Environment Diversiﬁcation with Multi-head Neural Network for Invariant Learning
Anonymous Author(s)
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Abstract
Neural networks are often trained with empirical risk minimization; however, ithas been shown that a shift between training and testing distributions can cause unpredictable performance degradation.
On this issue, a research direction, invariantlearning, has been proposed to extract causal features insensitive to the distributional changes.
This work proposes an invariant learning framework containing amulti-head neural network to absorb data biases.
We show that this framework doesnot require prior knowledge about the environment or strong assumptions aboutthe pre-train model.
We also reveal that the proposed algorithm has theoreticalconnections to recent studies discussing properties of variant and invariant features.
Finally, we demonstrate that empirically models trained with this framework aremore robust against distributional shift.
Introduction
Ensuring model performance on unseen data is a common yet challenging task in machine learning.
Awidely adopted solution would be empirical risk minimization (ERM), where training and testing dataare assumed to be independent and identically distributed.
However, data in real-world applicationscan come with undesired biases, causing a shift between training and testing distributions.
It hasbeen known that the distributional shifts can severely harm ERM model performance and evencause the trained model be worse than random predictions [10].
In this work, we focus on Invariant
Learning , which aims at learning causal features expected to be robust against distributional changes.
Invariant risk minimization (IRM) [1] has been proposed as a popular solution for invariant learning.
Speciﬁcally, IRM is based on an assumption that training data are collected from multiple sources orenvironments having distinct data distributions.
The learning objective is then designed as a standard
ERM loss function with a penalty term constraining the trained model (e.g. classiﬁer) to be optimalin all the environments.
IRM has shown to be effective; however, we note that IRM and many invariant learning methods relyon strict assumptions which limit the practical impacts.
The limitations are summarized as follows.
Prior knowledge of environments: IRM assumes training data are collected from environments andthe environment labels (i.e. which data instance belongs to which environment) are given.
However,the environment labels are often unavailable.
Moreover, the deﬁnition of environments can be implicit,making human labeling more difﬁcult and expensive.
To ﬁnd environments without supervision,
Creager et al.
[6] propose EIIL, a min-max optimization framework training the model of interestand inferring the environment labels.
Another work, HRM [20] (or the extension, KerHRM [21]),parameterizes the environments and proposes clustering-based approaches to estimate the parameters.
A recent method, ZIN [18], also learns to label data.
However, it relies on carefully chosen featuressatisfying a series of theoretical constraints, and thus human efforts are still required
Delicate initialization: EIIL is able to infer environments but requires an ERM model for initialization.
Crucially, the ERM model has to be underﬁtted and heavily depend on spurious correlations.
Asthe distributional shift is assumed to be unknown in the training stage, appropriate initialization isdifﬁcult to guarantee.
Efﬁciency issue: HRM and KerHRM, though do not possess the above two limitations, suffer fromthe efﬁciency issue.
Speciﬁcally, HRM is assumed to be trained by low dimensional data. KerHRMas
the extension of HRM avoids the issue of dimensions by adopting kernel methods.
However, thetraining cost of the kernel methods can be very high if the data or pre-trained model size is large.
This work proposes a novel framework, E nvironment D iversiﬁcation with multi-head N eural networkfor I nvariant L earning (EDNIL).
EDNIL is able to infer environment labels without supervisionand be jointly optimized with an invariant learning model.
The underlying neural network explicitlydiversiﬁes the inferred environments, which is consistent with recent studies [5, 20, 21] revealing thebeneﬁts of diverse environments.
Notably, the structure of the proposed neural network is similar to amulti-class classiﬁer and can be optimized efﬁciently.
The advantages of EDNIL are summarized as:
• We implement this framework using various pre-trained models such as Resnet [13] and
DistilBert [28], and evaluate it with diverse data types and varied biases.
The
results
show that
EDNIL can constantly outperform the existing state-of-the-art models.
