# MCDAL: Maximum Classifier Discrepancy for Active Learning

Jae Won Cho∗, Student Member, IEEE, Dong-Jin Kim∗, Member, IEEE,
Yunjae Jung, Student Member, IEEE, In So Kweon, Member, IEEE ∗ indicates equal
contributionJae Won Cho, Yunjae Jung, and In So Kweon are with the Department
of Electrical Engineering, Korea Advanced Institute of Science and Technology,
Daejeon, Republic of Korea. (E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>iskweon77@kaist.ac.kr})Dong-Jin Kim is with the Electrical Engineering and
Computer Science Department, University of California, Berkeley, CA. (E-mail:
<EMAIL_ADDRESS>
###### Abstract

Recent state-of-the-art active learning methods have mostly leveraged
Generative Adversarial Networks (GAN) for sample acquisition; however, GAN is
usually known to suffer from instability and sensitivity to hyper-parameters.
In contrast to these methods, we propose in this paper a novel active learning
framework that we call Maximum Classifier Discrepancy for Active Learning
(MCDAL) which takes the prediction discrepancies between multiple classifiers.
In particular, we utilize two auxiliary classification layers that learn
tighter decision boundaries by maximizing the discrepancies among them.
Intuitively, the discrepancies in the auxiliary classification layers’
predictions indicate the uncertainty in the prediction. In this regard, we
propose a novel method to leverage the classifier discrepancies for the
acquisition function for active learning. We also provide an interpretation of
our idea in relation to existing GAN based active learning methods and domain
adaptation frameworks. Moreover, we empirically demonstrate the utility of our
approach where the performance of our approach exceeds the state-of-the-art
methods on several image classification and semantic segmentation datasets in
active learning setups.111Our code is available at
https://github.com/chojw/Maximum-Classifier-Discrepancy-for-Active-Learning

###### Index Terms:

Deep learning, active learning, visual recognition, classifier discrepancy,
data issues.

##  I Introduction

In recent years, deep learning has made great advancements with the help of
large labeled datasets [1] that have been handcrafted and labeled through
various expensive methods [2, 3]. The cost of labeling and procuring such
datasets have frequently pushed many researchers to find ways to minimize the
need for methods to train powerful deep learning models that require fewer
labels such as semi-supervised learning [4], few-shot learning [5], or active
learning [6]. Active learning explores a setting where there is a large
dataset, but the budget to annotate the dataset is limited. In this setting, a
small set of labeled samples is first used to train a model, then this trained
model is used to decide which of the unlabeled samples would give the largest
performance gain if given labels to. In recent years, active learning has been
shown to be a practical approach to achieving high performance with relatively
fewer samples in computer vision tasks such as image classification [7, 8, 9],
semantic segmentation [10, 11, 12], and hyperspectral image classification
[13, 14].

Figure 1: Overall architecture of our model. $G(\cdot)$ shows the “Generator”
or the task model backbone, $F(\cdot)$ shows the “main classifier”
(classification layer). $F\circ G(\cdot)$ represents off-the-shelf deep neural
networks, and $F_{1}(\cdot)$ and $F_{2}(\cdot)$ are our auxiliary
classification layers. In the proposed method MCDAL, we propose to collect
unlabeled samples that shows the highest classifier discrepancy to add to a
labeled pool.

Most recent works in active learning have started to use a Generative
Adversarial Network [15] (GAN) based method where a Variational Auto-Encoder
(VAE) [16] is used to train latent space representations of both the labeled
and unlabeled datasets. The VAE learns to fool a discriminator that both the
labeled and unlabeled datasets are within the same pool. This discriminator is
then used to measure how likely a sample is within the labeled set during the
sampling stage, and samples that are not well represented in the labeled set
are given labels to [17, 18, 19]. Although these methods have shown promising
performance, adding an auxiliary GAN based network to the existing task models
is heavy and unstable [20, 21]. In addition, adversarial training is shown to
be heavily sensitive to hyper-parameter choices between the discriminator and
the generator [22].

In order to tackle this issue, we propose a novel minimally intrusive and easy
to implement method for training and sampling with less additional model
parameters for active learning. In this paper, we introduce a novel ad hoc
classifier _discrepancy_ based method for active learning that we call Maximum
Classifier Discrepancy for Active Learning (MCDAL). To make our model as
unobtrusive as possible, we leave the task model alone and add two new
identical classification layers that are exactly the same as the task model’s,
making our model easy to implement. Note that these auxiliary classification
layers do not effect the training of the task model (not a regularizer) as the
classification branches are detached from the task model and are removed after
the training process. In addition, as our method creates identical layers, our
method can be readily applied to other tasks and datasets. Then, using both
the labeled and unlabeled dataset, we train the additional layers using both
labeled cross-entropy loss and discrepancy loss in order for the ad hoc layers
to mimic the “main” classification layer and to have a tighter decision
boundary at the same time.

After training our model in this manner, we leverage these auxiliary layers
during the active learning sampling stage with our novel sampling method.
Because the classification layers are trained to have tighter decision
boundaries, intuitively, the samples that give the _largest discrepancy_ from
the outputs of all the classifiers would be both uncertain and far from the
labeled distribution. Thus, we hypothesize that labeling these samples would
give the greatest performance gain at each sampling stage. To the best of our
knowledge, our work is the first work that leverages classifier discrepancy
for sampling in active learning.

Our contribution in this work can be summarized as follows: (1) We propose a
novel active learning framework named Maximum Classifier Discrepancy for
Active Learning (MCDAL) which is lightweight, fast, and easy to implement
compared to the recent GAN based active learning methods. (2) We provide an
interpretation in several perspectives to understand the connection of MCDAL
with GAN based active learning methods and domain adaptation frameworks. (3)
Extensive experiments and analysis consistently show the performance
surpassing recent state-of-the-art active learning works with a noticeable
margin.

