Automatic Visual Instrumental Variable Learning for Confounding-Resistant Domain Generalization
Authors: Fuyuan CAO
Shichang Qiao
Kui Yu
Jiye Liang
Venue: NeurIPS 2025 poster ()
Keywords: Domain generalization
visual instrumental variable
causal representation learning
Primary area: general_machine_learning
TL;DR: We propose a novel approach, VIV-DG, that automatically learns visual instrumental variables to enable confounding-resistant domain generalization, even in the presence of unobserved confounders.

Abstract: Many confounding-resistant domain generalization methods for image classification have been developed based on causal interventions. However, their reliance on strong assumptions limits their effectiveness in handling unobserved confounders. Although recent work introduces instrumental variables (IVs) to overcome this limitation, the reliance on manually predefined instruments, particularly in the context of visual data, may result in severe bias or invalidity when IV conditions are violated. To address these issues, we propose a novel approach to automatically learning Visual Instrumental Variables for confounding-resistant Domain Generalization (VIV-DG). We observe that certain non-causal visual attributes in image data naturally satisfy the basic conditions required for valid IVs. Motivated by this insight, we propose the visual instrumental variable, a novel concept that extends classical IV theory to the visual domain. Furthermore, we develop an automatic visual instrumental variable learner that enforces IV conditions on learned representations, enabling the automatic learning of valid visual instrumental variables from image data. Ultimately, VIV-DG inherits the strengths of classical IVs to mitigate unobserved confounding and avoids the significant bias caused by violations of IV conditions in predefined IVs. Extensive experiments on multiple benchmarks verify that VIV-DG achieves superior generalization ability.

Automatic Visual Instrumental Variable Learning for Confounding- Reehinnttt Domain Generalization
Fuyuan Cao Shichang Qiao'; Yu', Jiye Liang'
'chooo of Computer and Information Technology, Key Laboratory of Computational
Intelligence and Chinese Information Processing of Ministry Education, Shanxi University, Taiyuan, China
Shanxi Taihang Laboratory. Taiyuan, China
Schooo of Computer Science and Information Engineering.
Hefei University of Technology. Hefei, China cfy@szu. edu. cn, qiaoshi chaossiccanngexuu edu. cn, yukui wufffthuu..... www...eeu.......m..mmmoommuueeee cn, ljy@azu. edu.
Abstract
Many seffoonnding-essistaan domain generalization methods for image classifica- tion have been developed based on causal interventions. However, their reliance on strong assumptions limits their effectiveness in handling unobserved confounders. Although recent work introduces instrumental variables (IVs) to overcome this limitation, the reliance on manually predefined instruments, particularly in the context of visual data, may result in severe bias or invalidity when IV conditions are violated. To address these issues, propose novel approach automat- ically learning Visual Instrumental Variables for cesooouddnngresisstai Domain Generalization (VIV-DG). We observe that certain non-causal visual attributes in image data naturally satisfy the basic conditions required for valid IVs. Moti- vated by this insight, we propose the visual instrumental variable, novel concept that extends classical IV theory to the visual domain. Furthermore, we develop an automatic visual instrumental variable learner that enforces IV conditions on learned representations, enabling the automatic learning of valid visual instrumental variables from image data. Ultimately, VIV-DG inherits the strengths of classical IVs to mitigate unobserved confounding and avoids the significant bias caused by violations of IV conditions in predefined IVs. Extensive experiments on multiple benchmarks verify that VIV-DG achieves superior generalization ability.
Introduction
In machine learning, enhancing model generalization under distribution shifts remains critical [ To tackle this issue, domain generalization (DG) has emerged as a prominent research area that aims extract knowledge from source domains and effectively generalize unseen target domains [3] 4). Recently, numerous DG methods have been proposed, such as adversarial learning (5). domain augmentation 66, 71 invariant representation learning (8) explicit feature alignment (9], and meta-learning 100 12) which have achieved significant progress. However, these methods
primarily rely on statistical correlations, which may still be insufficient for addressing domain shifts
[13]. This is because domain shifts are typically accompanied by confounders that simultaneously influence both features and labels, introducing confounding effects and creating spurious correlations between them. Consequently, methods based purely on statistical correlations may inadvertently model these spurious associations, compromising their generalization capability.
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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To avoid spurious correlations, researchers employ causal intervention techniques 14 (such as backdoor adjustment, frontdoor adjustment, and instrumental variable methods) to mitigate con- founding effects and learn cross-domain invariant causal representations. However, these techniques rely on strong assumptions and face significant limitations in real-world applications. For example, DIR-RelD [15] adopts backdoor adjustment to develop a more generalizable person re-identification framework. Yet, backdoor adjustment assumes that all confounders are observable, while unobserved confounders are commonly present in practice, limiting its effectiveness. FAGT [16) utilizes front- door adjustment to mitigate confounding effects, but depends on valid mediators, which are often diffieult to identify and validate in many tasks. In contrast, instrumental variables (IVs) provide an alternative solution that avoids explicitly modeling confounders or relying on mediators. IV-DG 17 incorporates the instrumental variable framework into domain generalization by adopting predefined strategy that treats data from one domain as instruments for another. However, such predefined instruments often violate the exclusion restriction. Although they originate from different domains, features extracted by pretrained encoders for images of the same category typically exhibit cross-domain discriminability. This violates the core assumption of IVs, rendering them weak or invalid, and ultimately impairing generalization performance. In addition, predefined IV' rely heavily on expert knowledge and require substantial human effort. Taken together, existing methods remain inadequate in mitigating confounding effects, especially those caused by unobserved confounders.
Moreover, shown in Figure Ia and Figure [bb] existing DG methods based on eliminating confounding effects of- ten rely on prior structural causal mod- els (SCMs) to partition inputs into causal factors (such as shape and con- tour) and non-causal factors (such background, style, and color), treating all non-causal factors as confounders to be suppressed. However, this coarse- grained division and the paradigm of uniformly treating all non-causal factors as confounders have notable limitations. In image data, there often exist non-causal features that are stable across domains (c.g.. the color and texture of the main object). If these features are indiscriminately treated as confounders along with domain-specific factors (such as background color), the model's discriminative capability may be undermined. For instance, although object color and texture may not be sufficient on their own to determine cate- gory labels (e.g.. yellow patches from a yellow cat and a yellow dog may look similar), they can significantly enhance discrimination when combined with shape and contour cues.
V. Noo caso-oassallaa Suctors
San-rooa fooues
Lahal (Dag)
Image
Creoi 000und
N: Crusol Tactoes
(Deg)
(Dug)
(a) Decomposition
(b) Prior SCM
(c)
Proposed SCM
Figure 1: Comparison of proposed and prior SCM.
In this paper, we propose a novel approach, VIV-DG (Visual Instrumental Variable for Domain Generalization), which effectively mitigates confounding effects and preserves valuable non-causal information by automatically learning visual instrumental variables from image data without relying on strong assumptions. However, obtaining valid IVs remains a significant challenge. Fortunately, we observe that image data typically contain some non-causal visual attributes that satisfy the conditions for IVs. Inspired by these observations, we unify the two critical instrumental variables and preserving discriminative non-causal features into a single learning objective: autonomously learning representations from images that satisfy the IV conditions and using IV regression to mitigate confounding effects. Specifically, as depicted in Figure Ic] categorize non-causal factors into confounders and IVs, and establish more refined SCM to analyze the causal relationships among variables. Furthermore, we construct a learning framework for IVs, composed of three alternately optimized subnetworks: (i) causal feature extractor that extracts causal representations and performs classification tasks; (ii) a visual instrumental variable learner that learns valid IV representations by enforcing relevance, independence. and exclusion constraints; (iii) a regression predictor that predicts causal factors unaffected by confounding effects. The entire framework is trained with an alternating optimization strategy, progressively yielding more refined IV and causal representations. Ultimately, VIV-DG inherits the advantages of classical IVs in mitigating unobserved confounding, while avoiding the significant bias that may arise when predefined IVs violate their conditions. Moreover, it reduces excessive reliance on expert knowledge and the high cost of manual design. Experiments multiple benchmarks show that VIV-DG significantly enhances domain generalization performance. To sum up. the contributions of our work are as follows:
iarennses-ddennffiinggn valid
we finely
2

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We propose VIV-DG, novel approach that automatically learns visual instrumental variables effectively mitigate the effects both observed and unobserved confounders, resulting in improved domain generalization.
We define the novel concept of visual instrumental variables and develop learner that auto- matically learns valid ones, effectively overcoming the severe bias caused by violations of IV conditions predefined approaches.
Extensive experiments on multiple real-world benchmarks verify the effectiveness and advantages of VIV-DG, demonstrating improved generalization ability.
2
Motivation
We draw inspiration from two essential observations, which collec- tively serve as the foundation for the development of our approach.
Intrinsic differences among non-causal factors. We observe that non-causal factors image data exhibit substantial intrinsic differences. As shown in Figure[2] local textures and color patches of object, although not directly determining the category, can provide discriminative features when combined with causal fac- tors such as shape and contour, aligning with human cognitive perception. In contrast, combining background elements like grass or flowers with object contours often leads to biased or inconsistent recognition.
Figure 2:
Non-causal factors
comparison.
Certain visual attributes serving as IVs. We further find that when causal factors such as object shape and contour are treated as treatment variables, intrinsic visual attributes of the object, such as color and texture, often satisfy the conditions for IVs [18]: : (i) Relevance: Object color and texture are biologically or physically associated with shape and contour; (ii) Independence: Object color and texture are primarily determined by inherent genetic or physical properties, and are thus independent of confounders; (iii) Exclusion: Object color and texture do not directly determine the category label but influence classification indirectly by affecting the object's 's shape and contour.
Methodology
We propose the VIV-DG approach and develop a multi-stage alternating optimization strategy to progressively refine both visual IV' and causal representations. The framework illustrated in Figure
Extracting Initial
Encoder
Factor Refinement Debiasing
Augmaration
MMD
Chocsiferr
Chaselfeer
encoder
Phase II: Visual Instrumental Variable Loarnar2, Cui.
Trainable
Frozen
Misk
Initial Causal Factors
Car
Visual IV's
Confounders
leamer
Eer Ralevance
Independence
Exclusion
Dehicsed Causal Factors
Training Rogressor IV-based Factor
U2 Phase Freeze Gi Inin TV learnan
Phase III: Frees G., 2,, train R,
Regrosor
R,
Regressor
R
Phase 84: Preeze 2, R2 fine tuee
Figure 3: Overview of the VIV-DG framework, which consists of multiple alternately optimized phases: (I) Initial causal factor extraction, (II) Visual IV learning under IV constraints, (III) Causal prediction robust to confounding, and (I+) Causal factor refinement and debiasing.

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3.1 Visual instrumental variable
Observing that certain visual attributes naturally satisfy instrumental variable conditions, we propose the concept of visual instrumental variable (Visual IV) to better exploit such valuable information.
Definition 1 (Visual instrumental variable). Suppose the visual space contains triplet (X, Y, S), where X denotes a visual object (e.g.. an image), Y is the label for a downstream task, and S represents specific causal factor in the visual object such that S -> Y. A visual attribute Z. is defined as visual instrumental variable (Visual IV) if it satisfies the following three conditions:
(i) Relevance: Z is significantly associated with the causal factor S, i.e., Z 14 S;
(ii) Independence: Z is independent of the confounders U, i.e., Z 11 U:
(iii) Exclusion: Z affects Y only through its inftuence on the causal factor S, i.e. Z 1L Y
3.2 Proposed SCM for DG with IVs
Building the Visual IVs, we establish a refined SCM that explicitly incorporates both IVs and unobserved confounders, shown in Figure 4 In this model, X represents the input data (e.g., images), and Y represents the class labels, while U, C. Z. and S collectively constitute the generative factors of
Input data
confounders
confounders
the input data. Specifically, S represents causal factors (e.g., the shape and contours of the object), while U, C, and Z are non-causal factors. Here, U denotes observed confounders (e.g.. IVs
Causal factors
Labels
the background and style of the current image); C' represents unobserved confounders (e.g., visual factors that vary across Figure 4: Proposed SCM for DG.
unseen domains); and Z denotes Visual IVs (e.g., object color and texture) that influence X but do not directly affect Y. All the detailed connections in the SCM are presented in the Appendix C
Notably, since the confounders U and C simultaneously affect both the causal factors S and the labels Y, they establish backdoor paths between S and Y: S ++ U + Y and S C + Y. As result, in addition to the causal relationship S Y. spurious correlations are introduced. leading to confounding effects 19 20]. Conventional estimation of P())S) results in a biased estimator that absorbs both the true causal dependence and spurious correlations, limiting generalizability.
3.3 Disentanglement of causal factors and confounders
Guided by the proposed SCM, we employ the widely used Fourier-based data augmentation to learn representations of the initial causal factors (mainly including the shape and contour information of the object) and confounders.
Causal factor extraction. The phase component obtained via the Fourier transform 21 of an image primarily encodes semantic (i.e., discriminative) information, whereas the amplitude component mainly captures style-related (i.e., non non-discriminative details [3] Based on this observation, we perturb the amplitude of the original image x, while preserving its phase, and generate augmented version denoted as x" We then learn the consistent representation of x, and x as the initial causal factor through a consistency constraint using maximum mean discrepancy (MMD) 22 Specifically, for an image X,, its Fourier transform F(x,) is represented as
F(x) A(x;) -xP((x))
(I)
where A(x;) and Pxx.) denote the amplitude and phase components, respectively. To generate the augmented image xf that emphasizes phase information, perturb the amplitude component as
A' (x.) (1 (xx) (x;),
where X, is randomly sampled from arbitrary source domain, and a (0.55, 1) controls the perturbation strength. We then reconstruct the image by combining the perturbed amplitude A'(x;) with the original phase P(x,) through inverse Fourier transform:
x = F- (A' (x,) x (xx))x-ixPPee xxxxx(()))

