LOG: Active Model Adaptation for Label-Efficient OOD Generalization
Authors: Jie-Jing Shao
Lan-Zhe Guo
Xiao-wen Yang
Yu-Feng Li
Venue: NeurIPS 2022 Accept (2022)
Keywords: out-of-distribution generalization
active learning
domain adaptation

Abstract: This work discusses how to achieve worst-case Out-Of-Distribution (OOD) generalization for a variety of distributions based on a relatively small labeling cost. The problem has broad applications, especially in non-i.i.d. open-world scenarios. Previous studies either rely on a large amount of labeling cost or lack of guarantees about the worst-case generalization. In this work, we show for the first time that active model adaptation could achieve both good performance and robustness based on the invariant risk minimization principle. We propose \textsc{Log}, an interactive model adaptation framework, with two sub-modules: active sample selection and causal invariant learning. Specifically, we formulate the active selection as a mixture distribution separation problem and present an unbiased estimator, which could find the samples that violate the current invariant relationship, with a provable guarantee. The theoretical analysis supports that both sub-modules contribute to generalization. A large number of experimental results confirm the promising performance of the new algorithm.

LOG: Active Model Adaptation for Label-Efficient OOD Generalization
Jie-Jing Shao, Lan-Zhe Guo, Xiao-Wer Yang, Yu-Feng Li* National Key Laboratory for Novel Software Technology Nanjing University, Nanjing 210023, China {shaojj, guolz, yangxu, liyf) (lyf)elamda.n)) edu. cn
Abstract
This work discusses how to achieve worst-case Out-Of-Distribution (OOD) gen- eralization for a variety of distributions based on relatively small labeling cost. The problem has broad applications, especially in non-i.idd open-world scenarios. Previous studies either rely on large amount of labeling cost or lack of guarantees about the worst-case generalization. In this work, we show for the first time that active model adaptation could achieve both good performance and robustness based on the invariant risk minimization principle. We propose LOG, an interactive model adaptation framework, with two aub-modules: active sample selection and causal invariant learning. Specifically, we formulate the active selection as a mixture distribution separation problem and present an unbiased estimator, which could find the samples that violate the current invariant relationship, with provable guarantee. The theoretical analysis supports that both sub-modules contribute generalization. A large number of experimental results confirm the promising performance the new algorithm.
1
Introduction
Machine learning models are typically trained and tested on the same data distribution. However, when these models are deployed in real task scenarios, they face much inapplicability, because the data distribution of the target task usually deviates from that of training. For example, the financial data prediction model based local users is often inaccurate predicting user behavior other regions. Similar examples include self-driving 42 speech recognition [16], influenza detection (30] etc.
To this problem, the recent research on 00D (Out- Of Di-tirrrbbutinn Generalization 23
Training Distributton CALIFORSEA
Testing Distrixtions
13
has given a series of technolo- gies, trying to obtain models with robustness, which have certain worst-case generalization guarantees on a variety of unseen distributions. Although they do not access target data, they need a large number of high-quality source data to remove the source- source-specific spurious correla- tion and obtain the generalizable invariant re- lationship, i.e., sufficient multi-source labeled data and accurate source information. However, these two requirements are still difficult to sat- isfy most practical tasks.
40
19 5
Develop
Deplog
Model
Data
NEW YORK
Bectify model
A Actively anapiate fee samples
Expest
Figure 1: Example of model adaptation to a variety of distributions. We would like to annotate a few samples from target testing distributions to rectify the model in the source iteratively.
Conresponding author
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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There are also some more traditional approaches to handle distribution shift, such as Domain Adaptation [2 9 20 They introduce unlabeled data from the target distribution, reducing the requirements on the source data relative to OOD generalization. However, they typically assume that target samples are from an isolated homogeneous distribution, which ignores the generalization in varied distributions. Recently, there are some works [10] [6 22 33 focusing on the adaptation to multiple latent domains. Nevertheless, the robustness, i.c., worst-case guarantees has not been addressed, especially without explicit domain labels.
To make a reasonable compromise between strongly labeled information dependence and robustness, Active Model Adaptation may be a feasible scheme. It gradually increases the labeling cost and actively annotates the data in the target task that is difficult to be robustly generalized. Such scheme could effectively reduce the strong dependence on label information. It has shown promising performance when the target distribution is an isolated homogeneous distribution (39 28
In this paper, we would like to derive a benefit from Active Model Adaptation to address robustness small labeling cost. We first propose the invariant risk minimization principle for active model adaptation. That is, we could tend to both overall performance and worst-case guarantee via maximal invariant predictor. Based on this, we further present an interactive framework to achieve label- efficient OOD generalization (LOG), composed of an actively querying module and an invariant learning module. The key challenge is to find unlabeled samples where the current invariant model does not hold. Based on the structural causal model assumption, we formulate it as a mixture distribution separation problem and present an unbiased estimator to address it. In theory, our actively querying module could accurately find the samples which violate the current invariant relationship. In the experiments on a series of tasks, the promising performance of our LoG has been confirmed.
