Training Debiased Subnetworks with Contrastive Weight Pruning Geon Yeong Park1Sangmin Lee2Sang Wan Lee1Jong Chul Ye123 1Bio and Brain Engineering2Mathematical Sciences3Kim Jaechul Graduate School of AI

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Training Debiased Subnetworks with Contrastive Weight Pruning

Geon Yeong Park1Sangmin Lee2Sang Wan Lee1∗Jong Chul Ye1,2,3∗

1Bio and Brain Engineering, 2Mathematical Sciences, 3Kim Jaechul Graduate School of AI

Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea

{pky3436, leeleesang, sangwan, jong.ye}@kaist.ac.kr

Abstract

Neural networks are often biased to spuriously corre-

lated features that provide misleading statistical evidence

that does not generalize. This raises an interesting ques-

tion: “Does an optimal unbiased functional subnetwork ex-

ist in a severely biased network? If so, how to extract such

subnetwork?” While empirical evidence has been accumu-

lated about the existence of such unbiased subnetworks,

these observations are mainly based on the guidance of

ground-truth unbiased samples. Thus, it is unexplored how

to discover the optimal subnetworks with biased training

datasets in practice. To address this, here we ﬁrst present

our theoretical insight that alerts potential limitations of

existing algorithms in exploring unbiased subnetworks in

the presence of strong spurious correlations. We then fur-

ther elucidate the importance of bias-conﬂicting samples

on structure learning. Motivated by these observations, we

propose a Debiased Contrastive Weight Pruning (DCWP)

algorithm, which probes unbiased subnetworks without ex-

pensive group annotations. Experimental results demon-

strate that our approach signiﬁcantly outperforms state-of-

the-art debiasing methods despite its considerable reduc-

tion in the number of parameters.

1. Introduction

While deep neural networks have made substantial

progress in solving challenging tasks, they often undesir-

ably rely on spuriously correlated features or dataset bias,

if present, which is considered one of the major hurdles in

deploying models in real-world applications. For example,

consider recognizing desert foxes and cats from natural im-

ages. If the background scene (e.g., a desert) is spuriously

correlated to the type of animal, the neural networks might

use the background information as a shortcut to classiﬁca-

tion, resulting in performance degradation in different back-

grounds (e.g., a desert fox in the house).

To investigate the origin of the spurious correlations, this

paper considers shortcut learning as a fundamental architec-

tural design issue of neural networks. Speciﬁcally, if any

available information channels in deep networks’ structure

could transmit the information of spuriously correlated fea-

tures (spurious features from now on), networks would ex-

ploit those features as long as they are sufﬁciently predic-

tive. It naturally follows that pruning weights on spurious

features can purify the biased latent representations, thereby

improving performances on bias-conﬂicting samples1. We

conjecture that this neural pruning may improve the gener-

alization of the network in a way that reduces the effective

dimension of spurious features, considering that the failure

of Out-of-Distribution (OOD) generalization may arise due

to high-dimensional spurious features [26,34].

Recently, Zhang et al. [37] has empirically demonstrated

the existence of subnetworks that are less susceptible to

spurious features. Based on the modular property of neu-

ral networks [5], they prune out weights that are closely

related to the spurious attributes. While [37] affords us

valuable insights on the importance of neural architectures,

the study has limitation in that such neural pruning requires

sufﬁcient number of ground-truth bias-conﬂicting samples.

Thus, how to discover the optimal subnetworks in practice

when the dataset is highly biased?

To address this, we ﬁrst present a simple theoretical ob-

servation that reveals the limitations of existing substruc-

ture probing methods in searching unbiased subnetworks.

Speciﬁcally, we reveal that there exists an unavoidable gen-

eralization gap in the subnetworks obtained by standard

pruning algorithms in the presence of strong spurious corre-

lations. Our analysis also shows that trained models may in-

evitably rely on the spuriously correlated features in a prac-

tical training setting with ﬁnite training time and a number

of samples.