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To learn consistent representation (i.e., the initial causal factor) between the images and x we employ a shared encoder Go()) extract their features: r, Go (x, and r Go ). We then leverage the MMD 22 to enforce representation consistency. The MMD loss is defined as
A1 n1
LMDD =
+
(x
(x
H
where and n2 denote the numbers of original and augmented samples, respectively. Here, = 72-- The function (-)) maps imputs to reproducing kernel Hilbert space (RKHS), denoted by H.
To capture class disermmiicriiieetiee information from both the original image x and the augmented image x², apply classifier / to the extracted features and r", producing predicted label distributions ÿ. The classification loss is computed using the standard cross-entropy:
Cels = E (x.x)))p(xx) | log om (f: 0)) + E (xe "=)""""))(>""""""
y-y" log om (f (x); (5)
where Fm is the softmax activation and f(-) denotes h o 9-)) ). To further encourage representation consistency, we incorporate the LMDD loss, and define the overall loss for learning causal factors as
Leausal Li MLMDD
(6)
where and are trade-off hyperparameters. The learned consistent representation is used as the initial causal representation, i.e., the initial causal factor s;, for the original image x,.
Confounder extraction. To extract confounders without relying domain labels, we also adopt the Fourier-basek data augmentation strategy. In contrast to causal factors, confounders typically capture domain-specific characteristies such as color and style, which are primarily reflected in the amplitude spectrum. Thus, we perturb the phase information and combine it with the original image's 's amplitude A(x;) generate the augmented image x; that retains the amplitude information:
Pl (x:) (1 22) (x.) )2P (8z).
x = F (xx)
(xx) (x-xxx(xx))
(8)
where P'(x;) denotes the perturbed phase components, and X2 UO0.5, 1) controls the perturbation strength. We then derive the confounder representation u; by extracting features from the augmented image X through the encoder Go i.e., 14 r Go (x²).
Remark 1. Unlike the correlation matrix loss used in CIRL 133, which enforces both consistency between original and augmented causal features and decorrelation across different samples, adopt the MMD loss to focus solely on the former. We observe that enforcing inter-sample independence may discard shared causal cues. For instance, zebras and horses share similar shapes, which are generally considered causal features (23]. In this case, emphasizing feature independence may weaken the representation of shape-related causal information.
3.4 Automatic Visual IV learning
3.4.1 Masked image construction for IV representation learning
To ensure that the learned Visual IV representations compensate for the texture and color features while excluding shape and contour information, we design mask generation module to preprocess the input data. This module consists of two main components: image difference computation and mask generation, and (ii) masked image construction. Specifically, this module preserves texture and color-related regions by computing pixel-wise differences between grayscale versions of the original and augmented images. It then masks out regions dominated by shape information. This targeted preprocessing procedure ensures that the learned Visual IV representations capture attributes complementary to causal features. We denote the masked image as x and extract its representation using the IV encoder Zg,, denoted by z == Z ). The details are provided in Appendix D
3.4.2 Relevance-constrriine learning
We enforce that Visual IV Z remains highly informative of the causal factors S by maximizing their mutual information I S); Mutual information is formally defined as 24
((Z:)) = Ep((.)) log p(2,s)
(9)
p(z)p(s))
5

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which is intractable in high dimensions because it requires the true joint and marginal densities. Drawing inspiration from classifier-based variational bounds [25], we employ an adversarial training scheme that leverages positive and negative sample pairs: Positive pairs {22,, 8)) are drawn from the same original sample. Negative pairs {(2,, sy)l are from different image samples. Furthermore, we construct a discriminator network T₁ IR2d + R and apply a sigmoid activation o,(+) E (0, 1) interpret the output the probability of positive pair. The mutual information admits the following lower bound based on the
divergence variational principle:
(1 + -T Ep(s)p(s) [- log (1 + =
(2; S) 2 Epp...)) log
(10)
We operationalize this bound through a minibalch-approximatee binary cross-entropy objective:
N logo, o, (T (2), 8:))
1 N
1 Σ N N
1
DMLZS
Σ log ao (Tμ (2i, 8,)))- (11)
The first term encourages o. (T, 8:)) to approach for positive pairs, while the second term drives as (T,TT 8y]) toward 0 for negative pairs. For notational clarity, we further express the objective in its expectation form:
LML2S (x..))))p( lo (T, - B(s.)))p(e))())) log (1 - (Tx s)))]-
(12)
By minimizing CML2s, we effectively maximize the variational lower bound on I(:: thereby satisfying the relevance condition of the Visual IV.
3.4.3 Independenne-coostraaane learning
To enforce statistical independence between the learned Visual IV and the confounders U, we aim to minimize their mutual information I(Z; U). Direct computation of I (ZZ is intractable in high- dimensional space, so we adopt an adversarial approach based on the Jensen- -haanno (JS) divergence. Specifically, introduce discriminator network Ty RRdd IK that distinguishes samples from the joint distribution p(2, u) and the product of marginals p(z)p(u). Positive pairs {(zg, ) are drawn from p(z,u) preserving the original correlation, while negative pairs {(zi, Uy)) Jayj are generated by randomly shuffling the confounder indices. confuse the discriminator and reduce mutual information, we define the empirical adversarial loss as
N
N
CMLZZ
log (Z,, ui))
Σ log(1 - oz log(1-o,(T) uz))), jfi
(13)
N
where (x(.) denotes the sigmoid function. The equivalent expectation form is
LMI_ZU = T (=(zz,))p((..)) [log ax ( (2, + ((c.u))=(x))(() (og({
[log(1 a, aT u)))]
(14)
Minimizing this loss with respect to the Visual IV encoder confuses the discriminator, thereby reducing [(Z U) and enforcing the independence condition for the Visual IV.
3.4.4 Exclusion-constrainee learning
To ensure the learned Visual IV Z satisfies the exclusion condition (i.e., Z affects the outcome Y solely through the causal factors S), we propose variational approximation minimize the conditional mutual information I (Y; Z S). Since direct computation in high- sigh--imensional spaces is intractable, we approximate it by minimizing the Kullback Leibler conditional distributions. Formally, the conditional mutual information is defined as
(KL) divergence between two
I(: S) = Ep(s.2) (pL s, p) s))] = H |S (Y S,,
(15)
where H-)) denotes conditional entropy and (.T.(-1-) denotes KL divergence. If Z remains depen- dent on Y given S, then HY | S, Z) < H(Y S), implying /(Y; Z S) > O. To enforce that Z carries predictive information beyond S, we minimize this divergence term. In practice, use the main classifier to produce differentiable proxies for the true conditional distributions:
Ps (uls) om (h(s)), Paas(u) 2,8) = am (h(( é s)),
(16)

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where @ denotes element wise addition (adopted instead of concatenation to avoid dimensionality explosion), and on represents the softmax function. the causal factor S sufficiently encodes information relevant to predicting Y, the ideal scenario satisfies
Praa(u|2, 8) = p=(ls)) Vz,
(17)
which yields r(Y; Z |S) () 0. Thus, we enforce this constraint by minimizing the KI. divergence between the two distributions:
Ps (y 2,
LMI_TY (ppees) DEY. (Pamallp.) E
Σ Pama (21 2, log
(18)
Pa 8)
Minimizing LMI_ zY is equivalent to maximizing H 2), which encourages Z to preserve the uncertainty of Y given S. This mechanism enforces the exclusion restriction guaranteeing that the instrument Z can only affect the outcome Y through the causal pathway mediated by S.
3.4.5 Overall objective function for Visual IV learning
The overall objective for Visual IV representation learning is thus formulated as
LiotaLIV a1Cxızz + azf a3CMıIZY:
(19)
where Qt,, Q22, and ag trade-off hyperparameters. During backpropagation, only the parameters of the Visual IV learner 2z. and the discriminator are updated, while the parameters of both the main encoder Go and the classifier h are kept frozen. Notably, although derived from the images, the learned Iss can still exhibit exogeneity through disentangled learning.
3.5 Regression predictor for causal factor refinement and debiasing
We construct lightweight regression predictor Rs (abbreviated as the regressor) that maps the learned Visual IV representation back to the original causal feature space, predicting new causal factors (causal representations) that are unaffected by confounding effects.
Regressor R. is implemented as three-layer MLP with LayerNorm, BatchNorm, GELU activation, and Dropout. Given the layer-normalized input It outputs the estimated reconstructed causal factors, 5 == R& RR,LLN (zN( where LN(.) denotes layer normalization. We train R.g. by minimizing the meea--sqared eтror between and the learned initial causal factors The loss is expressed as
(20) During this stage, both the main encoder Go and the Visual IV learner Zyp remain frozen; only the parameters regressor R. are updated. After training the regressor Rg, we freeze the parameters of both the IV learner Z₁ and the regressor R2g We then input the debiased causal factor 8, predicted by R2z, into the main classifier and compute the corresponding classification loss:
Lreg (((.2))(ss.z) [|] (R - -1
Cass (s.))))p((3)) [-y log hm)(
(21)
where n denotes the softmax function. Subsequently, unfreeze the main encoder Go and classifier h, and jointly optimize the classification loss Les on the original image x and its augmened images x", along with the MMD loss LMMD, obtain the refined causal representation learning loss:
t = LeL + B2Ca. + TLMDD-
(22)
By optimizing the above objective, we correct and debias the main encoder Ge, encouraging the model to learn more comprehensive causal representations and thus improve generalization.
3.6 Progressive optimization via four-stage alternating training
We adopt four-stage training protocol to optimize each component: (i) we first train the primary encoder Go to extract initial causal factors; (ii) we then freeze Go and train the IV learner Zg to learn Visual IVs; (iii) next, we freeze both Go and Zg,, and train the regressor R. establish mapping from Visual IVs to causal factors; (iv) finally, we freeze Z and Rg, and fine-tune Go to further refine causal representation learning. particular, the third and fourth stages reflect the causal logic of IV regression. Notably, although the model components are optimized in alternating stages, the overall framework end-to-end trainable. For clarity, the stage-wise configuration summarized in Appendix H.3.4 The pscudocode for VIV-DG is presented in Appendix F
7

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Remark (Limitation - To address the limitation that training the regression predictor increases computational cost when using larger backbones for the main encoder Gg and the Visual IV learner Zg, propose simplified alternative: a feature fusion stralegy that integrates the learned Visual features and the initial causal factors causorfcctors via addition in the representation space. We refer to this lightweight version as VIV-DG-Lite Notably, the fusion reduces training cost but compromises interpretabillty, whereas the regressor offers better interpretability at the expense of model complexity.
4 Theoretical analysis
We present the main theoretical guarantee, with formal proofs provided in Appendix
UKS.UUY, be four rundom variables, where X denotes observed images, S denotes the causal factors affecting Y, U denotes the confounders, and Y denotes the downstream labels. Assume that Hz {hr X 7 is a sufficiently expressive
Theorem 1 (Learnability of Visual IVs).
family of mappings (e.g., one that contains all smooth bijections latent subspace). Consider the objective
((w) (x))(( + (h) ah S),
(23)
with on, 02 03 > Then any global minimizer w yields
Z hw- (X),
(24)
which is learnable as defined in the learnability desinition (Definition in the Appendix.
5
Experiments
5.1
Datasets and settings
We evaluate VIV-DG on several real-world benchmarks: Digits-DG [26). PACS 277. Office-Home
28
and VLCS 29
The datasets and implementation details shown in Appendix H
5.2
Experimental results
5.2.1
Evaluation on benchmarks
We compare non-causal and causality-ins inspired DG methods. A detailed description of the baselines provided in Appendix H22 Table presents the results of our VIV-DG and its simplified version,
VIV-DG-Lite, on the Digits- DG, PACS 188, and Office-Home (ResNet- 18) datasets.
Table Leave-one-domain-oou accuracies (%) on Digits-DG, PACS, and Office- Home
Digits-DG
PACS (ResNet-18)
Office Office-Home (ResNet-18)
Methods
MN
MM
SVHN
SYN
Avg.
A
C
P
S
Avg.
C
P
R
Avg.
DeepAll
26
95.8
58.8
61.7
78.6
73.7
77.6
76.8
95.9
69.5
79.9
$7.9
52.7
73.5
74.8
64.7
CCSA 30
95.2
58.2
65.5
79.1
74.5
59.9
49.9
74.1
75.7
64.9
JiGien
31
96.5
61.4
63.7
74.0
73.9
79.4
75.3
96.01
71.4
80.5
53.0
47.5
71.5
72.8
61.2
RSC 22
83.4
80.3
96.0
80.9
85.2
58.4
47.9
71.6
74.5
63.1
CrossGrad
35
96.7
61.1
65.3
80.2
75.8
58.4
49.4
73.9
75.8
64.4
DDAIG 20
96.6
64.1
68.6
81.0
77.6
84.2
78.1
95.3
74.7
83.
59.2
52.3
74.6
76.0
65.5
MatchDXi
14
81.3
96.5
79.7
L2A-OT 35
96.7
63.9
68.6
78.1
83.3
78.2
96.2
73.6
82.8
60.6
50.1
74.8
77.0
65.6
CIRL 13
96.1
69.9
76.2
87.7
82.5
86.1
80.6
95.9
82.7
86.3
61.5
55.3
75.1
76.6
67.1
LRDG 30
81.9
80.2
95.2
84.7
85.5
61.7
52.4
73.0
75.9
65.8
FACT 7
97.9
65.6
72.4
90.3
81.5
85.9
79.4
96.6
80.9
85.7
60.3
54.9
74.5
76.6
66.6
IV-DG 77
83.4
78.8
96.9
78.7
84.4
60.4
47.7
72.6
76.1
64.2
FAGT 16
98.3
65.7
70.9
90.4
81.3
87.5
80.9
96.9
81.9
86.8
60.1
55.0
74.5
75.8
66.4
CDIM
32
98.7
64.0
74.1
92.9
82.4
83.6
77.6
95.5
78.2
83.7
VIV-DG-Lite
77.8
91.4
83,5
86.1
81.8
96.6
83.3
60.8
55.9
75.8
76.3
67.2
VIV-DG
97.6
67.1
77.4
83.4
86.6
81.5
96.9
83.9
87.2
61.1
56.3
75.3
76.8
67.4
second best are marked hold underlined, respectively, Avg. Average accuracy
Table 1 shows that our method exhibits two key advantages: (i) it outperforms existing causality- inspired DG methods, with significant improvements over the IV-DG [17] method, which relies on the predefined IV strategy; (ii) it demonstrates pronounced advantage under severe distribution

