# Adversarial Counterfactual Augmentation: Application in Alzheimer’s Disease
Classification

Tian Xia 1,∗, Pedro Sanchez 1, Chen Qin 1 and Sotirios A. Tsaftaris 1,2

###### Abstract

Due to the limited availability of medical data, deep learning approaches for
medical image analysis tend to generalise poorly to unseen data. Augmenting
data during training with random transformations has been shown to help and
became a ubiquitous technique for training neural networks. Here, we propose a
novel adversarial counterfactual augmentation scheme that aims at finding the
most effective synthesised images to improve downstream tasks, given a pre-
trained generative model. Specifically, we construct an adversarial game where
we update the input conditional factor of the generator and the downstream
classifier with gradient backpropagation alternatively and iteratively. This
can be viewed as finding the ‘weakness’ of the classifier and purposely
forcing it to overcome its weakness via the generative model. To demonstrate
the effectiveness of the proposed approach, we validate the method with the
classification of Alzheimer’s Disease (AD) as a downstream task. The pre-
trained generative model synthesises brain images using age as conditional
factor. Extensive experiments and ablation studies have been performed to show
that the proposed approach improves classification performance and has
potential to alleviate spurious correlations and catastrophic forgetting.
Code will be released upon acceptance.

##  1 Keywords:

Alzheimer’s Disease, Generative model, Classification, Counterfactuals, Data
efficiency

##  2 Introduction

Deep learning has been playing an increasingly important role in medical image
analysis in the past decade, with great success in segmentation, diagnosis,
detection, etc (Shen et al., 2017). Although deep-learning based models can
significantly outperform traditional machine learning methods, they heavily
rely on the large size and quality of training data (Chlap et al., 2021). In
medical image analysis, the availability of large dataset is always an issue,
due to high expense of acquiring and labelling medical imaging data (Shorten
and Khoshgoftaar, 2019). When only limited training data are available, deep
neural networks tend to memorise the data and cannot generalise well to unseen
data (Dietterich, 1995; Srivastava et al., 2014). This is known as over-
fitting (Dietterich, 1995). To mitigate this issue, data augmentation has
become a popular approach. The aim of data augmentation is to generate
additional data that can help increase the variation of the training data.

Conventional data augmentation approaches mainly apply random image
transformations, such as cropping, flipping, and rotation etc. to the data.
Even though such conventional data augmentation techniques are general, they
may not transfer well from one task to another (Cubuk et al., 2019). For
instance, color augmentation could prove useful for natural images but may not
be suitable for MRI images which are presented in greyscale images (Shorten
and Khoshgoftaar, 2019). Furthermore, traditional data augmentation methods
may introduce distribution shift, i.e., the change of the joint distribution
of inputs and outputs, and consequently adversely impact the performance on
non-augmented data during inference111 An example could be when training and
testing brain MRI data are already well-registered, traditional augmentations,
e.g. rotation, shift, etc., on the training data will hurt the performance of
the trained model on testing data. See Section 5.1.3 for more details. (i.e.,
during the application phase of the learned model) (Gong et al., 2021b).

Some recently developed approaches learn parameters for data augmentation that
can better improve downstream task, e.g. segmentation, detection, diagnosis,
etc., performance (Chen et al., 2021; Cubuk et al., 2019; Gao et al., 2021) or
select the hardest augmentation for the target model from a small batch of
random augmentations for each traning sample, (Gong et al., 2021a). However,
these approaches still use conventional image transformations and do not
consider semantic augmentation (Wang et al., 2021), i.e., creating unseen
samples by changing semantic information of images such as changing the
background of an object or changing the age of a brain image. Semantic
augmentation can complement traditional techniques and improve the diversity
of augmented samples (Wang et al., 2021).

One way to achieve semantic augmentation is to train a deep generative model
to create counterfactuals, i.e., synthetic modifications of a sample such that
some aspects of the original data remain unchanged (Zhang et al., 2020;
Shamsolmoali et al., 2021; Bowles et al., 2018; Oh et al., 2021; Dash et al.,
2022). However, these approaches mostly focus on the training stage of
generative models and randomly generate samples for data augmentation, without
considering which counterfactuals are more effective for downstream tasks,
i.e. data-efficiency of the generated samples . Wang et al. (2021); Li et al.
(2021); Chen and Su (2021) proposed to augment the data in the latent space of
the target deep neural network, by estimating the covariance matrix of latent
features obtained from latent layers of the target deep neural network for
each class (e.g., car, horse, tree, etc.) and sampling directions from the
feature distributions. These directions should be semantic meaningful such
that changing along one direction can manipulate one property of the image,
e.g. color of a car. However, there is no guarantee that the found directions
will be semantically meaningful, and it is hard to know which direction
controls a particular property of interest.

Figure 1: A schematic of the adversarial classification training. The pre-
trained generator $G$ takes a brain image $\mathbf{x}$ and a target age $a$ as
input and outputs a synthetically aged image $\hat{\mathbf{x}}$ that
corresponds to the target age $a$. The classifier $C$ aims to predict AD label
for a given brain image. To utilise $G$ to improve $C$, we formulate an
adversarial game between $a$ (in red box) and $C$ (in cyan box), where $a$ and
$C$ are updated alternatively and iteratively using $L_{1}$ and $L_{2}$,
respectively (see Sec. 3.3). Note $G$ is frozen.

In this work, we consider the scenario that we have a classifier which we want
to improve (e.g. an image-based classifier of Alzheimer’s Disease (AD) given
brain images). We are also given some data and a pre-trained generative model
that is able to create new data given an image as input and conditioning
factors that can alter corresponding attributes in the input. For example, the
generative model can alter the brain age of the input. We propose an approach
to guide a pre-trained generative model to generate the most effective
counterfactuals via an adversarial game between the input conditioning factor
of the generator and the downstream classifier, where we use gradient
backpropagation to update the conditioning factor and the classifier
alternatively and iteratively. A schematic of the proposed approach is shown
in Fig. 1.