• The learning algorithm of EDNIL has theoretical connections to recent studies [5, 18, 20, 21]
discussing conditions of ideal environments.
• EDNIL does not have the three limitations discussed above.
The comparison between EDNIL
and other models is shown in Table
[Table]
Preliminaries and related works
The goal of EDNIL is to tackle out-of-distribution problem with invariant learning in the absenceof manual environment labels.
In Section 2.1, background knowledge about out-of-distributiongeneralization and invariant learning are introduced.
In Section 2.2, we discuss recent studiesinvestigating ideal environments.
In Section 2.3, we introduce the existing environment inferencemethods that do not require human efforts.
2.1 Out-of-distribution Generalization and Invariant Learning
Following [1, 20, 21], we consider a dataset D
= { D e } e 2 supp( E tr ) with different sources D e ={ ( x ei , y ei ) } n e i =1 collected under multiple training environments e 2 supp( E tr ) .
Random variable E tr 65 represents the training environment labels.
For simplicity, X e , Y e and P e denote data, target anddistribution in environment e respectively.
With E all containing all possible indices of environments such that E all  E tr , the goal of out- 68 of-distribution generalization is to learn a predictor f ( · ) :
X !
Y as Equation 1.
R e :=
E e [ l ( f ( X e ) , Y e )] measures the risk under environment e , where l ( · , · ) is the loss function.
In general,for e 2 supp( E tr ) and e 0 2 supp( E all ) \ supp( E tr ) , P e 0 ( X e 0 , Y e 0 ) is different from P e ( X e , Y e ) .
[Equation]
Recently, several studies [1, 24, 26, 17, 4] attempt to tackle the generalization problem by discoveringinvariant relationships across supp( E all ) .
A commonly proposed assumption is the existence ofinvariant features X c and variant features X v .
Speciﬁcally, raw features X are assumed to becomposed of X c and X v , or X =
h ( X c , X v ) where h ( · ) is a transformation function.
Invariantfeatures X c are assumed to be equally informative for predicting targets Y across environmentse 2 supp( E all ) .
On the contrary, the distribution P e ( Y | X v ) can arbitrarily vary across e .
As a result,predictors relying on X v can have unpredictable performance in unseen environments.
To extract X c , IRM [1] assumes there is an encoder  for obtaining representations  ( X ) ⇡ X c .
The encoder is trained with a regularization term enforcing simultaneous optimality of the classiﬁerw   ( X ) with E tr , where dummy classiﬁer w = 1 . 0 is a ﬁxed multiplier for each output dimension:
[Equation]
As w is a dummy layer, the encoder  is also regarded as a predictor.
2.2 Ideal environments
As E tr is unavailable or sub-optimal in most applications, learning to ﬁnd appropriate environments(denoted by E learn ) is attractive.
However, the challenge is the lack of knowledge of valid environments.
Recently, Lin et al.
[18] propose Equation 3 and 4 as the conditions of ideal environments, where H isconditional entropy.
To satisfy the conditions, Lin et al.
[18] propose leveraging auxiliary informationfor model training, which requires extra human efforts to verify if the additional information is valid.
[Equation] [Equation]
Equation 4 can also be implied by empirical studies [5] where diversity is recognized as the key toobtaining effective IRM models.
More precisely, large discrepancy of spurious correlations betweenenvironments is favored.
Take the case of two environments for example.
When the spuriouscorrelations in the two environments are 1 and -1 respectively, the diversity is desirable for giving aclear indication of the variant feature.
On the other hand, when the spurious correlations are both 0.5,the diversity is insufﬁcient for identifying the variant feature.
Beyond IRM, HRM [20] and KerHRM[21] can also be regarded as optimizing diversity via clustering-based methods speciﬁcally.
2.3 Unsupervised environment inference
Here we provide a detailed introduction of the existing unsupervised methods identifying environments. The general idea is to jointly learn environment inference and invariant learning models.