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shifts. For instance, on the Digits-DG dataset, where SVHN and SYN exhibit large distribution gaps, VIV-DG and VIV-DG-Lite outperform the baselines by clear margin. Similarly, in PACS, where there is substantial distribution shift, our method consistently surpasses all baselines.
For more comprehensive validation, ResNet-50, to assess its scalability and robustness. presens the results on the VLCS dataset. It can be observed that both VIV-DG and VIV-DG-Lite achieve the same optimal average performance, surpassing recent causality-inspired methods with gains of 0.8% over iDAG and 0.5% over GMDG. Moreover, they demonstrate competitive performance across multiple domains. Table and Table 2 demonstrate that our methods effectively mitigate confounding effects and improve generalization.
evaluate method using larger backhone network,
5.3 Analysis of the distinct behaviors of VIV-DG and VIV-DG-Lite
As shown in Table and Table 2 the per- formance of VIV-DG and VIV-DG-Lite varies across target domains. To understand this dis- erepancy, we provide detailed analysis. Specif- ically, VIV-DG-Lite approximates the causal representations by combining the IVs with the initial causal representations, which retains more original discriminative information but is more susceptible to confounding. Therefore, it performs better in domains with smaller distribu- tion shifts, such as Product in Office-Home and Cartoon in PACS dataset. In contrast, VIV-DG predicts causal representations through a trained regressor for bias correction, which helps reduce confounding effects and thus performs better in do- mains with larger distribution shifts, such as Sketch in PACS dataset. Overall, VIV-DG demonstrates greater robustness compared to VIV-DG-Lite.
Table 2: The results on VLCS with ResNet-50
Methods
C
Avg.
RSC 32
98.0
67.2
70.3
75.6
77.8
MixStyle
98.6
64.5
72.6
75.7
77.9
SagNet
97.9
64.5
71.4
77.5
77.8
PCL
40
99.0
63.6
73.8
75.6
78.0
iDAG 41
98.1
62.7
69.9
77.1
76.9
RICE 42
98.3
69.2
74.6
75.1
79.3
GMDG 45]
98.3
65.9
73.4
79.3
79.2
VIV-DG-L.ite
99.2
67.4
74.4
77.6
79.7
VIV-DG
99.0
67.7
74.6
77.5
79.7
5.4
Analytical experiments
5.4.1
Parameter sensitivity
$1.25
87
86.4%
sson
85
83
82
0103055765 0467181316 0103e587h4 630355076
C1
C2
C4
C5
00.. (...1.0...5
(1..02.62) (1..04..0 (53,95, 1.00 (10.63.07)
Figure (Left) Univariate hyperparameter analysis. (Right) Combined hyperparameter analysis.
To evaluate the robustness of the VIV-DG under different hyperparameter settings, we perform sensitivity analysis on the PACS dataset using the ResNet-18. Appendix H.3.3 summarizes the hyperparameter settings for each stage. Since and 32 are adaptively adjusted, the analysis focuses on 31, a1, 02, and a33, We first perform univariate sensitivity analysis for each hyperparameter, followed by a combination analysis of a1, 022 and 033, keeping other hyperparameters at their default values. The value ranges and step sizes are listed in Appendix H.4 As shown in Figure 5 the model demonstrates satisfactory robustness across both individual and combined hyperparameter settings.
5.4.2 Ablation study
To evaluate the contribution of each component, we conduct ablation studies on Digits-DG, PACS, and Office-Home. As shown in Table removing both the Visual IV module and the regressor (w/o VIV & R₂,) leads to significant performance drop compared to the full model, indicating that relying solely on data augmentation and MMD for invariant representation learning is insufficient. This demonstrates that mitigating confounding effects crucial for effective domain generalization. Moreover, the results show that the exclusion loss is the most critical, as it effectively prevents the

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introduction of extra confounders IVs. Meanwhile, the relevance and independence losses help reduce confounding effects and enhance the expressiveness of causal representations.
Table 3: Ablation study (%) on Digits-DG, PACS, and Office-Home datasets
Digits-DG
PACS (ResNet-18)
Office-Homm (ResNet-18)
Methods
Setting
MN
MM
SY
Avg.
A
S
Avg.
A
R
Avg.
w/ VIV & R
96.7
62.3
12.8
89.7
80.4
82.7
78.1
93.6
80.5
83.7
58.7
54.6
74.2
76.1
65.9
w/o (2: S)
96.8
64.5
74.0
90.6
81.5
85.5
80.4
95.8
83.6
86.3
60.9
54.4
75.6
76.7
66.9
VIV-DG
((;4))
96.9
64.1
75.0
90.2
81.6
84.9
79.7
95.9
83.0
85.9
60.9
53.9
75.3
76.6
66.7
7(Y; (8)
96.7
90.5
84.6
79.1
60.7
75.4
76.5
Full Model
97.6
67.1
77.4
91.6
83.4
86.6
81.5
96.9
83.9
87.2
61.1
56.3
75.3
76.8
67.4
W/ VIV & R2
96.7
62.3
72.8
89.7
80.4
82.7
78.1
93.6
80.5
83.7
58.7
54.6
74.2
76.1
65.9
w/o/ (2:
97.1
64.4
74.9
90.4
81.7
85.6
79.7
95.4
82.9
85,9
60.5
53.9
75.4
75.9
66.4
VIV-DG-Lite
(2;U)
96.9
90.2
84.7
75.6
76.1
w/o (())
96.5
64.2
74.4
90.1
81.3
84.5
78.2
95.5
81.9
85.0
60.2
53.3
75.2
75.7
66.1
Full Model
97.8
66.8
77.8
91.4
83.5
86.1
81.8
96.6
83.3
87.0
60.8
55.9
75.8
76.3
67.2
The results "wo IV & R., are identical for both VTV-DG and VVV Lite under this setting. and are provided for both methods to facilitan direct comparison.
5.4.3
Visual explanation
To validate that VIV-DG learns causal representations consistent with human cognition, we conduct visual analysis using Grad-CAM on images with complex backgrounds from the Art- Art-Paintin domain. We deliberately choose this domain because the classification accuracies of VIV-DG and the baseline CIRL 113 are comparable, making any visual improvement in identifying causal regions meaningful contribution. We compare VIV-DG against both CIRL and an ablated version our model without the Visual IV learning module (Ours (w/o VIV)). As shown Figure 6 VIV-DG demonstrates superior ability to capture global features and focus on the core object structures. For instance, in images of dog, giraffe, and house, VIV-DG identifies broader, more holistic regions compared to the localized cues relied upon by CIRL. More importantly, the ablated model (Ours (w/o VIV)) exhibits significant deviations, especially in complex backgrounds (e.g., dog). Even in simpler scenes (e.g., horse and house), it is less accurate than the full model in capturing causally relevant regions. These results provide strong visual evidence that the Visual IV module is crucial, further supporting VIV-DG's effectiveness in leamingg cognitively consistent causal representations.
Dog
Elephant
Giraffe
Guftar
Horse
House
Person
Basaline
Ours (wlo VV)
Ours
Figure Grad-CAM visualization on the PACS dataset with Art-Painting as the target domain.
6
Conclusion
We propose a novel canfounding-resistane DG approach, termed VIV-DG. We break away from the conventional view that simply categorizes non-causal factors confounders and observe that certain visual attributes in image data satisfy the conditions of IVs. Building on this insight, we develop a framework that automatically learns valid Visual IVs and mitigates the significant bias arising from violations of IV conditions in predefined IVs. By mitigating confounding effects, including those from unobserved confounders, VIV-DG consistently achieves improved generalization.
10

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Acknowledgments
This work was supported by the National Science and Technology Major Project of China 10011D00118011,, the National Natural Science Foundation of China (U24A20323, 62376145, and 62376087), the Open Project Special Fund of Taihang Laboratory In Shanxi Province, China (THYT-KFKT-25000200)) the Key Technologies Program of Taihang Laboratory in Shanxi Province (THXX-J)ZZX20000700)),,, the Science and Technology Innovation Talent Team of Shanxi Province (202204051002) 1002016), and the Taiyuan City "Double hundred Research action" of the first batch project about "Leading the Charge with Open Competition" (2024TYJB0127)
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Appendix
This appendix includes technical and supplementary materials: related work, technical background. the VIV-DG pseudocode, experimental setup, additional results, and broader impacts.
A
Related work
A.1 Confounding effect mitigation for DG
Recently, numerous domain generalization methods have been developed to learn causal representa- tions by mitigating confounding effects. These methods are typically categorized into three causal intervention frameworks: backdoor adjustment, frontdoor adjustment, and instrumental variable (IV) methods. For instance, Zhang et al. [15] propose the DIR-ReID framework based on backdoor adjustment to learn domain-invariant representations for person re-identification. Similarly, Sui et al. 44 design causal attention mechanism using backdoor adjustment to improve robustness and interpretability. However, backdoor adjustment assumes access to all confounders, which is often unrealistic due the presence of unobserved confounders real-workh scenarios. Toan Nguyen et al. [16] propose the FAGT method based on frontdoor adjustment, achieving improved generalization. Nonetheless, relies on observable mediators, which are difficult to identify and validate in practice. Yuan et al. [17] introduce IV-DG, which treats data from one domain as instrumental variables for another. However, even if domains differ, features extracted from the same class may still share cross-domain discriminative information, potentially violating the exclusivity assumption of IVs.
A.2 Instrumental variable
Instrumental variables are exogenous variables that are associated with the treatment but have no direct causal effect on the outcome [18]. Even in the presence of unobserved confounders, IV methods can effectively eliminate confounding effects, which makes instrumental variables widely applicable for causal effect estimation and counterfactual prediction 45) 44). Recently, instrumental variables have gained increasing attention in machine learning. In addition to the previously mentioned domain generalization method II) instrumental variables are also applied in recommendation systems. For example, Si al. 47 propose the IV4Rec method, which leverages user search data as instrumental variables to effectively mitigate confounding bias caused by latent variables in recommendation models.
In our work, we introduce a domain generalization approach for image classification that directly learns instrumental variables from image data. Our method is not restricted by specific physical concepts but instead autonomously learns representations that satisfy instrumental variable conditions. The key advantage of our approach is that predefined instrumental variables may introduce severe biases if they fail to meet valid IV conditions, whereas our method effectively avoids this issue by autonomously learning instrumental variables.
B Preliminary: instrumental variables for confounding mitigation
Instrumental Variables (IVs) 18 are a statistical method that can effectively estimate causal effects even in the presence of unobserved confounders. The core idea is to introduce a exoge- nous variable that satisfies specific conditions, thereby blocking the confounding effect on the relationship between the treatment and the outcome Y. To illustrate the properties and role of IVs, we present a general SCM, as shown in Figure 7 A valid instrumental variable must satisfy three conditions: (i) Relevance: The variable Z must be correlated with the treatment T; (ii) Inde-
Confounders
IV's
Treatment
Outcome
Figure 7: The SCM with IVs.
pendence: The variable must be independent of the confounder O: (iii) Exclusion: The variable Z affects the outcome Y only through the treatment T (i.e., Z has direct effect on Y).
The most widely used IV-ba causal effect estimation method two-stage least squares regression (2SLS), which follows two-stage procedure. In the first stage, the treatment T' is regressed on the instrument 2, yielding the predicted treatment values T:. In the second stage, the outcome Y
21