Specifically, we choose the classification of AD as the downstream task and
utilise a pre-trained brain ageing synthesis model to improve the AD
classifier. The brain ageing generative model used in this paper is adopted
from a recent work (Xia et al., 2021), which takes a brain image and a target
age as inputs and outputs an aged brain image222Code is available at
https://github.com/xiat0616/BrainAgeing. We show that the proposed approach
can improve the test accuracy of the AD classifier. We also demonstrate that
it can be used in a continual learning333Deep models continuously learn based
on input of new data while preserving previously learnt knowledge. context to
alleviate catastrophic forgetting, i.e. deep models forget what they have
learnt from previous data when training on new given data, and can be used to
alleviate spurious correlations, i.e. two variables appear to be causally
related to one another but in fact they are not. Our contributions can be
summarised as follows:

1. We propose an approach to utilise a pre-trained generative model for a classifier via an adversarial game between conditional input and the classifier. To the best of our knowledge, this is the first approach that formulates such an adversarial scheme to utilise pre-trained generators in medical imaging.

2. We improve a recent brain ageing synthesis model by involving Fourier encoding to enable gradient backpropagation to update conditional factor and demonstrate the effectiveness of our approach on the task of AD classification.

3. We consider the scenario of using generative models in a continual learning context and show that our approach can help alleviate catastrophic forgetting.

4. We apply the brain ageing synthesis model for brain rejuvenation synthesis and demonstrate that the proposed approach has the potential to alleviate spurious correlations.

##  3 Methodology

###  3.1 Notations and problem overview

We denote an image as $\mathbf{x}\sim\mathcal{X}$, and a conditional
generative model $G$ that takes an image $\mathbf{x}$ and a conditional vector
${\mathbf{v}}$ as input and generates a counterfactual $\mathbf{\hat{x}}$ that
corresponds to $\mathbf{v}$: $\mathbf{\hat{x}}=G(\mathbf{x},\mathbf{v})$. For
each $\mathbf{x}$, there is a label $y\sim\mathcal{Y}$. We define a classifier
$C$ that predicts the label $\hat{y}$ for given $\mathbf{x}$:
$\hat{y}=C(\mathbf{x})$. In this paper, $\mathbf{x}$ is a brain image, $y$ is
the AD diagnosis of $\mathbf{x}$, and $\mathbf{v}$ represents the target age
$a$ and AD diagnosis on which the generator $G$ is conditioned. We select age
and AD status to be conditioning factors as they are major contributors to
brain ageing. We use a brain ageing generative model as $G$, and a VGG444A
popular deep learning neural network that has widely been used for
classification.-based (Simonyan and Zisserman, 2015) AD classification model
as $C$. Note that we only change the target age $a$ in this paper, thus we
write the generative process as $\hat{\mathbf{x}}=G(\mathbf{x},a)$ for
simplicity.

Suppose a pre-trained $G$ and a $C$ are given, the question we want to answer
is: “How can we use $G$ to improve $C$ in a (data) efficient manner”? To this
end, we propose an approach to utilise $G$ to improve $C$ via an adversarial
game with gradient backpropagation to update $a$ and $C$ alternatively and
iteratively.

###  3.2 Fourier encoding for conditional factors

The proposed approach requires backpropagation of gradient to the conditional
factor to find the hard counterfactuals. However, the original brain ageing
synthesis model (Xia et al., 2021) used ordinal encoding to encode the
conditional age and AD diagnosis, where the encoded vectors are discrete in
nature and need to maintain a certain shape, which hinders gradient
backpropagation to update these vectors.

To enable gradient backpropagation to update the conditional vectors, we
propose to use Fourier encoding (Tancik et al., 2020; Mildenhall et al., 2020)
to encode the conditional attributes, i.e., age and heath state (diagnosis of
AD). The effectiveness of Fourier encoding has been experimentally shown in
Tancik et al. (2020); Mildenhall et al. (2020), and we also achieved similar
synthetic results with Fourier encoding as the original model. The key idea of
Fourier encoding is to map low-dimensional vectors to a higher dimensional
domain using a set of sinusoids. For instance, if we have a d-dimensional
vector which is normalised into $[0,1)$, $\mathbf{v}\in[0,1)^{d}$, then the
encoded vector can be represented as Tancik et al. (2020):

$\mathbf{\gamma}(\mathbf{v})=[p_{1}\cos(2\pi\mathbf{b_{1}^{T}\mathbf{v})},p_{1}\sin(2\pi\mathbf{b_{1}^{T}}\mathbf{v}),...,p_{m}\cos(2\pi\mathbf{b_{m}^{T}}\mathbf{v}),p_{m}\sin(2\pi\mathbf{b_{m}^{T}}\mathbf{v})],$
(1)

where $\mathbf{b_{j}}$ can be viewed as the Fourier basis frequencies, and
$p^{2}_{j}$ the Fourier series coefficients.

In this work, the vector $\mathbf{v}$ represents the target age $a$ and the
health status (AD diagnosis), and $d=2$. In our experiments, we set
$p^{2}_{j}=1$ for $j=1,...,m$, and $\mathbf{b_{j}}$ are independently and
randomly sampled from a Gaussian distribution,
$\mathbf{b_{j}}\sim\mathcal{N}(\mu_{scale}*\mathbf{I},0)$, where $\mu_{scale}$
is set to 10. We set $m=100$ and the resulting $\gamma(\mathbf{v})$ is
200-dimensional. After encoding, the generator $G$ takes the encoded vector
$\gamma(\mathbf{v})$ as input.

The use of Fourier encoding offers two advantages. First, Xia et al. (2021)
encoded age and health state into two vectors and had to use two MLPs to embed
the encoded vectors into the model. This may not be a big issue when the
number of factors is small. However, extending the generative model to be
conditioned on tens or hundreds of factors will increase the memory and
computation costs significantly. With Fourier encoding, we can encode all
possible factors into a single vector, which offers more flexibility to scale
the model to multiple conditional factors. Second, Fourier encoding allows us
to compute the gradients with respect to the input vector $\mathbf{v}$ or
certain elements of $\mathbf{v}$, since the encoding process is
differentiable. As such, we replace the ordinal encoding with Fourier encoding
for all experiments. The generative model $G$ takes $\mathbf{v}$ as input:
$\hat{\mathbf{x}}=G(\mathbf{x},\mathbf{v})$, where $\mathbf{v}$ represents
target age and health state. Since we only change the target age $a$ in this
paper, we write the generative process as $\hat{\mathbf{x}}=G(\mathbf{x},a)$
for simplicity.