2 Related Work
Generalization works learning models that generalize well variety of unseen distributions. There are two main branches: Domain Generalization [23] 40) mostly focuses on computer vision problems predictions are prone to be affected by disturbance on images (e.g., style, background, etc). Causal & Invariant Learning starts from causal inference and explores causal variables to address generalization ability under covariate shift [ 35). Most of them rely on labeled data from multiple sources to remove the source- sourc--ppccffic spurious correlation. Recently, some works [19 5 attempt to mine the heterogeneity of assembled source without explicit prior division. Nevertheless, they commonly assume source data is sufficient to learn the invariance and ignore the exploration of data from target distributions.
OOD
Domain Adaptation works on addressing the domain shift between the training source and the
testing target, where no labeled data are available in the target domain 25 20 21). They typically assume the target samples are from single homogeneous domain. The most related subtopic to us is Latent Domain & Domaiin-gnostic Adaptation that focuses on the target distribution with multiple latent domains 226 22 33 Although they propose methods with good overall performance in the presence of multiple latent distributions, the worst-case guarantee has not been addressed.
Active Model Adaptation works active learning under distribution shift. Previous works have focused on two types of distribution shifts. 1) label space shift: adapt pre-trained model to a task with different label spaces [II] 2) domain shift: adapt source model target domain (39 28 34). To the best of our knowledge, all of them directly regard the target distribution as a homogeneous domain and ignore the generalization ability when massive distributions exist.
3 Problem and Analysis
Generally speaking, the above technologies are difficult to directly deal with the model generalization of variety of distributions under a small labeling cost. To deal with such a challenge, in this paper, focusing on active model adaptation, we first analyze the theoretical basis of the problem. Based on this, we present a new active adaptation principle, invariant risk minimization. We further put forward the corresponding algorithm, and show its effect on generalization ability.

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3.1 Problem Formulation
In the source training stage, the learner collects a sufficient labeled dataset Ds (D" leeEs. which is a mixture of data D E X' x y collected from the collection of training sources E Eg. X' and y denote an imput and output space, respectively. Source model fs X + + is well-trained on the source dataset Dy with a small risk R(Js; Ds). When the model is deployed in open-world scenarios, it needs to adapt to the distributions εr, i.e. a collection of varying testing distributions. Following 26 33,, we formulate the target distribution Dr x' as mixture of base distributions:
Dr =y=E DD where DDE is the distribution observed from the environment e and X represents the corresponding proportion. Note that in reality, data are firequetly assembled under implicit environment information e and >5 thus we do not depend on this information to develop algorithms. They are used only during evaluation. To fast adapt the source model to the target distribution, active learning has been introduced [39 28 g 34 Generally, their goal is:
Definition 3.1 (Performance Maximization). The performance goal is to minimize the generalization risk of model f on the overall target distribution Dr based on some queried samples Q:
min RIA ([s (ss, ;D)
(1)
Q.A
where A is a model adaptation algorithm to rectify fs: f + A A (Ds, Q}).
In addition the optimization of ideal overall generalization, we also need to comprehensively consider the robustness, i.c., worst-case generalization on Ey:
Definition 3.2 (Robustness Preservation). The robustness means we could maintain low risk even in the worst distribution D across Er, i.e. worst-case guarantee on each base distribution D
min max (Jss: (Ds, :D) eEA
Q and A are important to address active model adaptation. Previous works [I| focus on the Q address performance maximization. They do not consider the choice of A (simply take it as standard supervised learning or semi-supervised domain adaptation) and the robustness.
39 28
8
mainly
3.2 Invariant Risk Minimization
Without any prior knowledge or structural assumptions, it is impossible to adapt the source model to target distributions, since one cannot characterize the shift between Es and εy. Following the [1 2 24 38 [77, we consider the data generation via structural causal model under the covariate shift assumption (i.c., the P(y|z) is unchanged across the varying distributions):
Assumption 3.3. Consider structural causal model 4ll governing the random distribution P(X, Y) and the learning goal of predicting Y from X. Then the set of all distributions & indexes all the interventional distribution Pe (X+, Ye) obtainable by valid interventions
X S(Z., Zg), Y f(.)) L
(3)
Zg and 7 represent the semantic variable and intervention variable, respectively. The feature X could be regrad an observation S for Z, in the intervention variable e, influenced by the Zς. The label Y is caused by the semantic z and an independent noise term e. We assume the data generation process S is inevitable and there exists (S(())) Z, to recover the semantics for all Z. and Z
Remark 3.4. Following the above assumption about structural causal model, the ideal 4- satisfies: 1)
The margirial invariance Pa (*( = =pe' (8*(X) and conditional invariance P Po (b (X)) = pe' (Y (X)) hold for any e,e' E E. 2) It is sufficient to predict the target using de (X) as the
input: Y = f()((X)) +c, 1 X, across varying e E.