##  II Related Work

Active leaning for deep vision. Active learning has been widely studied with
several different approaches being introduced in recent years to tackle the
data issue. Several methods have been proposed to tackle the data issue such
as Bayesian Dropout [23] and interactive learning [24, 25, 26], however in
this work we focus on the task of active learning. Classical active learning
approaches either use pool-based or query synthesis methods. In query
synthesizing approaches, generative models are used to find the most
informative samples [27, 28, 29]. In pool-based methods, there are also
several categories: uncertainty-based [7, 30, 31, 32, 33, 34], representation-
based [9], and more recently a combination of these two [17, 19] and our work
belongs here.

In recent years, pool-based active learning has been successfully applied in
many deep vision tasks such as image recognition [9, 34], object detection
[35], and semantic segmentation [36]. Meanwhile, theoretical dropout-based
frameworks have also been used to measure uncertainty [8, 37]. More recently,
works have started to use latent space representations with the aid of
Variational Auto-Encoders (VAE) [16] in addition to adversarial training
methods [15] to find the informativeness of unlabeled samples [17, 18, 19].
Although these methods have shown to be promising, we choose to turn our
attention away from Generative Adversarial Network (GAN) [15] based methods
due to the instability and complexity in training and increase in network
parameters. Instead, we focus on classifier discrepancy based training methods
which introduce little addition parameters to the network. In the world of
using additional classifiers, works such as such as [38] incorporate
additional classifiers for fine-grained classification.

Unlike ensemble based method [7, 39], where [7] uses several randomly
initialized models for ensembling while [39] uses different dropout masks from
a single _model_ , creating additional computation time as several forward
passes are required, we share a backbone model and add auxiliary fully
connected _layers_ , while introducing significantly less parameters, being
extremely lightweight, and surpassing the state-of-the-art performance.

Maximum classifier discrepancy in domain adaptation. The goal of domain
adaptation is to improve the testing performance on an unlabeled target domain
while a model is trained on a related yet different source domain. Recent GAN
[15] based methods [40, 41, 42, 43] have leveraged a discriminator to
distinguish source and target domain by trying to reduce the discrepancy of
the distributions based on the theory proposed by [44]. However, distribution
aligning methods using GAN do not consider the relationship between target
samples and _decision boundaries_. To tackle these problems, Saito et al. [45]
proposed an approach to directly use task-specific classifiers as a
discriminator. In order to achieve this, the authors implement a two-
classifier system with discrepancy loss to maximize the disagreement between
the classifiers, leading to tighter decision boundaries. Also, they train a
generator to minimize this disagreement to align the source and target
features given the tight decision boundaries. The idea of maximizing the
classifier discrepancy is utilized in other tasks that require distribution
alignment such as out-of-distribution detection [46].

Similarly, we tackle the problem of existing GAN based active learning
approaches and propose a novel classifier discrepancy based active learning
method which only requires task-specific classification layers. Motivated by
the goal of domain adaptation, we propose to select unlabeled samples that
show the largest discrepancy when compared to the distribution of the labeled
dataset in the sample acquisition stage. Unlike [45], we introduce two
auxiliary classification layers alongside the main classifier without training
the generator. Instead, in the sample acquisition stage, we select the
unlabeled samples that have maximum classifier discrepancy. Although [47]
tried to leverage classifier discrepancy maximization as a semi-supervised
learning technique instead of active learning, we propose a novel active
learning method to directly leverage classifier discrepancy.

##  III Proposed Method

In this section, we introduce our method, Maximum Classifier Discrepancy for
Active Learning (MCDAL) in detail and the methods for both training and
sampling. For clarity, we first show Table I with the list of notations used
in this work.

TABLE I: Notations and their descriptions used throughout this work.

Notation | Description
---|---
$\mathcal{D}\textsubscript{L}=\\{x\\}$ | Labeled set
$\mathcal{D}\textsubscript{U}=\\{(x,y)\\}$ | Unlabeled set
$C$ | Total number of classes
$F(\cdot)$ | Backbone architecture
$G(\cdot)$ | Main classification layer
$\theta$ | Parameter for $F\circ G(\cdot)$
$F_{1}(\cdot)$ & $F_{2}(\cdot)$ | Auxiliary classification layers with parameter $\theta_{1}$ & $\theta_{2}$
$p(y|x)$ | Output probability prediction from the main classifiers
$p_{1}(y|x)$ & $p_{2}(y|x)$ | Probability prediction from the auxiliary classifiers
$S(\cdot)$ | Acquisition function
$D(x)$ | Total discrepancy metric given an image
$d(\cdot,\cdot)$ | L1 distance
$\eta$ | Learning rate

###  III-A Problem Definition : Active Learning

Given an image $x$, the general aim of active learning is finding the best
sample acquisition function $S(x)$ that assigns the highest scores to the most
_informative_ samples, which can be seen as samples that increase the
performance gain the most when labeled, given a fixed backbone architecture.
In this setup, a labeled set $\mathcal{D}\textsubscript{L}$ is first used to
train a model. After the model converges, the model is used to sample from the
unlabeled set $\mathcal{D}\textsubscript{U}$. Finally, the chosen samples are
moved from the unlabeled set $\mathcal{D}\textsubscript{U}$ to the labeled set
$\mathcal{D}\textsubscript{L}$ and are given labels. This step (otherwise
called a _stage_) is repeated several times depending on the experimental
setup. Recently, Generative Adversarial Networks (GAN) based methods [17, 19]
have shown state-of-the-art active learning performances by finding the
unlabeled samples that has the largest latent space gap with the labeled
sample distribution. However, GAN based methods usually suffer from difficulty
in training, such as instability or sensitivity to hyper-parameters between
the discriminator and the generator [20, 22, 21]. In light of this, we define
a novel acquisition function that takes the _discrepancy_ of the classifiers
into account. Our approach utilizes multiple auxiliary classification layers
and measures the discrepancy between the outputs of the classification layers.