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is regressed on the predicted values T. The coefficient obtained in the second-stage regression represents the local average treatment effect (LATE), providing an unbiased estimate of the causal effect of the T on Y. Notably, 2SLS is primarily designed for scenarios with linear relationships between variables.
C Connections of the proposed SCM
Figure 8 iillustrates our proposed fine- fineggrained SCM, with all the connections detailed below.
- U + X. (S. U. 2) + Each input image consists of causal factors (i.e.. object-specific factors) S and non-causal factors U and Z. Although both U and Z are non-causal factors, they differ fundamentally. As an inherent attribute of the object, Z remains relatively stable and, when combined with S, provides valuable information for category prediction.
C-Y. The observed confounders U should not be directly associated with Y. However, in prac- tice, when selection bias exists in the training data, pretrained encoders may learn shortcut features from non-causal factors and use them for prediction. As result, U influences Y, even though this relationship is non-causal. Additionally, due to natural or manual biases, S and U frequently co-occur in the data distribution, leading to a statistical association between them. We represent this correlation with a bidirectional arrow (S ++ U). Moreover, C' shares similarities with U but represents unobserved confounders that simultaneously influence both S and Y. Without loss of generality, we assume that C' represents confounders related to unknown domain styles or backgrounds. Since domains cannot be exhaustively enumerated, such unobserved confounders inevitably exist. However, these factors do not appear in the current observed input data or known source domains, and therefore do not exert direct causal effect on the input data X. For example, factors such as image style or background color may serve as potential confounders, but they are not always visible or relevant in specific samples. The art style of an image, as a confounder, may not affect the input of sketch image. Similarly, a background factor like yellow desert does not appear in an image of dog in the snow.
S + U---Y
Observed confouunerss
Unobservad
Inpot data
confounders
IVs
Causal factors
Labels
Figure 8: The proposed SCM.
Z-S-X Y. The factors and S may exhibit biological or physical association. For example, in animal images, this could reflect the co-evolution of an organism's coat color and shape. Additionally, Z does not directly influence the labels. For instance, it is not possible to determine the category of an object solely based on its local coat color. However, when combined with shape and contours, Z provides valuable predictive information.
Notably, since the confounders U and C simultaneously affect both the causal factors S and the labels Y. they establish backdoor paths between S and Y: S U- Y and S C' + Y. As result, in addition to the causal relationship S + Y, spurious correlations are introduced, leading confounding effects [19]. Conventional estimation of P(Y|S) results in biased estimator that absorbs both the true causal dependence and spurious correlations, limiting its generalizability.
D Mask generation module
The mask generation module consists of two main components: (i) image difference computation and mask generation, and (ii) masked image construction.
(i) Image difference computation and mask generation: We convert both the original image x; and its augmented counterpart x" to grayscale to reduce the influence of color. Then, we compute the absolute pixel-wise difference (p, q) at each location (p:
Ar q) (p. ( (p.
(25)
where Xi(g) and ( represent the transformed grayscale images, respectively. A binary mask is then generated using threshold 8 (empirically set to 30):
1, A (p.q) q) <
m (p. = 0,
8= = 30.
(26)
(p...) q)
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(ii) Masked image construction: We extend the binary mask m, q) to all RGB channels and apply the original image x, to obtain the masked input:
x (p, q, c) = (p, 9, c, x mp(p, q),
(27)
where e denotes the color channel. The masked image x;" is then fed into the IV learner to extract the IV representations.
E Theoretical foundations
E.1 Preliminaries and assumptions
We consider a tuple of random variables (X, Y) defined on a probability space (S2 F, P), where X denotes the observed images, S denotes the causal factors affecting Y. U denotes the confounders, and Y denotes the downstream labels. Let Z be a measurable representation space, and let Z = ((X) for some representation function X + Z. Without loss of generality, we do not distinguish between observed and unobserved confounders here.
Assumption 1 (Basic Regularity). The joint laws of all relevant random variables admit regular conditional distributions, and all mutual informations appearing below are finite. Moreover, all measurable maps and densities are taken modulo null sets (i.e., hold almost surely).
Assumption 2 (Parametric Representation Family; Continuity and Compactness). The representation family is parameterized as Hiz = {hw : X + 2 ween where (i) the parameter set S2 is compact, (ii) the map w + h is measurable, and for each of the mutual-information functionals
+ I( (X);
w (X);
w + (X) S),
(28)
we assume continuity (or at least the required lower/upper semicontinuity) so that objectives built from these functionals admit minimizers on Q by standard compactness arguments.
E.2 Identifiability conditions and learnability
lemma (Identifiability Conditions for Visual IVs via Mutual Information). Let (X,S,U.Y) be four random variables, where X denotes observed images, S denotes the causal factors affecting Y. U denotes the confounders, and Y denotes the downstream labels. A candidate variable Z is called a visual instrumental variable if it simultaneously satisfies: (i) Relevance: I(Z > 0, (ii)
Independence: 1(Z;U)- =0, O, and (iii) Exclusion: I(Z; Y 1S) =0
Proof. By definition of mutual information,
[(: = /I p(2,s) 1(2:))= p(z)p(s) and T(Z:5) > and only p(2,8) # p(=)p(s). ie.. 7 P. S. This is exactly the relevance requirement. Similarly,
p(2,8) dz ds,
(29)
T(Z:U) = /ʃ p(z, u) log p(z,u) dz du,
(30)
p(2)p(u)
and T(Z:U)- = if and only if p(z,, u) = p(=)p(u), i.e. 7 11 U. This is the independence requirement. Finally.
T(Z [) = ʃIʃ p(s, y, 2) log
ply. = |
p(y s)p(z \s) ds dy dz,
(31)
and I(Z;Y| |) = 0 if and only if p(y, 2 =P(y s)p(z s), i.e. Z 11 Y S. This is the exclusion requirement.
lemma (Recoverability of Visual IVs). Let hws : X + Z be a parameterized mapping (e.g. via a deep network), with the training objective
L(w) = -a ((2 S) + oz (2; + aag [S),
(32)
where 01, 022, 03 > Under the idealized assumption of sufficient network capacity, any global minimizer w' satisfies
()> U() = 0, I(Z 0
(33)
where Z = hh-((X), i.e., it exactly recovers visual instrumental variable.
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Proof. Since - 5Z) 2 sup.. (), I(2:0) 2 0, and (2: S) 2 0. any global min- imizer w" must simultaneously (i) maximize (Z: (ii) drive [Z; U) to zero, and (iii) drive
T(Z;Y) S) to zero; otherwise one could reduce L further.
By the zero mutual information conditions, we get Z Il U and Z 11. Y S,, and by the maximization, [[2; S) > 0. These three facts are exactly the three requirements of Lemma
E,3
Identifiability theory
Definition 2 (Learnability of Visual IVs). Let (X.Y be four random variables, where X denotes observed images, S denotes the causal factors affecting Y, denotes the confounders, and Y denotes the downstream labels. Let Hz = =(h: X - 7 be a family of representation functions. We say that visual instrumental variable is learnable Hz if:
(1) (Existence) There exists at least one h- E Hz whose output Z = =X*(X) satisfies the three conditions of Lemma 1 I(Z';U)=0, S(z
T( Y =
(34)
(2) (Uniqueness modulo invertible transforms) For any two h1, h2 E Hz whose oulputs both satisfy those three conditions and maximize (Z: (i.e., Z is a sufficient statistic for S), their outputs are related by an invertible reparumeterization:
(x2(X) (X)h((X)), q: 2
2 + bijective.
(35)
Proposition 1 (Uniqueness of Visual IVs). Lel Hz =(h: X + Z} be a sufficiently rich function family (containing all smooth bijections). Suppose some h* E H2 satisfies
:)) > 0, I(Z" =0, I =)
(36)
where Z (X), and SS) is maximized (i.e., Z is a sufficient statistic for S). Then the induced variable Z is unique up to invertible reparameterization.
Proof. Let 21 h2 (X) and Z, h2(X) both satisfy the conditions. Since I(Z1;U) 0 and 1[Z1; Y S) = 0, Zi is independent of U and conditionally independent of Y given S. Therefore, Zi can only influence Y through Similarly, Zz can only influence Y through S. Since [[ S) and T (Z2 8) are both maximized, Zı and Z2 are both sufficient statistics for S. This means there exist functions 91 and 82 such that S 91 (Z1) and = (Z)) almost surely (in an appropriate sense). Consequently, Z₁ and Zz determine each other through S: there exist mappings f12 and fei satisfying Z₃ = f1) = f12 (S)) and Z f21 (Zz) = f92 (S)) Since the function family Hz is sufficiently rich (containing all smooth bijections), f12 and f21 can taken bijective transformations. Thus, Z and Zg are equivalent up to an invertible reparameterization.
Theorem 2 (Learnability of Visual IVs). Let (X,S,U,Y) be four random variables, where X denotes observed images, S denotes the causal factors affecting Y, U denotes the confounders, and Y denotes the downstream labels. Assume that Hz = (hw- X + Z} is a sufficiently expressive family of mappings (e.g.. one that contains all smooth bijections objective
latent subspace). Consider the
((w) Ia)I(h(())) + ah (X);U) agI(h S),
(37)
with 0n, a2 a3 > Then any global minimizer w yields
Z" h (X),
(38)
which is learnable as defined in Definition
Proof.
(1) Existence follows directly from Lemma 2, which shows that any global minimizer w of L enforces
I(z"; > 0, I(Z'U))=0 = 0, I S) =.
(39)
(2) Uniqueness follows from Proposition 1 Since the objective function contains -I(Z; S). the global minimizer simultaneously maximizes I (Z; S), making Z a sufficient statistic for S. Therefore, any other representation satisfying the three conditions must be related to 7" by an invertible map.
Together, these two facts establish full learnability of the visual IV under Definition 2
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Remark 3. In practice finite positive weights (a1, az., as) induce a off among the three mutual information terns, so a global minimizer may allow nonzero penalties if the relevance gain oufweighs them.
Pseudocode of the VIV-DG
Algorithm Visual Instrumental Variables for Domain Generalization (VIV-DG)
Input: Data loaders (train val test), device, args
Output: Accuracy on the test set
1: Initialize: Main encoder Go. Classifier h, IV-learner Zv,, Regressor Rs.
2: Data loaders: Parin v Dust
3: Stage epochs: (E1, E2, E3, Es), Total epochs: E Σ E
4: for epoch e = 1 to E do
5: if E, then
6: 861
7: else e S Ei + Eg then
8:
4-2 2
9:
else e < Ej + E2 + Eg then
10:
3
11:
else
12:
4
13:
end
14: if s
1 or then
Stage I: Main Training
15:
Unfreeze Gn. h; freeze 2 Rss. for each batch in Dirain do
17:
Compute Leausal 31 Lecss + T LMMD if then
18:
19:
Compute S = 31 + B2C + LMMD
Stage I+: Causal Correction
end if Backpropagate and update Go. h end for
21:
23:
else if s = then
b. Stage II: IV Encoder Training
Freeze Go. h; unfreeze Z for each batch in Dirain do
25:
Compute Ltotal IV = a' LMI 7s + a2fx1_z0 + a3LMu Backpropagate and update 2.p end for
27:
29:
else if s 3 then
Stage III: Regressor Training
30:
Freeze Go. Zei unfreeze R. for each batch in Dirain do
31:
32:
Compute regression loss: Greg = E (x.s) (LN(2)) - -s|[2] Backpropagate and update Rs
33:
34:
end for
35:
if e E, + Ez Ey then
36:
Freeze R.
end if 38: end
37:
end for
40:
Evaluate on val / test set
Return: Accuracy on the test set
G Computational cost
The computational cost of our VIV-DG is justified by the improved scalability and automation. The main computational cost of VIV-DG comes from the automatic visual IV learning module. Although this introduces additional training cost, substantially reduces the manual effort required to design instrumental variables, which is often necessary in predefined IV-based methods. It also avoids
25

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reliance on domain knowledge or expert heuristics. Despite achieving only moderate gains over some baselines, it consistently outperforms predefined IV-based methods like IV-DG in both performance and practicality. Therefore, we consider this a reasonable trade-off that improves both flexibility and reduces human effort. Using ResNet-50 as the backbone, training on a single A800 GPU requires approximately 40-50 GB of memory and runs stably.
The overall time complexity of our training framework is T' = O(N E Pmax), where N denotes the number of training samples, E' is the total number of training epochs, and maxx represents the maximum computational cost among the modules trained in each stage.
H Experimental setup
H.1 Dataset details
We evaluate method four widely used domain generalization benchmarks: Digits-DG (26).
PACS (27). Office-Home (28), and VLCS 229 Detailed descriptions of these datasets are provided
below:
Digits-DG 26 is a benchmark dataset for digit recognition, consisting of four domains: MNIST 48]. MNIST-M 49), SVIN [50]. and SYN [49]. These domains differ significantly in font style, background, and stroke color. Following 26).. we randomly select 600 images per class in each domain, using 80% for training and 20% for validation.
a domain generalization benchmark that includes four domains with different visual styles: Art-Puinting, Cartoon, Photo, and Sketch. It contains total of 9,991 images across 7 categories: dog, elephant, giraffe, guitar, house, horse, and person. For fair comparison, we use the original training-validationn split provided by [27].
PACS 26
Office-Home 28 consists of four domains: Art, Clipart, Product, and Werl-World. Each domain includes 65 object categories related to office and home environments, with a total of 15,588 images. Following 31]. we use 90% of the data for training and 10% for validation.
VLCS (99 contains four domains: Caltech, LabelMe, Sun, and Pascal. Each domain shares the same 5 categories, with total of 10,729 images.
H.2 Baseline methods
We categorize the compared baselines into two groups: non-causal methods and causality-inspired methods. The detailed descriptions are as follows:
(i) Non-causal methods primarily rely on statistical correlations to learn camain-invariant features. FACT (7] constructs a consistency loss through data augmentation methods based on Fourier transform and co-teaching regularization. RSC (32) iteratively discards dominant features activated on training data and forces the network to activate remaining features that are correlated with labels. DeepAll 26 maps source domain training data to unseen domains via a label classifier, domain classifier, and a domain transformation network, thereby enhancing the robustness of the label classifier to unknown domain variations. L2A-OT (35) employs a data generator to synthesize pseudo-novel domain data for augmenting the source domain, directly increasing the diversity of training domains and improving model generalization. LRDG (36) learns domain-invariant model by tactically removing domain-specific features from the input images.
(ii) Causality-nnspire approaches explicitly model causal relationships to address distribution shifts. MatchDG [34], a matching- method, aligns inputs from the same latent object across domains enhance out-of-domain performance. CIRL [13 utilizes data augmentation techniques and causal intervention methods to learn causal representations. IV-DG [17] leverages instrumental variables to eliminate confounding effects and considers that data from one domain can serve as instrumental variables for another domain. FAGT [16) eliminates confounding effects through style transfer and the front-door adjustment method. iDAG [1 extracts invariant graph structures as proxies for causal structures to enhance representation generalization. GMDG [43 jointly learns domain-invariant conditional features and maximizes the posterior, providing a flexible framework that generalizes and thcoretically explains existing multi-domain generalization methods.
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H.3
Implementation details
H.3.1 Basic details
In experiments, used an NVIDIA A800 80GB PCIe GPU. We adopt the same backbone for both the primary encoder and the instrumental variable encoder across all benchmarks. For Digits-DG, we follow the architecture used in [35] training from scratch with batch size of 8 and an initial learning rate of 0.001, which decays by a factor of 0.1 after 80% of the total epochs. For ResNet- 18 and ResNet-50 [ST] backbones
PACS and office-Home, we use ImageNet-pretraine with a batch size of 5. VLCS follows the same configuration using ResNet-50. All models are trained using SGD optimizer with momentum of 0.9. All reported results are averaged over 3 runs. Following the DomainBed protocol, we adopt the official data splits, model selection strategy, and evaluation settings to ensure the fairest possible comparisons. Except for the batch size constrained by GPU memory and the model-specific training epochs, other configurations, including dataset splits, evaluation protocols, backbone architectures, and the woree-oon-domii-oooo strategy, remain identical to the DomainBed defaults. For fair comparison, our main extraction network (initial training and bias correction fine-tuning) is trained for 50 epochs, consistent with the compared methods.
H.3.2 Method-specific details
We design a stage-wise training scheme for both VIV-DG and VIV-DG-Lite, with detailed configura- tions summarized in Table and T Notably, the main encoder Go is trained for total of 50 epochs, consisting of the initial training Phase and the fine-tuning stage in Phase I+.
Table 4: Epoch configuration for stage-wise training of VIV-DG
Stage
Training Objective
DG
PACS (ResNet-18) 18)
Office- (ResNet- 18)
Phase
Train main encoder Ge
30
30
30
Phase II
Freeze Ge; train IV encoder 2x
25
25
25
Phase III
Freeze Go. Zo; train regressor Rsg
25
25
25
Phase I+
Freeze Ze. Ry; fine-tune Go
20
20
20
Phose It refers to fine-tuning stage built upon Phase I.
Table 5: Epoch configuration for stage-wise training of VIV-DG-Lite
Stage
Training Objective
VLCS (ResNet-50)
PACS (ResNet-50)
Office-Home (ResNet-50)
Phase
Train main encoder Go
30
Phase II
Freeze Go; train IV encoder 2g
20
20
20
Phase
Freeze Z,;; fine-tune Go
Phase I+ refers to fine-tuning stage built upon Phase I.
H.3.3 Hyperparameter settings
The hyperparameter settings are summarized in Table 6 encompassing the various datasets and training stages employed in our experiments.
The loss function Leausal incorporates two hyperparame- ters: 31 and т. 331 is set per dataset (Digits-DG- = 1, PACS = 0.5, Office-Home = 0.1, VI.CS = 0.1), while 7 adapts dynamically according the number of training epochs. This scheme ensures rapid convergence across datasets and stabilizes causal factor extraction in carly training.
Phase I (Initial causal factor extraction):
Phase II (Visual IV learning) & Phase III (Regressor training): To reduce tuning complexity, Phases II and III adopt identical hyperparameter settings across all datasets. Specifically. in Phase II's total loss LiotalIv, set a1 1, 0% 0.5, and ag 0.5. In Phase III, the regressor loss Lreg carries a default weight of to maintain stability.
Phase I+ (Causal factor refinement and debiasing): The loss L reuses 31 and from Phase I while introducing a dynamically adjusted 32. This adaptive strategy enables the model flexibly respond to distributional shifts and further refine debiasing.
27