To acquire such (X)), a branch of invariant risk minimization proposals I Invariant Features and the corresponding Maximal Invariant Predictor, defined as:
3
19
5
finds the
Definition 3.5 (Invariant Features). The set of invariant features I with respect to E is defined as: (XX) _
Ie = (x)) HY (X)]
where H|- is Shannon entropy of a random variable. The corresponding Maxiaml Invariant Predictor (MIP) of Ie is defined as: "* arg max T(Y; () where T(::-) measures Shannon mutual information between two random variables.
3

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Here we prove that we could achieve both performance and robustness based on the MIP d"
Theorem 3.6. For predictor 4" (X) in the collection &, the solution (*(X) P(Y could achieve both performance maximization ; S mmnn R(f: Dr) and worst-case robustness maX.Es R(f"; D*) I miny maX.eg Rff:D").
Although the ideal MIP has shown a favorable theoretical guarantee, we find its generalization is heavily dependent on the heterogeneity of source distributions collection Es. We further analyze the connection between generalization and source heterogeneity and propose an active model adaptation framework expand the heterogeneity and trend to the ideal MIP.
3.3 Active Heterogeneity Expansion
Given source collection Es and the target collection Ey, denote the corresponding invariant features set Is and Ir respectively. For &s S Êr, the corresponding invariant features set satisfies Ir S Is [9] It indicates the generalization of Is could be improved by extending the source &s and promoting the Iş to ideal Σr. Here, we present the generalization condition for &s to unseen distribution e and then justify that the generalization of 4g is heavily dependent on the heterogeneity of &s.
Theorem 3.7. For distribution pe' (x Y"), if ((X') ) = (b(X)) Ig is equal to the invari- ance set constrained by Eg {e'). The optimal source model fs could generalize on the distribution pe', The generaliable distributions: εg = {e' [Is Iegufer) = (e') (Xx)), Es}.
Particularly in the linear structural causal model, we further assume the semantics Z, takes values in RR, intervention Z takes values in R and S (dx((++)) Let $- Rdxd, we have:
Eg = 1
2,6 G E Es, x E ker($))
(4)
where dim (ker(O)) = dim((span(( (eEEs)) 1
Remark 3.8. Theorem 3.7 propose the generalization condition for any unseen é'. Compared with previous analysis 1 [27] in linear cases, the Equation further indicates quantitative dependence of generalization on source heterogeneity. In words, the freedom of &g and the nullspace kcr () is limited by the maximal linearly independent system (heterogeneity) from source collection Es-
Theorem 3.7 motives to expand collection Eg via querying samples from a distribution pe where & (X*') * (XX) )eEEs- Formally, we have the following proposition:
Proposition 3.9 (Active Heterogeneity Expansion). To address performance maximization and robustness, we could guide the active model adaptation via the heterogeneity expansion:
A)fs:Q) - f" P(Y
(5)
promoting the 4g of fs to ideal ọr- At each active data collection stage, we query and collect the samples from \EG expand the data heterogeneity.
Theorem 3.10. Under the linear structural causal model, when we collect distribution e' where @(X + ((X") and update tb + & rank(( rank ($) holds. The & will converge the ideal Ø* most w iterations, where the freedom of the intervention variable Z E R
Theorem 3.10 indicates our proposed active heterogeneity expansion framework shares favorable convergence rate. In words, each time we collect a distribution from &y\Eg, we can remove one degree of freedom in the space of the variant intervention factor. It is noteworthy that although the convergence analysis based on the linear condition, our proposed framework is still applicable in non-linear cases where the condition Eg [c'((((X* = (xx),, ), E Es} in Theorem 3.7 holds.
4 Algorithm
Following the above analysis, we would like to query the samples from εr\Eg to achieve the ideal Ir. In this work, propose our active model adaptation method LOG, with two interactive modules:
1) Query Strategy Mq: given the unlabeled data pool Xr and the current invariant relationship I, actively query the representative samples Q from the un-generalizable distributions Er EErYEG.

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2) Model Adaptation MA given the newly queried samples Q, update the invariant relationship I via invariant learning.
The whole framework is iterative so that the mutual promotion between active exploration and invariance exploitation can be leveraged.