EIIL [6] proposes formulating invariant learning as a min-max optimization problem.
Speciﬁcally,
EIIL is composed of two objectives, Environment Inference ( EI ) and Invariant Learning ( IL ), where EI is optimized by maximizing the training loss via labeling the data, and IL is optimized byminimizing the training loss given the data labeled by EI .
The two-stage algorithm bypasses thedifﬁculty of deﬁning environments; however, the training result heavily relies on the initialization ofthe EI optimization.
Empirically, the initialization demands an underﬁtted and biased ERM model,or EIIL can have a signiﬁcantly weaker performance.
Another method, HRM [20], proposes a clustering-based method for learning plausible environments.
HRM assumes that spurious correlations in each environment can be modeled by a parameterizedfunction and the dataset is generated by the mixture of the functions.
The parameters are thenestimated by employing EM algorithm.
Additionally, HRM equips a joint learning framework whichalternatively learns invariant predictors and improves quality of clustering results.
A known issue of
HRM is an assumption that the data are represented by raw features; namely, data such as images andtext requiring non-linear neural networks to obtain representations are beyond the capability.
To extend HRM the [21] proposes KerHRM.
The main idea is to adopt the Neural Tangent Kernel [14] method whichtransforms non-linear neural network training into a linear regression problem on the proposed Neural
Tangent Features space.
As a result, KerHRM elegantly resolves the shortcoming of HRM and isshown to be more robust and effective.
However, the kernel method also brings additional trainingcost depending on data size and pre-trained models.
In particular, for applications favoring largepre-trained models such as Resnet [13] and BERT [7], KerHRM may not be an affordable option.
Methodology
In this section, we propose a general framework to learn invariant model without manual environmentlabels.
As shown in Figure 1, our proposed method consists of two models, M EI for environmentinference and M IL for invariant learning.
Given the pooled data ( X, Y ) , M EI generates environments
E learn satisfying Condition 3 and 4, and M IL is an invariant model trained with the inferred environments.
Our framework is jointly optimized with alternating updates.
The learned M IL can provideinformation of invariant features to M EI , so that Condition 3 and 4 can be fulﬁlled simultaneously.
3.1 The environment inference model
The target of environment inference is to partition data into environments E learn satisfying Condition3 and 4.
In this regard, we propose a graphical model, which is a sufﬁcient condition for Condition 3and 4 (the proof is in the supplementary ﬁle Section A), as our foundations of inference model andlearning objectives.
The graph is shown in Figure 2, where the data generation process follows theproposed example in [1].
The inference model, M EI , is a deep neural network learning to realize the underlying graphicalmodel.
Following the idea of HRM [20] and KerHRM [21], we assume the mapping relation between
X and Y in environment e can be modelled by a function f e ( ( X )) , where ( X ) is learnedrepresentations expected to encode variant features X v .
Instead of taking f e ( ) as a center of acluster, we propose combining the environmental functions to build a multi-head network as shownin Figure 3.
Speciﬁcally, a cluster in HRM [20] or KerHRM [21] corresponds to a single-head neuralnetwork in M EI with shared parameters.
The training procedure of M EI can be divided into inferencestage and learning stage .
3.1.1 Inference stage of M EI
The goal is to infer a environment label for each training data.
As in the graphical model, theassignments depend on variant relationships.
Inspired by multi-class classiﬁcation problem, wepropose Equation 5, where the probability P ( e | X, Y ) is estimated via a softmax of negativel ( f e ( ( X ) , Y )) with temperature ⌧ .
The function l is expected to be the commonly used crossentropy or mean squared error loss, measuring the discrepancy between Y and f e ( ( X )) for eachenvironment e .
[Equation]
3.1.2 Learning stage of M EI
The goal is to update the neural network to improve the quality of inference.