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Table 6: Hyperparameter settings
Stage
Loss Function
Hyperparameters
Settings
Phase
Leaneal
$1.7
B:: Digis--G==1; PACS=055; Office-Home-ll VLCS=0.1; adaptively adjusted
Phuse 11
IV
01.02.00
1; 0,5; 0.5
Phose III
Long
Regression weight set to by default
Phase I+
causal
B1 92.
$1, 7: same as Phase I; fg: adaptively adjusted
Phase refers to fine-tuning stage bailt upon Phase I.
H.3.4 Stage-wise training configuration of VIV-DG and VIV-DG-Lite
Table 7 illustrates the stage-wise training procedures for both VIV-DG and its simplified variant, VIV-DG-Lite. Notably, Phase III is omitted in VIV-DG-Lite, as the regressor training is replaced by feature fusion strategy, as described Remark
Table 7: Stage-wise training configuration of VIV-DG and VIV-DG-Lite
Stage
Training Objective
VIV-DG
VIV-DG-Lite
Phase
Truin main encoder Go
Phase
Freeze Co: train IV leamer 2
Phase III
Freeze So. Ze: train regressor R2
Phase I+
Freeze 2, and optionally R. fine-tune
V indicates that the phase is included in the method. Phase It refers to fine-tuning stage based on Phase
H.4 Hyperparameter sensitivity analysis design
Table 8 summarizes the value ranges and step sizes of the hyperparameters involved in each training stage for sensitivity analysis.
Table 8: Hyperparameter sensitivity analysis design
Parmmeter(S)
Value Range (Step Size)
Defauit Value
Single-variable Sensitivity
8,
[0.1. 0.3, 0.5, 0.7, 0.9) (linear)
(0.4, 0.7, 1.0, 1.3, 1.6] (linear) [0.1. 0.3, 0.5. 0.7, 0.9) (lincar)
1.0
02
03
[0.1, 0.3, 0.5, 0.7, 0.9] (linear)
0.5
Multi-variable Comhinations of 01,, O2g and oig
C1: (0.5, 1.0, 0.5)
0,0/
C2: 0.2)
1.0/0.5
d1 +0g
C3: (1.0, 0.5, 0.5)
1.00.55/5
C4: (0.5, 0,5, 1.0)
1.0/0.5/05
C5: (1.0, 0.3, 0.7)
1.0/0.5/..
I Additional results
I.1 Evaluation PACS and Office-Home using ResNet-50
Table 9 and Table 10 present the experimental results on PACS and Office- using ResNet-50, respectively. These results further demonstrate the effectiveness of our method, and in particular, the results in Table I0 show that VIV-DG exhibits significant superiority Office-Home.
28

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Table 9: The results PACS with ResNet-50
Methods
A
Avg.
SagNet A9
87.4
80.7
97.1
80.0
86.3
MatchDG
84
85.6
82.1
97.9
78.8
86.1
SAGM
87.4
CIRL 15
90.7
84.3
97.8
87.7
90.1
FACT 2
90.9
89.7
RICE
87.8
84.3
84.7
88.4
iDAG 31
90.8
83.7
98.0
82.7
88.8
GMDG
VIV-DG--Lite
90.3
83.8
97.6
86.7
89.6
VIV-DG
90.5
83.6
97.8
86.7
89.7
The best and second hest results are marked in hold and underlined, respec- lively. Avg. Average accuracy (%)
Table 10: The results on Office-Home with ResNet-50
Methods
Avg.
RSC
32
60.7
51.4
74,8
75.1
65.5
MixStyle
SagNet
39
63.4
54.8
75.8
78.3
68.1
PCI. 40
67.3
59.9
78.7
80.7
71.6
SAGM
65.4
57.0
78.0
80.0
70.1
FAGT 16
66.3
59.5
77.6
79.0
70.6
GMDG +
68,9
56.2
79.9
82.0
70.7
VIV-DG-L.ite
67.8
60.7
79.7
81.4
72.4
VIV-DG
68.4
62.3
79.4
81.2
73.0
The best and second best results marked in bold and underlined, respec- tively. Avg. = Average accuracy (%).
1.2 Sensitivity to mask threshold
To verify the robustness of the parameter threshold 5 used in the mask generation process (Sec. D) conduct sensitivity experiments on for VIV-DG using the Digits-DG and PACS datasets, with PACS experiments conducted using the ResNet-18 18 backbone. The results are shown in Figure P We
vary from 20 to in steps (20, 25, 30, 35, and 40) and observe minimal performance variation, indicating that VIV-DG is robust with respect to the parameter d.
Sensitivity to on Digits-DG
Sensitivity to å PACS
100
100
90
Accuracy (%5
Accurscy (95)
85
75
60
25
30
Figure 9: Sensitivity analysis of mask threshold 8: (Left) Digits-DG; (Right) PACS.
I.3 Visual explanation
The visualization results on the PACS dataset, with Cartoon, Photo, and Skeich as the target domains, are shown in Figure TO Figure I1 and Figure 12] respectively. It can be observed that VIV-DG attends broader and more holistic discriminative regions, whereas CIRL. 13 tends to rely on more localized cues. This demonstrates the effectiveness of our method in learning causal representations.
29

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Dog
Elephant
Giraffe
Gultar
Horse
House
Person
Baseline
Ours
URL
Figure 10: Grad-CAM visualization the PACS dataset with Carloon the target domain.
Dog
Elephant
Giraffe
Guitar
Horse
House
Person
Baseline
Ours
Figure 11: Grad-CAM visualization on the PACS dataset with Photo as the target domain.
Dog
Elephant
Giraffe
Guitar
Horse
House
Person
Baseline
Qurs
Figure 12: Crad-CAM visualization on the PACS dataset with Sketch as the target domain.
L.4 Mutual information analysis
To verify whether the representations learned by the visual instrumental variable learner satisfy the instrumental variable conditions, estimate three key mutual information terms S); T(Z:Ú), and T(Z: Y|S)) on the PACS dataset. For better interpretability and consistent scaling, all mutual information values normalized the range [0. As shown in Figure 13] the results indicate that the learned representation Z exhibits high mutual information with the causal factor S, low mutual information with the confounder U, and low conditional mutual information with the potential outcome Y given These findings demonstrate that the learned Visual IVs satisfy the relevance, exclusion, and independence constraints, thereby validating their effectiveness.
Vonalized MonalieedMMooi Mutual Information
0.4
0.2
Domains
KZYS
Figure 13: Mutual information analysis.
30

---

1,5 Convergence analysis
We analyze the convergence behavior on the Art-Painting domain, with the corresponding curves presented in Figure and Figure 13| Specifically, Figure [ demonstratess that the loss decreases steadily across all stages. Figure I revelss that VIV-DG converges rapidly during the early phase of causal representation learning, with accuracy further improving in Phase I+ (causal correction and debiasing) compared Phase Notably, during Phase and Phase III, while training the instrumental variable learner and the regressor, the main network parameters remain unchanged, so the accuracy shown Figure for these two phases stays the same as the final epoch of Phase I.
E. Main Training
Phase IV-itammer Training
Phase III: Regressor Training
Phase F+: Causal Correction
0.7
0:007 End
Stam E.7g
0.69
Stan: 0:42
13-
Roladion 89.5%
6.70
Rodacion 30.2%
0.50
Rolactoo 28.7%
Redacioo 28.4%
Lost
Toal Loss Loss
0.4
0.60
0.5
0.30
0.2
10
20
Epoch
Epoch
Epoch
Epoch
Figure 14: Total loss convergence by stage.
95
Ascuracy (50)
85
75
20
40
60
160
Phase
Phase II
Ppoch
Phase III
Phase I+
Phane Toe:
Paass
Plans Toat
Phanel Tea
Plans
Pass Val
Phone 11
Phau
Figure 15: Overall test/validation accuracy convergence.
J
Broader impact
Our work leverages the exogeneity of visual instrumental variables to mitigate confounding effects, achieving robust domain generalization even in the presence of unobserved confounders, which demonstrates significant societal value. By mitigating confounding effects, it enhances model robustness in open environments, reducing the influence of environmental changes classification results and ensuring more reliable performance in real-world image classification tasks. Furthermore, the improved cross-domain generalization capability facilitates broader applications in medical imaging, autonomous driving, and security surveillance, thereby amplifying the technology's 's societal value. However, this research may also involve certain potential risks that warrant further academic investigation. Specifically, the reliability of the system's 's output could be undermined when deployed in real-world scenarios that diverge significantly from the training distribution.
31