4.1 Query Strategy Module MM
Following the problem formulation, we have the XT 8XG + +1 (1-8)Xy, where XG P(X(EG).
Xr P(X The X reperesents the obversation from the un-generalizable &o = &
Based Proposition 3.9 goal could be formulated as:
query few instances Q from Xy promote ((Ds.Q) + $7. This
Ma([X,, Vel. [Xq. UQl) + MM((X,, Vel. [Xu,yc)
We first detect the samples from Xy and then find the representative instances Xę for X to query. Based on the connection (Xx)) = 4 (Xa,, we could transform the detection of X as a mixture distribution separation problem. We denote [1;;; XG) to indicate if is observed from the &g. The indicator is essentially binary classifier g (x) X' + {1, -1}. Given a sample æ, when go ((x) 1, is observed from &g. and the current model could give a correct prediction, otherwise it will violate the current invariance & and is risky for the current model.
One of the main challenges learning g is that do not have information about the marginal distribution of unseen Ey. We handle this problem by using the risk rewriting technique 12 43 with the unlabeled data from ΣT.
P(b(Xu)) = -OP -
-1
(6)
Proposition 4.1. For all measurable funtrrr we could rewrite the original risk as:
R(g) = XE [6[g o (x), (1 (0-00) Ex [[[g ((x)),())))
0E [6 (xx) fg ((x). -1]] E (xx) -1]]
Since the risk equals the ideal risk, its empirical estimator R(g) is unbiased over the target distribution. We can thus perform the standard empircal risk minimization. Particularly when the binary loss satisfying {(=) ((-2) = -, for all E IR the empirical risk R(g) is convex w.r.t. g 12 It leads to a convex optimization problem, for which the globally optimal solution can be obtained.
Here we further analyze the estimation error of indictor g based on [12]. Formally, the empirical estimator I has the estimation error bound:
Theorem 4.2 (Estimation Error Bound). Suppose that inf geg R(g) 2 a > 0 and G is closed under negation, iee. g E 9 iff -ge E G. Then for any 8 > O, with probability at least
1
R(g) inf R(g) I Op( gEg
VNs
For a better presentation, we use the Op-notation to keep the dependence on Ng and NT only.
Theorem 4.2 shows that the estimation error of the estimated 9 decreases with growing number of source data Ns and unlabeled target data Ny. our problem, we have the plentiful source data Ds from Es and unlabeled data XT from Eq, which helps to estimate 9 accurately.
Notice that the implementation of our algorithm requires the knowledge of the mixture proportion 8, where plenty of works 31 [2 have been explored to estimate 0 from the unlabeled data. We adopt the mixture proportion estimation method of (29] in our implementation and omit the details here. All of our empirical studies are conducted by the estimated mixture proportion.
Based on the distribution inference module g, we could get the probability pa( pl 1 lx; y () that x belongs to Ey. Here we further consider to obtain the limited subset Q covering the empirical set Xu = Pa() Xx. Specifically, we follow the core-set objective proposed in [32]
min max min/Iæ x' æ'112 < 5Q
X ree
Many popular loss functions satisfy the condition, such logistic loss, square loss and double hinge loss.
5

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Informally, we are trying to find subset Q to query labels that are close to the raw candidates Xu- Although this problem is Na-hard 4 we could obtain an approximate solution efficiently using the greedy iterative approarch:
1) Get arg max ming' Pa (a()))] 2/122
2) Add sample to the subset Q: Q {x)
4.2 Model Adaptation Module AM
Given the newly queried samples Q (Xq: MQ ). we have environments & [(Xs, vs), (Xq, UQ)] Then we could update the invariant feature via standard environment based invariant learning. feature level, simple but general setting. We further obtain & through feature selection: ((X) = M @ X.
Following [14 36 15 19), mine the invariance
The objective function of MA with M [00, 10, is:
min M.J
Σ DE,CEE
MDD", subject to f ara min Ri D))
However, the optimization of hard feature selection with binary mask M suffers from high variance, we use the soft feature selection with gates taking continuous value in [0, 1].
LOG is applicable broadly invironment-basee invariant learning objectives through the different choices of MA. A In this paper, we choose IGA [I3] which has guaranteed to achieve the maximal invariant predictor with respect to given environments.
4.3 Interactive Promotion
Theorem 4.3 (Interactive Promotion). Given the newly queried samples from Ev. invariant learning module MA could promote the invariance set Is to ideal Ir. Given the updated invariance set I' Is. have better generalization: Egg 3 Eg and reduced the candidates for actively sampling.