In addition, we show that sampling more bias-conﬂicting

data makes it possible to identify spurious weights. Speciﬁ-

cally, bias-conﬂicting samples require that the weights as-

1The bias-aligned samples refer to data with a strong correlation be-

tween (potentially latent) spurious features and target labels (e.g., cat in

the house). The bias-conﬂicting samples refer to the opposite cases where

spurious correlations do not exist (e.g., cat in the desert).

arXiv:2210.05247v3 [cs.LG] 26 Jun 2023

Figure 1. Concept: We demonstrate an inevitable generalization gap of subnetworks obtained by standard pruning methods including [37].

Based on these observations, we design a novel subnetwork probing framework by fully exploiting unbiased samples.

sociated with spurious features should be pruned out as

the spurious features do not help predict bias-conﬂicting

samples. Our theoretical observations suggest that balanc-

ing the ratio between the number of bias-aligned and bias-

conﬂicting samples is crucial in ﬁnding the optimal unbi-

ased subnetworks.

In practice, the dataset may severely lack diversity for

bias-conﬂicting samples due to the potential pitfalls in data

collection protocols or human prejudice. Since it is of-

ten highly laborious to supplement enough bias-conﬂicting

samples, we propose a novel debiasing scheme called De-

biased Contrastive Weight Pruning (DCWP) that uses the

oversampled bias-conﬂicting data to search unbiased sub-

networks.

As shown in Fig. 1, DCWP is comprised of two stages:

(1) identifying the bias-conﬂicting samples without expen-

sive annotations on spuriously correlated attributes, and (2)

training the pruning parameters to obtain weight pruning

masks with the sparsity constraint and debiased loss func-

tion. Here, the debiased loss includes a weighted cross-

entropy loss for the identiﬁed bias-conﬂicting samples and

an alignment loss to further reduce the geometrical align-

ment gap between bias-aligned and bias-conﬂicting samples

within each class.

We demonstrate that DCWP consistently outperforms

state-of-the-art debiasing methods across various biased

datasets, including the Color-MNIST [23,27], Corrupted

CIFAR-10 [13], Biased FFHQ [21] and CelebA [25], even

without direct supervision on the bias type. Our approach

improves the accuracy on the unbiased evaluation dataset

by 86.74% →93.41%,27.86% →35.90% on Colored-

MNIST and Corrupted CIFAR-10 compared to the second

best model, respectively, even when 99.5% of samples are

bias-aligned.

2. Related works

Spurious correlations. A series of empirical works have

shown that the deep networks often ﬁnd shortcut solutions

relying on spuriously correlated attributes, such as the tex-

ture of image [10], language biases [12], or sensitive vari-

ables such as ethnicity or gender [7,28]. Such behavior is

of practical concern because it deteriorates the reliability of

deep networks in sensitive applications like healthcare, ﬁ-

nance, and legal services [4].

Debiasing frameworks. Recent studies to train a debi-

ased network robust to spurious correlations can be roughly

categorized into approaches (1) leveraging annotations of

spurious attributes, i.e., bias label [29,36], (2) presuming

speciﬁc type of bias, e.g., texture [1,9] or (3) without using

explicit kinds of supervisions on dataset bias [22,27]. The

authors in [15,29] optimize the worst-group error by using

training group information. For practical implementation,

reweighting or subsampling protocols are often used with

increased model regularization [30]. Liu et al.; Sohoni et

al. [24,32] extend these approaches to the settings without

expensive group annotations. Goel et al.; Kim et al. [11,21]

provide bias-tailored augmentations to balance the major-

ity and minority groups. In particular, these approaches

have mainly focused on better approximation and regular-

ization of worst-group error combined with advanced data

sampling, augmentation, or retraining strategies.

Studying impacts of neural architectures. Recently,

the effects of deep neural network architecture on gener-

alization performance have been explored. Diffenderfer et

al. [6] employ recently advanced lottery-ticket-style prun-

ing algorithms [8] to design the compact and robust network

architecture. Bai et al. [2] directly optimize the neural ar-

chitecture in terms of accuracy on OOD samples. Zhang et

al. [37] demonstrate the effectiveness of pruning weights on

spurious attributes, but the solution for discriminating such

spurious weights lacks robust theoretical justiﬁcations, re-

sulting in marginal performance gains. To fully resolve the

above issues, we carry out a theoretical case study, and build

a novel pruning algorithm that distills the representations to

be independent of the spurious attributes.