###  III-B Network Architecture

Let us denote $G(\cdot)$ as the network backbone that generates features from
an input $x$. Then, with the “main” classification layer $F(\cdot)$, a class
prediction is computed ($p(y|x)=F\circ G(x)\in\mathbb{R}^{C}$ where $C$ is the
total number of classes). The combined network $F\circ G(\cdot)$ can be
implemented with typical image classifiers [2, 3]. The main model is trained
with cross-entropy loss with the ground truth label $y$:

$\begin{split}&\min_{F,G}\mathcal{L}_{CE}(F,G),\text{ where }\\\
&\mathcal{L}_{CE}=\mathop{\mathbb{E}}_{(x,y)\in\mathcal{D}\textsubscript{L}}\bigg{[}-\sum_{c=1}^{C}\mathbbm{1}[c=y]\log
p^{c}(y|x)\bigg{]},\end{split}$ (1)

where $p^{c}(y|x)$ denotes a probability element of $p$ for class $c$ and
$\mathbbm{1}[a]$ is the indicator function, which is 1 if predicate $a$ is
true and 0 otherwise.

In order to measure the classifier discrepancy, we introduce two additional
classification layers $F_{1}(\cdot)$ and $F_{2}(\cdot)$. The extra layers
introduced share the identical shape as $F(\cdot)$, meaning that our method
can be easily applied to any network and any task. In addition, note that we
detach the output from $G(\cdot)$ before we pass it through $F_{1}(\cdot)$ and
$F_{2}(\cdot)$ in order to not affect the task-model and to objectively test
our method. Although these classification layers are still trained with the
typical cross-entropy loss $\mathcal{L}_{CE}$, we do not consider the
performance of the auxiliary classifiers as their main goal is not to classify
for the task correctly, but to learn tighter decision boundaries for sample
acquisition.

Input: Labeled pool $\mathcal{D}\textsubscript{L}$, and unlabeled pool
$\mathcal{D}\textsubscript{U}$, Initialized models where $F\circ
G(\cdot),F_{1}(\cdot),\text{ and }F_{2}(\cdot)$ with parameters
$\theta$,$\theta_{1}$, and $\theta_{2}$ respectively

Input: Hyper-parameters : learning rate $\eta$, maximum number of epochs
$MaxEpoch$

Output: $F\circ G(\cdot),F_{1}(\cdot),\text{ and }F_{2}(\cdot)$ with trained
$\theta$,$\theta_{1}$, and $\theta_{2}$

begin

for  _$e=1$ to MaxEpoch_ do

sample $(x_{l},y_{l})\in\mathcal{D}\textsubscript{L}$;

Compute $\mathcal{L}_{CE}$;

Update the model parameters:

$\theta\leftarrow\theta-\eta\nabla\mathcal{L}_{CE}$

$\theta_{1}\leftarrow\theta_{1}-\eta\nabla\mathcal{L}_{CE}$

$\theta_{2}\leftarrow\theta_{2}-\eta\nabla\mathcal{L}_{CE}$

sample $(x_{u})\in\mathcal{D}\textsubscript{U}$;

Compute $\mathcal{L}_{dis}$;

Update the model parameters:

$\theta_{1}\leftarrow\theta_{1}+\eta\nabla\mathcal{L}_{dis}$

$\theta_{2}\leftarrow\theta_{2}+\eta\nabla\mathcal{L}_{dis}$

end for

end

Algorithm 1 Training MCDAL

###  III-C Training with Discrepancy Losses

Let us denote $p_{1}$ and $p_{2}$ as the output probability computed from the
auxiliary classification layers $F_{1}(\cdot)$ and $F_{2}(\cdot)$,
$p_{1}(y|x)=F_{1}\circ G(x)\in\mathbb{R}^{C}$, $p_{2}(y|x)=F_{2}\circ
G(x)\in\mathbb{R}^{C}$. The auxiliary layers $F_{1}(\cdot)$ and $F_{2}(\cdot)$
are trained with the following objective:

$\min_{F_{1},F_{2}}\mathcal{L}_{CE}(F_{1},G)+\mathcal{L}_{CE}(F_{2},G)-\mathcal{L}_{dis}(F_{1},F_{2},G).$
(2)

The objective function consists of the cross-entropy loss and the additional
discrepancy loss which is defined as follows:

$\mathcal{L}_{dis}=\mathop{\mathbb{E}}_{x\in\mathcal{D}\textsubscript{U}}\bigg{[}d(p_{1},p)+d(p_{2},p)+d(p_{1},p_{2})\bigg{]},$
(3)

which is the sum of the discrepancy values between all pairs of probabilities.

As a discrepancy measure, we utilize the absolute values of the difference
between the classifiers’ output probabilities (L1 distance between the
probability vectors):

$d(p_{1},p_{2})=\frac{1}{C}\sum_{c=1}^{C}{\Big{|}p_{1}^{c}(y|x)-p_{2}^{c}(y|x)\Big{|}},$
(4)

where the $p_{1}^{c}(y|x)$ and $p_{2}^{c}(y|x)$ denote probability elements of
$p_{1}$ and $p_{2}$ for class $c$ respectively. The choice for L1-distance is
based on the theory proposed by [48]. We tried other distance or divergence
metrics such as L2 and KL-Divergence (KL) in the experiment in Fig. 8, and we
empirically find that L1 is the best option in our given setting. Unlike other
works that train a _separate_ feature generator [45, 46], note that we do not
train our feature generator $G(\cdot)$ with the discrepancy loss as the scope
of active learning is sampling and not training the task model with additional
methods. Since the discrepancy loss is computed with unlabeled samples,
including the discrepancy loss into the feature generator would result in an
unfair comparison to other active learning methods as this can be viewed as
semi-supervised training. We show our pseudo-code for training in Algorithm 1
for ease of understanding.