Based on the structureof the graphical model, three losses are designed for minimization, i.e. Environment Diversiﬁcation
Loss ( L ED ), Label Dependency Loss ( L LD ) and Invariance Preserving Loss ( L IP ).
In particular, L EDand L IP correspond to the concepts of Condition 4 and 3 respectively.
Environment Diversiﬁcation Loss ( L ED )
We consider maximizing H ( Y | X v )  H
( Y | X v , E learn ) 154 to satisfy Condition 4 and capture diverse variant relationships.
Given the estimated P (
E learn | X, Y ) , 155 L
ED selects the most probable environment and its corresponding network for optimization:
[Equation]
Although only one environment e i = arg max e P ( e | x i , y i ) is selected for the minimization, thesoftmax simultaneously propagates gradient to maximize l ( f e 0 ( ( x i ) , y i )) for e 0
= e i .
The networklearns to involve E learn in optimizing the performance of predicting Y , which is consistent with themaximization of H ( Y | X v )  H ( Y | X v , E learn ) .
In practice, we utilize scaling weight w i inversely 160 proportional to the size of e i .
The importance of smaller environments will be thus enhanced withinsummation.
Label Dependency Loss ( L LD )
With d-separation [23], E learn is independent of Y in the graphicalmodel.
Hence, L LD constraints their dependency measured by the mutual information
I ( Y ; E learn ) .
Inpractice, L LD prevents a trivial solution that environments are determined purely by labels Y withoutconsidering input features X , which is undesirable for invariant learning.
To minimize I ( Y ; E learn ) , itcan be veriﬁed that it is equivalent to minimizing Equation 7 given P ( Y ) is known.
[Equation]
Invariance Preserving Loss ( L IP )
As M IL learns some invariant features after several training steps(Section 3.2), L IP can be included to prevent existing invariant relationship from being diversiﬁedfor Condition 3.
Speciﬁcally, designed as the contrary of L ED , L IP restricts E learn to inducing smallvariance of expected loss from invariant predictor  (in M IL ) across environments (Equation 8).
Similar idea can be found in [16].
However, instead of training invariant model given knownenvironments, we freeze the invariant predictor and regularize the adjustment of E learn (i.e. the updatesof M EI ) here.
[Equation]
All in all, the training loss of M EI can be summarized as Environment Inference Loss ( L EI ).
Thestrength of each loss can be controlled by hyper-parameters  and  :
[Equation]
In addition, before minimizing L EI , we pre-train our and one arbitrary f e with ERM.
In general, itempirically facilitates better feature extraction.
Unlike [6] taking ERM as reference model heavilyrelying on variant features, EDNIL performs consistently under various choices of ERM.
In particular,it is not conﬁned to under-ﬁtted ERM training.
We verify the argument in Section 4.
3.2 The invariant learning model
Invariant Learning identiﬁes invariance across given environments.
For our model, M IL , IRM[1] is selected as our base algorithm to learn invariant predictor  .
We assign environment labele 2 supp( E learn ) with largest P ( e | x i , y i ) to each data ( x i , y i ) .
However, it is inevitable that there existsome noises in automatically inferred environments, especially in the beginning of joint optimization.
To reduce the impact of immature environments on invariant learning, we calculate the conﬁdencescore
c e for each environment e 2 supp( E learn ) , i.e. P ( e | X e , Y e ) .
Our training objective is modiﬁedto minimize Invariant Learning Loss ( L IL ) that considers the weighted average of environmentallosses:
[Equation]
[Equation] [Equation]
Experiments
We empirically validate the proposed methods on four biased datasets, Adult-Confounded, CMNIST,
Waterbirds and SNLI.
The generation of spurious correlations mainly follow the protocols proposed by
In Section 4.1, Adult-Confounded and CMNIST are tested with multilayer perceptron(MLP).
In Section 4.2, two more complex datasets, Waterbirds and SNLI, are taken for evaluating theintegration of representation learning.
Modern deep models with pretrained weights will serve as theencoders of variant and invariant features.