3. Theoretical insights

3.1. Problem setup

Consider a supervised setting of predicting labels Y∈ Y

from input samples X∈ X by a classiﬁer fθ:X → Y

parameterized by θ∈Θ. Following [37], let (Xe, Y e)∼

Pe, where Xe∈ X and Ye∈ Y refer to the input random

variable and the corresponding label, respectively, and e∈

E={1,2,...E}denotes the index of environment, Peis

the corresponding distribution, and the set Ecorresponds to

every possible environments. We further assume that Eis

divided into training environmments Etrain and unseen test

environments Etest, i.e. E=Etrain ∪ Etest.

For a given a loss function ℓ:X × Y × Θ→R+, the

standard training protocol for the empirical risk minimiza-

tion (ERM) is to minimize the expected loss with a training

environment e∈ Etrain:

θERM = arg min

E(Xe,Y e)∼ˆ

Peℓ(Xe, Y e;θ),(1)

where ˆ

Peis the empirical distribution over the training data.

Our goal is to learn a model with good performance on

OOD samples of e∈ Etest.

3.2. Motivating example

We conjecture that neural networks trained by ERM in-

discriminately rely on predictive features, including those

spuriously correlated ones [34].

To verify this conjecture, we present a simple binary-

classiﬁcation example (Xe, Y e)∼Pe, where Ye∈ Y =

{−1,1}represents the corresponding target label, and a

sample Xe∈ X ={−1,1}D+1 ∈RD+1 is constituted

with both the invariant feature Ze

inv ∈ {−1,1}and spu-

rious features Ze

sp ∈ {−1,1}D, i.e. Xe= (Ze

inv,Ze

sp).

Suppose, furthermore, Ze

sp,i denote the i-th spurious feature

component of Ze

sp. Note that we assume D≫1to simulate

the model heavily relies on spurious features Ze

sp [26,37].

We consider the setting where the training environment

e∈ Etrain is highly biased. In other words, we suppose that

inv =Ye, and each of the i-th spurious feature compo-

nent Ze

sp,i is independent and identically distributed (i.i.d)

Bernoulli variable: i.e. Ze

sp,i independently takes a value

equal to Yewith a probability peand −Yewith a prob-

ability 1−pe, where pe∈(0.5,1],∀e∈ Etrain. Note

that pe→1as the environment is severely biased. A

test environment e∈ Etest is assumed to have pe= 0.5,

which implies that the spurious feature is totally indepen-

dent with Ye. Then we introduce a linear classiﬁer fpa-

rameterized by a weight vector w= (winv,wsp)∈RD+1,

where winv ∈Rand wsp ∈RD. In this example, we

consider a class of pretrained classiﬁers parameterized by

w(t) = ˜winv(t),˜wsp,1(t),..., ˜wsp,D (t), where t<Tis

a ﬁnite pretraining time for some sufﬁciently large T. Time

twill be often omitted in notations for simplicity.

Our goal is to obtain the optimal sparse classiﬁer with a

highly biased training dataset. To achieve this, we introduce

a binary weight pruning mask mas m= (minv,msp)∈

{0,1}D+1 for the pretrained weights, which is a signiﬁ-

cant departure from the theoretical setting in [37]. Specif-

ically, let minv ∼Bern(πinv), where πinv and 1−πinv

represents the probability of preserving (i.e. minv = 1)

and pruning out (i.e. minv = 0), respectively. Simi-

larly, let msp,i ∼Bern(πsp,i),∀i. Then, our optimiza-

tion goal is to estimate the pruning probability parameter

π= (π1, . . . , πD+1)=(πinv, πsp,1, . . . , πsp,D), where

m∼P(π)is a mask sampled with probability parameters

π. Accordingly, our main loss function for the pruning pa-

rameters given the environment ecan be deﬁned as follows:

ℓe(π) = 1

2EXe,Y e,m[1 −Yeˆ

Ye]

2EXe,Y e,mh1−Ye·sgn ˜wT(Xe⊙m)i,

(2)

where ˆ

Yeis the prediction of binary classiﬁer, ˜wis the pre-

trained weight vector, sgn(·)represents the sign function,

and ⊙represents element-wise product.