Reviews:
[{"review_id": "eGccfh7iPA", "reviewer": "Reviewer_Z3TL", "summary": "This paper introduces VIV-DG, a novel method for confounding-resistant domain generalization in image classification. The central contribution is the concept of a \"Visual Instrumental Variable\" (Visual IV), which proposes to automatically learn representations from image data that satisfy the conditions of an instrumental variable (IV). This method refines the causal model by partitioning non-causal factors into confounders and useful IVs. The framework consists of three main components trained via a multi-stage alternating optimization: 1) a causal feature extractor, 2) a visual IV learner that is explicitly optimized to enforce the IV conditions of relevance, independence, and exclusion using mutual information objectives, and 3) a regression predictor to generate debiased causal features. The authors claim this approach effectively mitigates the effects of unobserved confounders and outperforms existing state-of-the-art methods.", "questions": "1. The paper uses proxy losses to enforce the IV conditions but only evaluates success via downstream accuracy. Can you provide a more direct verification that the learned representation Z actually satisfies the IV conditions post-training? For example, please report the estimated values of I(Z;S), I(Z;U), and I(Y;Z|S) on a held-out set and compare them to a baseline without these explicit constraints. A satisfactory answer here would provide direct evidence for the central claim of the paper.\n2. The Grad-CAM visualizations (Fig. 6) are confusing because I do not observe a noticeable difference between baseline and the author's method. It also does not provide definitive proof that the model is learning more \"causal\" representations. Can you provide a toy example showing the effectiveness of the proposed method on learning causal representations?", "limitations": "The authors acknowledge the computational cost of the regressor (Remark 2) and briefly mention generic failure modes (Appendix H).", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### **Strengths**:\n\n1. The core idea of *learning* an instrumental variable representation directly from visual data, rather than relying on predefined, potentially invalid instruments (as in IV-DG), is a significant and novel conceptual contribution to the causal DG literature. The proposed structural causal model in Figure 1c, which explicitly separates non-causal factors into confounders and IVs, provides a more nuanced and promising theoretical framing for the problem than the standard approach of discarding all non-causal information.\n\n2. The translation of the three classical IV conditions into differentiable loss functions using mutual information neural estimation and KL-divergence is technically sound and well-motivated. The paper is also thorough in its experimental evaluation, testing the proposed method across four standard DG benchmarks (Digits-DG, PACS, Office-Home, VLCS) and comparing it against a wide range of baselines.\n\n### **Weaknesses**\n\n1. The \"automatic\" IV learner depends critically on a heuristic masking procedure to separate IV-relevant features (texture/color) from causal features (shape). This mask is generated by thresholding the pixel-wise difference between grayscale images (Appendix D). This is a brittle, handcrafted step at the core of an otherwise \"automatic\" learning pipeline. There is no analysis of its robustness (e.g., to the choice of threshold δ) or verification that it truly isolates the intended features.\n\n2. The primary motivation for this work is to improve upon methods like IV-DG that use predefined instruments. However, the comparison to IV-DG is limited, and the authors fail to include it on the Digits-DG benchmark. A more direct and analytical comparison showing why the learned IV succeeds where the predefined one fails is needed, rather than just reporting aggregate scores.\n\n3. The reported performance gains over the strongest baselines are marginal on several key datasets. For instance, the improvement on PACS is approximately 0.4% over FAGT, and on Office-Home, it is 0.3% over CIRL. While SOTA is achieved, these modest gains, coupled with the model's immense complexity and questionable assumptions, fail to provide a compelling conceptual takeaway. The Grad-CAM visualizations (Fig. 6) are confusing because I do not observe a noticeable difference between baseline and the author's method. It also does not provide definitive proof that the model is learning more \"causal\" representations.", "was_revised": true, "final_justification": "I thank the authors for the detailed additional experiments and theoretical analysis. Most of my earlier concerns have been satisfactorily addressed.\n\nThe only remaining major concern is the dependence on handcrafted masking: Although the δ sensitivity experiment on Digits-DG shows reasonable robustness, the grayscale-difference threshold mask remains a core dependency, which partially conflicts with the “automatic” positioning of the IV learning process. I suggest highlighting this limitation more prominently in the final version and prioritizing fully automated mask generation in future work.\n\nOverall, I find that the rebuttal addresses most issues, so I would raise my score to accept.", "rebuttal": "We sincerely appreciate your support of the novelty of our work and your valuable comments. The suggestions help improve our work, and we clarify and address the raised concerns as follows.\n\n**Questions:**\n\n**Q1: Direct verification that the learned IV representation satisfies the IV conditions.**\n\n**R1: We provide validation from both experimental and theoretical perspectives:** \n\n**(1) Experiments on the estimated values of mutual information:** We conduct additional experiments to estimate and quantify the mutual information terms $I(Z;S)$, $I(Z;U)$ and $I(Y;Z|S)$. The results indicate that the learned Visual IV satisfies the relevance, independence, and exclusion conditions, confirming its validity.\n#### **Tab. 1 Estimated values of mutual information on PACS**\n||A|C|\n|-|-|-|\n|I(Z;S)|0.86|0.84|\n|I(Z;U)|0.21|0.25|\n|I(Z;Y\\|S)|0.25|0.27|\n\n**(2) Ablation studies on the three mutual information loss:** We supplement ablation studies on the three mutual information loss terms (**relevance**, **independence**, and **exclusion**) for both versions of our model on the PACS dataset, as shown below. The results show that the **exclusion** loss is the most critical, as it effectively prevents the introduction of extra confounders as IVs. Meanwhile, the **relevance** and **independence** losses help reduce confounding effects and enhance the expressiveness of causal representations. These results, together with those from other datasets, will be included in the revised version.\n#### **Tab. 2 Ablation study of VIV-DG on PACS (ResNet18)**\n|Settings|A|C|P|S| Avg.|\n|-|-|-|-|-|-|\n|w/o I(Z;S)|85.5|80.4|95.8|83.6|86.3|\n|w/o I(Z;U)|84.9|79.7|95.9|83.0|85.9|\n|w/o I(Z;Y\\|S) |84.6|79.1|95.3|82.4|85.4|\n|VIV-DG |86.6|81.5|96.9|83.9|87.2|\n\n#### **Tab. 3 Ablation study of VIV-DG-Lite on PACS (ResNet18)**\n|Settings|A|C|P|S| Avg.|\n|-|-|-|-|-|-|\n|w/o I(Z;S)|85.6|79.7|95.4|82.9|85.9|\n|w/o I(Z;U)|84.7|78.4|95.5|82.4|85.3|\n|w/o I(Z;Y\\|S) |84.5|78.2|95.5|81.9|85.0|\n|VIV-DG-Lite|86.1|81.8|96.6|83.3|87.0|\n\n**(3) Theoretical Support:**\n\nWe supplement a theoretical analysis to support the learnability and identifiability of Visual IVs. Below we summarize the main theoretical results and proof ideas; full details will be included in the revised version:\n\n- **Lemma  (Learnability of Visual IVs):** Let $h_\\psi:\\mathrm{Image}\\to Z$ be a parameterized mapping (e.g., via a deep network), with the following training objective\n\\begin{equation}\n\\mathcal{L}(\\psi)\n\\=\\\n-\\alpha_1 I(Z;S\\bigr)\n\\+\\\\alpha_2 I(Z;U\\bigr)\n\\+\\\\alpha_3 I(Z;Y\\mid S\\bigr),\n\\end{equation}\nwhere $\\alpha_1,\\alpha_2, \\alpha_3 >0$.  Then any global minimizer $\\psi^*$ satisfies\n\n$I(h_{\\psi^*}(\\mathrm{Image});S)>0$, \n\n$I(h_{\\psi^*}(\\mathrm{Image});U)>0$,\n\n$I(h_{\\psi^*}(\\mathrm{Image});\\,Y\\mid S)>0$,  \ni.e. it exactly recovers a Visual IV.\n\n**Proof idea:** We define an objective function that combines the three mutual information terms, and prove that any global minimizer must satisfy all three conditions, thereby yielding a valid Visual IV.\n\n- **Theorem (Identifiability for Visual IVs):** Let $(S, U, Y)$ be three random variables, where\n$S$ denotes the causal factor affecting $Y$,\n$U$ is the confounder,\nand $Y$ is the downstream label. Assume that $\\mathcal H_{Z}=\\{h_\\psi:\\mathrm{Image}\\to Z\\}$  is a sufficiently expressive family of mappings (e.g., one that contains all smooth bijections on a latent subspace).\nThe optimize the objective\n\\begin{equation}\n\\mathcal{L}(\\psi)\n\\=\\\n-\\alpha_1 I(h_\\psi(\\mathrm{Image});S\\bigr)\n\\+\\\\alpha_2 I(h_\\psi(\\mathrm{Image});U\\bigr)\n\\+\\\\alpha_3 I(h_\\psi(\\mathrm{Image});Y\\mid S\\bigr),\n\\end{equation}\nwhere $\\alpha_1,\\alpha_2, \\alpha_3 >0$.  Then any global minimizer $\\psi^*$ yields\n\n\\begin{equation}\nZ^* = h_\\psi(\\mathrm{Image})\n\\end{equation}\nwhich is identifiable in accordance with the definition of Vsual IV we defined.\n\n**Proof idea:** We further prove that, under these constraints, the learned visual IV is identifiable in the function space, with uniqueness guaranteed up to invertible transformations.\n\n**Q2: Explanation of Visualizations and Toy Example.**\n\n**R2: (1) Explanation for the Grad-CAM visualizations:** We apologize for the unclear explanation of the visualization experiments, and we clarify the details as follows. We compare our VIV-DG with CIRL, a strong causal domain generalization baseline, in the visualization experiments. We intentionally select the *Art Painting* domain—where the classification accuracies of the two methods are similar—to demonstrate that our method better captures causal regions. Even if the improvement is not visually striking, it still shows progress in identifying causal cues.\n\nTo further demonstrate the causal representation learning ability of VIV-DG, we extend the Grad-CAM visualizations to the *Cartoon* and *Sketch* domains. (We cannot display the images here.) The results further show that VIV-DG focuses more on the central structure and semantically relevant object regions, which is consistent with the superior performance of VIV-DG over CIRL in these domains.\n\n**Additional visualization evidence:** To verify that the learned IVs alleviate confounding effects, we visualize the Grad-CAM maps of causal features before and after IV-based correction. The results show that the model captures more robust and semantically meaningful regions after using the visual IVs.\n\n**(2) Toy example:** Consider an image of a dog on the grass, where the dog represents the causal feature $S$ and the grass serves as the confounder $U$. Methods like CIRL aim to disentangle causal features but do not explicitly remove confounding bias, so the learned representations may still retain spurious signals (e.g., grass color).\nIn contrast, our method learns a visual IV $Z$ that satisfies the IV conditions and uses it to predict a new causal feature $\\hat{S}$ via a regressor, which is unbiased by confounders. Since $Z$ is explicitly independent of $U$ and influences $Y$ only through $S$, the predicted $\\hat{S}$ is free from confounding interference. It is determined solely by $Z$ and guided by supervision from $Y$, which encourages $\\hat{S}$ to capture mainly causal information. This results in more robust and accurate causal representations.\n\n**Weaknesses:**\n\n**W1: Robustness analysis of the masking process.** \n\n**R1:** We add a sensitivity experiment on the Digits-DG dataset using threshold  δ ranging from 20 to 40, as shown below, and observe minimal performance variation, indicating good robustness. \n#### **Tab. 4 Sensitivity to δ on Digits-DG**\n|δ|MN|MM|SVHN|SYN| Avg.|\n|-|-|-|-|-|-|\n|20|97.2|66.5|76.9|90.4|82.8|\n|25|97.0|66.7|76.3|90.8|82.7|\n|30|97.6|67.1|77.4|91.6|83.4|\n|35|97.4|67.3|77.0|91.8|83.4|\n|40|97.0|66.5|76.6|90.6|82.7|\n\nWe use a grayscale difference δ of 30 to generate the mask, which is a common empirical value in image processing. **This simple strategy helps reduce computational cost while maintaining efficiency.** In future work, we plan to develop an automated mask generation mechanism to further enhance robustness and automation. \n\n**W2: Lack of comparison with IV-DG on the Digits-DG dataset and analysis of their respective advantages and disadvantages.** \n\n**R2: (1) Explanation for not including a comparison with IV-DG on Digits-DG:** Following common practice, we report IV-DG results directly from the original paper for fair comparison. IV-DG is only evaluated on PACS and Office-Home in its original paper, and not on other benchmarks such as Digits-DG. Therefore, we do not include the results of IV-DG on the Digits-DG benchmark. Notably, compared to IV-DG, our method achieves improvements of +2.6% on PACS and +3.0% on Office-Home, demonstrating a clear advantage. \n\n**(2) Learnable IVs vs. Predefined IVs:** Regarding the limitations of predefined IVs: Classical IV assumptions—especially the exclusion condition—are strict and often difficult to verify in practice. For instance, IV-DG selects data from one domain as the IV for another, which can easily violate the exclusion condition. Consider using a cartoon image of a dog as the IV for an art-style image of the same dog. Although the domains differ, the semantic label is the same, and features extracted from the cartoon image may directly predict the label \"dog\", thus violating the exclusion requirement and leading to weak or invalid IVs. \nIn contrast, our learnable IV approach explicitly enforces the IV conditions during training, thereby avoiding such violations and yielding more reliable causal representations.\n\n**W3: The performance improvements are relatively modest, the model introduces additional complexity, and the Grad-CAM visualizations do not clearly illustrate whether more causal representations are being learned.**\n\n**R3: (1) Computational cost.** **We consider the computational cost of VIV-DG to be a reasonable trade-off for achieving both automation and scalability.** The main computational cost of VIV-DG comes from the automatic visual IV learning module. Although this introduces additional training cost, it substantially reduces the manual effort required to design IVs, which is often necessary in predefined IV-based methods. It also avoids reliance on domain knowledge or expert heuristics. Despite achieving only moderate gains over some baselines, it consistently outperforms predefined IV-based methods like IV-DG in both performance and practicality. \n\nMoreover, the computational cost is also a limitation we mention in the paper. To address this, we provide a simplified version, VIV-DG-Lite, which removes one training stage, significantly reducing the computational cost while still maintaining competitive accuracy.\n\n**(2) Theoretical Support:** We provide a theoretical analysis of the identifiability of visual IV, please see our response to Question 1. This further supports the learnability and validity of our approach.\n\n**(3) Grad-CAM visualizations:** For clarification and analysis of the Grad-CAM visualizations, please see our response to Question 2."}, {"review_id": "r0LKgYfef3", "reviewer": "Reviewer_ts1f", "summary": "The paper proposes a novel framework named VIV-DG, which can learn visual instrumental variables in causal domain generalization. It identifies key visual attributes and instrumental variables, in order to retain useful non-causal information and ensures the retention through three different learning paradigms. Results showcase improved performance on causality-driven domain generalization methods and on other baselines.", "questions": "* Could the authors explain why the results vary in their experiments with both versions of the model and on how many iterations or different seeds the results were evaluated?\n* Are there any ablations done on why the Lite version of the model performs as good as the normal model on certain domains?\n* How much VRAM does it take to train this framework and how could it scale on larger backbones for domain generalization training? Are the training resources worth the increase in performance or model explanation?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n* The identification of instrumental variables is novel and interesting, as encoding critical information in a way that is not detrimental towards the generalization of the model seems to provide more information.\n* The pipeline descriptions is sound, providing clear explanations on how the framework can identify causal variables and non-causal variables.\n\nWeaknesses:\n* The results on the task seem a bit random. Authors cite that in PACS that the framework surpasses all baselines, when in actuality it is equal or lower in three out of four domains, despite being higher on average. \n* No explanation is given on why the Lite version of the model is performing better on certain domains. There is also no experiment with the normal version of the model on VLCS.\n* The framework is said to increase computational costs with larger backbones, but the performance gain on the reported datasets seems marginal for the amount of training resources.", "was_revised": true, "final_justification": "Authors have sufficiently answered my questions. Provided they include the new experiments on the manuscript and provide the explanations, I am raising my score to 4. I still think the method is a bit heavy on the compute reesources, but it has merit for causal domain generalization.", "rebuttal": "We sincerely appreciate your support of the novelty of our work and your valuable comments. Your suggestions help strengthen our paper. Here, we clarify and address these issues accordingly.