By training the auxiliary layers to maximize the discrepancy between their
respective outputs, the decision boundaries of the classifiers become tighter
as illustrated in Fig. 2, resulting in the region between the auxiliary
decision boundaries in the feature space to become larger. As the supervised
cross-entropy loss is also applied to train the auxiliary layers, we expect
most of the labeled samples to be located within the boundaries, whereas some
of the labeled samples would be located out of the boundaries (we call this
region the “sampling region”). In the sample acquisition stage, our key idea
is to collect the unlabeled samples that are located in this sampling region.
The detailed acquisition process is depicted in the following subsection.

###  III-D Sampling with MCDAL

By maximizing the discrepancy between auxiliary classification layers, we
obtain two additional tight decision boundaries which means that we have a
large region between the two decision boundaries (we call this region the
“sampling region”). The unlabeled samples that are (1) difficult to train and
that are (2) far from the labeled data distribution will be located in this
“sampling region” which meets the condition that would be helpful if labeled.
In this regard, we again utilize the discrepancy between the auxiliary
classification layers by defining the sample acquisition function $S(\cdot)$
as $S(x_{u})=D(x_{u})$ for $x_{u}\in\mathcal{D}\textsubscript{U}$, where
$D(x)=d(p_{1},p_{2})$ (note $p_{1}$ and $p_{2}$ in this equation are
interchangeable with $p$ where needed). In order to further leverage the
difference between the labeled set and unlabeled set, our final acquisition
function compares the average classifier discrepancy of the labeled samples:

$S(x_{u})=\Bigg{|}D(x_{u})-\frac{1}{|\mathcal{D}\textsubscript{L}|}\sum_{x_{l}\in\mathcal{D}\textsubscript{L}}{D(x_{l})}\Bigg{|},\text{for
}x_{u}\in\mathcal{D}\textsubscript{U}.$ (5)

This in turn measures how large the classifier discrepancies for the given
unlabeled samples are compared to the average classifier discrepancies of the
labeled samples. For the final design of $S(\cdot)$, we use
$D(x)=d(p_{1},p_{2})+d(p,p_{1})+d(p,p_{2})$ to leverage the discrepancy
between all present classifiers and include the pseudo-code in Algorithm 2 for
ease of understanding. Although we consider using the discrepancy value
directly (i.e., $S(\cdot)=D(\cdot)$), we do not do so and explain in further
detail in the following subsection.

Figure 2: Illustration of training our auxiliary classification layers based
on the discrepancy in feature space. The two dotted red lines each are for
each of the auxiliary classifiers. (a) shows the early stages of the training
where all the classifiers are similar and (b) shows when after training with
the discrepancy loss (Eq. (3)), the classifiers are pushed apart to maximize
discrepancy and forms a space that we to consider the “sampling region.”

Input: Labeled pool $\mathcal{D}\textsubscript{L}$, Unlabeled pool
$\mathcal{D}\textsubscript{U}$, Number of samples collecting for each stage
$b$, Models $F\circ G(\cdot),F_{1}(\cdot),\text{ and }F_{2}(\cdot)$

Output: Updated $\mathcal{D}\textsubscript{L}$ and
$\mathcal{D}\textsubscript{U}$

begin

From $\mathcal{D}\textsubscript{U}$, collect a set of samples $\\{x_{s}\\}$
with
$\max_{b}S(x_{u})=\max_{b}{\Bigg{|}D(x_{u})-\frac{1}{|\mathcal{D}_{L}|}\sum_{x_{l}\in\mathcal{D}_{L}}{D(x_{l})}\Bigg{|}},$
where $D(x)=d(p_{1},p_{2})+d(p,p_{1})+d(p,p_{2}),$ where
$d(p_{1},p_{2})=\frac{1}{C}\sum_{c=1}^{C}{|p_{1}^{c}(y|x)-p_{2}^{c}(y|x)|}$;
$\\{(x_{s},y_{s})\\}=ORACLE(\\{x_{s}\\})$;

$\mathcal{D}\textsubscript{L}\leftarrow\mathcal{D}\textsubscript{L}\cup\\{(x_{s},y_{s})\\}$;

$\mathcal{D}\textsubscript{U}\leftarrow\mathcal{D}\textsubscript{U}-\\{x_{s}\\}$;

end

Algorithm 2 Sampling with MCDAL

###  III-E Discussion

Connection with GAN based approaches. The goal of GAN based active learning
works [17, 19] is to train a discriminator to distinguish labeled and
unlabeled samples so that they can collect the unlabeled samples that are the
furthest from (not well represented in) the distribution of the labeled data.
However, these approaches do not explicitly consider the class-wise behavior
of the samples. Sinha et al. [17] does not consider the model’s final output,
and Zhang et al. [19] only adds output uncertainty into the GAN model, but
they also do not leverage class-aware behavior based in their uncertainty
measure. In our case, what we sample in the acquisition stage are the
unlabeled samples that are both uncertain (as the uncertainty is large in the
“sampling region”) and far from the distribution of the labeled data (as the
labeled samples are trained to be outside the “sampling region”). Therefore,
our approach has a more general effect in comparison to the existing GAN based
active learning approaches.

Domain adaptation perspective. As classifier discrepancy has been used in
several domain adaptation research [45], our approach can also be interpreted
in the domain adaptation perspective. In particular, the labeled and unlabeled
data pools in active learning can be seen as source and target domain in
domain adaptation, respectively.

