R R= (
w(Fo: b,, $10.
e1 (() 0, 00.
It follows from Assumption 1 with probability least it holds supse(mo.o) [l((x,) f(xi)ll2 s
Craia
we conclude that
KSo: 00 I. Dy P10, 2)1 cararo log(())))oo((1/)
holds with probability at least 1 - 6. Bound on 0g,, L(fo. 31, @) an. i(fo: b,, B10. Ø)
Using the taylor expansion, we have
ie i(fo 310. 2) I bı, 310, Ø) (6r b,, Do, 0o L( bi, 31, 2))
m(oo,b))
R = 0, 0 1
61 by. Ob.
((oo, 310, Ø)
=u(() (1 (x(x) - -22)-
= δ (b, b1)
w(fu bn, Bru. 2)
m(For b1)
Ex(()) 310 + {1 Eo (x) by) 20 310)
w(foi by, (10.
w(fo, by B10.0)
22

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Published in Transactions on Machine Learning Research (04/2026)
Assume B10
0 and - bj > 0, ie., there exist C2 such that
(1 (1)
u(Jo.ri)
"xoo ()))))(---)) (00
< C2. This implies the fol-
b/ B10.0)
310..1
w(So.bI.B10.0)
lowings: we have
-So(x)07 - (0 - b,)-
E (b1 b1)
w(Jo: b1 310: 2
p(fo, b1)
-co(x)0 310 + {1 fo(x) 2 (10)
w(Jo: bi, 110.00
w(fo, by 310,07
S 02 bb b1/-
We apply Hoeffing's concentration inequality for a sample mean of i.id. sub-gaussian random variable Y; and obtain a C3 > 0 such that for any with probability at least 8 it holds
log(1/6)
bb = ppr(- 1 1, 0) pr(Y 1 1, 1 0) VVV
11-0
Bound on 03,0 f(fo, b 310. 2) 03 [(fo: bi, F10. e) Using the taylor expansion, have
Op L(fo, b1 310, @) 0g.. I bi 01, 0) (Ø 0.0,0sm I(fo. 310. Ø))
b1) (0- 0) (ufo, b1) 5x) -1-b)-
E
R 0,A 0
w² b1, 310, ②)
Assune > B10 >
0 and - b
0, ie., there exist cs > such that
(11 b² (10.00
< c4. This implies the followings: we have
E 0 0) u(fo. - (0(x) w²( b1. 310 2)
of
R = 0, 0,
< c Ø/-
We apply Hoeffing's concentration inequality for sample mean of i..d. sub-gaussian random variable A, and obtain a cs > 0 such that for any with probability at least - 8 it holds
log(1/8)
el 1prAA 0|R 0) pr(A 0|R 0) cs
no
The term 03 T. 02 310 2): We have
0% i(So, Bio, @) 03 I[ bi, Bio, e) Op bi,
0g,. ī(fo. b,, 310. o) + (pb b,l + +10 (
Now, we study the term 01 L( b, Br0 0), use strong convexity of L(Fo, b, Broi e) with B10 and the convergence of the loss that.
sup (0)
[[ by, S10. Ø) L(fo, b1, S10. 0)
no +00
for B10 E (0. in Wellner et al. (2013) (sce Corollary 3.2.3) to conclude that 310 - B10 in probability and hence Bio consistent estimator for 310-
23

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Published in Transactions on Machine Learning Research (04/2026)
Following the consistency of $10 that for sufficiently large no-o, have 1810 - 310l - وو(وg is chosen bound by 2 We define empirical process
Sa (سسي with probability at least - and on the event it holds: 310 E [310 - 63, 310 + 6z].
Z
sup SE[ 10.310
for which we shall provide high probability upper bound. We denote Zoo (3) OgI(fo.bl. 3,0)) igL(fo,br, 3, e) and notice that
OzL(fo. b1, 3.0) (1,
E
b1) R R- 0 E w((fo, 3) w((f. 3)
p(fo,br) b1)
R = 0, 100,A-0 A(B)
where to bound A(3) we notice that ((fo.b))
W(o..b,.f.g. areiidd. and bounded by co(3 (0 Bio e-110 S co for all x E X) and hence sub-gaussian. We apply Hoeffding's concentration inequality for sample mean if i..d. sub- sub-gaussian random variables and obtain a constant > 0 such that for any ó > 0 with probability at least 1 - it holds
A(3) = E { M(Jo, bn)
R = 0, A -.0.==0- E' m(oo,br bn) W(Jo b1, 3)
R= 0, of
W(Fo, bi, B,
log
I Eouoo
nuo00
Use chained arguments for & with interval length 26g we obtain uniform bound as the following: there exists constant 07 such that for any 8% > 0 with probability at least 1 if it holds
log(1/0)
sup BElB.o -5 +5x1
A(B) I COCOCT
no.0
Therefore, with probability at least 8, we have
log(1/6)
Zmoo S coc607
no-0
Returning the first order optimality condition for estimating In we notice that
0 (B10 - 310) { sa.. [(fo. by 310 2) + (Fo fo. fo 0z L(F. by B10. 2)}}
(B10 - B10) { (sn, i( by 310 2 - 03.0 i( by B10. e) + 02 i by, 310, 2)
+(Fo fo. o 0n L by 3 Ø)
(B10 - B10)0 L( by 310. 2)
+810 310) 10% T. Tn 310, 2 0310 To by Fro, e) + Zno (210)
+(Fo fo. or 8r L b, 3 }))}
We combine it with the first order optimality condition for B to obtain
(310 - 310) { (d L(fo. b, Pio. e) - 0g.0 L(fo, by 310 310.e) e²}
+ +B10 - Buo) { (0 i 8n Br0. 2) 03.0 i(fo. by Bro, e) + Znoo (Bro)
+ +io fo. 0 0n B[,o, x) = 0,
24

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Published in Transactions on Machine Learning Research (04/2026)
which can be rewritten
- (10l { (0 L(fo,1, P1o. 2) 03.0 [[o.[[oo.... b, 910 e²
(17)
(B10 B10) { 08 [(Jo, by 3i0, 2 0m I[Jo, by B10, 2) + Znoo (310)
+(Fo - fo. R Br L(o by B10. Ø))}
Using the strong convexity of function -L at B10, we obtain that the left-hand side in the above equation is lower bounded as
- B10) { (8 L(oo b,, B10. e) 03 b1, 810 2)} u(3i0 B10)²
(18)
Let & be the event which the following hold:
1810 S10l S 63.
Iifo - fu, 05 22,0 [(o, 3,0 2)) & c o (logmmo))log(1/))
Znoa S c006erV
(o5(1/8)
no-
10310 L(fo, B10. 2)
Sro I(oo Bi0. 0)1 s (c203 + cgcs)
log
n1.
We notice that the event & has probability 1 56. Under the event there exists 08 > such that the right-hand side in 17 is upper bounded as
(8io B1o) {on [(oo B1, @) 0: Li b1, 2) Zng (B10)
(19)
+iFo fo. oy Gg [ by 310 e)
1ro 31l { 103,00 i by 310 0) 0310 [n by 310, 2)1 Znno. (310)
s
+ [fo - fo. 0 0 I(Jo Bio. 2y
log(L/6)
log(1/6)
log(L/o)
<
os mm omoo (ogg
Gau Szul-
711-0
no
nu-o
Combining the bounds E it holds
and 19 for left and right hand sides we obtain a Cιo > 0 such that on the event
log(1/o) n1-00
log(1/0)
log(1/0)
1i00 Biol S cio
Toio v /o) +
+
n1
no
Further, since A, is bound random variable, then can obtain constant Co 0 such that for any ō > 0 with probability at least 1 - it holds
log(1/6)
1300 3ool = 100 - 310) (e - B10) S le el 1310 - Biol s co
+ 1310 - Biol-
In summary, have constant C10 such that for any 8 > 0 with probability at least 1 - 66 it holds
log(1/8)
log(1/6)
log(1/6)
|| 111 s cso
Teio (log(non))og(11)++
+
n1-o
n7p
no-o
25

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Published in Transactions on Machine Learning Research (04/2026)
Lemma 3. (Derivatives). The following equations hold:
(56) 25o(1 Eu);
ds Moluuoo. b.)} 0 (G0)(bb + (1- b1)
Sso (wotw bI, B10, 0)} = 0% (Eo) (60) 107 310 (1
Oso {
u(fo.b1)
(50(1-E0)
01 + -b)-))=- (( 810 (1-b1) '00 BIo)]
(1100,s, (10
(10001
Proof.
0 0% cto te- - eto (efo te e (efo - -c- = =25o(1 En);
elo
(elo + + 50)2
0% fu(fo, b1)) = = (Eo) {0j + (1-b,) b,)
0 (W b1, 310, 0)} 0 (Eo (6o))) 3/0 (5o)(()( - -1))~((- -B10)
0 (b) 310 (1 ( B10)}.
Thus,
p(fo,bi)
0 (Jo B10
= S (5o) + (1 - b1) b1) (w bI, 0) - 11 bi b1 310 -1--b)) (0-100000)) -10)
w² b1, B10,
(Eo) w(fo, b1. 310. Ø)
b1)
B10 (1 b1) b 310)}
b1 + (1 (1-bb)-
b1)
w(fo b1 810 2)
= 25o(1 E0)
01 + (1 - 61) bifo.bi)(0 B10 (1 b1) ( - 310)}
w(Fo, b1, 310,0 e)
bi B10, 2)
Proof of Proposition
Define w(y) = pr(vlA=0,R=00)
R
(h(
= / ((((()) Yl(xx), 0)x
I
p(x, y 20:
- R
0, 1)xy
/ -r pr(: y/ 0- 1)xy
pr(y)1 0,
0, R = =: T., (h,
Let Lh) E. ( R 1], then have
Lo( - Lo(h) = Li w) - (h, w.)
Li w) E, (, w:) + Ei w) - 2,, 1)
(20)
(a)
(6)
+ 2, 1) - (n t ()+ + Ei 0) - Er w) + 22 w) - L1 (h, w)
<0
(0)
(d)
where πn = Nz.
26

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Published in Transactions on Machine Learning Research (04/2026)
Uniform bound on (a) To control (a) in 20 we establish concentration bound the following gener- alization error
sup(Lig,, - (igg,
gEF
E' (()))) 0, R- [elg(X), -B (((()) =0 =1]}
sup SEF
= : F(1mm1o)))
where, for i > 1 we denote Zi = (Z),... Zz) and Z, = (X,, Yi). First, we use modification of McDi- armid concentration inequality to bound (Zi in terms its expectation and a Op(1//71.)) term, as elucidated the following lemma.
Lemma There erists constont 6j > such that with probability at least ô the following holds
log(1/6)
F(ZimLo) S ((F(ZI + 01
(21)
11.
The proof is similar Lemma A.3 of Maity et al. 2022). so we omit it.
Next, symmetrization argument (sce Wellner al 2013). Chapter 2, Lemma 2.3.1) to bound the expectation E E(F(Z by the Rademacher complexity of the hypothesis class G, i.e.,
E{F(Zim,o)} S o
(22)
Combining 21 and 22
obtain
log(1/0)
(a) < o + eit
(23)
71-00
with probability at least 1 1 8.
Uniform bound on (b) and (c) Denoting 2, = (X,, Y,) and (Z.)) = (lo(X.), Yi) we notice that for any F we have
(g Li(g,
1 u1.o
[(((o(X), Y) (w(Y) ((Y)) A= R, = 1]15
Σ
(w(ys)) ((v:)l-
71.0
Since w(y) ((y) is sub-gaussian random variable, we use sub-gaussian concentration to establish that for some constant C2 0,
log(1/0)
for any g E F,1 Ei(g,00 s 18 Ey|wu(V) - a(Y)) + 02
111-0
with probability at least - 8. This provides a simultancous bound (on the same probability event) for both
(b) and (c) with g h and g h. Further, by Lemma for some constants C₁ and C2 and any g F,, have
EL w) - Ei(g. x)
(24)
log(1/0) 710
log(1 (1/8)
log (1/8)
< lo 11113 - B111 Co
+
+ 02
71
71.00
27

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Published in Transactions on Machine Learning Research (04/2026)
with probability at least 1 56.
Uniform bound on (d) We note that
[i w) (i(,,
[(e)((x), Y (Y) 0, 0 (3()) -
where (e)h(X;), [=) are i..d sub-gaussian random variables. Using Hoeffding concentration bound conclude that there exists constant Cg 0 such that for any å >> 0 the following holds with probability at least 1 1-6. 6,
log(1/8)
w)) Li(h, w) sco
(25)
n1.0
Finally, using 23
(a) (which is true
event of probability 2 1--), 24 on (b) and (c) (simultancously
true on an event of probability 1 - 56), and (25 on (d) (holds on an event of probability 2 6) we conclude that with probability at least 1 - 75 the following holds
log(1/0)
log(1/6)
log(1/5)
Lo(T Lo(h) < (
B||1 +
11-00
m
where c == = c 111llgla (C2 + 02) c3.
Lemma 5. Assume /30//a0 S B1 for any = 0. 1. There erist conslants C, c1, 02, such that with probabilily at least 1 - 48,
(V
/log(1/0) 110
log(1/0)
(ū(y) w<vv)l S CB B|| CBi +22)
n1
Proof.
pr(y 0, R= 0)
O, 0)
(w()) - w(y)
pr(y) A 0, R 1)
pr(y = 0, 11
pr(y, A = 0 0) pr(A 0 =1
pr(y, A 0|R 0) pr(A= 1)
pr(y, A 0 1) (r = 0 = 0)
pr(y, A= 0| 1)
pr(y, R| 0)
Ar =
=| = 0) 5r = [|AA0RR=]
pr(y, A 0 1)
pr(y, A = 0 = 1)
pr(A = 0 0)
pr(y, 0 0)
pr(A = R 1)
pr(A 0 1)
pr(y. A = O = 1)
pr(A = R 0)
pr(A = 0 = 0)
For the first term, we have
3y0 8y0 n1-o no
ayoā0
no-0
(Byu B00 + Byolago
ayy)) n1-0 no
ayoā0
na-00 T&
youo
N1-0 nu
310 31ol +
(1 - y)ayo 71-o no
300 - Bool
no.0 ag
n
agoāvo
noo00
y/g0 71.o ng
(1 B) N1-o no
(010 - 2n0) +
(000 200)
agaayo No.0
ayoāgo
nu-00
< C B11
100 aool
| onn)))
28

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Published in Transactions on Machine Learning Research (04/2026)
Here C' is some constant. Since Y, and A, are sub-gaussian random variables, we use sub- sub-gaussian concen- tration to establish that for some constant >> 0,
log(1/6) 11
T.0o - Cool + [010 a10l & c
with probability at least 25.
For the second term, we have
Byo pr(A- R = 1
Ay0
pr(A 0 = 0 0)
Syo
pr(A = 0//2 = 1)
pr A 0 0)pp(A 0 R (pr(A 0| 0)
0))
Oy0
Syo
pr(A = 0/R 0)
0/ = 0)pr( A 0 = 0) (pr(A 0 1) pr 0 1)}
0'
pr( A
Syo
P1-0(1 m)
|(pr(A 0 0) pr(A 0 0)}
0y
Po.OTTRO00
Byo
no
+
І(r{A 0 1) (A 0 1)}
0yy
Rg-00
(M R = pr(A 0/R 0)} Kpr(A = 0| 1) - pr 0 1)))-
Here C is some constant. Since A, sub- sub-gaussian random variable, we use sab--gaussian concentration establish that for some constant 0,
Hpr(A- 0 =) pr(A 0 0)} + (pr(A = 0|R 1) pr 0 1)}
log(1/0)
log(1/6)
n2
71
with probability at least 1 25.
29


Reviews:
[{"review_id": "7bghoybQn6", "reviewer": "Reviewer_C8uy", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors have focussed on a quite specific problem in domain adaptation. They have access to features $X$, binary labels $Y$ and binary background variables $A$ in training. They want to train a classifier for $Y$ based on $X$ and $A$, so both $A$ and $X$ are available for the test set.  The crux is that the combination $A=Y=1$ is never observed in the training set. This poses problems of identifiability and estimation.\n\nThe authors lean on methods similar to positive-unlabeled learning to develop a distribution matching estimator. They derive theoretical properties of this estimator under extra assumptions. They compare the estimator to a ‘naive baseline’.", "claims_and_evidence_yn": "No", "claims_explanation": "While the work is initially quite compelling, I think it needs more work to be clear and convincing.\n\nThe ‘naive baseline’ proposed, as per equation (5), has to be constantly zero. Since it is $\\\\xi_1(x)=pr(Y=1|X=x,A=1,R=1)$ but we also, by construction, know that $pr(Y=1|A=1,R=1)=0$ . Because of this, I wonder what the whole point of section 4.1 is. It also makes the benchmarking method of the experimental section quite strange. We benchmark against the null predictor and call it the “standard benchmark”. I feel this is misleading. I also realize addressing this problem is exactly the point of the paper, but since it is never spelled out, and almost half a page is given for this ‘naive benchmark’, I feel this is not communicated clearly enough.\n\nThe writing is a bit unclear. Trhoughout, the word ‘model’ is used in an imprecise way. Often within statistics, a ‘model’ means a parametric family. In the ML community, ‘model’ often means an algorithm to fit a function to data (as in “XGBoost is a model that …”) or a function that was obtained after using such an algorithm. In this work, ‘model’ seems to be synonymous with conditional expectation functions or pdfs. I think the authors should make their wording more precise to not be misunderstood. This is even more so a problem in Proposition 1, where three ‘prediction models’ are derived. These seems not to be models, but just analytical expressions for certain pdfs.\n\nIn section 4.5, a result about risk (expected loss) estimation is squeezed in. This result is interesting, but poorly motivated. Notation for a ‘loss function’ and some classification function ‘h’ is introduced without much context. It is revisited in proposition 2, and maybe the article would be clearer by just moving that to the theoretical results section?\n\nIn chapter 6 on synthetic data studies, the nuiscance parameters $\\\\xi$, $\\\\xi_0$, $\\\\tau$ etc are all assumed known. The authors write “we utilize the known data-generating distribution to compute the conditional probability models required” . While it can be good to isolate sources of variability it also hides the problem that inverse probability estimators like equation (5) can be very sensitive to these nuiscance parameters. I think you should supplement this analysis with one where the nuiscance parameters are estimated. You do this in the waterbird study, but I would like you to do that in the synthetic setting as well.