\n\n**Questions:**\n\n**Q1: Could the authors explain why the results vary in their experiments with both versions of the model and on how many iterations or different seeds the results were evaluated?**\n\n**R1: (1) Analysis of result variations between the two versions:** The performance differences between the two model versions mainly arise from their different strategies for predicting causal factors from Visual IVs. \n\n**VIV-DG-Lite** approximates the causal factors by combining the instrumental variables with the initial causal features, which retains more original discriminative information but is more affected by confounders. Therefore, it performs better in domains with smaller distribution shifts, such as *Product* in Office-Home and *Cartoon* in PACS. \n\nIn contrast, **VIV-DG** predicts causal factors via a trained regressor for bias correction, which reduces confounding effects and thus performs better in domains with larger distribution shifts, such as *Sketch* in PACS. \n**Overall, VIV-DG is more robust than VIV-DG-Lite.**\n\n**(2) Explanation regarding how many iterations or different seeds the results were evaluated.** We provide a brief report of the training epochs for VIV-DG and VIV-DG-Lite in Tables 4 and 5 of the appendix. Here, we offer a more detailed explanation for clarity: VIV-DG is trained for a total of 100 epochs, while VIV-DG-Lite is trained for 70 epochs in total, as it removes the regression stage compared to VIV-DG. Specifically, for fair comparison, we fix the number of training epochs of the main encoder (initial training + bias correction fine-tuning) to 50 for both models, which is consistent with the baseline methods. All reported results are averaged over 3 runs with different random seeds. We will clarify these settings in the revised version.\n\n**Q2: Are there any ablations done on why the Lite version of the model performs as good as the normal model on certain domains?**\n\n**R2:** We supplement ablation studies on the three mutual information loss terms (**relevance**, **independence**, and **exclusion**) for both versions of our model, VIV-DG and VIV-DG-Lite, on the PACS dataset, as shown below. \n\n#### **Tab. 1 Ablation study of VIV-DG on PACS (ResNet18)**\n|Settings| A|C|P|S| Avg.|\n|-|-|-|-|-|-|\n|w/o I(Z;S)|85.5|80.4|95.8|83.6|86.3|\n|w/o I(Z;U)|84.9|79.7|95.9|83.0|85.9|\n|w/o I(Z;Y\\|S) |84.6|79.1|95.3|82.4|85.4|\n|VIV-DG |86.6|81.5|96.9|83.9|87.2|\n\n#### **Tab. 2 Ablation study of VIV-DG-Lite on PACS (ResNet18)**\n|Settings| A|C|P|S| Avg.|\n|-|-|-|-|-|-|\n|w/o I(Z;S)|85.6|79.7|95.4|82.9|85.9|\n|w/o I(Z;U)|84.7|78.4|95.5|82.4|85.3|\n|w/o I(Z;Y\\|S) |84.5|78.2|95.5|81.9|85.0|\n|VIV-DG-Lite|86.1|81.8|96.6|83.3|87.0|\n\nThe results show that the **exclusion** loss is the most critical, as it effectively prevents the introduction of extra confounders as instrumental variables. Meanwhile, the **relevance** and **independence** losses work together to reduce confounding effects and enhance the expressiveness of causal representations. These three loss terms collaboratively contribute to the strong performance of both the full and Lite versions. These results, together with those from other datasets, will be included in the revised version.\n\nThe Lite version shares the same mutual information constraint components as the full version. **Notably, the Lite version derives new causal factors by aggregating the instrumental variables with the initial causal representations, which retains more of the original discriminative information and enables it to achieve comparable performance to the full version in certain domains.**\n\n**Q3: VRAM usage, Scalability, Training cost vs. performance.**\n\n**R3: (1) It takes how much VRAM:** By using ResNet-50 as the backbone, single-GPU training on an A800 GPU requires about 40+ GB of memory and runs stably.\n\n**(2) Scalability to larger backbones:** We provide theoretical analyses of the identifiability and learnability of visual IVs (as shown below), showing that any extracted features that satisfy the IV conditions can serve as approximate visual IVs. **This theoretical grounding ensures that our method remains valid regardless of the backbone architecture. As a result, VIV-DG scales effectively to larger backbones in domain generalization training while still producing reliable visual IVs.**\n\n- **lemma[Learnability of Visual IVs]** \nLet $h_\\psi:\\mathrm{Image}\\to Z$ be a parameterized mapping (e.g., via a deep network), with the following training objective\n\\begin{equation}\n\\mathcal{L}(\\psi)\n\\=\\\n-\\alpha_1 I(Z;S\\bigr)\n\\+\\\\alpha_2 I(Z;U\\bigr)\n\\+\\\\alpha_3 I(Z;Y\\mid S\\bigr),\n\\end{equation}\nwhere $\\alpha_1,\\alpha_2, \\alpha_3 >0$.  Then any global minimizer $\\psi^*$ satisfies\n\n$I(h_{\\psi^*}(\\mathrm{Image});S)>0$, \n\n$I(h_{\\psi^*}(\\mathrm{Image});U)>0$,\n\n$I(h_{\\psi^*}(\\mathrm{Image});\\,Y\\mid S)>0$,  \ni.e. it exactly recovers a Visual IVs.\n\n**Proof idea:** Since $-I(Z;S)\\ge -\\sup_\\psi I(Z;S)$,  $I(Z;U)\\ge0$,  $I(Z;Y\\mid S)\\ge0$, any global minimizer $\\psi^*$ must simultaneously (i) maximize $I(Z;S)$, (ii) drive $I(Z;U)$ to zero, and (iii) drive $I(Z;Y\\mid S)$ to zero; otherwise one could reduce $\\mathcal{L}$ further. By the zero‐mutual‐information conditions we get $Z\\perp\\ U$ and $Z\\perp\\ Y\\mid S$, and by the maximization $I(Z;S)>0$. These three facts are exactly the three requirements of the definition of Visual IVs.\n\n- **Theorem (Identifiability for Visual IVs):** Let $(S, U, Y)$ be three random variables, where\n$S$ denotes the causal factor affecting $Y$,\n$U$ is the confounder,\nand $Y$ is the downstream label. Assume that $\\mathcal H_{Z}=\\{h_\\psi:\\mathrm{Image}\\to Z\\}$  is a sufficiently expressive family of mappings (e.g., one that contains all smooth bijections on a latent subspace).\nThe optimize the objective\n\\begin{equation}\n\\mathcal{L}(\\psi)\n\\=\\\n-\\alpha_1 I(h_\\psi(\\mathrm{Image});S\\bigr)\n\\+\\\\alpha_2 I(h_\\psi(\\mathrm{Image});U\\bigr)\n\\+\\\\alpha_3 I(h_\\psi(\\mathrm{Image});Y\\mid S\\bigr),\n\\end{equation}\nwhere $\\alpha_1,\\alpha_2, \\alpha_3 >0$.  Then any global minimizer $\\psi^*$ yields\n\n\\begin{equation}\nZ^* = h_\\psi(\\mathrm{Image})\n\\end{equation}\nwhich is identifiable in accordance with the definition of Vsual IV we defined.\n\n**Proof idea:** We further prove that, under these constraints, the learned visual IV is identifiable in the function space, with uniqueness guaranteed up to invertible transformations.\n\n\n**(3) Trade-off between training cost, interpretability, and automation:** \n\n**We consider the computational cost of VIV-DG to be commensurate with its higher degree of automation, enhanced interpretability, and improved scalability.** The main computational cost comes from the automatic Visual IV learning module. Although this introduces additional training cost, it significantly reduces the manual effort required to design instrumental variables and avoids reliance on domain knowledge or expert heuristics.\n\nImportantly, we supplement our work with theoretical analysis proving that the automatically learned Visual IVs strictly satisfy the IV conditions, which enhances the **interpretability** of the model.\n\nWhile the performance improvements over some baseline methods are moderate, our approach consistently outperforms predefined instrumental variable methods such as IV-DG in terms of accuracy and practical applicability. **Therefore, we consider the computational cost a reasonable trade-off.**\n\nMoreover, the computational cost is also a limitation we mention in the paper. To address this, we provide a simplified version, VIV-DG-Lite, which removes one training stage, significantly reducing the computational cost while still maintaining competitive accuracy.\n\n**Weaknesses:**\n\n**W1: The results seem a bit random. Authors cite that in PACS that the framework surpasses all baselines, when in actuality it is equal or lower in three out of four domains.**\n\n**R1: (1) Explanation of Experimental Results:** At first glance, the experimental results may seem less favorable on individual domains. However, it is important to note that our two versions, VIV-DG and VIV-DG-Lite, are also compared against each other. As shown in Table 1 of the paper, VIV-DG achieves the best performance only on the *Photo* domain in the PACS dataset, which may suggest limited advantage at first glance. However, **a closer examination reveals that VIV-DG is mainly outperformed by its simplified version, VIV-DG-Lite, in the domains where it ranks second.**\n\nIf VIV-DG-Lite is excluded from the comparison, VIV-DG achieves the best average performance on PACS, ranks first in two domains, and second in the remaining two. \n\nFurthermore, beyond PACS, **VIV-DG achieves the best results on two domains of Digits-DG with a notable improvement in average performance, and outperforms all baselines on three out of four domains in Office-Home.**\n\n**W2: Lack of explanation for why the Lite version outperforms in some domains, and missing results of the normal version on VLCS.**\n\n**R2: (1)  Analysis of result variations between the two versions:** The performance differences and detailed analysis between VIV-DG-Lite and VIV-DG across domains are provided in our response to Question 1.\n\n**(2) Results of normal version VIV-DG on VLCS:** We thank the reviewer for pointing this out. This was our oversight. **We supplement the results of VIV-DG on the VLCS dataset**, as shown below, further demonstrating the effectiveness of VIV-DG. We will incorporate these results into Table 2 in the revised version.\n#### **Tab. 3 Results of VIV-DG on VLCS**\n\n|Methods|C|L|S|P|Avg.|\n|-|-|-|-|-|-|\n|VIV-DG-Lite|**99.2**|67.4|74.4|**77.6**| **79.7** |\n|**VIV-DG**|99.0 |**67.7**|**74.6** |77.5|**79.7** |\n\n**W3: Marginal gains vs. Computational cost.**\n\n**R3:** We provide a detailed explanation; please refer to our response to Question 3."}, {"review_id": "D8DELZ4N8x", "reviewer": "Reviewer_ubcT", "summary": "This paper proposes a novel confounding-resistant DG method termed VIV-DG. The VIV-DG automatically learns visual instrumental variables to effectively mitigate the effects of both observed and unobserved confounders and improve domain generalization. However, obtaining valid IVs remains a challenge. The paper defines the novel concept of visual instrumental variables and develops an IVs learning framework composed of three alternately optimized subnetworks that automatically learns valid ones.  Extensive experiments verify the effectiveness of VIV-DG on generalization.", "questions": "See the Weaknesses.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The paper is well-structured, and its concepts are clearly defined.\n2. The motivation is clear. Considering the intrinsic differences among non-casual factors and identifying the valid instrumental variables to improve generalization is reasonable and promising.\n3. The visual explanation well illustrated the effectiveness of VIV-DG in capturing the core object structures.\n\nWeaknesses:\n1. The ablation study is not complete. The ablation study should provide a more in-depth analysis of each component in Visual IV learning and the regression predictor, such as relevance-constrained learning.\n2. Missing experimental results on other common backbones, such as Vision Transformer.\n3. Missing classification experiments on large-scale datasets (such as ImageNet), as well as experiments on other OOD generalization datasets like (ImageNet-Sketch).\n4. Missing theoretical proofs of the VIV-DG, as well as analysis and proofs of the generalization's lower and upper bounds.\n5. Missing experiments and analysis of the VIV-DG 's efficiency.", "was_revised": true, "final_justification": "I have already read all the comments and discussions about this paper and have finally decided to keep my initial score.", "rebuttal": "We sincerely appreciate your support of the novelty of our work and your valuable comments. Here, we clarify and address these issues accordingly.\n\n**Q1: Ablation study:** \n\n**R1: (1) Ablation studies on the three mutual information loss:** We supplement ablation studies on the three mutual information loss terms (**relevance**, **independence**, and **exclusion**) for both versions of our model, VIV-DG and VIV-DG-Lite, on the PACS dataset, as shown below. \n\nThe results show that the exclusion loss is the most critical, as it effectively prevents the introduction of extra confounders as IVs. Meanwhile, the relevance and independence losses help reduce confounding effects and enhance the expressiveness of causal representations. These results, together with those from other datasets, will be included in the revised version.\n#### **Tab. 1 Ablation study of VIV-DG on PACS (ResNet18)**\n| Settings| A|C|P|S| Avg.|\n|-----------------|-------|-------|-------|---------|--------|\n|w/o I(Z;S)|85.5|80.4|95.8|83.6|86.3|\n|w/o I(Z;U)|84.9|79.7|95.9|83.0|85.9|\n|w/o I(Z;Y\\|S) |84.6|79.1|95.3|82.4|85.4|\n|VIV-DG |86.6|81.5|96.9|83.9|87.2|\n\n#### **Tab. 2 Ablation study of VIV-DG-Lite on PACS (ResNet18)**\n| Settings| A|C|P|S| Avg.|\n|-----------------|-------|-------|-------|---------|--------|\n|w/o I(Z;S)|85.6|79.7|95.4|82.9|85.9|\n|w/o I(Z;U)|84.7|78.4|95.5|82.4|85.3|\n|w/o I(Z;Y\\|S) |84.5|78.2|95.5|81.9|85.0|\n|VIV-DG-Lite|86.1|81.8|96.6|83.3|87.0|\n\n**(2) Ablation analysis of the regression predictor:**\nThe ablation analysis of the regression module is essentially reflected in the comparison between VIV-DG and VIV-DG-Lite: VIV-DG incorporates a regressor to predict causal representations, while VIV-DG-Lite does not. As shown in Table 1 of the paper, the performance of VIV-DG and VIV-DG-Lite varies across target domains. We provide further analysis to explain this: **Overall, the regression predictor effectively prevents the learned causal representations from being affected by confounders.**\n\nSpecifically, VIV-DG-Lite approximates the causal representations by combining the instrumental variables with the initial causal representations, which retains more original discriminative information but is more susceptible to confounding. Therefore, it performs better in domains with smaller distribution shifts, such as *Product* in Office-Home and *Cartoon* in PACS dataset. \n\nIn contrast, VIV-DG predicts causal representations through a trained regressor for bias correction, which helps reduce confounding effects and thus performs better in domains with larger distribution shifts, such as *Sketch* in PACS dataset. Overall, VIV-DG demonstrates greater robustness compared to VIV-DG-Lite.\n\n**Q2: Missing experimental results on other common backbones.**\n\n**R2:** The baseline methods we compare with mainly use ResNet18 and ResNet50. For fair comparison, we adopt the same backbones and do not conduct experiments on Vision Transformer. In fact, we use three different backbones in total: ResNet18, ResNet50, and a lightweight backbone with the same architecture as existing works on the Digits-DG dataset.\n\nIn addition, *Lemma 2* and the *Theorem* provided in our response to Question 4 support that our method can automatically learn instrumental variables across different backbone architectures.\n\n**Q3: Experiments on large-scale datasets.**\n\n**R3:** We supplement experiments on the very large-scale domain generalization dataset DomainNet (586,575 images, 345 classes, and 6 domains). Due to the dataset’s size and long training time, we currently report results with *Sketch* as the target domain where distribution shift is large, and the other five domains as source domains. The results are as follows:\n\n#### **Tab. 3 Results on DomainNet (ResNet50)**\n||SagNet |RSC |iDAG|VIV-DG|\n|-----------------|-------|-------|------|------|\n|Sketch |48.8|47.8|56.9|56.9|\n\n**Q4: Theoretical proofs of the VIV-DG.**\n\n**R4:** We supplement a theoretical analysis to demonstrate both the learnability and identifiability of Visual IVs, thereby establishing the effectiveness and theoretical soundness of our VIV-DG approach. Below we summarize the main theoretical results and proof ideas; full details will be included in the revised version:\n\n- **Lemma 1 (Identifiability Conditions for Visual IVs via (Conditional) Mutual Information):** Let $(S, U, Y)$ be three random variables, where\n$S$ denotes the causal factor affecting $Y$,\n$U$ is the confounder,\nand $Y$ is the downstream label.\nA candidate variable $Z$ is called a visual instrument variable if it simultaneously satisfies: (i) Relevance: $I(Z;S)>0$; (ii) Independence: $I(Z;U)=0$; (iii) Exclusion: $I(Z;Y\\mid S)=0$. Here $I(\\cdot;\\cdot)$ and $I(\\cdot;\\cdot\\mid\\cdot)$ denote (conditional) mutual information.\n\n**Proof idea:** Based on the integral definition of mutual information, we establish the connection between IV conditions and mutual information. Each condition is verified individually to show that the three requirements for IVs can be rigorously expressed in terms of mutual information. We demonstrate that the three classical IV conditions—relevance, independence, and exclusion—can be rigorously characterized using (conditional) mutual information.\n\n- **Lemma 2 (Learnability of Visual IVs):** Let $h_\\psi:\\mathrm{Image}\\to Z$ be a parameterized mapping (e.g., via a deep network), with the following training objective\n\\begin{equation}\n\\mathcal{L}(\\psi)\n\\=\\\n-\\alpha_1 I(Z;S\\bigr)\n\\+\\\\alpha_2 I(Z;U\\bigr)\n\\+\\\\alpha_3 I(Z;Y\\mid S\\bigr),\n\\end{equation}\nwhere $\\alpha_1,\\alpha_2, \\alpha_3 >0$.  