We ﬁrst derive the upper-bound of the training loss ℓe(π)

to illustrate the difﬁculty of learning optimal pruning pa-

rameters in a biased data setting. The proof can be found in

Supplementary Material.

Theorem 1. (Training and test bound) Assume that pe>

1/2in the biased training environment e∈ Etrain. Deﬁne

w(t)as weights pretrained for a ﬁnite time t<T. Then

the upper bound of the error of training environment w.r.t.

pruning parameters πis given as:

ℓe(π)≤2 exp −2πinv + (2pe−1) PD

i=1 αi(t)πsp,i2

4PD

i=1 αi(t)2+ 1 ,

(3)

where the weight ratio αi(t) = ˜wsp,i(t)/˜winv (t)is

bounded below some positive constant. Given a test envi-

ronment e∈ Etest with pe=1

2, the upper bound of the

error of test environment w.r.t. πis given as:

ℓe(π)≤2 exp −2π2

inv

4PD

i=1 αi(t)2+ 1,(4)

which implies that there is an unavoidable gap between

training bound and test bound.

The detailed proof of Theorem 1is provided in the sup-

plementary material. This mismatch of the bounds is at-

tributed to the contribution of πsp,i on the training bound

(3). Intuitively, the networks prefer to preserve both ˜winv

and ˜wsp,i in the presence of strong spurious correlations due

to the inherent sensitivity of ERM to all kinds of predictive

features [17,34]. This behavior is directly reﬂected in the

training bound, where increasing either πinv or πsp,i, i.e.,

the probability of preserving weights, decreases the train-

ing bound. This inertia of spurious weights may prevent

themselves from being primarily pruned against the spar-

sity constraint.

We note that the unintended reliance on spurious features

is fundamentally rooted to the positivity of the weight ratio

αi(t). In the proof of Theorem 1in Supplementary Ma-

terial, we show some intriguing properties of αi(t): (1) If

inﬁnitely many data and sufﬁcient training time is provided,

the gradient ﬂow converges to the optimal solution which is

invariant to Ze

sp, i.e., αi(t)→0. In this ideal situation,

the gap between training and test bound is closed, thereby

guaranteeing generalizations of obtained subnetworks. (2)

However, given a ﬁnite time t<Twith a strongly biased

dataset in practice, αi(t)is bounded below by some positive

constant, resulting in an inevitable generalization gap.

Theorem 1implies that the classiﬁer may preserve spu-

rious weights due to the lack of bias-conﬂicting sam-

ples, which serve as counterexamples that spurious features

themselves fail to explain. It motivates us to analyze the

training bound in another environment ηwhere we can sys-

tematically augment bias-conﬂicting samples. Speciﬁcally,

consider Xη= (Zη

inv,Zη

sp), where Zη

inv =Yηand mix-

ture distribution of Zη

sp given Yη=yis deﬁned in an ele-

ment wise as follows:

Pη

mix(Zη

sp,i |Yη=y) =ϕP η

debias(Zη

sp,i |Yη=y)+

(1 −ϕ)Pη

bias(Zη

sp,i |Yη=y),

(5)

where ϕis a scalar mixture weight,

Pη

debias(Zη

sp,i |Yη=y) = (1,if Zη

sp,i =−y

0,if Zη

sp,i =y(6)

is a debiasing distribution to weaken the correlation be-

tween Yηand Zη

sp,i by setting the value of Zη

sp,i as −Yη,

and

Pη

bias(Zη

sp,i |Yη=y) = (pη,if Zη

sp,i =y

1−pη,if Zη

sp,i =−y(7)

is a biased distribution similarly deﬁned in the previous en-

vironment e∈ Etrain. Given this new environment η, the

degree of spurious correlations can be controlled by ϕ. This

leads to a training bound as follow:

Theorem 2. (Training bound with the mixture distribution)

Assume that the deﬁned mixture distribution Pη

mix is biased,

i.e., for all i∈ {1, . . . , D},

Pη

mix(Zη

sp,i =−y|Ye=y)≤Pη

mix(Zη

sp,i =y|Yη=y).