The following four competitors are selected as our competitors: Empirical Risk Minimization(ERM), Environment Inference for Invariant Learning (EIIL [6]),
Kernelized Heterogeneous Risk
Minimization (KerHRM [21]) and Invariant Risk Minimization (IRM [1], Equation 2).
EIIL and
KerHRM are invariant learning methods with unsupervised environment inference, which share thesame settings as EDNIL.
HRM [20] is replaced by KerHRM for non-linearity.
For IRM who requiresenvironment partitions, we re-label E oracle on given biased training set, which diversiﬁes the spuriousrelationships to elicit upper-bound performance of IRM.
For hyper-parameter tuning, we split 10% of training data as in-distribution validation set.
In eachdataset, several testing environments with different distributions are listed to evaluate the robustnessof each method, and we mainly take worst-case performance for assessment.
† will be marked if thedifference is statistically signiﬁcant (p-value less than 0.05) comparing with the 2nd-best result.
4.1 Simple datasets with MLP
This section includes two simple datasets, Adult-Confounded and CMNIST, where spurious correlations are synthetically produced with the pre-deﬁned strength.
For all competitors, MLP is takenas the base model and full-batch training is implemented.
Since KerHRM performs unstable overrandom seeds, we ﬁrst average the results after 10 runs as its ﬁrst score, and select top-5 among themas the second one, which will be marked with an asterisk ( ⇤ ) in each table.
4.1.1 Discussion on Adult-Confounded dataset
We take UCI Adult dataset [15] to predict binarized income levels (above or below $50,000 peryear) based on demographic information 1 .
Following [6], individuals are re-sampled according tosensitive features race and sex to simulate spurious correlations.
Speciﬁcally, with binarized race(Black/Non-Black) and sex (Female/Male), four possible subgroups are constructed: Non-black Males(SG1), Non-black Females (SG2), Black Males (SG3), and Black Females (SG4).
Keeping originaltrain/test split and subgroup size from UCI Adult, we sample data for the given label distributions ineach sensitive subgroup as shown in Table 2.
In this task, MLP with one hidden layer of 96 neuronsis considered.
For IRM, four environments comprise E oracle , where the correlations between variantfeatures ( race, sex ) and target Y are distributed without overlapping.
More details are provided inthe supplementary ﬁle Section B.
Results The results are shown in Table 3.
First of all, notsurprisingly we found ERM adopts poorly to distribution shift.
Among all invariant learning methods without prior environment labels, EDNIL can perfectly identify variant featuresand generate ideally diversiﬁed environments.
Therefore, itachieves the most invariant performance over different testingdistributions.
EIIL can improve consistency to some degree,but not as strong as EDNIL.
A possible reason is that empirically trained reference model is not guaranteed to be purelyvariant [6].
For KerHRM, it performs inconsistently acrossrandom seeds, which can be reﬂected on large standard deviation.
In some cases, the performance hardly improves overiterations, as also observed in Liu et al.
Ablation Study for M EI
We ﬁrst claim the importance ofconstraining label dependency by setting the coefﬁcient  of
L LD to zero.
As discussed in Section 3, the resulting environments are split purely by labels,which leads to inferior performance for invariant learning as shown in Table 3.
Figure 4 showsthe effectiveness of joint optimization.
The regularization L IP promotes environment inference, sothat the worst-case performance improves and remains stable over iterations
If the coefﬁcient  is
Confounded.
IID shares spurious correlations with train set.
IND has no bias on race and sex.
OOD deﬁnes the worst- performance.
[Table]
Three subsets are deﬁned in Table 2.
= {+
+} and = mean the removal of L LD and L when
[Table]
[Table]
turned off, feedback generated by M IL will be ignored and the effect of invariant learning risks beingundesirable.
In Table 3, the degradation occurs on all subsets.
It indicates that some invariant featuresare eliminated since Condition 3 cannot be realized in the splitting of environments.