The core of our LoG framework is the mechanism for MQ and MA to mutually promote each other. Here we theoretically justify the positive feedback. It indicates that our active exploration could help the reduction of the current invariance set Is. On the other hand, the better invariance set I could help to expand the generalizable distributions and reduce the querying candidates.
5 Empirical Study
In this section, we provide extensive results to evaluate Log and compared methods for both benchmark simulation and a series of real-world tasks.
Competing Methods We firstly consider learning only on source: including ERM baseline, and 2 state-of-the-art OOD generalization methods: HRM [19] and EIII. (5), which mine the latent
heterogeneity without prior division label e E Es.. Then we consider active model adaptation, including randomly querying and CoreSet 32 baselines, 3 state-of-the-art active model adaptation
methods: AADA (39] CLUE 28 DBAL
6).
We use Logistic Regression and Linear Regression as the base models for classification and regression, respectively. Our feature selection weights M could be derived directly from the parameters of them.
5.1 Simulation Data
Synthetic data are important tools to simulate explainable and controllable distributional shifts. As indicated by [2] 37]. it is necessary to introduce such simple but challenging data, which can reflect whether and what extent an algorithm can resist certain kinds of distributional shifts. Generally speaking, the covariates X are divided into X = [S, VJ corresponding to the invariant and variant parts inside the data. The P(Y|S) remains invariant across distributions, The P(Y[V) is perturbed with different mechanisms, which brings distribution shift.

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ERM
EXM
HRM
HORM
KIBL
KIIL
RAS
CheeSet
Cordict
8.30
CLUB
0.25
NADAA
AADA
Eastraamonc F
Revtnemsnnce
(a) Error under Anti-Causal Effect Shifts
(b) Error under Anti-Causal Effect Shifts (Imbalanced)
1
2.5
2.8
8.0
2.5
1.5
3.
Oveall
3.0
Oveell
(c) Error under Selection Bias Shifts
(d) Error under Selection Bias Shifts (Imbalanced)
Figure 2: Results on varying base distributions (under 10% labeling budgets).
Classification with Anti-Causal Effect Arjovsky et al. 2019) introduce anti-causal relationship
change VVY Specifically, each distribution is characterized by its bias rate (0, 11, measureing the strength and direction of the spurious correlation between Y {1, 0} and E {1. 0). To be detailed, bias rate represents that for 100 x r% data, A = Y, and for the other 100 * (1 r)% data, Y. Then invariant S and variant V are generated as:
S ~ N(Y1, VIA ~ ((1,
We generate 1000 samples from r, = 0.9 as source data Ds. 5000 samples from uniform environ- ments & with [0.9. 0.7, 0.5, 0.3, 0.1] unlabeled data pool X, (1000 samples for each r). We carry out the procedure 10 times and report the average results in Figure ((a)
Regression with Selection Bias Kuang et al. (2220) propose selection bias mechanism intro- duce distributional shifts, and similar settings are also adopted in [19 [8] The data are generated as: Y = =(S) + where I V. The selection probability of certain data point (x,
P(z,8) II |r| bx1--=ll-sgg()>>ee sign())
(10)
where |rl Intuitively, controls the strengths and direction of the spurious correlation between V and Y. The larger Ir] means the stronger spurious correlation between V and Y. > O means positive correlation and vice versa. Therefore, we can adopt different r to simulate varying distributions.
We generate 2000 samples from r, = 2.0 as source data Ds. 3000 samples from uniform environ- ments with E &r [3.0,2.0 1.5, 11.5, -2.0, -3.0] unlabeled data pool Xy (500 samples for each r). We carry out the procedure 10 times and report the average results in Figure 2(c)
Imbalanced Mixture In the real world, there is natural phenomenon that empirical data follow a pawer-law distribution. Here we further simulate an imbalanced situation where the source distribution dominates the target distribution collection Ey. Specifically, samples from r's and generate the other from different r. In this case, is more difficult to query the target-specific samples than in the uniform case. We report the results in Figure 2(b) and 2(d)
generate half of the target
From the results, HRM and EIII. could improve worst-case robustness over the ERM baseline. With- out sufficient multi- source labeled data, their robustness is still limited because source heterogeneity

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Oheralt Chaeettttii Krees
0.35
1RM
Ovgrill Cleyiifiatiin Errer
8.18
Ouraal Mean Squuree Kimir
E1II
KIII
XXIL
RAN
0,005
0.5
DRAL
8.16
DIAL
0L005
DIAL
XADA
AADA
10G
1.06
1004
LOG
100
109
40
100
Labeling Madget
Labeting Btddget
Labeling Ulddeet
(a) Overall error on Insurance
(b) Overall error on Income
(c) Overall error on House
8:15
NJGS
0.36
8:40
HRM
BRM
BRM
1.58
8.5
AA25
CareSet
Cursel
CunSel
0.30
8.25
0.20
AADA
0.36
LOG
AADA
0000
AADA
8.15
20
40
69
80
108
0.34
20
401
99
100
20
40
60
80
100
Labeling Bufgget
Labeling Budget
Labeling Budget
(d) Robustness error on Insurance
(c) Robustness error Income
(f) Robustness error on House
Figure 4: Results on real-world tasks. The average results for 10 times produces are reported.
sufficient support varied unseen distributions. In contrast, active adaptation methods which introduce the few labeled data have achieved more significant improvement.