Since the classifier discrepancy based domain adaptation works [45] are
motivated by the theory in domain adaptation proposed by Ben-David et al.
[48], we show the relationship between our method and the theory in this
section. Ben-David et al. [48] proposed the theory that bounds the expected
error on the target samples, $R_{\mathcal{T}}(h)$, by using three terms: (1)
expected error on the source domain, $R_{\mathcal{S}}(h)$; (2)
$\mathcal{H}\Delta\mathcal{H}$-distance
($d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{S},\mathcal{T})$), which is
measured as the discrepancy between two classifiers; and (3) the shared error
of the ideal joint hypothesis, $\lambda$. $\mathcal{S}$ and $\mathcal{T}$
denote source and target datasets respectively. The theory can be explained as
follows: let $\mathcal{H}$ be the hypothesis class. Given two domains S and T,
we have

$\centering\begin{split}\forall h\in\mathcal{H},R_{\mathcal{T}}(h)&\leq
R_{\mathcal{S}}(h)+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{S},\mathcal{T})+\lambda,\\\
\end{split}\@add@centering$ (6)

where $d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{S},\mathcal{T})$ is defined
as:

$2\sup_{(h,h^{\prime})\in\mathcal{H}^{2}}\Bigg{|}\mathop{\mathbb{E}}_{x\in\mathcal{S}}\mathbbm{1}[h(x)\neq
h^{\prime}(x)]-\mathop{\mathbb{E}}_{x\in\mathcal{T}}\mathbbm{1}[h(x)\neq
h^{\prime}(x)]\Bigg{|}.$ (7)

Here, $\lambda=\min|R_{\mathcal{S}}(h)+R_{\mathcal{T}}(h)|$,
$R_{\mathcal{T}}(h)$ is the error of hypothesis $h$ on the target domain, and
$R_{\mathcal{S}}(h)$ is the corresponding error on the source domain.
$\lambda$ is a constant which is considered sufficiently low to achieve an
accurate adaptation. We will show the relationship between our method and
$\mathcal{H}\Delta\mathcal{H}$-distance.

In our case $h$ and $h^{\prime}$ share the feature extractor $G(\cdot)$, we
decompose the hypothesis $h$ into $G(\cdot)$ and $F_{1}(\cdot)$, and
$h^{\prime}$ into $G(\cdot)$ and $F_{2}(\cdot)$. $G(\cdot)$, $F_{1}(\cdot)$
and $F_{2}(\cdot)$ correspond to the network in our method. Substituting these
notations into the
$\sup_{(h,h^{\prime})\in\mathcal{H}^{2}}|\mathbb{E}_{x\in\mathcal{S}}\mathbbm{1}[h(x)\neq
h^{\prime}(x)]-\mathbb{E}_{x\in\mathcal{T}}\mathbbm{1}[h(x)\neq
h^{\prime}(x)]|$ and for fixed $G(\cdot)$, the term becomes:

$\begin{split}\sup_{F_{1},F_{2}}&\Bigg{|}\mathop{\mathbb{E}}_{x\in\mathcal{S}}\mathbbm{1}[F_{1}\circ
G(x)\neq F_{2}\circ G(x)]\\\
&-\mathop{\mathbb{E}}_{x\in\mathcal{T}}\mathbbm{1}[F_{1}\circ G(x)\neq
F_{2}\circ G(x)]\Bigg{|}.\end{split}$ (8)

Furthermore, if we replace the expectations
$\mathbb{E}_{X\in\mathcal{S}}[\cdot]$ and
$\mathbb{E}_{X\in\mathcal{T}}[\cdot]$ as the empirical means in the labeled
and unlabeled datasets $\mathcal{D}\textsubscript{L}$ and
$\mathcal{D}\textsubscript{U}$ respectively, and if we replace $\sup$ with
$\max$, we obtain:

$\begin{split}\max_{F_{1},F_{2}}&\Bigg{|}\frac{1}{|\mathcal{D}\textsubscript{L}|}\sum_{x_{l}\in\mathcal{D}\textsubscript{L}}{\mathbbm{1}[F_{1}\circ
G(x_{l})\neq F_{2}\circ G(x_{l})]}\\\
&-\frac{1}{|\mathcal{D}\textsubscript{U}|}\sum_{x_{u}\in\mathcal{D}\textsubscript{U}}{\mathbbm{1}[F_{1}\circ
G(x_{u})\neq F_{2}\circ G(x_{u})]}\Bigg{|},\end{split}$ (9)

which is similar to our final acquisition function in Eq. (5). Moreover, in
order to increase the tractability in training stage, we introduce further
assumptions. As $h$ and $h^{\prime}$ are expected to classify labeled samples
perfectly, we can assume the term
$\frac{1}{|\mathcal{D}\textsubscript{L}|}\sum_{x_{l}\in\mathcal{D}\textsubscript{L}}{\mathbbm{1}[F_{1}\circ
G(x_{l})\neq F_{2}\circ G(x_{l})]}$ to be close to zero. In other words, $h$
and $h^{\prime}$ should agree on their predictions on labeled samples. Thus,
Eq. (9) can be approximated as:

$\centering\max_{F_{1},F_{2}}\frac{1}{|\mathcal{D}\textsubscript{U}|}\sum_{x_{u}\in\mathcal{D}\textsubscript{U}}{\mathbbm{1}[F_{1}\circ
G(x_{u})\neq F_{2}\circ G(x_{u})]}.\@add@centering$ (10)

This equation is similar to the optimization problem we solve in our
discrepancy loss in Eq. (3), where classification layers are trained to
maximize their discrepancy on unlabeled samples. Although we must train all
branches to minimize the classification loss on labeled training data, we can
see the connection to the theory proposed by [48].

While [45] additionally requires minimizing the equation similar to Eq. (10)
with respect to $G(\cdot)$, which makes the optimization unstable, we do not
minimize the approximated objective Eq. (10) because it is not fair to
optimize $G(\cdot)$ with the loss from unlabeled samples as previously
mentioned. Instead, through our acquisition function Eq. (5), we indirectly
minimize the non-approximated objective Eq. (9) by sampling the unlabeled
sample that gives a large difference of discrepancy with the labeled data in
the sample acquisition stage, and learn from the sample with supervised cross-
entropy loss (which corresponds to the formulation in our final acquisition
function in Eq. (5)).