4.1.2 Discussion on CMNIST dataset
We report our evaluation on a noisy digit recognition dataset, CMNIST.
Following [1], we ﬁrst assign
Y = 0 to those whose digits are larger than 5 and Y = 1 to the others.
Next, we apply label noise byrandomly ﬂipping Y with probability 0.2.
Finally, the digits are colored with color label C , which isgenerated by randomly ﬂipping Y with probability e .
For training, e is set to 0.15, which is equivalentto merging two training environments ( e = 0 .
and e = 0 . 2 ) from [1].
For testing, three situationsare considered when e is 0.1, 0.5 or 0.9 respectively.
Note that when e = 0 .
1 , the spurious correlationis much aligned with training set.
On the other hand, e = 0 .
9 deﬁnes the most challenging scenariosince the correlation shifts most dramatically from training.
[Table]
For all competitors except KerHRM, weselect MLP with two hidden layers of390 neurons, and consider the wholedataset (50,000 samples) for training.
Due to massive computing resources required by KerHRM, we follow the settings recommended by [21].
Speciﬁcally,we randomly select 5,000 samples andtrain MLP with one hidden layer of 1024neurons.
To construct ideally diversiﬁed
E oracle for IRM, we pack all examples 267 with C =
Y into one environments, and
C
= Y into the other.
Results The results are shown in Table 4.
With large spurious correlation at train time, ERMobtains high accuracy as the correlation remains aligned; however, its generalization to other testingdistributions is limited.
Among all invariant learning methods without manual environment labels,
EDNIL gets closest to IRM with E oracle , achieving consistent and robust performance in this dataset.
As shown in Figure 6, EDNIL provides almost ideally diversiﬁed E learn for invariant learning.
Number of Environments As shown in Figure 5, EDNIL is not sensitive to predeﬁned numberof environments.
Speciﬁcally, when the environment number is larger than the oracle (i.e. 2), someenvironment classiﬁers become redundant.
Each of them owns a high constant loss, taking up a ﬁxedand ignorable space in softmax function.
The comparison between E oracle and E learn with size of 5 isvisualized in Figure 6.
Additionally, training EDNIL with more environments is much more efﬁcientthan clustering-based methods, such as the one proposed in KerHRM.
4.2 Complex datasets with pre-trained deep learning models
This section extends MLP to deep learning models with pre-trained weights on more complex data.
With mini-batch ﬁne-tuning, we consider all competitors except KerHRM due to its limitation todeep representation learning [21].
In Section 4.2.1, image dataset, Waterbirds [27], with controlledspurious correlations is selected for evaluating the generalization on more high-dimensional images.
In Section 4.2.2, a real-world NLP dataset, SNLI [3], is considered.
The biases in SNLI are naturallyderived from the procedure of data collection, and we deﬁne biased subsets for evaluation following
Dranker et al.
4.2.1 Discussion on Waterbirds dataset
In Waterbirds [27], each bird photograph, from CUB dataset [29], is combined with one backgroundimage, from Places dataset [30].
Both birds and backgrounds are either from land or water, and ourtarget is to predict the species of birds.
At train time, landbirds and waterbirds frequently present inland and water backgrounds respectively.
Therefore, empirically trained models are prone to learncontext features, and fails to generalize as background varies [2, 6, 10, 19, 27].
To split validation set so that its distribution is i.i.d.
to training set, we merge original trainingand validation data 2 and split 10% for hyper-parameter tuning.
For testing, we observe all fourcombinations of birds and backgrounds in the original testing set.
Among them, the minor subgroup(waterbirds on land) contributes the most challenging scenario.
In this task, Resnet-34 [13] ischosen with mini-batch ﬁne-tuning.
Given two binary labels ( target, background ) , we distributetarget = background and target 6 = background into two different environments and applybalanced class weights for the oracle settings of IRM.