(8)

Then, ϕsatisﬁes 0≤ϕ≤1−1

2pη. Then the upper bound

of the error of training environment ηw.r.t. the pruning

parameters is given by

ℓη(π)≤

2 exp −2(πinv + (2pη(1 −ϕ)−1) PD

i=1 αi(t)πsp,i)2

4PD

i=1 αi(t)2+ 1 !.

(9)

Furthermore, when ϕ= 1 −1

2pη, the mixture distribution is

perfectly debiased, and we have

ℓη(π)≤2 exp −2π2

inv

4PD

i=1 αi(t)2+ 1,(10)

which is equivalent to the test bound in (4).

The detailed proof is provided in the supplementary ma-

terial. Our new training bound (31) suggests that the sig-

niﬁcance of πsp,i on training bound decreases as ϕprogres-

sively increases, and at the extreme end with ϕ= 1 −1

2pη,

it can be easily shown that Pη

mix(Zη

sp,i |Yη=y) = 1

2for

both y= 1 and y=−1so that Zη

sp,i turns out to be ran-

dom. In other words, by plugging ϕ= 1−1

2pηinto (31), we

can minimize the gap between training and test error bound,

which guarantees the improved OOD generalization.

4. Debiased Contrastive Weight Pruning

Our theoretical observations elucidate the importance

of balancing between the bias-aligned and bias-conﬂicting

samples in discovering the optimal unbiased subnetworks

structure. While the true analytical form of the debiasing

distribution is unknown in practice, we aim to approximate

such unknown distribution with existing bias-conﬂicting

samples and simulate the mixture distribution Pη

mix with

modifying sampling strategy. To this end, we propose a De-

biased Contrastive Weight Pruning (DCWP) algorithms that

learn the unbiased subnetworks structure from the original

full-size network.

Consider a Llayer neural networks as a function

fW:X → RCparameterized by weights W=

{W1,...,WL}, where C=|Y| is the number of classes.

Analogous to the earlier works on pruning, we introduce bi-

nary weight pruning masks m={m1,...,mL}to model

the subnetworks as f(·;m1⊙W1,...,mL⊙WL). We de-

note such subnetworks as fm⊙Wfor the notational simplic-

ity. We treat each entry of mlas an independent Bernoulli

variable, and model their logits as our new pruning param-

eters Θ={Θ1,...,ΘL}where Θl∈Rnland nlrepre-

sents the dimensionality of the l-th layer weights Wl. Then

πl,i =σ(Θl,i)denotes the probability of preserving the i-

th weight of l-th layer Wl,i where σrefers to a sigmoid

function. To enable the end-to-end training, the Gumbel-

softmax trick [18] for sampling masks together with ℓ1reg-

ularization term of Θis adopted as a sparsity constraint.

With a slight abuse of notations, m∼G(Θ)denotes a

set of masks sampled with logits Θby applying Gumbel-

softmax trick.

Then our main optimization problem is deﬁned as fol-

lows:

min

Θℓdebias{(xi, yi)}|S|

i=1;˜

W,Θ+λℓ1X

l,i

|Θl,i|,(11)

where Sdenotes the index set of whole training sam-

ples, λℓ1>0is a Lagrangian multiplier, ˜

Wrepresents

the pretrained weights and ℓdebias is our main objective

which will be illustrated later. Note that we freeze the

pretrained weights ˜

Wduring training pruning parameters

Θ. We interchangeably use ℓdebias{(xi, yi)}|S|

i=1; Θand

ℓdebiasS; Θin the rest of the paper. For comparison with

our formulation, we recast the optimization problem of [37]

with our notations as follows:

min

Θℓ{(xi, yi)}|S|

i=1;˜

W,Θ+λℓ1X

l,i

|Θl,i|,(12)

where [37] uses the cross entropy (CE) loss function for ℓ.

Bias-conﬂicting sample mining In the ﬁrst stage, we

identify bias-conﬂicting training samples which empower

functional modular probing. Speciﬁcally, we train a bias-

capturing model and treat an error set Sbc of the index

of misclassiﬁed training samples as bias-conﬂicting sample

proxies. Our framework is broadly compatible with vari-

ous bias-capturing models, where we mainly leverage the

ERM model trained with generalized cross entropy (GCE)

loss [39]:

ℓGCE (xi, yi;WB) = 1−pyi(xi;WB)q

q,(13)

where q∈(0,1] is a hyperparameter controlling the de-

gree of bias ampliﬁcation, WBis the parameters of the

bias-capturing model, and pyi(xi;WB)is a softmax out-

put value of the bias-capturing model assigned to the target

label yi. Compared to the CE loss, the gradient of the GCE

loss up-weights the samples with a high probability of pre-

dicting the correct target, amplifying the network bias by

putting more emphasis on easy-to-predict samples [27].

To preclude the possibility that the generalization per-

formance of DCWP is highly dependent on the behavior

of the bias-capturing model, we demonstrate in Section 5

that DCWP is reasonably robust to the degradation of ac-

curacy on capturing bias-conﬂicting samples. Details about

the bias-capturing model and simulation settings are pre-

sented in the supplementary material.

Upweighting Bias-conﬂicting samples After mining

the index set of bias-conﬂicting sample proxies Sbc, we treat

Sba =S\Sbc as the index set of majority bias-aligned sam-

ples. Then we calculate the weighted cross entropy (WCE)

loss ℓW C E {xi, yi}|S|

i=1;˜

W,Θas follows:

ℓW C E S;˜

W,Θ:= Em∼G(Θ)λupℓbc(Sbc;m,˜

W)+

ℓba(Sba;m,˜

W),

(14)

where λup ≥1is an upweighting hyperparameter, and

ℓbc(Sbc;m,˜

W) = 1

|Sbc|X

i∈Sbc

ℓCE (xi, yi;m⊙˜

W),(15)

where ℓCE denotes the cross entropy loss. ℓba is deﬁned as

similar to ℓbc.

The expectation is approximated with Monte Carlo esti-

mates, where the number of mask msampled per iteration

is set to 1 in practice. To implement (14), we oversample

the samples in Sbc for λup times more than the samples in

Sba. This sampling strategy is aimed at increasing the mix-

ture weight ϕof the proposed mixture distribution Pη

mix in

(5), while we empirically approximate the unknown bias-

conﬂicting group distribution with the sample set Sbc.

Note that although simple oversampling of bias-

conﬂicting samples may not lead to the OOD generalization

due to the inductive bias towards memorizing a few coun-

terexamples in overparameterized neural networks [30],

such failure is unlikely reproduced in learning pruning pa-

rameters under the strong sparsity constraint. We sam-

ple new weight masks mfor each training iteration in a

stochastic manner, effectively precluding the overparame-

terized networks from potentially memorizing the minority

samples. As a result, DCWP exhibits reasonable perfor-

mance even with few bias-conﬂicting samples.

Bridging the alignment gap by pruning To fully uti-

lize the bias-conﬂicting samples, we consider the sample-

wise relation between bias-conﬂicting samples and major-

ity bias-aligned samples. Zhang et al. [38] demonstrates

that the deteriorated OOD generalization is potentially at-

tributed to the distance gap between same-class representa-

tions; bias-aligned representations are more closely aligned

than bias-conﬂicting representations, although they are gen-

erated from the same-class samples. We hypothesized that

well-designed pruning masks could alleviate such geomet-

rical misalignment. Speciﬁcally, ideal weight sparsiﬁca-

tion may guide each latent dimension to be independent of

spurious attributes, thereby preventing representations from

being misaligned with spuriously correlated latent dimen-

sions. This motivates us to explore pruning masks by con-

trastive learning. (Related illustrative example in appendix)

Following the conventional notations of contrastive

learning, we denote fenc

W:X → RnL−1as an encoder

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TrainingDebiasedSubnetworkswithContrastiveWeightPruningGeonYeongPark1SangminLee2SangWanLee1∗JongChulYe1,2,3∗1BioandBrainEngineering,2MathematicalSciences,3KimJaechulGraduateSchoolofAIKoreaAdvancedInstituteofScienceandTechnology(KAIST),Daejeon,Korea{pky3436,leeleesang,sangwan,jong.ye}@kaist.ac.krAbst...

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