Semi-supervised learning perspective. Our approach has similar motivation to
that of consistency based semi-supervised learning works [49, 4] which is to
utilize the difference between the multiple outputs when training with
unlabeled samples. The goal of consistency based semi-supervised learning is
to directly reducing the discrepancy between the model outputs with different
conditions (e.g., augmentations) for the unlabeled samples:

$\min_{F,G}\mathbb{E}_{x\in\mathcal{D}\textsubscript{U}}[d(p_{1},p_{2})].$
(11)

On the other hand, as our task is active learning, we do not apply additional
losses other than supervised cross-entropy loss for fairness of testing.
Instead, we find the unlabeled training samples that have large discrepancy
between the outputs from different branches so that we can exploit that
samples with a supervised loss. Therefore, we verify that our approach
theoretically in the perspective of well-explored semi-supervised learning
approaches.

##  IV Experiments

In this section, we explain the implementation details and the experiments
performed while providing detailed discussions of our results. We evaluate
MCDAL against various recent state-of-the-art methods in addition to a random
annotation oracle on the image classification and segmentation task. For our
active learning setup, we start with an initial pool of 10% of the training
set. At the sampling stage, the oracle annotates from the unlabeled set, and
each sampling stage increases the dataset by 5%, and we repeat this step until
the final stage where our labeled data consists of a total of 40% of the
training set. To verify the performance of our sampling algorithm, we use the
average of five runs and initialize the task model at every stage.

###  IV-A Image Classification

Datasets. For the image classification task, we test our method on the
classical image classification datasets CIFAR-10 [50], CIFAR-100 [50], and
Caltech-101 [51] following the footsteps of some recent works. CIFAR-10 and
CIFAR-100 consists of 50,000 training images and 10,000 test images. CIFAR-10
has 6,000 images per class while CIFAR-100 has 600 images per class.
Caltech-101 has a total of 9,146 images with 101 classes and 1 background
class. Each class in Caltech-101 has a different number of images ranging from
40 to 800 and we believe that this setup shows a more real-world-like setup.
These different distributions of datasets show the different situations of
active learning and how the different algorithms perform under these varying
dataset conditions.

Figure 3: Comparison with existing baselines on CIFAR-10. Our method, MCDAL
shows the best performance compared to the state-of-the-art methods in a
noticeable margin, especially in the late stages.

Comparison baselines. We compare MCDAL to various recent-state-of-the-art
approaches including Core-set [9], Learning Loss (LL) [34], VAAL [17], and
SRAAL [19]. Although DAAL [18] is the most recent work, the implementation
details in the paper for their VAE and discriminator is not clear enough to
implement with no official repository, in addition to the performance being
similar to SRAAL. Therefore, we do not compare to DAAL in this paper. We also
include random sampling as a baseline. As most of the baselines do not have an
official repository, we implement the baselines to the best of our ability
using the information present in the respective papers. We do not compare with
classical methods (MC-dropout [39], DBAL [8], Ensemble [7]) in this paper as
several previous works [17, 19] have shown that the classical method shown
similar or worse performance in comparison to the Random sampling approach.

Implementation details. Following recent active learning works such as [34,
19], we also adopt the Resnet-18 [2] architecture and use a publicly available
code 222https://github.com/weiaicunzai/pytorch-cifar100 as the base
architecture for all image classification tasks and use the same image
augmentation strategy from the aforementioned code. Since we use Resnet-18, we
use the same classification layer (final linear layer) for the respective
tasks. The features that pass through the auxiliary classification branches
are detached so as not to affect the task model. We use Stochastic Gradient
Descent as an optimizer for both the task model and auxiliary layers with the
same learning rate of $1\times 10^{-1}$ with a multi-step learning rate
scheduler with a decay of $2\times 10^{-1}$ at 30%, 60%, 80% of the total
training epochs. We train the CIFAR-10 and CIFAR-100 for 100 epochs and train
Caltech-101 for 50 epochs as it is a much smaller dataset. We use budget sizes
2500 for CIFAR-10 and CIFAR-100, and 450 for Caltech-101.

Figure 4: Comparison with existing baselines on CIFAR-100. Our method, MCDAL
shows the best performance compared to the state-of-the-art methods overall.

Performance on CIFAR-10. Fig. 3 shows the performance of MCDAL in comparison
to recent state-of-the-art methods. Among the baseline methods, SRAAL shows
the highest performance in the early stage while LL shows the best performance
in the late stage. However, our method, MCDAL shows the best performance
compared to the state-of-the-art methods in all the stages with a noticeable
margin, especially in the late stages. In our setting, using 100% of the
labeled dataset gives 94.51% accuracy and MCDAL shows the highest performance
of 93.92% with standard deviation of $\pm 0.24\%$ at 40% labeled, which is
only 0.59% off. Although in the early stages of active learning VAAL, SRAAL,
and LL shows similar performance as MCDAL, MCDAL consistently outperforms at
all stages and shows large performance gain at the final stages. This is worth
noting because the performances of other methods start to saturate in the
later stages whereas MCDAL shows a continuously increasing trend.

Performance on CIFAR-100. As CIFAR-100 has 10 times more classes than
CIFAR-10, it is a considerably more difficult dataset. As seen from Fig. 4,
even previous works do not show much difference among themselves. At 20%
labeled data, LL shows slightly better performance than MCDAL. Overall, MCDAL
shows the best performance across most sampling stages and shows a
considerably higher performance at 40% labeled data, with an accuracy of
67.16% and standard deviation of $\pm 0.28$. Also, similar to the result in
CIFAR-10 dataset, the performance gap between MCDAL and the baseline methods
is especially large in the later stages. Moreover, MCDAL is also shown to be
advantageous when the amount of labeled data is low (15% dataset labeled).
Even with the minimal differences in some stages of active learning, MCDAL is
shown to be promising by showing noticeable performance gap with the state-of-
the-art methods especially in the early and late stages.