Results The results are shown in Table 5.
As observed in [6, 27], ERM suffers in the hardest scenario(i.e.
waterbirds on land).
EIIL also performs poorly in this scenario.
With more sophisticated learningframework, EDNIL narrows the gaps between subgroups and uplifts the worst-case performance.
The results show that EDNIL is more resistant to distribution shifting.
Choice of Initialization Both EIIL and EDNIL take ERM as initialization.
As mentioned in Section1, heavy dependency on initialization is risky when testing distribution is unavailable.
Therefore, wetake ERM with different training steps for EIIL and EDNIL to verify the stability.
The results areshown in Figure 7.
As suggested by [6], EIIL works only with underﬁtted reference model (trainingsteps fewer than 400).
If the reference model is well-ﬁtted, the performance of EIIL will greatly decline since ERM learns beyond variant features.
One can be misled into undesirable choice for
EIIL when seeking hyper-parameters with in-distribution validation set.
For instance, the validation score of EIIL with 500-step reference model (96.9%) is much higher than that with 100-step (94.9%),which is not consistent with their performances on testing.
In comparison, EDNIL remains consistentacross different pre-trained steps, which accentuates our strength of insensitive initialization.
4.2.2 Discussion on SNLI dataset
The target of SNLI [3] is to predict the relation between two given sentences, premise and hypothesis.
Recent studies [12, 22, 25] reveal hypothesis bias in SNLI, which is characterized by patterns in
However
The subgroup (Water, Land) contributes the worst-case performance.
[Table]
hypothesis highly correlated with a speciﬁc label.
One can achieve low empirical risk withoutconsidering premises during prediction.
However, as the bias no longer holds, the performancedegradation occurs [11, 22].
We sample 50,000 examples and select two major classes, entailment and contradiction , for ourexperiment.
Following [9], we deﬁne three subsets, unbiased, bias aligned and bias misaligned, bytraining a biased model with hypothesis as its only input.
Speciﬁcation of the subsets is as follows:
• Unbiased: Examples whose predictions from biased model are ambiguous.
Aligned: Examples that biased model can predict correctly with high conﬁdence. Misaligned: Examples that biased model can predict incorrectly with high conﬁdence.
The proportions of the three subsets are 22%, 67% and 11% respectively.
Due to the minority ofmisaligned bias subset, it is more likely to be ignored and thus deﬁnes the worst-case performance.
[Table]
For all methods, DistilBERT [28] is taken as thepre-trained model for further mini-batch ﬁnetuning.
For E oracle , we assign the aligned biassubset to the ﬁrst environment, and the misaligned bias subset to the second.
In order tomake bias prevalence equal in the two environments, samples in the unbiased subset are scattered according to their sizes.
Results Table 6 shows our results with thethree subsets.
As reported by [9], ERM receivesremarkable score on the bias aligned subset, but it fails on the bias misaligned case.
ited.
Among allinvariant learning methods without environment labels, only EDNIL improves signiﬁcantly on thebias misaligned subset.
Even though the deﬁnition of biases are in high level, EDNIL is still capableof encoding and diversifying possible variant features.
Limitation
The learning algorithm of EDNIL is based on the graphical model plotted in Figure 2.
As data arenot necessarily generated by the presumed process, there can exist biases that cannot be captured bythe proposed neural network.
In the paper, we provide empirical studies of model effectiveness ondiverse datasets, while we are aware that a stronger guarantee of performance is required.
Conclusion
This work proposes EDNIL for training models invariant to distributional shifts.
To infer environmentswithout supervision, we propose a multi-head neural network structure with a two-stage learningprocedure to identify and diversify plausible environments.
With joint optimization, the trainedmodels are shown to be more robust than existing solutions on data having distinct characteristicsand different strengths of biases.
We attribute the effectiveness to the underlying learning objectivesconsistent with recent studies of ideal environments.
Additionally, we note that the classiﬁer-
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