###  3.3 Adversarial counterfactual augmentation

Suppose we have a conditional generative model $G$ and a classification model
$C$. The goal is to utilise $G$ to improve the performance of $C$. To this
end, we propose an approach consisting of three steps: pre-training, hard
sample selection and adversarial classification training. A schematic of the
adversarial classification training is presented in Fig. 1. Algorithm 1
summarises the steps of the method. Below we describe each step in detail.

Pre-training. The generative model is pre-trained using the same losses as in
Xia et al. (2021) except that we use Fourier encoding to encode age and AD
diagnosis. Consequently, we obtain a pre-trained $G$ that can generate
counterfactuals conditioned on given target ages $a$:
$\mathbf{\hat{x}}=G(\mathbf{x},{a})$.

The classification model $C$ is a VGG-based network (Simonyan and Zisserman,
2015) trained to predict the AD diagnosis from brain images, optimised by
minimising:

$L_{pre-
train}=\mathbb{E}_{\mathbf{x}\sim\mathcal{X}_{train},\mathbf{y}\sim\mathcal{Y}_{train}}L_{s}(C(\mathbf{x}),y),$
(2)

where $L_{s}(.)$ is a supervised loss (binary cross-entropy loss in this
paper), $\mathbf{x}$ is a brain image, and $y$ is its ground-truth AD label.
To note that if the pre-trained $G$ and $C$ are available in practice, we
could avoid the pre-training step.

Hard sample selection. Liu et al. (2021); Feldman and Zhang (2020) suggested
that training data samples have different influence on the training of a
supervised model, i.e., some training data are harder for the task and are
more effective to train the model than others. Liu et al. (2021) propose to
up-sample, i.e. duplicate, the hard samples as a way to improve the model
performance. Based on these observations, we propose a similar strategy to Liu
et al. (2021) to select these hard samples: we record the classification
errors of all training samples for the pre-trained $C$ and then select $N=100$
samples with the highest errors. The selected hard samples are denoted as
$D_{hard}$: $\\{X_{hard},Y_{hard}\\}$.

Adversarial classification training. (Bowles et al., 2018; Frid-Adar et al.,
2018a; Dar et al., 2019) augmented datasets by randomly generating a number of
synthetic data with pre-trained generators. Similar to training samples, some
synthetic data could be more effective for downstream tasks than others. Here
we assume that if a synthetic data sample is hard, then it is more effective
for training. We propose an adversarial game to find the hard synthetic data
to boost $C$.

Algorithm 1 Adversarial counterfactual augmentation with a pre-trained $G$.

Input: Training set $D_{train}$; hyperparameter $k$, $N$; a pre-trained $G$;
$C$.

Pre-training:

1\. Train the classifier $C$ on $D_{train}$ (Eq. 2).

Hard sample selection:

2\. Select $N$ samples from $D_{train}$ that result in highest classification
errors

for $C$, denoted as $D_{hard}$.

Adversarial classification training:

3\. Randomly initialize target ages $a$, and obtain initial synthetic data.

For $k$ do

4\. Update $a$ in the direction to maximise classification error (Eq. 4).

5\. Obtain synthetic images with $D_{hard}$ and the updated $a$, denoted as
$D_{syn}$.

6\. Update $C$ to optimise Eq. 5 on $D_{train}\cup D_{syn}$ for one epoch.

Specifically, let us first define the classification loss for synthetic data
as:

$L_{C}=\mathbb{E}_{\mathbf{x}\sim X_{hard},{y}\sim
Y_{hard}}L_{s}(C(\mathbf{\hat{x}}),y),$ (3)

where $\mathbf{\hat{x}}$ is a generated sample conditioned on the target age
$a$: $\mathbf{\hat{x}}=G(\mathbf{x},a)$, and $y$ is the ground-truth AD label
for $\mathbf{{x}}$. Here we assume that changing target age does not change
the AD status, thus $\mathbf{{x}}$ and $\mathbf{\hat{x}}$ have the same AD
label.

Since the encoding of age $a$ is differentiable (see Section 3.2), we can
obtain the gradients of $L_{C}$ with respect to $a$ as:
$\nabla_{a}L_{C}=\nabla_{a}[L_{s}(C(G(\mathbf{x},a)),y)]$, and update $a$ in
the direction of maximising $L_{C}$ by:
$\tilde{a}=a+\gamma_{a}\nabla_{a}L_{C}$, where $\gamma_{a}$ is the step size
(learning rate) for updating $a$. Formally, the optimization function of $a$
can be written as:

$L_{1}=\max_{a}\operatorname{\mathbb{E}}_{\mathbf{x}\sim X_{hard},{y}\sim
Y_{hard}}L_{s}(C(\mathbf{\hat{x}}),y).$ (4)

Then we could obtain a set of synthetic data using the updated $\tilde{a}$:
$\hat{\mathbf{x}}_{syn}=G(\mathbf{x},\tilde{a})$ where $\mathbf{x}\sim
X_{hard}$, denoted as $D_{syn}:\\{X_{syn},Y_{syn}\\}$.

The classifier $C$ is updated by optimising:

$L_{2}=\min_{C}\operatorname{\mathbb{E}}_{\mathbf{x}\sim X_{combined},y\sim
Y_{combined}}L_{s}(C(\mathbf{x}),y),$ (5)

where $D_{combined}$: $\\{X_{combined},Y_{combined}\\}$ is a combined dataset
consisting of the training dataset and synthetic dataset:
$\\{X_{combined},Y_{combined}\\}=\\{X_{train}\cup X_{syn},Y_{train}\cup
Y_{syn}\\}$. Similar to Liu et al. (2021), we update $C$ on $D_{combined}$
instead of $D_{syn}$ as we found updating $C$ only on $D_{syn}$ can cause
catastrophic forgetting (Kirkpatrick et al., 2017).