Under the anti-causal shift, all of these active adaptation methods have improved the overall perfor- mance of target distributions Ey. Nevertheless, they show significant performance degradation base distribution compared to the source distribution. Our LoG has error which is close to the source error, showing superiority in robustness.
Under selection bias shift, note that although these adaptation methods have achieved improvement on overall performance, their performance on base distributions (3.0, 2.0, 1.5) close to the source is weaker than source-only methods. To obtain a more clear explanation, we further visualize the dependence of them invariant variables (SO-S6) and variant variables (V0-V2). One plausible reason is that the source- only methods directly fit the source- specific correlation, overfitting the source distribution. As illustrated in Figure 3 previous methods focus on source-specific correlation V2 leads to performance degradation in shifted distributions. In contrast, our method pays more attention in- variant variables (SO-S6) and excludes the variant variables (V0-V2).
Pu Crilt 100
Coc
81
V1
82
V1
(a) Uniform Case
(b) Imbalanced Case
Figure 3: Feature importance for each method.
5.2 Real-world Data Sets
We further evaluate our method on three real-world tasks, including car insurance prediction, people income prediction, and house price prediction, with diverse shift types: region, person, and time.
To evaluate both the overall performance and worst-case robustness, we employ the overall error: error(f: Dr) and worst-case enror: max.ee, emrrr(f; D*).

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Car Insurance Prediction In this task, we use a real-world dataset for car insurance prediction (Kaggle). It a classification task to predict whether 381 281109 persons will buy car insurance based on related information, such as vehicle damage, and vehicle age We split the dataset into sub- distributions, according to the region of these persons. 50% samples from the first region are split in the source labeled data Ds and the rest is regarded as unlabeled data pool. We report the results under varying labeling budgets in Figure ((a) and and [(d)
People Income Prediction In this task, we use the Adult dataset to predict personal income levels as above or below $50000 per year based on personal details. There are 48842 instances in this dataset. we split them into 10 groups according to demographic attributes sex and race. 50% samples from the first group are split in the source labeled data Ds. and the rest is regarded as unlabeled data pool. We report the results under varying labeling budgets in Figure 4(b) and 4(e)
House Price Prediction this task, we use a real-world regression dataset of house sales prices from King County, US The target variable is the transaction price of the house. Each sample contains 17 predictive variables, such as the built year, number bedrooms, square footage of the home, etc. Since it is fairly reasonable to assume the relationships between predictive variables and the target vary along the time (for example, the pricing model may change along the time), there exist distributional shifts in the price prediction task concerning the build year of houses. Here, 50% samples of houses built in [1900, 1990] are split in the source labeled data Ds. and the rest samples regarded unlabeled data pools. We evaluate model on base distributions through time intervals: [1991, 1995], [1996, 2000], [2001, 2005] and [2006, 2020], to obtain the observations under time shift. We report the results under varying labeling budgets in Figure (icure4(c)aan [nd 4())
Analysis From the results, have the following observations and analyses: The unstable perfor- mance of HRM and EIIL indicates the difficulty of generalizing to varied target distributions only by the heterogeneous nature of the source. Randomly querying could perform better than source-only methods at a small labeling cost. In the income task, the source person group dominates the whole distribution, being imbalanced case. Randomly querying more samples increases the risk of robustness and shows poor worst-case errors. Nevertheless, our L.G has shown stable robustness. CoreSet and CLUE achieve performance gain under distribution shift but are still not satisfying. An interesting phenomenon is that in the early stage of adaptation, DBAL achieved high improvement in the income task but a significant drop in the house task. One plausible reason that these samples queried by discrepancy measures are much different from the source data. While they bring rich heterogeneous information, they also increase the risk of the model adaptation part. In contrast, our Log consistently achieves improvement in overall performance and worst-case robustness, showing the effectiveness of our framework.
5.3 Ablation Study
To evaluate the mutual promotion be- tween two sub-modules MA and Mo. further make the ablation study here. Specifically, we remove MA and Me respectively for comparative exper- iments. In Table we report the overall error and worst-case eτror on these real- world tasks. It demonstrates that we could tend to both overall performance and robustness through our interaction between active exploration and invari- ance exploitation.