\n\nFinally, I think you should present some more interesting benchmark for both synthetic and waterbird study than the simple null predictor of section 4.1.", "audience_yn": "Yes", "audience_explanation": "The problem of structured missingness is highly interesting. Using a distributional matching method like in this work could be a viable way forward, and I think this work holds some promise.", "requested_changes": "Clarify that $\\\\xi_1$ is always zero out of necessity. Connect this to the identifiability problems discussed in section 4.3\n\nClarify terminology and notation. ‘model’. The function ‘h’.\n\nImplement some other benchmark that has any chance of providing interesting results.", "broader_impact_concerns": "none", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ncOa4n4uKX", "reviewer": "Reviewer_iLYK", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies an unsupervised domain adaptation problem where somehow subpopulation of instances are unobservable.  The core idea is that “accurate estimation of the parameter in the target domain enables valid model adaptation across domains”, and the authors reformulate the estimation of such parameters as a constrained distribution matching problem. The authors use synthetic data to verify the effectiveness of the proposed methods, and put into practice on Waterbirds dataset. \n\nThe major contributions are summarized as follow:\n1. By modeling the conditional probabilities, the authors reformulate the estimation of target domain parameters as a constrained distribution matching problem.\n2. The proposed theoretical framework is being properly proven and verified by synthetic data.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The proposed theoretical framework is being properly proven and verified by synthetic data.", "audience_yn": "Yes", "audience_explanation": "This paper studied domain adaption problem in the setting of binary classification. In practice, dedicated binary classification models are widely used, and the domain adaptation problem is also prevalent. The theory proposed in this paper is meaningful for situations where the target domain is relatively clear and does not change frequently.", "requested_changes": "1.\tIn introduction, A and Y should be explained or exampled before introducing.\n2.\tExplain the difference between “unobservable subgroup ” and relevant topics such as “class imbalance”, “zero-shot”, etc.\n3.\tAssumption “Under a mild assumption that the distribution of features X given (Y,A) stays the same across domains,” in introduction may not hold in general domain adaption settings. If this assumption is made, the applicable scope of the domain adaptation method proposed in this paper will be narrowed.\n4.\tThis paper studied domain adaption with unobservable subgroups, yet the related works focus on OOD detection. Maybe correlated, but not strong enough. The readers would like to see “does unobservable subgroups truly an important issue” or “did anyone tried to solve it”.\n5.\tWhy is that “the distribution p(x | R = 0,A = 0) is identifiable from the target population.” In section 4.4?\n6.\tThe experiment is weak. Resnet-18 and Waterbirds dataset are outdated, other baseline method is missing. Modeling such conditional probabilities with modern pretrained models like ViTs COULD MIGHT result in promising results, but experiment is needed.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Fsf56YlUve", "reviewer": "Reviewer_JJgV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper tackles a challenging UDA scenario where a specific subpopulation, defined by a combination of a binary label ($Y$) and a binary background ($A$), is completely missing from the source (training) domain. The authors' primary contribution is demonstrating that despite this \"structured missingness,\" target domain predictions can still be recovered. They derive closed-form expressions for target prediction models under an assumption of conditional feature invariance, propose a distribution matching method using KL divergence to estimate unknown subpopulation proportions in the target domain, and develop a practical algorithm that avoids the need to model complex, high-dimensional feature distributions directly, instead relying on standard classification techniques.\n\nStrengths\n\nS1. Practical Relevance: The problem of unobserved subpopulations is common in real-world data collection, such as in healthcare or ecological studies, where certain groups are systematically excluded.\n\nS2. Theoretical Rigor. The authors provide solid theoretical guarantees, including proofs for model identifiability and upper bounds on prediction error.\n\nS3. Algorithmic Efficiency: By using KL divergence for distribution matching, the method sidesteps the difficult task of generative modeling in high dimensions.