The adversarial game is formulated by alternatively and iteratively updating
$a$ and classifier $C$ via Eqs. 4 and 5, respectively. In practice, to prevent
$a$ from going to unsuitable ages, we clip it to be in [60, 90] after every
update.

Updating $a$ vs. updating $G$. Note here the adversarial game is formulated
between $a$ and $C$, instead of $G$ and $C$. This is because training $G$
against $C$ allows $G$ to change its latent space without considering image
quality, and the output of $G$ could be unrealistic. Please refer to Section
5.1.2 for more details and results.

Counterfactual augmentation vs. conventional augmentation. Here we choose to
augment data counterfactually instead of applying conventional augmentation
techniques. This is because that the training and testing data are already
pre-processed and registered to MNI 152, and in this case conventional
augmentations do not introduce helpful variations. Please refer to Section
5.1.3 for more details and results.

###  3.4 Adversarial classification training in a continual learning context

Most previous works (Bowles et al., 2018; Antoniou et al., 2018; Frid-Adar et
al., 2018a, b; Shin et al., 2018; Dar et al., 2019) that used pre-trained deep
generative models for augmentation focused on generating a large number of
synthetic samples, and then merged the synthetic data with the original
dataset and trained the downstream task model (e.g. a classifier) on this
augmented dataset. However, this requires training the task model from
scratch, which could be inconvenient. For instance, if we suddenly decided to
generate some new synthetic data for augmentation, we would have to retrain
the task model from scratch. Furthermore, if the size of the original dataset
is large, then the number of synthetic samples can be huge, which would make
the training process extremely expensive and time-consuming. Thus, in
practice, we need to consider cases where we aim to improve a pre-trained
classifier with synthetic data but without retraining the whole model from
scratch. We design the proposed procedure in such a way that allows us to use
the pre-trained $G$ to improve $C$ flexibly.

In Section 3.3, after we obtain the synthetic set $D_{syn}$, we choose to
update the classifier $C$ on the augmented dataset $D_{syn}\cup D_{train}$,
instead of $D_{syn}$ (stage 6 in Algorithm 1). This is because re-training the
classifier only on the $D_{syn}$ would result in catastrophic forgetting
(Kirkpatrick et al., 2017), i.e. a phenomenon where deep neural networks tends
to forget what it has learnt from previous data when being trained on new data
samples. To alleviate catastrophic forgetting, efforts have been devoted to
developing approaches to allow artificial neural networks to learn in a
sequential manner (Delange et al., 2021; Parisi et al., 2019). These
approaches are known as continual learning (Delange et al., 2021; Chaudhry et
al., 2019; Lopez-Paz and Ranzato, 2017), lifelong learning (Chen and Liu,
2018; Aljundi et al., 2017), sequential learning (McCloskey and Cohen, 1989;
Aljundi et al., 2018), or incremental learning (Chaudhry et al., 2018;
Gepperth and Karaoguz, 2016). Despite different names and focuses, the main
purpose of these approaches is to overcome catastrophic forgetting and to
learn in a sequential manner.

Algorithm 2 Adversarial classification learning with $D_{store}$.

Input: Training dataset $D_{train}$; hyperparameter $M$, $N$, $k$; a pre-
trained generator $G$; a pre-trained classifier model $C$.

Construct $D_{store}$:

1\. Randomly select $M\%$ data from $D_{train}$, denoted as $D_{store}$.

Hard sample selection

2\. Select $N$ samples from $D_{store}$ that result in highest classification
errors for $C$, denoted as $D_{hard}$.

Adversarial training:

3\. Randomly initialise target ages $a$, and obtain initial synthetic data.

For $k$ do

4\. Update $a$ in the direction to maximise classification error (Equation 4).

5\. Obtain synthetic images with $D_{hard}$ and the updated $a$, denoted as
$D_{syn}$.

6\. Update $C$ to minimise the classification error on $D_{store}\cup D_{syn}$
(Equation 5).

If we consider the generated data as new samples, then the update of the pre-
trained classifier $C$ can be viewed as a continual learning problem, i.e. how
to learn new knowledge from the synthetic set $D_{syn}$ without forgetting old
knowledge that is learnt from the original training data $D_{train}$. To
alleviate catastrophic forgetting, we re-train the classifier on both the
synthetic dataset $D_{syn}$ and the original training dataset $D_{train}$.
This strategy is known as memory replay in continual learning (Robins, 1995;
van de Ven et al., 2020) and was also used in other augmentation works (Liu et
al., 2021). The key idea is to store previous data in a memory buffer and
replay the saved data to the model when training on new data. However, it
could be expensive to store and revisit all the training data, especially when
the data size is large (van de Ven et al., 2020). In Section 5.2, we perform
experiments where we only provide a portion ($M\%$) of training data to the
classifier when re-training with synthetic data (to simulate the memory
buffer). We want to see whether catastrophic forgetting would happen or not
when only a portion ($M\%$) of training data is provided, and if so, how much
it affects the test accuracies. Algorithm 2 summarises the steps of the method
in the continual learning context.

##  4 Experimental setup

Data. We use the ADNI dataset (Petersen et al., 2010) for experiments. We
select 380 AD and 380 CN (control normal) T1 volumes between 60 and 90 years
old. We split the AD and CN data into training/validation/testing sets with
260/40/260 volumes, respectively. All volumetric data are skull-stripped using
DeepBrain555https://github.com/iitzco/deepbrain, and linearly registered to
MNI 152 space using FSL-FLIRT (Woolrich et al., 2009). We normalise brain
volumes by clipping the intensities to $[0,V_{99.5}]$, where $V_{99.5}$ is the
$99.5\%$ largest intensity value within each volume, and then rescale the
resulting intensities to the range $[-1,+1]$. We select the middle 60 axial
slices from each volume and crop each slice to the size of $[208,160]$,
resulting in 31200 training, 4800 validation and 9600 testing slices.

Implementation. The generator is trained the same way as in Xia et al. (2021),
except we replace ordinal encoding with Fourier encoding. We pre-train the
classifier for 100 epochs. The experiments are performed using Keras and
Tensorflow. We train pre-trained classifiers $C$ with Adam with a learning
rate of 0.00001 and decay of 0.0001. During adversarial learning, the step
size of $a$ is tuned to be 0.01, and the learning rate for $C$ is 0.00001. The
experiments are performed using a NVIDIA Titan X GPU.

Comparison methods. We compare with the following baselines:

1\. Naïve: We directly use the pre-trained $C$ for comparison as the lower
bound.

2\. RSRS: Random Selection + Random Synthesis. We randomly select $N=100$
samples from the training set $D_{train}$, denoted as $D_{rand}$, and then use
the generator $G$ to randomly generate $N_{synthesis}=5$ synthetic samples for
each sample in $D_{rand}$, denoted as $D_{syn}$. Then we train the classifier
on the combined dataset $D_{train}\cup D_{syn}$ for $k=5$ steps. This is the
typical strategy used by most previous works (Bowles et al., 2018; Frid-Adar
et al., 2018a; Dar et al., 2019).

3\. HSRS: Hard Selection + Random Synthesis. We select $N=100$ hard samples
from $D_{train}$ based on their classification errors of $C$, denoted as
$D_{hard}$, and then use the generator $G$ to randomly generate
$N_{synthesis}=5$ synthetic samples for each sample in $D_{hard}$, denoted as
$D_{syn}$. Then we train the classifier on the combined dataset $D_{train}\cup
D_{syn}$ for $k=5$ steps.

4\. RSAT: Random Selection + Adversarial Training. We randomly select $N=100$
samples from the training set $D_{train}$, denoted as $D_{rand}$, and then use
the adversarial training strategy to update the classifier $C$, as described
in Sec. 3.3.. The difference between RSAT and our approach is that we select
hard samples for generating counterfactuals, while RSAT uses random samples.

5\. JTT: Just Train Twice (Liu et al., 2021) record samples that are
misclassified by the pre-trained classifier, obtaining an error set. Then they
construct an oversampled dataset $D_{up}$ that contain examples in the error
set $\lambda_{up}$ times and all other examples once. Finally, they train the
classifier on the oversampled dataset $D_{up}$. In this paper, we set
$\lambda_{up}=2$ as we found large $\lambda_{up}$ results in bad performance.
We also found the original learning rate (0.01) used for the second training
stage results in very bad performance and set it to be 0.00001.

##  5 Results and Discussion

###  5.1 Improving the performance of classifiers.

####  5.1.1 Comparison with baselines.

We first compare our method with baseline approaches by evaluating the test
accuracy of the classifiers. We set $N=100$ and $k=5$ in experiments. We pre-
train $C$ for 100 epochs and $G$ as described in Section 4. The weights of the
pre-trained $C$ and the pre-trained $G$ are the same for all methods. For a
fair comparison, the total number of synthetically generated samples is fixed
to 500 for RSRS, HSRS, RSAT and our approach. For JTT, there are 2184 samples
mis-classified by $C$ and oversampled. We initialize $a$ randomly between real
ages of original brain images $\mathbf{x}$ and maximal age (90 yrs old).

Table 1: Average test accuracies of models trained via our procedure and baselines. We first present the average test accuracies for different age groups with AD (column 2-4) or CN (column 5-7) and then present the average test accuracies for the whole testing set (column 8). For each method, the worst-group performance is shown in italic. For each age group, i.e. each column, the best performance is shown in bold. We also report the number of testing images for each age group.  Acc % | CN | AD | All
---|---|---|---
Age group | 60-70yrs | 70-80yrs | 80-90yrs | 60-70yrs | 70-80yrs | 80-90yrs | overall
Test group size | 1540 | 1600 | 1660 | 1720 | 1540 | 1540 | 9600
Naïve | 85.2 | 91.5 | 70.7 | 92.5 | 94.2 | 97.1 | 88.4
RSRS | 86.0 | 90.4 | 73.8 | 87.3 | 95.1 | 90.0 | 87.0
HSRS | 85.6 | 91.1 | 80.4 | 89.8 | 93.8 | 96.9 | 89.5
RSAT | 86.1 | 93.1 | 81.5 | 91.8 | 96.0 | 95.7 | 90.6
JTT | 83.9 | 94.2 | 80.1 | 92.8 | 90.8 | 93.7 | 89.2
Proposed | 86.4 | 93.7 | 83.4 | 91.5 | 96.5 | 95.7 | 91.1

From Table 1 we can observe that our proposed procedure achieves the best
overall test accuracy, followed by baseline RSAT. This demonstrates the
advantage of adversarial training between the conditional factor (target age)
$a$ and the classifier. On top of that, it shows that selecting hard examples
for creating augmented synthetic results helps, which is also demonstrated by
the improvement of performance of HSRS over Naïve. We also observe that JTT
(Liu et al., 2021) improves the classifier performance over Naïve, showing the
benefit of up-sampling hard samples. In contrast, baseline RSRS achieves the
lowest overall test accuracy, even lower than that of Naïve. This shows that
randomly synthesising counterfactuals from randomly selected samples could
result in synthetic images that are harmful to the classifier.

Furthermore, we observe that for all methods, the worst-group performances are
achieved on the 80-90 CN group. A potential reason could be: as age increases,
the brains shrink, and it is harder to tell if the ageing pattern is due to AD
or caused by normal ageing. Nevertheless, we observe that for this worst
group, our proposed method still achieves the best performance, followed by
$RSAT$. This shows that adversarial training can be helpful to improve the
performance of the classifier, especially for hard groups. The next best
results are achieved by HSRS and JTT, which shows that finding hard samples
and up-sampling or augmenting them was helpful to improve the worst-group
performance. We also observe the improvement of worst-group performance for
RSRS over Naïve, but the improvement is small compared to other baselines.

We also report the precision and recall for all methods, as presented in Table
2. We can observe that our approach achieves the highest overall precision and
recall rersults.

Table 2: The test precision and recall values for all methods. We first present the precision for different age groups (column 2-4) and all testing data (column 5), and then present the recall for different age groups (column 6-8) and all testing data (column 9). For each group, the best results are shown in bold.  Age Range | 60-70 | 70-80 | 80-90 | Overall | 60-70 | 70-80 | 80-90 | Overall
---|---|---|---|---|---|---|---|---
Metrics | Precision | Recall
Naive | 0.875 | 0.914 | 0.761 | 0.842 | 0.925 | 0.942 | 0.971 | 0.945
RSRS | 0.874 | 0.905 | 0.768 | 0.844 | 0.873 | 0.951 | 0.900 | 0.906
HSRS | 0.874 | 0.910 | 0.826 | 0.866 | 0.898 | 0.938 | 0.969 | 0.933
RSAT | 0.881 | 0.930 | 0.832 | 0.877 | 0.918 | 0.960 | 0.957 | 0.943
JTT | 0.865 | 0.938 | 0.822 | 0.868 | 0.928 | 0.908 | 0.960 | 0.924
Proposed | 0.883 | 0.936 | 0.848 | 0.885 | 0.915 | 0.965 | 0.965 | 0.945

In summary, the quantitative results show that it is helpful to find and
utilise hard counterfactuals for improving the classifier.

####  5.1.2 Train G against C

We choose to formulate an adversarial game between the conditional generative
factor $a$ (the target age) and the classifier $C$, instead of between the
generator $G$ and the classifier $C$. This is because we are concerned that an
adversarial game between $G$ and $C$ could result in unrealistic outputs of
$G$. In this section, we perform an experiment to investigate this.

Specifically, we define an optimization function:

$L_{G}=\max_{G}\operatorname{\mathbb{E}}_{\mathbf{x}\sim
X_{train},\mathbf{y}\sim Y_{train}}L_{s}(C(G(\mathbf{x},a)),y),$ (6)

where we aim to train $G$ in the direction of maximising the loss of the
classifier $C$ on the synthetic data $G(\mathbf{x},a)$.

After every update of $G$, we construct a synthetic set $D_{syn}$ by
generating 100 synthetic images from $D_{train}$, and update $C$ on
$D_{train}\cup D_{syn}$ via Equation 5. The adversarial game $G$ vs. $C$ is
formulated by alternatively optimising Equation 6 and 5 for 10 epochs.

Figure 2: The synthetic results for a healthy (CN) subject $x$ at age 70: (a)
the results of the pre-trained $G$, i.e. before we train $G$ against $C$; (b)
the results of $G$ after we train $G$ against $C$. We synthesise aged images
$\hat{x}$ at different target ages $a$. We also visualise the difference
between $x$ and $\hat{x}$, $|\hat{x}-x|$. For more details see text.

In Figure 2, we present the synthetic brain ageing progression of a CN subject
before and after the adversarial training of $G$ vs. $C$. We can observe that
after the adversarial training, the generator $G$ produces unrealistic
results. This could be because there is no loss or constraint to prevent the
generator $G$ from producing low-quality results. The adversarial game only
requires the generator $G$ to produce images that are hard for the classifier
$C$, and naturally, images of low quality would be hard for $C$. A potential
solution could be to involve a GAN loss with a discriminator to improve the
output quality, but this would make the training much more complex and require
more memory and computations. We also measure the test accuracy of the
classifier $C$ after training $G$ against $C$ to be $81.6\%$, which is much
lower than the Naïve method ($88.4\%$) and our approach ($91.1\%$) in Table 1.
The potential reason is that $C$ is misled by the unrealistic samples
generated by $G$.

####  5.1.3 Effect of conventional augmentations for registered brain MRI
data.

In this section, we test the effect of applying conventional augmentations,
e.g. rotation, shift, scale and flip, to the training of the AD classifier.
These are typical conventional augmentation techniques applied to computer
vision classification task. Specifically, we train the classifier the same way
as Naïve, except we augment training data with conventional augmentations.

Interestingly, we find that after applying rotation (range 10 degrees), shift
(range 0.2), scale (range 0.2), and flip to augment the training data, the
accuracy of the trained classifier drops from $88.4\%$ to $71.6\%$. We then
measure accuracies when trained with each augmentation to be $74.1\%$
(rotation), $87.1\%$ (shift), $82.9\%$ (scale), and $87.8\%$ (flip). This
could be because both training and testing data are already pre-processed,
including registered to MNI 152, and these conventional augmentations do not
introduce helpful variations to the training data but distract the classifier
from focusing on subtle differences between AD and CN brains.

We also tried to train the classifier with MaxUp (Gong et al., 2021a) with
conventional augmentations. The idea of MaxUp is to generate a small batch of
augmented samples for each training sample and train the classifier on the
worst-performance augmented sample. The overall test accuracy is $57.7\%$.
This could be because that MaxUp tends to select the augmentations that
distract the classifier from focusing on subtle AD features the most.

The results with conventional aumentations (+MaxUp) suggest that for the task
of AD classification, when training and testing data are pre-processed well,
conventional data augmentation techniques seem to not help improve the
classification performance. Instead, these augmentations distract the
classifier from identifying subtle changes between CN and AD brains. By
contrast, the proposed procedure augment data in terms of semantic
information, which could alleviate data imbalance and improve classification
performance.

###  5.2 Adversarial counterfactual augmentation in a continual learning
context

####  5.2.1 Results when re-training with a portion ($M\%$) of training data

Suppose we have a pre-trained classifier $C$ and a pre-trained generator $G$,
and we want to improve $C$ by using $G$ for data augmentation. However, after
pre-training, we only store $M\%$ ($M\in(0,100]$) of the training dataset,
denoted as $D_{store}$. During the adversarial training, we synthesise $N$
samples using the generator $G$, denoted as $D_{syn}$. Then we update the
classifier $C$ on $D_{store}\cup D_{syn}$, using Equation 5 where
$D_{combined}=D_{store}\cup D_{syn}$. The target ages are initialised and
updated the same way as in Section 5.1. Algorithm 2 illustrates the procedure
in this section.