Table I: Overall and worst-case error under 100 labels.
MA
MQ
Insurance
Income
House
.1312.008
.154:.003
296:.009
O.
.128
.153+.014
.344+.205
.123+.000
.150+.002
.266=.001
.185:1.028
.2774.027
.9494.190
W.
.168
.368-L.199
.578:1391
V
.164:.000
.2355.006
.4391.052

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6 Conclusion
In this paper, we study the active model adaptation to rectify a known model adapting to a variety of distributions. To our best knowledge, we first introduce the invariant risk minimization principle to guide active adaptation, which leads us to optimal performance and worst-case robustness. Based on the structural causal model assumption, we find the generalization could be significantly improved by expanding the heterogeneity of training data. It motivated us to actively expand the data heterogeneity. We further propose algorithm LoG that integrates query strategy and invariant model adaptation, with an unbiased estimator to detect the un-generalizable samples. We theoretically justify the mutual promotion relationship between
sub-modules, resonating with the joint process. A series of empirical studies validate the effectiveness of our algorithm in terms of performance and robustness.
This work focuses on the active model adaptation and provides promising perspective on label- efficient OOD generalization. Our framework mainly focuses on the raw-level feature selection and the corresponding empirical studies conducted tabular tasks. The current proposal is not applicable to the -igh-dimensional data modality, such as image data. We will put efforts to integrate the power of representation learning capabiliiiess of neural networks and explore broader applications.
Acknowledgement
This research was supported by the National Science Foundation of China (62176118, 61921006). and the Nanjing -uaver Joint Research Program. We are grateful to the anonymous reviewers for their helpful comments.
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12


Reviews:
[{"review_id": "zJd-Ryvr3FT", "reviewer": "Reviewer_1tYQ", "summary": "This paper presents the invariance minimization principle, to address to handle distribution shift at a relatively small labeling cost. For this purpose, this paper designs an interactive model adaptation framework including active sample selection and invariant relationship learning. The authors demonstrate the effectiveness of the method both theoretically and experimentally.", "questions": "What is the physical meaning of the semantic variable and intervention variable and how these two variables are reflected in the experiment？", "limitations": "The author does not mention the limitations of the work.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "On one hand, the strengths include: \n1.\tThe paper is well-written.\n2. The theoretical work of this paper is sufficient, which improves the value of the paper.\n3.\tThe paper has several novelties: i)\n i) This paper proposes the invariance minimization principle for active model adaptation; ii) For this purpose, The authors present an interactive framework Robust Active model Adaptation (RA2); iii) This paper demonstrates the effectiveness in an extensive empirical study and theoretical analysis.\n\nOn the other hand, for me, the main weakness mainly lies in that the paper is not easy to follow. I understand that the authors have put necessary proofs and details into the supplementary material, but still Sections 3 are not easily understandable. The author explains some symbols unclearly, e.g., what is the physical meaning of the semantic variable and intervention variable and how these two variables are reflected in the experiment？ In addition, the use of formulas is not uniform, e.g., why does the formula \\Phi(S(X^{e})) use observation S in line 118, but not in line 120?\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "p43xOQCbbk3", "reviewer": "Reviewer_nvJG", "summary": "This paper considers Out-of-Distribution generalization via actively querying samples at a relatively small labeling cost. The authors propose an invariant minimization principle and the corresponding active heterogeneity expansion to implement it in an interactive framework. The proposed RA2 framework is theoretically sound and well-validated in sufficient experiments. More specifically, the authors present the generalization condition and a quantitative dependence on source heterogeneity under the linear causal structure. The convergence of RA2 is provided under the linear assumption. The experiments include both benchmark simulation and a series of real-world tasks, supporting the effectiveness of RA2. ", "questions": "1.\tCould the whole framework be trained in end-to-end optimization? \n2.\tA resource-constrained perspective on OOD generalization is interesting. This paper mainly considers the limited labeled information. What about the limited supervision (such as label noise), limited storage, and limited computation. I think a more comprehensive discussion will further improve this paper and have a wider impact.\n", "limitations": "The authors provide the discussion about negative societal impact. I suggest the authors make a further discussion about more powerful feature learning technologies with the proposed RA2 framework.\n\n", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1.\tThe problem is interesting and the novelty is valuable. Authors consider the limitation of the existing OOD research and theoretically justify their dependence on a large number of labeled data. This paper introduces active learning and addresses OOD generalization via interactive learning. It may inspire some consideration about OOD generalization under limited resources, such as limited label information. \n2.\tThe authors present the generalization analysis and design an active heterogeneity expansion to achieve the ideal maximal invariant predictor. Such a framework is intuitive and theoretically sound. \n3.