Performance on Caltech-101. Although the number of classes of Caltech-101
dataset is similar to that of CIFAR-100, the dataset is considerably smaller
and each class contains a vastly different number of images. Therefore, we
believe this is an even more challenging setup and is a good testing grounds
for a real-world setting where there is an imbalance of data. We can see from
Fig. 5 that in this setup, MCDAL far outperforms previous methods at all
stages and has the highest performance at 40% labels with 90.81% accuracy with
standard deviation of $\pm 0.35$. MCDAL shows the the maximum accuracy gap of
3.84% compared to the second-best method, Core-set approach, at 35% labeled
data. Note that other than the Core-set approach and MCDAL show almost similar
performance to that of the Random baseline.

Figure 5: Comparison with existing baselines on Caltech-101. Our method, MCDAL
shows the best performance compared to the state-of-the-art methods in a large
margin.

###  IV-B Semantic Segmentation

Dataset. For the image semantic segmentation task, we test our method on the
Cityscapes [52] dataset to compare with previous state-of-the-art works. The
Cityscapes dataset consists of 3475 frames for training with instance
segmentation annotations recorded in street scenes. Following the setup of
prior works [17, 19], we convert the annotations into 19 different classes.

Comparison baselines. We compare MCDAL to recent state-of-the-art methods
including Query-By-Committee (QBC) [10], Core-set [9], VAAL [17], and SRAAL
[19]. We also include a random sampling baseline as a standard to compare our
method. To evaluate the semantic segmentation, we use mean IoU (mIoU) as the
evaluation metric to measure the performance of all methods listed.

Implementation details. We adopt the Dilated Residual Network (DRN) [53] as
our baseline architecture. Since we use DRN, which predicts masks for each
instance, our auxiliary classification layers are the same as decoder layers
of DRN. In order to apply our method, the dimension is reduced from (N,C,H,W)
to (N,C) by averaging the mask. Similar to our Image Classification models,
the auxiliary branches are detached so as to not affect the task model. We use
Stochastic Gradient Descent as the optimizer, and set the learning rate to
$1\times 10^{-3}$ with a decay of $1\times 10^{-1}$ for every 100 epochs. We
train our method on Cityscapes for 150 epochs with a batch size 8, and we set
the budget size to be 150.

Segmentation performance. Fig. 6 illustrates our results of image semantic
segmentation on Cityscapes compared to previous approaches. From the figure,
we can clearly see that MCDAL outperforms all other methods in all sampling
stages, and MCDAL shows the highest performance at 57.9 mIoU with standard
deviation of $\pm 0.16$ at 40% of the labeled data. Even though semantic
segmentation is a more challenging setup compared to simple image
classification, our method still shows the best performance compared to
existing state-of-the-art methods by a large margin. In addition, we also show
in Fig. 7 representative qualitative results of MCDAL in comparison to the
Random baseline. We show in the white bounding boxes the regions where MCDAL
is able to refine the segmentation results as the stage progresses.

Figure 6: Comparison with existing baselines on Cityscapes dataset. MCDAL
consistently shows the best performance compared to the state-of-the-art
methods in a noticeable margin. Figure 7: Qualitative results of MCDAL on
Cityscapes. We show representative results in white bounding boxes to show
that MCDAL is able to further refine the segmentation regions as the stages
progress.

###  IV-C Additional Analysis

Analysis on sampling time and number of parameters. As active learning
frameworks introduce additional sampling stages and auxiliary networks,
sampling has to be fast and efficient. In other words, one of the ultimate
goals of active learning is to sample as fast as the Random baseline while
being as light as possible. Table II shows the sampling time required by MCDAL
at _stage_ 1 in comparison to recent methods on CIFAR-10 datset. We test all
methods on the CIFAR-10 dataset using a single NVIDIA TITAN Xp. As Core-set
uses an iterative method to sample, it is drastically slower than all other
methods (15.8 times the sampling time required compared to MCDAL). VAAL and
SRAAL both have VAEs and discriminators, and although faster than Core-set, is
still considerably slower than our MCDAL (2.6 times and 2.8 times the sampling
time required compared to MCDAL). LL shows the fastest time, which is slightly
faster than ours, as LL does not compare the labeled set to the unlabeled set.
As for number of network parameters, Core-set does not introduce any extra
parameters and has the same number of parameters as a vanilla Resnet-18
network (11.17M), so we can use this as the baseline. Table II shows that
MCDAL introduces the least number of parameters by a large margin even when
compared to LL, while VAAL and SRAAL both show a significant increase in
parameters. In other words, we conclude that MCDAL shows the best performance
with the least increase in parameters while having a reasonable sampling time.

TABLE II: Time required to sample at _stage_ 1 as tested on CIFAR-10 and
number of parameters of each method. MCDAL requires a reasonable sampling
time. Note that GAN based state-of-the-art approaches [17, 19] require
considerably longer sampling time in addition to a large increase in number of
parameters. Note that Core-set is simply Resnet-18.

Method | Sampling Time (Sec.) | No. of Parameters
---|---|---
Core-set [9] | 114.73 | 11.17M
LL [34] | 6.53 | 11.29M
VAAL [17] | 18.94 | 34.29M
SRAAL [19] | 20.48 | 45.58M
MCDAL (Ours) | 7.25 | 11.19M

Comparison on different distance measure. Based on the theory by [48], we
chose our discrepancy measure for $d(\cdot,\cdot)$ to be the L1 distance as in
Eq. (4); however, we perform an ablation study on CIFAR-10 with other
discrepancy measures such as KL-divergence (KL), which is broadly used in
knowledge distillation [54, 55, 56, 57, 58], and L2 distance with our final
design choice of L1. Fig. 8 shows that KL and L2 perform poorly and shows a
similar performance to the Random baseline, showing that using these distance
measures is meaningless as the Random baseline is much simpler and faster in
every way. Empirically, we show that L1 is the only viable design choice among
the three options evaluated.