Table 3 presents the test accuracies of our approach and baselines when $M$
changes. For Naïve-100, the results are then same as in Table 1. For JTT, the
original paper Liu et al. (2021) retrained the classifier using the whole
training set. Here we first randomly select $M\%$ training samples as
$D_{store}$ and find misclassified data $D_{mis}$ within $D_{store}$ to up-
sample, then we retrain the classifier on the augmented set. We can observe
that when $M$ decreases, catastrophic forgetting happens for all approaches.
However, our method suffers the least from catastrophic forgetting, especially
when $M$ is small. With $M=20\%$ of training data for retraining, our approach
achieves better results than Naïve. This might be because the adversarial
training between $a$ and $C$ tries to detect what is missing in $D_{store}$
and tries to recover the missing data by updating $a$ towards those
directions. We observe that RSAT achieves the second best results, only
slightly worse than the proposed approach. Moreover, HSRS and JTT are more
affected by catastrophic forgetting and achieve worse results. This might be
because the importance of selecting hard samples declines as $M$ decreases,
since the $D_{store}$ becomes smaller.

Table 3: Test accuracies of our approach and baselines when the ratio of the size $D_{store}$ vs. the size of $D_{train}$ changes. We can observe the decreases of test accuracies when $M$ decreases, which was due to the effect of catastrophic forgetting.  Acc % | $M\%$
---|---
Methods | 1 | 10 | 20 | 50 | 100
Naïve | N/A | N/A | N/A | N/A | 88.4
HSRS | 75.6 | 81.4 | 84.5 | 87.4 | 89.5
RSAT | 84.2 | 85.8 | 87.2 | 88.6 | 90.6
JTT | 77.3 | 82.3 | 85.1 | 88.1 | 89.2
Proposed | 84.8 | 86.8 | 88.5 | 89.4 | 91.1

These results demonstrate that our approach could alleviate catastrophic
forgetting. This could be helpful in cases where we want to utilise generative
models to improve pre-trained classifiers (or other task models) without
revisiting all the training data (a continual learning context).

####  5.2.2 Results when number of samples used for synthesis ($N$) changes

We also performed experiments where we changed $N$, i.e. the number of samples
used for generating counterfactuals. Specifically, we set $M=1$, i.e. only
$1\%$ of original training data are used for re-training $C$, to see how many
synthetic samples are needed to maintain good accuracy, especially when there
are only a few training data stored in $D_{store}$. This is to see how
efficient the synthetic samples are in terms of training $C$ and alleviating
catastrophic forgetting. The results are presented in Table 4.

Table 4: Test accuracies when $N$ changes ($M=1$) of our approach and baselines. acc % | N
---|---
Methods | 1 | 10 | 50 | 100
HSRS | 65.4 | 71.0 | 73.4 | 75.6
RSAT | 81.3 | 82.1 | 83.2 | 84.2
Proposed | 82.1 | 82.9 | 84.1 | 84.6

From Table 4, we can observe that the best results are achieved by our method,
followed by RSAT. Even with only one sample for synthesis, our method could
still achieve a test accuracy of $80\%$. This is probably because the
adversarial training of $a$ vs. $C$ guides $G$ to generate hard
counterfactuals, which are efficient to train the classifier. The results
demonstrate that our approach could help alleviate catastrophic forgetting
even with a small number of synthetic samples used for augmentation. This
experiment could also be viewed as a measurement of the sample efficiency,
i.e. how efficient a synthetic sample is in terms of re-training a classifier.

###  5.3 Can the proposed procedure alleviate spurious correlations?

Spurious correlation occurs when two factors appear to be correlated to each
other but in fact they are not (Simon, 1954). Spurious correlation could
affect the performance of deep neural networks and has been actively studied
in computer vision field (Sagawa et al., 2020; Liu et al., 2021; Sagawa et
al., 2020, 2019; Youbi Idrissi et al., 2021; Goel et al., 2021) and in medical
imaging analysis field (Mahmood et al., 2021; DeGrave et al., 2021). For
instance, suppose we have an dataset of bird and bat photos. For bird photos,
most backgrounds are sky. For bat photos, most backgrounds are cave. If a
classifier learns this spurious correlation, e.g. it classifies a photo as
bird as long as the background is sky, then it will perform poorly on images
where bats are flying in the sky. In this section, we investigate if our
approach could correct such spurious correlations by changing $a$ to generate
hard counterfactuals.

Here we create a dataset where 7860 images between 60 and 75 yrs old are AD,
and 7680 images between 75 and 90 yrs old are healthy, denoted as
$D_{spurious}$. This is to construct a spurious correlation: $young\rightarrow
AD$ and $old\rightarrow CN$ (in reality older people have higher chances of
getting AD (Goedert and Spillantini, 2006)). Then we pre-train $C$ on
$D_{spurious}$. The brain ageing model proposed in Xia et al. (2021) only
considered simulating ageing process, but did not consider brain rejuvenation,
i.e., the reverse of ageing. To utilise old CN data, we pre-train another
generator in the rejuvenation direction, i.e.,generating younger brain images
from old ones. As a result, we obtain two generators that are pre-trained on
$D_{train}$, denoted as $G_{ageing}$ and $G_{rejuve}$, where $G_{rejuve}$ is
trained to simulate the rejuvenation process. Fig. S2 in Supplementary
material shows visual results of $G_{rejuve}$. After that, we select 50 CN and
50 AD hard images from $D_{spurious}$, denoted as $D_{hard}$ and perform the
adversarial classification training using $G_{rejuve}$ for old CN samples and
$G_{ageing}$ for young AD samples. The target ages $a$ are initialized as real
ages of $\mathbf{x}$.

Figure 3: Example results of brain rejuvenation for an image ($x$) of a 85
year old CN subject. We synthesise rejuvenated images $\hat{x}$ at different
target ages $a$. We also show the differences between $\hat{x}$ and $x$,
$\hat{x}-x$. For more details see text.