Then any global minimizer $\\psi^*$ satisfies\n\n$I(h_{\\psi^*}(\\mathrm{Image});S)>0$, \n\n$I(h_{\\psi^*}(\\mathrm{Image});U)>0$,\n\n$I(h_{\\psi^*}(\\mathrm{Image});\\,Y\\mid S)>0$,  \ni.e. it exactly recovers a Visual IVs.\n\n**Proof idea:** We define an objective function that combines the three mutual information terms, and prove that any global minimizer must satisfy all three conditions, thereby yielding a valid Visual IV. Since $-I(Z;S)\\ge -\\sup_\\psi I(Z;S)$,  $I(Z;U)\\ge0$,  $I(Z;Y\\mid S)\\ge0$, any global minimizer $\\psi^*$ must simultaneously (i) maximize $I(Z;S)$, (ii) drive $I(Z;U)$ to zero, and (iii) drive $I(Z;Y\\mid S)$ to zero; otherwise one could reduce $\\mathcal{L}$ further. By the zero‐mutual‐information conditions we get $Z\\perp\\ U$ and $Z\\perp\\ Y\\mid S$, and by the maximization $I(Z;S)>0$. These three facts are exactly the three requirements of the definition of Visual IVs.\n\n- **Theorem (Identifiability for Visual IVs):** Let $(S, U, Y)$ be three random variables, where\n$S$ denotes the causal factor affecting $Y$,\n$U$ is the confounder,\nand $Y$ is the downstream label. Assume that $\\mathcal H_{Z}=\\{h_\\psi:\\mathrm{Image}\\to Z\\}$  is a sufficiently expressive family of mappings (e.g., one that contains all smooth bijections on a latent subspace).\nThe optimize the objective\n\\begin{equation}\n\\mathcal{L}(\\psi)\n\\=\\\n-\\alpha_1 I(h_\\psi(\\mathrm{Image});S\\bigr)\n\\+\\\\alpha_2 I(h_\\psi(\\mathrm{Image});U\\bigr)\n\\+\\\\alpha_3 I(h_\\psi(\\mathrm{Image});Y\\mid S\\bigr),\n\\end{equation}\nwhere $\\alpha_1,\\alpha_2, \\alpha_3 >0$.  Then any global minimizer $\\psi^*$ yields\n\n\\begin{equation}\nZ^* = h_\\psi(\\mathrm{Image})\n\\end{equation}\nwhich is identifiable in accordance with the definition of Vsual IV we defined.\n\n**Proof idea:** We further prove from the perspectives of existence and uniqueness that under these mutual information constraints, the learned visual IV is identifiable in the function space and guaranteed to be unique under invertible transformations.\n\nIn addition to the theoretical support provided by the above theorems, we will also include a theoretical analysis of generalization performance in the revised version.\n\n**Q5: Experiments and analysis of the VIV-DG 's efficiency.**\n\n**R5:** Under the experimental settings described in the appendix, we supplement the analysis and experiments to evaluate the efficiency of VIV-DG, including convergence behavior, computational complexity, and GPU memory usage.\n\n- **Convergence Behavior Experiment:** We record the convergence behavior during training and plot the corresponding convergence curves.  The results show that VIV-DG converges quickly in the initial stage of causal representation learning. In the fourth stage (bias correction), the model consistently improves upon the accuracy achieved in the first stage. Additionally, in the second and third stages, the loss decreases rapidly, indicating stable and efficient training across all phases.\n\n- **Computational Complexity:** The overall time complexity of our training framework is: $T = \\mathcal{O}(N \\cdot E \\cdot P_{\\max})$ where $N$ denotes the number of training samples, $E$ is the total number of training epochs, and $P_{\\max}$ represents the maximum computational cost among the modules trained in each stage. This complexity does not involve exponential or polynomial growth, making it computationally tractable.\n\n- **GPU Memory Usage:** Using ResNet-50 as the backbone, single-GPU training on an A800 GPU requires approximately 40+ GB of memory and runs stably without memory overflow or interruption.\n\n**The primary computational cost arises from the automatic visual IV learning module. However, we consider this a reasonable and acceptable trade-off, given the improvements in automation and generalization capability that it brings.**"}, {"review_id": "gCkeRHBqDT", "reviewer": "Reviewer_z3cX", "summary": "This work proposes Visual Instrumental Variable Learning for confounding-resistant domain generalization, which introduces instrumental variables to address the challenge of identifying confounders. The optimization process is decoupled into three stages:\n(i) The first stage utilizes frequency-based augmentation to learn causal factors under the constraints of MMD and classification loss;\n(ii) The second stage aims to learn implicit visual instrumental variables using three designed loss functions;\n(iii) The third stage trains a regressor to learn causal factors responsible for classification.\nExperiments are conducted on PACS, VLCS, Office-Home, and Digits-DG.", "questions": "1. Why is VIV-DG not included in Table 2?\n2. How are the positive and negative samples generated? How are the confounder indices identified in Sections 3.4.2 and 3.4.3?", "limitations": "yes", "rating": 2, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n(i) The proposed visual instrumental variable learning is theoretically effective and interesting for causal representation learning.\n(ii) The design of the three loss functions adheres to the principles of instrumental variable learning.\n(iii) The experiments outperform some existing methods.\nWeaknesses: There are several issues that need to be addressed:\n(i) Although the theory is interesting, the modeling may not be appropriate. The causal graph (i.e., Figure 4) appears incorrect—why do unobserved confounders not affect the input data? Although the appendix provides an explanation for the partial arrows in the figure, further clarification and examples are needed, as causal theory is inherently abstract.\n(ii) The experimental results are not convincing. The method is not evaluated on the standard DomainBed benchmark, and comparisons with more other causal learning-based methods are required.\n(iii) The optimization process is complex. How does its time complexity compare with other methods?\n(iv) The example involving only color and texture may not be sufficient to identify confounders. More comprehensive examples are needed.\n(v) The paper should clarify what can be visualized in the original figure or visualize the features learned for the instrumental variable and causal features SSS.\n(vi) The ablation studies are simple and lack sufficient investigation into the three designed loss functions.", "was_revised": true, "final_justification": "During the rebuttal phase, the authors still have not provided experiments on additional benchmarks from DomainBed, which makes it difficult to fully demonstrate the effectiveness of their approach. DomainBed includes more than the three datasets used in the paper.\nIn W5, the visualization cannot demonstrate the IV and causal features effectively. Furthermore, the response to the point, \"However, these factors do not appear in the current observed input data or known source domains, and therefore do not exert a direct causal effect on the input data,\" is not accurate. In the causal graph, there must be a connection between C and the input. The current explanation does not address this issue appropriately. Therefore, I maintain my original score", "rebuttal": "We are grateful for your recognition of the theoretical effectiveness and conceptual interest of our proposed visual IV learning framework. Here, we clarify and address these issues accordingly.\n\n**Questions:**\n\n**Q1: Why is VIV-DG not included in Table 2?**\n\n**R1:** We thank the reviewer for pointing this out. This was our oversight. We now supplement the results of VIV-DG on the VLCS dataset, further demonstrating the effectiveness of VIV-DG. \n#### **Tab. 1 Results of VIV-DG on VLCS**\n|Methods|C|L|S|P|Avg.|\n|-|-|-|-|-|-|\n|VIV-DG-Lite | **99.2** | 67.4 | 74.4 | **77.6**|**79.7**|\n|VIV-DG|99.0 |**67.7** |**74.6** |77.5|**79.7**|\n\n**Q2: Positive and negative samples. Confounder indices.**\n\n**R1: (1) Explanation of positive and negative samples generated.** In Section 3.4.2, positive sample pairs consist of visual instrumental variables $Z$ and causal representations $S$ from the same original sample, denoted as ${(z_i, s_i)}_{i=1}^N$, where both $z_i$ and $s_i$ originate from the image $x_i$. Negative sample pairs are formed by combining $Z$ and $S$ from different samples, $(z_i, s_j)$, where $i \\neq j$, created by cross-sample indexing. Similarly, in Section 3.4.3, positive sample pairs consist of $Z$ and confounders $U$ from the same image, while negative sample pairs combine $Z$ with shuffled confounders $U$ from different samples. This construction of positive and negative samples effectively approximates mutual information estimation during mini-batch training.\n\n**(2) Explanation of confounder indices:** The indices of confounders are automatically identified and separated by the model through the positive-negative sample contrastive mechanism during training, rather than being explicitly annotated or manually defined.\n\n**Weaknesses:**\n\n**W1: Explanation of the rationality of the causal graph.**\n\n**R1:** We apologize for the misunderstanding caused by our unclear explanation and appreciate the opportunity to clarify. In Fig. 4, the unobserved confounder $C$ does not have a direct causal link to the input image $X$, which is an intentional modeling choice. Without loss of generality, we assume that $C$ represents confounders related to **unknown domain** styles or backgrounds. Since domains cannot be exhaustively enumerated, such unobserved confounders inevitably exist. However, these factors do not appear in the current observed input data or known source domains, and therefore do not exert a direct causal effect on the input data $X$. Conversely, adding a direct edge from $C$ to $X$ would introduce ambiguity by implying that $C$ is the observed confounder.\n\nIt is worth noting that both known and unknown domains share the same causal features $S$ and potential outcome $Y$. In the unseen domain where the confounder $C$ exists, $C$ also affects $S$ and $Y$. Therefore, we model this by adding causal edges from $C$ to both $S$ and $Y$ in Fig. 4.\n\n\n**W2: Not evaluated on DomainBed and comparisons with other causal methods.**\n\n**R1: (1) Standard DomainBed benchmark:** Although we do not run our experiments directly on the DomainBed codebase, our experimental setup strictly follows the standard DomainBed protocol, including dataset splits, backbones, and the leave-one-domain-out strategy. For fair comparison，our main extraction network (initial training + bias correction fine-tuning) is trained for 50 epochs, consistent with the compared methods.\n\n**(2) More causal methods:** We further investigated and included several causal methods. The results are as follows. Our method consistently outperforms these approaches.\n\n#### **Tab. 2 Results on Digits-DG**\n|Methods|MN|MM|SVHN|SYN|Avg.|\n|-|-|-|-|-|-|\n|CDIM[1]|98.7|64.0|74.1|92.9|82.4|\n|VIV-DG|97.6|67.1|77.4|91.6|83.4|\n\n#### **Tab. 3 Results on PACS (ResNet18)**\n|Methods|A|C|P|S|Avg.|\n|-|-|-|-|-|-|\n|CDIM[1]|83.6|77.6|95.5|78.2|83.7|\n|VIV-DG|86.6|81.5|96.9|83.9|**87.2**|\n\n#### **Tab. 4 Results on PACS (ResNet50)**\n|Methods|A|C|P|S|Avg.|\n|-|-|-|-|-|-|\n|CSRDN[2]|88.3|84.5|97.5|84.9|88.8|\n|RICE[3]|87.8|84.3|96.8|84.7|88.4|\n|VIV-DG-Lite|90.3|83.8|97.6|86.7|**89.6**|\n\n#### **Tab. 5 Results on VLCS (ResNet50)**\n|Methods|C|L|S|P|Avg.|\n|-|-|-|-|-|-|\n|CSRDN[2]|98.8|67.0|74.2|77.1|79.3|\n|RICE[3]|98.3|69.2|74.6|75.1|79.3|\n|VIV-DG-Lite | 99.2 | 67.4 | 74.4 | 77.6|**79.7**|\n|VIV-DG|99.0 |67.7 |74.6 |77.5|**79.7**|\n\n[1] Unbiased Semantic Representation Learning Based on Causal Disentanglement for Domain Generalization. (2024)\n\n[2] A Causality-Aware Perspective on Domain Generalization via Domain Intervention. (2024)\n\n[3] Out-of-distribution Generalization with Causal Invariant Transformations. (2022)\n\n**W3: Time complexity.**\n\n**R3: (1) Computational Complexity:** The overall time complexity of our training framework is: $T = \\mathcal{O}(N \\cdot E \\cdot P_{\\max})$ where $N$ denotes the number of training samples, $E$ is the total number of training epochs, and $P_{\\max}$ represents the maximum computational cost among the modules trained in each stage. This complexity does not involve exponential or polynomial growth, making it computationally tractable.\n\n**(2) The primary computational cost arises from the automatic visual IV learning module. However, we consider this a reasonable and acceptable trade-off, given the improvements in automation and generalization capability that it brings.**  Specifically, the training process of VIV-DG is slightly more complex and time-consuming than some baseline methods, mainly due to its multi-stage Visual IV learning. However, unlike predefined IV methods, our approach does not rely on domain experts for IV design and selection, avoiding high human costs and prior bias. It also offers better adaptability and scalability. \n\n**W4: More comprehensive examples.**\n\n**R4:** We appreciate the suggestion and will include more representative examples of confounders in the revised version, such as scene background, co-occurrence bias, image style, lighting conditions, and camera angles. They are typically entangled and difficult to disentangle in the absence of labeled supervision. These confounders tend to exhibit similar patterns in the Fourier amplitude information of images, such as color, style, and brightness. Therefore, we propose a feature extraction method based on reconstructing images from amplitude information to model the confounders. Our Visual IV method mitigates the influence of both observed and unobserved confounders without explicitly modeling them.\n\n**W5: The paper should clarify what can be visualized in the original figure or visualize the features learned for the IV and causal features.**\n\n**R5:** We apologize for the unclear explanation of the visualization experiments, and we clarify the details as follows. We compare our VIV-DG with CIRL, a strong causal domain generalization baseline, in the visualization experiments. We intentionally select the *Art Painting* domain—where the classification accuracies of the two methods are similar—to demonstrate that our method better captures causal regions. Even if the improvement is not visually striking, it still shows progress in identifying causal cues.\n\nTo further demonstrate the causal representation learning ability of our method, we extend the Grad-CAM visualizations to the *Cartoon* and *Sketch* domains. (We cannot display the images here.) The results further show that VIV-DG focuses more on the central structure and semantically relevant object regions, which is consistent with the superior performance of VIV-DG over CIRL in these domains.\n\n**Additional visualization evidence:** To verify that the learned instrumental variables alleviate confounding effects, we visualize the Grad-CAM maps of causal features before and after IV-based correction. The visual results show that the model captures more robust and semantically meaningful regions after using the visual IVs.\n\n**Quantitative verification through mutual information:** Since visual results cannot be displayed here, we also provide more direct and quantitative evidence of IV effectiveness. We compute the estimated mutual information values of $I(Z;S)$, $I(Z;U)$, and $I(Y;Z|S)$ to verify whether the learned representation $Z$ satisfies the relevance, independence, and exclusion conditions of an instrumental variable. The results indicate that the learned representation approximately satisfies the IV conditions and thus serves as an effective guidance for mitigating confounding bias. We will include these results in the revised version.\n#### **Tab. 6 Estimated values of mutual information on PACS**\n||A|C|\n|-|-|-|\n|I(Z;S)|0.86|0.84|\n|I(Z;U)|0.21|0.25|\n|I(Z;Y\\|S)|0.25|0.27|\n\n**W6: Ablation studies.**\n\n**R6:** We supplement ablation studies on the three mutual information loss terms (**relevance**, **independence**, and **exclusion**) for both versions of our model, VIV-DG and VIV-DG-Lite, on the PACS dataset, as shown below. \n\n#### **Tab. 7 Ablation study of VIV-DG on PACS (ResNet18)**\n| Settings| A|C|P|S| Avg.|\n|-|-|-|-|-|-|\n|w/o I(Z;S)|85.5|80.4|95.8|83.6|86.3|\n|w/o I(Z;U)|84.9|79.7|95.9|83.0|85.9|\n|w/o I(Z;Y\\|S) |84.6|79.1|95.3|82.4|85.4|\n|VIV-DG |86.6|81.5|96.9|83.9|87.2|\n\n#### **Tab. 8 Ablation study of VIV-DG-Lite on PACS (ResNet18)**\n|Settings| A|C|P|S| Avg.|\n|-|-|-|-|-|-|\n|w/o I(Z;S)|85.6|79.7|95.4|82.9|85.9|\n|w/o I(Z;U)|84.7|78.4|95.5|82.4|85.3|\n|w/o I(Z;Y\\|S) |84.5|78.2|95.5|81.9|85.0|\n|VIV-DG-Lite|86.1|81.8|96.6|83.3|87.0|\n\nThe results show that the **exclusion** loss is the most critical, as it effectively prevents the introduction of extra confounders as instrumental variables. Meanwhile, the **relevance** and **independence** losses work together to reduce confounding effects and enhance the expressiveness of causal representations. These three loss terms collaboratively contribute to the strong performance of both the full and Lite versions. These results, together with those from other datasets, will be included in the revised version."}]

Author rebuttal:

Meta-review: The paper introduces the novel concept of Visual IVs, which is both innovative and impactful for domain generalization. The motivation is clear, and the framework is logically sound. The authors made strong efforts during rebuttal, including new experiments and theoretical clarifications. I suggest adding the DomainBed results in the final version. Overall, I recommend acceptance.
Decision: Accept (poster)