\n\nS4. Empirical Success: Experiments on both synthetic and real-world datasets show that the proposed method consistently outperforms naive benchmarks that ignore the missing subpopulation.\n\nWeaknesses\n\nW1. Missing Literature: While the paper correctly identifies gaps in standard UDA, it would benefit from a more explicit discussion of how \"Global Feature Alignment\" methods - such as Adversarial Discriminative Domain Adaptation [1] - behave in the presence of missing subpopulations. Discussing whether adversarial alignment would \"collapse\" the unique target subpopulation into existing source clusters would better motivate the need for the authors' specific recovery formulas.\n\nW2. Strong Reliance on Conditional Invariance: The methodology hinges on the assumption $P(X | Y, A)$ being invariant across domains. In real-world scenarios, the background $A$ itself might undergo a shift (e.g., different sensors or environments), which would violate the core assumptions of the paper. A discussion on the sensitivity of the model to such violations is missing.\n\nW3. Potential for Error Propagation: The practical implementation relies on a \"naive\" initial estimator $\\xi(x)$. If this initial model is significantly biased due to the missing subpopulation, those errors may propagate through the KL-matching step and into the final target predictions ($\\eta_0, \\eta_1$).\n\nW4. Scalability Constraints: The current mathematical derivation is elegantly optimized for the binary-binary case (label and background). The paper lacks a roadmap for how this framework might scale to multi-class settings or continuous environmental variables.\n\nThe paper presents a compelling solution to a significant and practical problem in machine learning. My overall assessment is that the paper is strong and theoretically rigorous, addressing a specific gap in UDA where standard techniques often fail. I recommend acceptance.\n\n[1] Tzeng et al, 2017, Adversarial Discriminative Domain Adaptation, https://arxiv.org/abs/1702.05464", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper presents theoretical and empirical evidence. The math looks solid and non-trivial. The empirical evidence includes both synthetic and real-world datasets.", "audience_yn": "Yes", "audience_explanation": "This paper addresses a specific intersection of problems that would highly interest several distinct subgroups within the TMLR audience robustness and OOD researchers, algorithmic fairness and AI ethics community, causal inference and invariance practitioners.", "requested_changes": "Critical for Acceptance\n\nC1. Expanded Literature Discussion: The authors should add a discussion ADDA and possibly related works. The discussion should address the theoretical failure mode of these methods in the \"structured missingness\" setting (i.e., how adversarial alignment might \"collapse\" the unobserved target subpopulation into an incorrect source cluster).\n\nC2. Addressing Violations of Conditional Invariance: The entire recovery framework rests on the assumption $P(X | Y, A)$ is invariant. The authors must include a section (or a dedicated paragraph in the Discussion) addressing what happens if the background/environment $A$ itself undergoes a shift in the target domain. This is a common real-world occurrence and constitutes a primary challenge for the proposed method.\n\nC3. Clarification of the \"Naive\" Benchmark: The current experiments compare the proposed method to a \"naive\" model. The authors need to explicitly define what this naive model is (e.g., is it simply a model trained on the biased source data without any adaptation?). Providing a more competitive, yet mis-specified, baseline - even if just in the literature discussion - is necessary to prove the value of the derived formulas.\n\nRecommendations:\n\nR1. Sensitivity Analysis of KL-Matching: The distribution matching step relies on estimating proportions ($\\alpha, \\beta, \\gamma$). It would strengthen the paper to include a sensitivity analysis showing how errors in these proportion estimates (or the choice of the initial estimator $\\xi(x)$) propagate to the final classification accuracy.\n\nR2. Connection to Causal Invariance Literature: The paper’s assumptions align closely with invariant causal prediction (e.g, [2]). Explicitly citing these works would provide a stronger theoretical pedigree for the \"structured conditional invariance\" assumption used in Section 3.\n\nR3. Roadmap for Generalization: Briefly discussing how the binary $(Y, A)$ framework could be extended to multi-class or continuous settings would help researchers understand the broader potential of this work beyond the specific binary case presented.\n\n[2] Peters et al., Causal inference using invariant prediction: identification and confidence intervals. JRSS B, 2016.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)