After we obtain $G_{ageing}$ and $G_{rejuvenation}$, we select 50 CN and 50 AD
images from $D_{spurious}$ that result in highest training errors, denoted as
$D_{hard}$. Note that the selected CN images are between 75 and 90 yrs old,
and the AD images are between 60 and 75 yrs old. Then we generate synthetic
images from $D_{hard}$ using $G_{rejuvenation}$ for old CN samples and
$G_{ageing}$ for young AD samples. The target ages $a$ are initialized as
their ground-truth ages. Finally, we perform the adversarial training between
$a$ and the classifier $C$. Here we want to see if the adversarial training
can detect the spurious correlations purposely created by us, and more
importantly, we want to see if the adversarial training between $a$ and $C$
can break the spurious correlations.

Table 5 presents the test accuracies of our approach and baselines. For Naïve,
we directly use the classifier $C$ pre-trained on $D_{spurious}$. For HSRS, we
randomly generate synthetic samples from $D_{hard}$ for augmentation. For JTT,
we simply select mis-classified samples from $D_{spurious}$ and up-sample
these samples.

Table 5: Test accuracies for our procedure and baselines when $C$ pre-trained on $D_{spurious}$. We first present the average test accuracies for different age groups with CN diagnosis (column 2-3) or AD (column 4-5), and then present the average test accuracies for the whole testing set (column 6). For each method, the worst-group performance is shown in italic. For each age group, i.e. each column, the best performance was shown in bold. For more details see text.  Acc % | CN | AD |
---|---|---|---
Methods | 60-75yrs | 75-90yrs | 60-75yrs | 75-90yrs | Overall
Naïve | 40.9 | 81.6 | 95.1 | 45.7 | 67.0
HSRS | 60.7 | 85.3 | 81.1 | 67.2 | 75.0
JTT | 50.5 | 88.4 | 85.5 | 40.7 | 67.9
proposed | 73.1 | 83.4 | 81.5 | 75.8 | 79.0

We can observe from Table 5 that the pre-trained $C$ on $D_{spurious}$ (Naïve)
achieves much worse performance ($67.0\%$ accuracy) compared to that of Table
1 ($88.4\%$ accuracy). Specifically, it tends to misclassify young CN images
as AD and misclassify old AD images as CN. This is likely due to the spurious
correlations that we purposely create in $D_{spurious}$: $young\rightarrow AD$
and $old\rightarrow CN$. We notice that for Naïve, the test accuracies of AD
groups are higher than that of CN groups. This is likely due to the fact we
have more AD training data, and the classifier is biased to classify a subject
to AD. This can be viewed as another spurious correlation. Overall, we observe
that our method achieves the best results, followed by HSRS. This shows that
the synthetic results generated by the generators are helpful to alleviate the
effect of spurious correlations and improve downstream tasks. The improvement
of our approach over HSRS is due to the adversarial training between $a$ and
$C$, which guides the generator to produce hard counterfactuals. We observe
JTT does not improve the test accuracies significantly. A potential reason is
that JTT tries to find ‘hard’ samples in the training dataset. However, in
this experiment, the ‘hard’ samples should be young CN and old AD samples
which do not exist in the training dataset $D_{spurious}$. By contrast, our
procedure could guide $G$ to generate these samples, and HSRS could create
these samples by random chance.

Figure 4 plots the histograms of the target ages $a$ before and after the
adversarial training. From Figure 4 we can observe that the adversarial
training pushes $a$ towards the hard direction, which could alleviate the
spurious correlations. For instance, in $D_{spurious}$ and $D_{hard}$ the AD
subjects are all in the young group, i.e. 60-75 yrs old, and the classifier
learns the spurious correlation: $young\rightarrow AD$, but in Figure 4 (a) we
can observe that the adversarial training learns to generate AD synthetic
images in the range of 75-90 yrs old. These old AD synthetic images can help
alleviate the spurious correlation and improve the performance of $C$.
Similarly, we can observe $a$ are pushed towards young for CN subjects in
Figure 4 (b).

Figure 4: Histograms of target ages $a$ before and after adversarial training:
(a) the histogram of $a$ for the 50 AD subjects in $D_{hard}$; (b) the
histogram of $a$ for the 50 CN subjects in $D_{hard}$. Here we show histograms
of $a$ before (in orange) and after (in blue) the adversarial training.

##  6 Conclusion

We presented a novel adversarial counterfactual scheme to utilise conditional
generative models for downstream tasks, e.g. classification. The proposed
procedure formulates an adversarial game between the conditional factor of a
pre-trained generative model and the downstream classifier. We improved a
recent brain ageing synthesis model as the generative model and focused on AD
classification. We presented quantitative results showing that our approach
can improve the performance of the AD classifier. Furthermore, we showed that
our method could alleviate the effect of catastrophic forgetting and spurious
correlations. There are several future directions. The proposed approach can
be potentially applied to other generators on other datasets. The way we
updated the conditional factor (target age) could be improved. Instead of a
continuous scalar (target age), we can consider extending the proposed
adversarial counterfactual augmentation to update other types of conditional
factors, e.g., discrete factor or image. The strategy that we used to select
hard samples may not be the most effective and could be improved.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

TX, PS, CQ and SAT contributed to the conceptualization of this work. TX, PS,
CQ and SAT designed the methodology. TX developed the software tools necessary
for preprocessing and analysing images files and for training the model. TX
drafted this manuscript. All authors reviewed the manuscript.

## Acknowledgments

This work was supported by the University of Edinburgh, the Royal Academy of
Engineering and Canon Medical Research Europe via PhD studentships of Pedro
Sanchez (grant RCSRF1819/825). This work was partially supported by the Alan
Turing Institute under the EPSRC grant EP//N510129//1. S.A. Tsaftaris
acknowledges the support of Canon Medical and the Royal Academy of Engineering
and the Research Chairs and Senior Research Fellowships scheme (grant
RCSRF1819/8/25).

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