Comparison on number of classifiers. In addition, we perform an ablation study
to see how changing the number of auxiliary layers set $|F_{\\{\cdot\\}}|$
would affect the active learning performance of our method. As our auxiliary
classification layers are in pairs, we test the different number of the
auxiliary layers in pairs to see the affect on the performance, two ($F_{1\sim
2}=\\{F_{1},F_{2}\\}$), four ($F_{1\sim 4}=\\{F_{1},F_{2},...,F_{4}\\}$), six
($F_{1\sim 6}=\\{F_{1},F_{2},...,F_{6}\\}$), and eight ($F_{1\sim
8}=\\{F_{1},F_{2},...,F_{8}\\}$). We test this ablation on the CIFAR-10
dataset and the result is illustrated in Fig. 9. Fig. 9 shows that increasing
the number of classification layers has little to no effect on performance.
Regardless of the number of auxiliary layers we use, the performance is
similarly favorable compared to other active learning methods. This shows that
our method already functions favorably with 2 classifiers, and we conjecture
that using more classifiers seems to introduce a form of redundancy and not
introduce any new information in regards to finding more informative samples,
therefore, we set the number of auxiliary classification layers to two
($\\{F_{1},F_{2}\\}$), due to the time complexity and memory cost of training.

Figure 8: Results of the variants of our method with different distance
measure for $d(\cdot,\cdot)$ (L1, L2, KL-divergence). L2 and KL-divergence
show similar performance as the Random baseline, showing that it cannot be
considered as a design choice. Figure 9: Results of our method with varying
number of the auxiliary classification layers, two ($F_{1\sim 2}$), four
($F_{1\sim 4}$), six ($F_{1\sim 6}$), and eight ($F_{1\sim 8}$) show similar
performances. In light of this finding, for the other experiments, we select
the number of the auxiliary layers to be _two_ due to the model complexity.
Figure 10: Results of ablation of with and without discrepancy loss, as the
first stage is the same and to show the standard deviation of each stage more
clearly, we show only from 15% to 40%. Our method without the discrepancy loss
shows similar average performance, but the addition of the discrepancy loss
aids heavily in the stability of MCDAL.

Ablation of the discrepancy loss. To understand the effects of the discrepancy
loss, we additionally perform an ablation of MCDAL with and without the
discrepancy losses and show our findings in Fig. 10. For better visualization
of the standard deviation error bars and as the first stage is fixed, we start
the graph at 15% labeled data. We find that the average performance of our
method without the discrepancy loss is similar to that of ours with the loss.
However, including the loss significantly increases stability especially in
the earlier stages as shown in the figure. Hence, we propose that the
discrepancy loss is a helpful addition.

##  V Conclusion

In this paper, we introduce a novel active leaning framework, Maximum
Classifier Discrepancy for Active Learning (MCDAL), that does not rely on
Generative Adversarial Networks (GAN) and still outperforms the recent state-
of-the-art models with lesser parameters, easy implementation, and fast
sampling. We show in our experimental results that our model is easily and
readily applicable to any and all different kinds of datasets and tasks. We
show through our findings that although GAN methods may seem like the way
forward, our method shows promise and might be an alternative for future
active learning research.

Acknowledgements This work was supported in part by the Institute of
Information and Communications Technology Planning and Evaluation (IITP) Grant
funded by the Korea Government (MSIT) (Artificial Intelligence Innovation Hub)
under Grant 2021-0-02068, and also under the framework of international
cooperation program managed by the National Research Foundation of Korea
(NRF2020M3H8A1115028, FY2021).

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|  Jae Won Cho received the B.S. degree in Electrical Engineering from Georgia
Institute of Technology, Atlanta, GA, USA in 2018,. He is currently a Ph.D.
candidate in Electrical Engineering at the Korea Advanced Institute of Science
and Technology (KAIST) under the supervision of Professor In So Kweon. He was
awarded a bronze prize from Samsung Humantech paper awards. His current
research interests include deep learning topics such as active learning and
high-level computer vision application such as vision and language. He is a
student member of the IEEE.
---|---
|  Dong-Jin Kim received the B.S. degree, M.S. degree, and Ph.D. degree in
Electrical Engineering from Korea Advanced Institute of Science and Technology
(KAIST), Daejeon, South Korea, in 2015, 2017, and 2021, respectively. He is
currently a Postdoc in EECS at UC Berkeley. He was a research intern in the
Visual Computing Group, Microsoft Research Asia (MSRA). He was awarded a
silver prize from Samsung Humantech paper awards and Qualcomm Innovation
awards. His research interests include data issues in computer vision
especially in high-level computer vision problems. He is a member of the IEEE.
---|---
|  Yunjae Jung received the B.S. degree in Electrical Engineering from Sogang
University, Seoul, South Korea in 2018 and reeived M.S. degree in Electrical
Engineering from Korea Advanced Institute of Science and Technology (KAIST),
Daejeon, South Korea, in 2019. He is currently working towards the Ph.D.
degree in Electrical Engineering at KAIST. He was awarded a honorable mention
from Samsung Humantech paper awards. His research interests include high-level
computer vision such as video scene understanding and vision and language. He
is a student member of the IEEE.
---|---
|  In So Kweon received the B.S. and M.S. degrees in mechanical design and
production engineering from Seoul National University, South Korea, in 1981
and 1983, respectively, and the Ph.D. degree in robotics from the Robotics
Institute, Carnegie Mellon University, USA, in 1990. He was with the Toshiba
R&D Center, Japan, and he is currently a KEPCO chair professor with the
Department of Electrical Engineering, since 1992. He served as a program co-
chair for ACCV 07’ and ICCV 19’, and a general chair for ACCV 12’. He is on
the honorary board of IJCV. He was a member of “Team KAIST,” which won the
first place in DARPA Robotics Challenge Finals 2015. He is a member of the
IEEE and the KROS.
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