\tThe experiments include three common distribution shifts: region shift, person shift, and time shift. The experiment results seem sufficient to support the main claim. \n\nWeaknesses:\n1.\tThe invariant learning module in the proposed RA2 is dependent on the previous environmental-based invariant learning methods. The connection between the two sub-modules seems weak. Could the whole framework be trained in end-to-end optimization?\n2.\tThe mask-based feature selection may be weak to more complicated data input, such as high-dimensional images. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["Responsible Research Practice (e.g., IRB, documentation, research ethics)"]}, "extra_text": {}}, {"review_id": "je-HM8Gdfpb", "reviewer": "Reviewer_WaQo", "summary": "This paper studies active learning with Out-of-Distribution (OOD) generalization where target distribution may differ from the source distribution. To address it, the authors present the invariant minimization principle to guide the active learning, interactively learn the invariant relationship and actively query the current un-generalizable samples. Authors formulate it as an Active Heterogeneity Expansion process, and propose the RA2 method, with the corresponding theoretical support. Moreover, the experiment results from the benchmark simulation and real-world tasks clearly verify that the proposed RA2 outperforms SOTA active and OOD methods. Overall, the paper is technically sound and well supported by the corresponding theoretical analysis and empirical evaluation. ", "questions": "1. About the combination with representation learning\n2. About the complexity analysis \n", "limitations": "N/A", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1.\tThe problem of model adaptation under changed distributions is an important problem and has not been well addressed yet. The invariant minimization principle presented by the authors provides a novel view of active learning to address both overall performance and worst-case guarantees. It may motivate researchers to understand active adaptation from causal views and promote robust deployment in real-world applications. \n2.\tThe proposed method RA2 is well-motivated by solid theoretical analysis. Theorem 3.7 presents the generalization conditions under the structural causal model, and indicates the heterogeneity of labeled data is critical to generalization performance. Theorem 3.10 shows the proposed Active Heterogeneity Expansion process could efficiently improve generalization via active interaction. \n3.\tThe experiments seem comprehensive. In both benchmark simulation and real-world tasks, RA2 outperforms the SOTA active adaptation and OOD methods. The effectiveness of the proposal has been clearly verified. \n4.\tThe paper is well organized and easy to follow. The related active adaptation and OOD literature have been comprehensively reviewed and discussed. The presentation of RA2 is generally clear. \n\nWeaknesses:\n1.\tThe proposed invariant learning module (Sec. 4.2) focuses on mask selection and raw-level features. The former framework (Line 167-174, Sec. 4) seems not limited to raw-level selection. There is also a discussion about representation learning in the appendix. I think the feature selection, presented in Section 4.2, could be further improved, with consideration of representation learning. \n2.\tThere are two interactive modules in the proposed RA2. Compared to previous active adaptation methods, which are designed on a specific metric, it introduces more computation processes. How about the complexity compared with previous methods? \n3.\tIllustration: The text in Figures 2, and 4 is too small. It should be adjusted to the same size as Figures 1, and 3. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Ib59l4YyOdJ", "reviewer": "Reviewer_8KWW", "summary": "The paper presents a method for active adaptation of a model trained on source data, when target data from changed distribution is available. It builds on the invariance minimization principle and extends the so called Maximal Invariant Predictor (MIP) for determining where to sample points from the unlabeled target data. The authors show that increasing the heterogeneity of training data improves generalization, and hence queries from samples where the current invariant model does not hold. Representative points are sampled from these using coreset. The model is updated by using a new invariance predictor obtained from the updated data.  \n\nThe experiments include comparison with relevant baselines. Results are shown on synthetic data (along with the case where the target is an imbalanced mixture), and on three real world datasets.     ", "questions": "The active adaptation baselines included in the paper show results on DIGITS, OFFICE, etc datasets -- a comparison on those datasets is imperative. Also, the current results are using a fixed budget size. It would be helpful to add the effect of different budget size on the method, and the time complexity of querying samples (w.r.t size/classes/budget of target data). \n \nThe method requires a knowledge of mixture proportion in the unlabeled target data. Does this limit the quality of the approach in different scenarios?\nA more thorough contextualization with related active adaptation methods would be helpful for readers. ", "limitations": "The limitations of the work have not been described. Potential societal impact of the work has been discussed.  ", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper provides an interesting extension of invariant minimization principle for active adaptation. The overall idea of the paper makes sense. The results on synthetic and small real data shows improvements in overall and worst-case error. Feature importance results and ablation studies further shows efficacy of the method on these datasets.    \n\nComparisons on standard datasets used in other active adaptation methods are missing. Discussion/experiments to help understand the effect of different practical choices would be helpful.  \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept