# Learning Pain from Action Unit Combinations:
A Weakly Supervised Approach via Multiple Instance Learning

Zhanli Chen  Department of Electrical and
Computer Engineering
University of Illinois at Chicago
Email<EMAIL_ADDRESS>Rashid Ansari  Department of Electrical and
Computer Engineering
University of Illinois at Chicago
Email<EMAIL_ADDRESS>Diana J. Wilkie  College of Nursing
University of Florida
Email<EMAIL_ADDRESS>
###### Abstract

Patient pain can be detected highly reliably from facial expressions using a
set of facial muscle-based action units (AUs) defined by the Facial Action
Coding System (FACS). A key characteristic of facial expression of pain is the
simultaneous occurrence of pain-related AU combinations, whose automated
detection would be highly beneficial for efficient and practical pain
monitoring. Existing general Automated Facial Expression Recognition (AFER)
systems prove inadequate when applied specifically for detecting pain as they
either focus on detecting individual pain-related AUs but not on combinations
or they seek to bypass AU detection by training a binary pain classifier
directly on pain intensity data but are limited by lack of enough labeled data
for satisfactory training. In this paper, we propose a new approach that
mimics the strategy of human coders of decoupling pain detection into two
consecutive tasks: one performed at the individual video-frame level and the
other at video-sequence level. Using state-of-the-art AFER tools to detect
single AUs at the frame level, we propose two novel data structures to encode
AU combinations from single AU scores. Two weakly supervised learning
frameworks namely multiple instance learning (MIL) and multiple clustered
instance learning (MCIL) are employed corresponding to each data structure to
learn pain from video sequences. Experimental results show an $87\%$ pain
recognition accuracy with $0.94$ AUC (Area Under Curve) on the UNBC-McMaster
Shoulder Pain Expression dataset. Tests on long videos in a lung cancer
patient video dataset demonstrates the potential value of the proposed system
for pain monitoring in clinical settings.

###### Index Terms:

FACS, Action Unit Combinations, Pain, MIL.

##  I Introduction

Assessing pain is a difficult but important task in clinical settings, which
in practice relies on self-report by patients through simple subjective pain
assessment measures like visual analog scale (VAS). Research has shown that
facial expressions can serve as reliable indicators of pain across human
lifespan [6] and there is also good consistency of facial expressions
corresponding to pain stimuli. The Facial Action Coding System (FACS) is
widely used in pain analysis because it provides an objective assessment to
score and recognize Action Units (AUs), which represent the muscular activity
that produces momentary changes in facial appearance [8]. Several studies [7],
[17] using FACS have identified a collection of core Action Units, which are
specific to pain and that occur singly or in combination as summarized in
Table I. These results are also confirmed in the study of facial expressions
of pain suffered by cancer patients [23]. Note that although AU27 (mouth
stretch) is included in [23], we did not include it in this study as it occurs
infrequently. Facial expression annotation of videos using FACS is generally
performed offline by trained experts who closely examine the video of a
patient’s face. A long video is typically divided into multiple subsequences
of fixed length duration and AUs are coded at each time step (i.e. each video
frame) within the video subsequence. Pain is assessed across the entire
sequence based on the occurrence and frequency of pain-related AUs. However
the Action Unit coding via human observations is very time consuming, which
makes its real-time clinical use prohibitive [13],[1]. Therefore, the
development of an efficient real-time automated FACS-based pain detection
would be a significant innovation for enhanced patient care and clinical
practice efficiency.

TABLE I: Action Unit Definition and Pain-Related AU combinations AU | Description | Pain-Related Combinations
---|---|---
4 | eye brow lower | 6/7
6 | cheek raiser | 20
7 | eye lid tightener | 4+6/7/43
9 | nose wrinkler | 4+9/10
10 | upper lip raiser | 4+26
20 | lip stretcher | 9/10+26
26 | jaw drop |
43 | eyes closed |

Progress in computer vision and machine learning (CVML) techniques over time
has led to significant development of general Automated Facial Expression
Recognition (AFER) systems, although limited effort has been reported for its
application in pain analysis. One major challenge is the difficulty in
establishing a comprehensive annotated dataset with sufficient examples of
pain-related expressions. Most existing video datasets containing pain-related
facial expressions are developed for targeted studies. These datasets are
typically not publicly accessible. They are mostly small in size and lack
sufficient diversity to train a robust automated system in general. One
development that has facilitated the research on spontaneous expressions
recognition is the introduction of UNBC-McMaster Shoulder Pain Archive, which
is the only publicly available comprehensive pain-oriented facial expression
dataset. This dataset contains recordings of subjects experiencing shoulder
pain in a clinical setting with complete labeling at both the frame level
(AUs) and the sequence level (OPI, VAS, etc.), and it has been widely employed
by FACS-based AFER research as a standard dataset for performance evaluation.
On the other hand, shoulder pain involved in UNBC-McMaster dataset is acute
pain, and we are not aware of any research suggesting that this dataset is
beneficial to study other type of pain, such as chronic pain caused by cancer.

An important observation that motivated our research is that existing research
on FACS-based automated AU recognition focuses on detection of single AUs. We
note that pain-related AUs occur in conjunction with other AUs to form
combinations irrelevant to pain. Therefore inference based only on occurrence
of individual AUs is not sufficient for pain identification. While the ground
truth of facial expressions and action units in video databases is available
at frame level, the ground truth about pain is typically available at sequence
level and only via self-report, which is an example of ’weakly labeled’ data.
In early attempts of automated pain analysis, proposed approaches [13][1]
employed an averaging paradigm by assigning the sequence label of pain
occurrence to each frame and training a support vector machine (SVM) on the
frame-level label. Pain is declared to occur in a video if the average output
of frame-level pain score exceeds a threshold. However, pain-related frames
may constitute a small fraction of all frames in a long video. In this case
averaging the output score could therefore severely attenuate the signal of
interest. Recent research [20] suggests that video-based pain detection can be
formulated as a weakly supervised learning problem, and multiple instance
learning (MIL) is an effective machine learning tool to handle this problem.
In the study [20], a binary pain classifier is trained directly via the high-
dimensional features extracted from video frames without going through the AU
coding procedure. Although encouraging results are reported from experiments
on UNBC-McMaster dataset, this setting is vulnerable to performance
degradation for trans-dataset application, due to interference from person-
specific features and demographic variations encoded in the the high-
dimensional features.

In a commonly used procedure for the manual detection of pain, FACS-certified
coders first perform AU coding for every video frame and then infer the
sequence level pain label from the occurrence and frequency of pain-related AU
combinations. In general, AU occurrence is highly correlated with the
appearance of pain in facial expressions, and the reliability of pain
detection strongly depends on the accuracy of AU coding. However, facial
expressions of pain are more likely to appear when the pain intensity level is
high or when intensity of pain changes to a higher level. Due to the sparsity
of pain expressions, video captured in clinical settings usually lacks
sufficient positive samples to train a reliable pain classifier by learning a
direct mapping between high-dimensional facial features and self-report pain
labels. On the other hand, state-of-the-art AFER systems can be effectively
trained on millions of online image [16], which possess sufficient
adaptability to all kinds of video datasets including pain-oriented ones.
Inspired by these observations, we propose a new FACS-based automated pain
detection framework comprised of two independent machine learning networks to
infer the presence of Action Unit (AU) combinations that signify pain. The two
networks are trained independently and are linked by two proposed novel AU-
based data structures – compact or clustered- which are created by mapping
single AU measurements per frame into pain-related low-dimensional feature
vectors representing AU combinations. The first machine learning network is a
generic AU detector that takes advantage of current progress in computer
vision and machine learning techniques to handle challenges from person-
specific variation and natural illumination characteristic of clinical
settings. The second machine learning network, called the Multiple Instance
Learning (MIL), solves pain detection as a weakly supervised learning problem
and performs analysis in the low-dimensional AU combination feature space.
This decoupled structure of two independent machine learning networks
facilitates data fusion from different pain-oriented video datasets and helps
to develop a commercial, robust, and generic automated pain analysis system.
The decoupled structure mimics the strategy used by human coders in clinical
settings. To our best knowledge, this is the first work on automated pain
analysis based on a complete set of pain-related AU combinations.

The rest of the paper is organized as follows, Section 2 provides a review of
the literature on automated pain detection and analysis. Section 3 gives a
brief overview of two pain-oriented datasets involved in this work: the UNBC-
McMaster dataset (acute shoulder pain) and Wilkie’s dataset (chronic cancer
pain). Section 4 presents the pain detection framework in a decoupled
structure, with focus on the feature representation based on AU combinations
and corresponding learning tools (MIL and MCIL). Section 5 provides a
demonstration of the advantages of the proposed framework based on the results
of testing our method on both datasets described in Section 3. Finally Section
6 presents the conclusion of the work.

##  II Related Past Work On Pain Detection

In the past decades, significant progress in computer vision and machine
learning techniques (CVML) has boosted the development of the AFER system.
With increasing demand for facial expression applications, the focus of AFER
research shifted from posed expressions obtained in a controlled setting to
spontaneous expressions evoked in a natural settings. Further details can be
found in survey papers [3][5][25]. A robust AFER system is capable of handling
perturbations due to demographic difference, environment variation, and rigid
motions, and is applicable to different datasets without a need for
retraining. Emotient [12] and Affectiva [16] are two well-known examples of
state-of-the-art AFER systems which are commercially available as platforms
that facilitate the development of applications based on facial expression.
However, the output of such systems is more appropriate as an intermediate
result in the form of general AU scores and further processing and
customization is required for advanced applications including pain analysis.
On the other hand, while popular CVML tools are widely used in most AFER
research, very limited effort are reported on exploring machine learning based
pain interpretation from facial expressions. Ashraf et al. [1] studied the
UNBC-McMaster dataset and proposed three feature types that are extracted from
the Active Appearance Model (AAM) to train SVM pain classifiers. They used an
averaging scheme to generate sequence-level labels. In their subsequent
research[15][14], the same set of features are used to train a binary
classifier for each single pain-related AU at frame level together with a pain
intensity classifier at sequence level using the OPI labels. Chen _et al_ [4]
used a simple rule-based method to model temporal dynamics of AUs to study
pain of patients suffering from lung cancer in the Wilkie’s dataset. Sikka _et
al_ [19] employed a CVML-based model to assess pediatric postoperative pain on
a video dataset of neurotypical youth. A total of $14$ single AUs are
extracted under $3$ statistics to form a $42$-dimensional descriptor for each
pain event which serves as the input to logistic regression models of both
binary pain classification and pain intensity estimation. Later, Sikka _et al_
[20] modeled video sequences from UNBC-McMaster dataset with the Bag of Words
representation and applied Multiple Segment Multiple Instance Learning (MS-
MIL) for jointly detecting and localizing frames containing pain using
sequence-level ground truth (OPI). We note three shortcomings observed in
previous research : (1) AU recognition and pain recognition are treated as
separate problems which are handled by an AU classier and a pain classifier
respectively, without giving sufficient attention to their relationship; (2)
pain analysis is more focused on single AU detection rather than pain relevant
AU combination; (3) The difficulty in having access to video datasets with
chronic pain. And also insufficient effort on developing advanced
comprehensive automated analysis tools targeted to clinical applications.

##  III Available Pain Expression Datasets

The UNBC-McMaster Shoulder Pain Expression Archive Dataset contains $200$
video sequences captured from patients suffering from shoulder pain and
spontaneous facial expressions are triggered by moving their affected and
unaffected limbs. All frames are coded by certified FACS coders for $10$
single pain-related AUs and the frame-level pain score is rated by the
Prkachin and Solomon Pain Intensity (PSPI). A sequence-level pain label is
assigned by self-reported Visual Analog Scale (VAS) and Observer-rated Pain
Intensity (OPI). In addition, $66$-point facial landmarks from Active
Appearance Model (AAM) are also provided for each frame to facilitate the
development of a user-customized AFER system. It is the only publicly
available pain-oriented facial expression video dataset in which the
spontaneous facial expressions are evoked solely by acute pain.

One major distinguishing feature of this study is that in addition to the UNBC
McMaster dataset we also conduct research on a unique dataset created by D.
Wilkie [23], containing videos of $43$ patients suffering from chronic pain
caused by lung cancer. The videos are captured in natural settings in the
subjects’ homes. The patients were required to repeat a standard set of
randomly ordered actions as instructed such as sit, stand up, walk, and
recline, in a $10$-minute video with a camera focused on the face area to
record their facial expressions. Each video was partitioned into $30$ equal-
duration of $20$ seconds per subsequence. The subsequences were reviewed and
scored for 9 AUs possibly occurring in combination by three trained human FACS
coders independently, and the results were entered in a scoresheet that served
as ground truth. Pain was scored in one video subsequence if at least two
coders agree with each other on the occurrence of a set of specific AU
combinations listed on the scoresheet. The intensity of pain for the entire
video was measured on the total number of subsequences that were associated
with pain labels. Due to the issue in illumination and video quality
degradation, only $1164$ out of the total $1290$ video subsequences were
amenable for processsing by Emotient and of these about $600$ subsequences are
suitable for pain analysis.

In this paper, we use the UNBC-McMaster dataset to train a MIL based pain
detector and then test it on Wilkie’s video dataset [23] with chronic cancer
pain. We hypothesize that the pain detector can capture the prevalent feature
of pain from the low-dimensional feature vectors of AU combinations that we
introduce in this paper.

##  IV Proposed Automated Pain Detection Framework

Pain detection is one of the applications of spontaneous facial expression
recognition. It is very beneficial for patient care to develop a pain
detection system for clinical settings if we take advantage of existing
progress on spontaneous AFER research. Recent state-of-the-art generic AFER
systems [9][16] have proved to be robust to context variations, especially
illumination and rigid motions, and have shown good performance on new
datasets with different demographics. This is a key motivation for us to adopt
the commercialized Emotient system in the front end of our proposed method. On
the other hand, the generic system is not optimized for application to pain
recognition，so that deterministic decisions made directly from the AU scores
with a preset threshold could cause a high false alarm rate. Therefore we
investigate a framework that uses a second layer of machine learning to refine
the single AU scores and give a more reliable prediction of pain.

The proposed automated pain detection system is comprised of two-level machine
learning systems, an Automated Facial Expression Recognition (AFER) system
that computes frame-level confidence scores for single AUs (shown in the left
side of Fig.1.) and a MIL system that performs sequence-level pain prediction
based on contributions from a pain-relevant set of AU combinations (shown in
the right side of Fig.1.). MIL is well-suited to handle the ’weakly labelled’
pain data that can be conveniently represented by a bag of word (BOW)
structure, and details will be covered in the following chapters. AU
combinations are described by two distinct low-dimensional novel feature
structures : the compact structure encodes all AU combinations of interest
into a single vector, which is analyzed in the MIL framework; the clustered
structure uses a sparse representation to encode each AU combination
separately followed by analysis in the multiple clustered instance learning
(MCIL) framework, which is an extension of MIL. To our knowledge, this is the
first work that applies MCIL in facial expression-related research. Both
machine learning systems are trained on the UNBC-McMaster dataset
independently with frame-level and sequence-level labels, as illustrated by
the three circular blocks in the center of Fig.1.

Figure 1: The Decoupled Pain Detection Framework

AU coding relies on observable facial muscular movements or facial
expressions, whereas pain is more like a latent variable and is not always
manifested itself in facial expressions, especially in the scenario of chronic
pain. Due to the limited number of positive labeled samples, it is often
difficult to learn a reliable mapping between pain and high-dimensional facial
features in a direct manner. On the other hand, the decoupled framework
utilizes AU codings as the intermediate results and performs pain analysis in
a low-dimensional AU score space, which would be more justifiable and
efficient than the direct method. In addition, the decoupled framework
alleviates the difficulty in training large scale dataset-specific pain
datasets by facilitating data fusion from different pain-labelled video
datasets, which may potentially lead to a generic pain detector for analyzing
multiple types of pain.

###  IV-A Automated Facial Expression Recognition

An AFER system can be conveniently described with four key blocks consisting
of face detection, feature representation, feature selection, and
classification, as shown in Fig.1. The first block identifies face area with a
rectangular boundary box in every video frames. The second block aligns the
detected face areas and employs various descriptors to extract features from
the facial images. The third block selects features most relevant to the non-
rigid motions caused by facial expressions, and applies dimension reduction
techniques to compress the feature vector to a size that is tractable for the
classifier. The fourth block contains a set of one-versus-all classifiers that
are trained on the refined feature vectors for each AU of interest, and the
output could be either binary decisions or soft scores that reflect the
probability or confidence about the target AUs. Existing research
[2][11][15][10] on spontaneous facial expression recognition shows AFER
systems are highly customizable, and more blocks could be added to this
framework to boost performance depending on the application.

One example of such an AFER system is the computer expression recogntion
toolbox (CERT) [12], which is the core of the Emotient [9] system. In the
system setup, face detection is based on an extentsion of classic Viola-Jones
approach. Ten facial feature points are tracked using GentleBoost and the
detected face area is aligned to a canonical template patch through an affine
warp estimated from the feature positions. A Gabor filter bank is then applied
to extract features in $8$ directions and $9$ spatial frequencies and the
filter outputs are concatenated into a single feature vector. The feature
vector is then fed into separate linear support vector machines (SVM) for
individual AU recognition.

We use Emotient to track and label a set of AUs $\\{4,6,7,9,10,20,26,43\\}$,
that are commonly used in most pain-oriented research. The processing results
are represented by the flow of Evidence numbers, where an Evidence number,
ranging between $-2$ to $2$ [9], is assigned to each AU on every frame. The
Evidence output for an expression channel represents the odds in logarithm
(base $10$) scale of a target AU being present. For example: Evidence = $2$
($10^{2}=100$) means the observed expression is $100$ times more likely to be
categorized by an expert human coder as target AU and Evidence = $0$ means the
chances that the expression to be categorized by an expert human coder as
target AU or not are equal. The Evidence scores can be conveniently
transformed to Probability measurements by the following equation:

$Probability=\frac{1}{1+10^{-Evidence}}$ (1)

The probability measurements derived from the Evidence scores are used as the
features of pain analysis framework. The evidence/probability score profile of
every AU can be viewed as a $1D$ time domain signal.

###  IV-B Action Unit Combination Encoding

####  IV-B1 Compact Structure Vs. Clustered Structure

In practice, Action Units $6/7$ are the most frequently observed pain-related
AUs in FACS coding [17]. Multiple AU combinations could also be activated in a
video segment for pain evalutation. Furthermore, the Evidence number produced
by AFER suggests uncertainty about AU coding, where higher uncertainty could
frequently be associated with spontaneous facial expressions due to their low
intensity. Hence when we define feature vectors based on AU combinations, it
is important to have a comprehensive characterization in terms of frequency
extent of individual AU contribution, and confidence of measurement for each
AU combination. Therefore, in the task of designing feature vectors for pain
analysis, we not only consider the activation of individual AU combination but
also take into account the correlation among activation of multiple AU
combinations. Two different feature vector structures based on AU combination
scores, which are referred to as compact or clustered, are proposed as
follows,

Compact Structure: Let $A=\\{4,6,7,9,10,20,26,43\\}$ be the set of single
pain-related AUs. The AU combination feature vector for frame $i$ of a video
sequence $S$ is an $11$ dimensional column vector

$\displaystyle v_{i}=$
$\displaystyle[P_{AU_{6}}\,P_{AU_{7}}\,P_{AU_{20}}\,P_{AU_{4\oplus
6}}\,P_{AU_{4\oplus 7}}\,P_{AU_{4\oplus 43}}$ $\displaystyle P_{AU_{4\oplus
9}}\,P_{AU_{4\oplus 10}}\,P_{AU_{4\oplus 26}}\,P_{AU_{9\oplus
26}}\,P_{AU_{10\oplus 26}}]^{T}$

in which each entry of the feature vector is the probability estimate of
corresponding AU or AU combination and the $\oplus$ operator denotes the co-
occurrence of AUs. The probability estimate of the combination $AU(i\oplus j)$
depends on the smaller probability between $AU_{i}$ and $AU_{j}$ so that
$P_{AU_{i\oplus j}}=min(P_{AU_{i}},P_{AU_{j}}),\forall i,j\in A$. As a result,
the pain information about frame $i$ is conveyed by the probability
measurement of all pain-related AU combinations that are compressed in a
single low-dimensional vector, as shown in Fig. 3(a)

Figure 2: AU combination structure: (a)Compact Structure, (b)Clustered
Structure

Clustered Structure: Here we follow Wilkie’s coding strategies [23] to group
the $11$ pain-related single AU and AU combinations into clusters according to
two criteria, 1) there is a common AU shared by the combinations in the
cluster and 2) the AU combinations within a cluster are actuated in adjacent
area on the face. Six clusters are formed in this way, including
$\\{AU_{6/7}\\},\\{AU_{20}\\},\\{AU_{4\oplus 6/7/43}\\},\\{AU_{4\oplus
9/10}\\},\\{AU_{4\oplus 26}\\}$ and $\\{AU_{9/10\oplus 26}\\}$. The AU
combination feature representation of a video frame $i$ under the clustered
structure is composed of an $8\times 6$ matrix, where the column $j$ is
highlighted by all the single AUs involved in the combinations belonging to
cluster $j$, where $j\in\\{1,2,3,4,5,6\\}$. The non-zero entry for column $j$
is the probability measurement of the highlighted single AU in cluster $j$ and
all the remaining entries of the feature matrix are set to zero, which results
in a sparse representation of features, as shown in Fig. 2(b).

####  IV-B2 Bag of Words representation

A patient’s self-report is still the golden rule for pain evaluation in
patient care. A pain label is commonly available for video segment but not for
every single frame. Such a situation is encountered frequently in computer
vision since it is easier to obtain group labels for the data rather than
individual labels, and is known as ’weakly supervised’ problem. On the other
hand, although temporal dynamics of spontaneous facial expressions have good
intra-dataset consistency, it could vary significantly under clinical settings
depending on the level and type of pain a patient is suffering. While AUs
evoked by acute pain last less than a second, those evoked by chronic cancer
pain could last for minutes. Hence conventional temporal modeling with fixed
moving window or a preset duration parameter[13][11] is inadequate to handle
practical applications. To address the challenges from weakly labeled data and
complicated temporal dynamics, we employ the bag of word (BOW) representation
as suggested in [21][20].

A video sequence $S_{i}$ in a dataset can be represented as a bag that
contains a number of segments generated from $S_{i}$. The bag is defined as
$\\{s_{ij}\\}{{}_{j=1}^{N_{i}}}$， where $s_{ij}$ is the $j$th segment in the
bag containing $N_{ij}$ frames i.e.
$s_{ij}=\\{f{{}_{i}^{m_{j}}},f{{}_{i}^{m_{j}+1}},\ldots,f{{}_{i}^{N_{ij}-m_{j}+1}}\\}$.
$s_{ij}$ contains only contiguous frames and $N_{ij}$ is the total number of
frames in $s_{ij}$ taken from $S_{i}$ and $f{{}_{i}^{m_{j}}}$ is the $m_{j}$th
frame in $S_{i}$ and the $1$st frame in ${s_{ij}}$. The bags are then
associated with the label of sequence $S_{i}$ as
$B=\\{S_{i},y_{i}\\}{{}_{i=1}^{N}},\;y_{i}\in\\{-1,\,1\\}$， which defines two
kinds of bags, positive and negative. A positive bag contains at least one
positive instance, while a negative bag contains no positive instances.
Adopting this representation for the pain detection problem, a positive bag
refers to a video sequence that contains pain-related facial expressions, and
a negative bag refers to a video sequence not containing any pain-related
facial expression. Practically, pain-related AU temporal segments occupy only
a small portion of the entire video sequence. The sparsity of positive
training samples fits well in the context of BOW structure, which is another
motivation to adopt this type of data structure. It takes three steps, S1-S3
as described below, to generate a BOW representation from the feature space.

(S1) Feature Extraction at Frame Level: Define a mapping $\phi_{F}:R^{m\times
n}\rightarrow R^{d}$ as the feature extraction process on frame level that
maps a frame of size $m\times n$ in image space to a $d$-dimensional feature
vector, where $d==11$ for the compact situation. In the case of clustered
structure, the mapping is defined as $\phi_{F}:R^{m\times n}\rightarrow
R^{D}$, where $R^{D}$ refers to the space of $8\times 6$ sparse feature
matrices. Feature vectors are typically of very high dimension in existing
unified framework. However, $R^{d}$ or $R^{D}$ in our case is simply the low-
dimensional AU combination feature vector space in the proposed decoupled pain
detection framework.

(S2) Multiple Segment Generation: The instances in a bag are video segments
containing consecutive frames belonging to the sequence. The bound of each
segment can be generated conveniently in two ways. A typical way is to run
overlapping temporal scanning windows at multiple scales known as Sc-Wind. A
parallel way is clustering the frames in a sequence using normalized cuts
(Ncuts). Each element of the weight matrix of Ncut algorithm is obtained by a
similarity measure between frames $f{{}_{i}^{u}}$ and $f{{}_{i}^{v}}$ in
sequence $i$ measured by

$\begin{split}W(u,v)=&\exp{(-|\frac{\phi_{F}(f{{}_{i}^{u}})-\phi_{F}(f{{}_{i}^{v}})}{\sigma_{f}}|^{2})}\\\
&+\exp{(-|\frac{t_{u}-t_{v}}{\sigma_{t}}|^{2})}\end{split}$ (2)

where $t_{u}$ refers to frame index of $f{{}_{i}^{u}}$, and $\sigma_{f}$ and
$\sigma_{t}$ are constants selected for feature domain and time domain
respectively. Details of Ncuts are provided in[18] [20].

(S3) Feature Representation at Segement Level: The feature representation of a
video segment is denoted by the mapping $\phi_{S}:S\rightarrow R^{d}$ that
transform video segment in sequence space $S$ to a $d$ dimensional feature
vector. This mapping is specified by a max-pooling strategy from the feature
representation of all the frames in the segment as:

$\phi_{S}(s_{ij})=\mathop{max}_{k}(\phi_{F}(f_{i}^{k})\,|\,f_{i}^{k}\in
s_{ij})$ (3)

The instance in a bag is now represented by a single feature vector with the
same dimension $d$ as the frame-level feature vector. After associating the
pain label to the bag, a multiple instance learning(MIL) framework can be
trained on the BOW data for automated pain detection.

###  IV-C Multiple Instance Learning

The general idea for solving machine learning problems is to establish a
classifier and optimize it with respect to a loss function. Viola _et al._[22]
first solved the MIL problem with a boosting framework, which is known as
MILboost, and discussed its application in object detection from images. In
this section, we give a brief overview of MILboost and how it can be
customized for pain detection. The decision on the presence of pain is based
on the probability of bags been positive. The posterior probabilities of bags
and instances are defined as:

$\displaystyle p_{i}$ $\displaystyle=\mathcal{P}(y_{i}=1|S_{i})$ (4)
$\displaystyle p_{ij}$ $\displaystyle=\mathcal{P}(y_{i}=1|s_{ij})$ (5)

The only available ground truth is the label of the bag, and all the instances
in a bag carry the same label as the bag.

A classifier $H_{T}:R^{d}\rightarrow R$ is trained on the feature vectors of
instances, and a posterior probability is assigned to each instance based on
the classifier output, $s.t.$

$p_{ij}=\sigma(H_{T}(\phi_{s}(s_{ij})))$ (6)

where $\sigma()$ is a sigmoid function $s.t.$

$\sigma(x)=\frac{1}{1+\exp(-x)},\>\forall x\in R$ (7)

The loss function is defined by negtive log-likelihood, which is the same as
that used in the logistic regression problem:

$\mathcal{L}=-\sum_{i}^{N}(r_{i}*p_{i}+(1-r_{i})(1-p_{i}))$ (8)

where $r_{i}=1$ if $y_{i}=1$ and $r_{i}=0$ if $y_{i}=-1$. Since a positive bag
contains at least one positive instance, the probability of bag to be positive
$p_{i}$ depends on the probability of the instance that is most likely to be
classified as positive, i.e.

$p_{i}=\mathop{max}_{j}(p_{ij})$ (9)

The MILboost uses the boosting procedure to construct a strong classifier
$H_{T}(s_{ij})$ by iteratively combining a set of weak classifiers
$h_{t}(s_{ij})$ as,

$H_{T}(s_{ij})=\sum_{t=1}^{T}\alpha_{t}h_{t}(s_{ij})$ (10)

where $H_{T}$ denotes the classifier constructed in the $T^{th}$ iteration and
all weak classifiers are the same type of learners that are generated from
space $\mathcal{H}$. Note $H_{T}(s_{ij})$ and $h_{t}(s_{ij})$ are simplified
notations of $H_{T}(\phi_{s}(s_{ij}))$ and $h_{t}(\phi_{s}(s_{ij}))$
respectively, and similar notation will be used in the following derivations.

The boosting algorithm updates the weight of instances at the end of each
iteration by taking the gradient of the loss function $\mathcal{L}$ $w.r.t$
$H_{T}(s_{ij})$

$\omega_{ij}=-\frac{\partial\mathcal{L}}{\partial
H_{T}(s_{ij})}=-\frac{\partial\mathcal{L}}{\partial p_{i}}\frac{\partial
p_{i}}{\partial p_{ij}}\frac{\partial p_{ij}}{\partial H_{T}(s_{ij})}$ (11)

The instance weights are then normalized as
$\omega_{ij}^{\prime}=\frac{|\omega_{ij}|}{\sum_{ij}|\omega_{ij}|}$.
Misclassified instances will be assigned higher weights and a weak classifier
$h_{T+1}(s_{ij})$ is trained on the reweighted data and added to $H_{T+1}$ in
the $(T+1)^{th}$ iteration

$h_{T+1}=\arg\max_{h\in\mathcal{H}}\sum_{i}^{N}\sum_{j}^{N_{i}}\omega_{ij}^{\prime}h(s_{ij})$
(12)

where $N$ is the total number of bags and $N_{i}$ is the number of instances
in the $i$th bag. However, since the $\max$ function is not differentiable, a
soft-max function $g$ is used as an approximation. Define a set
$\textbf{p}_{i}=\\{p_{i1}\,p_{i2}\,...\,p_{iN_{i}}\\}$, then the softmax
function with respect to the subscript $j,\,j\in\\{1,2,...,N_{i}\\}$ is given
by:

$p_{i}=g(\textbf{p}_{i})=g_{j}(p_{ij})\approx\mathop{max}_{j}(p_{ij})$ (13)

Among all the options for soft-max function, the generalized mean (GM) is a
preferable model as suggested by past research [20]. For the
instances$\\{s_{ij}\\}{{}_{j=1}^{N_{i}}}$ in the bag of $S_{i}$, the GM
approximation is given by:

$g_{GM}(\textbf{p}_{i})=g_{j}(p_{ij})=(\frac{1}{N_{i}}\sum_{j}p_{ij}^{u})^{\frac{1}{u}}$
(14)

where $u$ is the parameter that controls sharpness and accuracy in GM model
$s.t.$ $g_{GM}(\textbf{p}_{i})\rightarrow\max_{j}(p_{ij})$ as
$u\rightarrow\infty$. Now the gradient of the GM soft-max is given by:

$\frac{\partial p_{i}}{\partial
p_{ij}}=p_{i}\frac{p_{ij}^{(u-1)}}{\sum^{N_{i}}_{s=1}p_{is}^{u}}$ (15)

###  IV-D Multiple Clustered Instance Learning

In the compact structure feature settings, scores of all AU combinations are
encoded in one feature vector, which can be conveniently handled by the
original MIL framework. However, it may be desirable to distinguish the
contribution of individual AU combinations for more precise analysis on
different types of pain. Practically, the clustered representation is a more
natural way that is used by human coders and a positive decision on any of the
clusters is sufficient to identify pain. Multiple Clustered Instance Learning
(MCIL) proposed by Xu _et al._[24] is an extension of MIL that was proposed to
provide patch-level clustering of $4$ subclasses of cancer tissues, and
facilitates both image-level classification and pixel level segmentation
(cancer vs. non-cancer). Based on the structural similarity of the problems,
we have adapted the MCIL framework to handle the clustered feature structure
for pain recognition.

MCIL assumes there are $K$ clusters in a positive bag and an associated hidden
variable $y_{ij}^{k}\in\\{-1,1\\}$ that indicates whether the instance
$s_{ij}$ belongs to the $k$th cluster. An instance could be considered
positive if it belongs to one of the $K$ clusters and a bag is labeled as
positive bag only if it contains at least one positive instance. The goal of
MCIL is to learn one boosting classifier $H_{T}^{k}(s_{ij}^{k})$ for each of
the $K$ clusters. In our clustered data representation settings, each video
frame is encoded by a $8\times 6$ matrix. If we treat each column vector as an
independent instance, all the instances in one bag will form six clusters
naturally. A six-cluster MCIL learner can be trained and the overall decision
is based on the cluster classifier that gives maximum output:

$H_{T}(s_{ij})=\max_{k}(H_{T}^{k}(s_{ij}))$ (16)

Similar to the core of MIL, the posterior probability of bag $i$ is given by,

$p_{i}=\max_{j}\max_{k}(p_{ij}^{k})$ (17)

where $j\in\\{1,2,...,N_{i}\\}$ and $k\in\\{1,2,...,K\\}$. The max function is
approximated by a soft-max function,

$p_{i}=g_{jk}(p_{ij}^{k})=g_{j}(g_{k}(p_{ij}^{k}))$ (18)

Note that the order of soft-max functions is interchangeable, i.e.

$g_{j}(g_{k}(p_{ij}^{k}))=g_{jk}(p_{ij}^{k})=g_{k}(g_{j}(p_{ij}^{k}))$ (19)

Taking GM model as an example, the proof of (19) goes as follows,

$\begin{split}g_{k}(g_{j}(p_{ij}^{k}))&=(\frac{1}{K}\sum_{k}(g_{j}(p_{ij}^{k}))^{u})^{\frac{1}{u}}\\\
&=(\frac{1}{K}\sum_{k}((\frac{1}{N_{i}}\sum_{j}(p_{ij}^{k})^{u})^{\frac{1}{u}})^{u})^{\frac{1}{u}}\\\
&=(\frac{1}{KN_{i}}\sum_{k,j}(p_{ij}^{k})^{u})^{\frac{1}{u}}=g_{j}(g_{k}(p_{ij}^{k}))\end{split}$
(20)

Finding the strong classifier of a cluster follows a standard boosting
procedure and all the classifiers are trained on the same set of BOW
instances. However, the weight of instances are updated respectively as per
cluster,

$\omega_{ij}^{k}=-\frac{\partial\mathcal{L}}{\partial
H_{T}^{k}(s_{ij})}=-\frac{\partial\mathcal{L}}{\partial p_{i}}\frac{\partial
p_{i}}{\partial p_{ij}^{k}}\frac{\partial p_{ij}^{k}}{\partial
H_{T}^{k}(s_{ij})}$ (21)

The partial derivative of $\frac{\partial p_{i}}{\partial p_{ij}^{k}}$ for the
GM model is given by,

$\frac{\partial p_{i}}{\partial
p_{ij}^{k}}=p_{i}\frac{(p_{ij}^{k})^{(u-1)}}{\sum_{s=1}^{N_{i}}\sum_{t=1}^{K}(p_{is}^{t})^{u}}$
(22)

For the remaining two items in the partial derivative of the weight update
expression, $\frac{\partial\mathcal{L}}{\partial p_{i}}$ is the same as in
MILboost and $\frac{\partial p_{ij}^{k}}{\partial
H_{T}^{k}(s_{ij})}=p_{ij}^{k}(1-p_{ij}^{k})$, which is the derivative $w.r.t$
a sigmoid function.

##  V Experimental Results

The pain detection system is first tested on the UNBC-McMaster dataset, where
video sequences with OPI ratings$\geq 3$ are treated as positive samples and
those with OPI$=0$ are treated as negative samples. This yielded the same set
of 147 sequences from 23 subjects as in [20]. Video sequences are processed by
Emotient and the output of single AU Evidence score dataflows are encoded with
the compact and clustered AU combination structures separately. These two type
of BOW features are used to train MIL (refered to as Compact-MIL) and MCIL
(referred to as Clustered-MCIL) learners respectively. Instances in a bag are
generated by two temporal aggregation methods, Sc-wind and Ncuts. We set the
multiple scaling window size at $30,40,50$ for Sc-wind. The size of segment in
a cluster is limited between $21$ and $81$ for Ncuts, and
$\sigma_{f}=0.1,\>\sigma_{t}=30$ in computing the frame correlation matrix.
The GM soft-max function is adopted for the approximation of $\max$ in MIL
training. The performance of MIL and MCIL learners are evaluated by accuracy
and area under curve (AUC). We use the work in [20], referred to as MS-MIL,
for comparison and the results based on a $10$-fold cross-validation are
summarized in Table II.

TABLE II: Comparison of the decoupled framework with MS-MIL [20] Framework | Sc-wind | Ncut
---|---|---
Accuracy($\%$) | AUC | Accuracy($\%$) | AUC
MS-MIL | 83.7 |  | 82.99 |
Compact-MIL | 83.96 | 0.875 | 85.04 | 0.9
Clustered-MCIL | 85.18 | 0.92 | 86.84 | 0.94

The best performance is achieved by the Clustered-MCIL framework in
conjunction with Ncuts. The improvement on classifier accuracy may be
attributed to the advantage of the decoupled structure. AU coding is a ’hard’
problem involving learning a mapping from a very high-dimensional pixel space
to the low-dimensional AU space in a complex environment. However, due to the
fact that AUs are evoked by certain facial muscular movement, reliable AU
coding can be achieved by a robust AFER system if trained on sufficient data.
On the other hand, pain is a subjective measure and can be viewed as a latent
variable. It will be easier to synthesis similarity from low-dimensional
features that are highly correlated to pain under the impact of problem
uncertainty. In addition, the improvement on AUC between Clustered-MCIL and
Compact-MIL could be attributed to the feature sparsity from the clustered
representation, which not only increases the margin on features but also
follows more naturally to a human coder’s decisions.

Next, we employ the Clustered-MCIL settings with GM approximation to train a
pain detector on UNBC-McMaster dataset and test it on video sequences from
selected patients in Wilkie’s dataset. Each test sequence is divided into $30$
subsequences and each subsequence has a duration of $20$ seconds. Three human
expert coders performed FACS coding on each subsequence and pain is identified
if any pain-related AU combination is detected by at least two human expert
coders in the original research. However, since the human coding does not
reveal pain intensity information and coders do not always agree with each
other, this ground truth is more suitable for qualitative analysis. $393$
subsequences from $27$ patients are selected for this experiment, where at
least $50\%$ of each subsequence is analyzable by Emotient. A subsequence is
considered as positive (pain) if AU combinations are coded by at least two
coders (more credible) or only one coder (less credible). A subsequence is
negative (no pain) if no AU combination is scored by any coder. As a result,
$82$ subsequences are identified as positive by at least two coders, $121$
subsequences are identified as negative by only one coder, and the rest $190$
subsequences are identified as negative samples. The automated system is used
as an independent coder and it checks the consistency between machine
prediction and human coder decisions and the results are summarized in Table
III. Note that the system has no prior knowledge about the test dataset. In
general, observe that the decisions of automated system are highly correlated
with that of a majority of human coders on both pain and no pain videos.
Additionally, a $68.6\%$ consistency rate is observed between the system and
the only coder, however this is more likely due to the ambiguity in less
credible videos rather than to the accuracy degradation. As a result, the
system shows its potential in pain assessment for long videos under clinical
settings and we will conduct further tests with focus on patient monitoring in
future research.

TABLE III: Comparison of machine prediction with human coder decision on subsequences in Wilkie’s dataset[23] | Subsequences
---
Label
| Pain
---
($2+$ coders
scored)
| Pain
---
($1$ coder
scored)
| No Pain
---
($3$ coders
agreed)
Human Coder | $82$ | $121$ | $190$
Automated System | $68$ | $83$ | $169$
Consistency Rate | $82.9\%$ | $68.6\%$ | $88.9\%$

##  VI Conclusion

This paper proposed and investigated the performance of automated pain
detection via spontaneous facial expressions in the context of clinical
applications. The proposed framework mimics the decision strategy of FACS-
certified human coder by following the procedures of _Facial Expression_
$\rightarrow$_AU_ $\rightarrow$_AU Combination_ $\rightarrow$_Pain_. To
address the challenge in accessing sufficient pain annotated video data, we
proposed a decoupled structure for automated pain detection task: 1) the AU
coding from facial expressions takes full advantage of the AFER development,
2) pain detection is based on simply low-dimensional features and handled by
MIL as a weakly supervised problem. The proposed system not only demonstrates
improvement on existing state-of-the-art work, but also shows adaptivity on
trans-dataset learning and long video analysis. In future work, we will
conduct comprehensive test on new pain oriented videos datasets and
investigate practical methodologies that facilitate clinical pain analysis
using our system.

## Acknowledgment

This research and publication was made possible in past by Grant Number P30
NR010680-S1 from the National Institutes of Health, National Institute of
Nursing Research. Its contents are solely the responsibility of the authors
and do not necessarily represent the official views of the National Institute
of Nursing Research.

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# On the Robustness of Monte Carlo Dropout Trained with Noisy Labels

Purvi Goel
Facebook
<EMAIL_ADDRESS>Li Chen
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<EMAIL_ADDRESS>
###### Abstract

The memorization effect of deep learning hinders its performance to
effectively generalize on test set when learning with noisy labels. Prior
study has discovered that epistemic uncertainty techniques are robust when
trained with noisy labels compared with neural networks without uncertainty
estimation. They obtain prolonged memorization effect and better
generalization performance under the adversarial setting of noisy labels. Due
to its superior performance amongst other selected epistemic uncertainty
methods under noisy labels, we focus on Monte Carlo Dropout (MCDropout) and
investigate why it is robust when trained with noisy labels. Through empirical
studies on datasets MNIST, CIFAR-10, Animal-10n, we deep dive into three
aspects of MCDropout under noisy label setting: 1. efficacy: understanding the
learning behavior and test accuracy of MCDropout when training set contains
artificially generated or naturally embedded label noise; 2. representation
volatility: studying the responsiveness of neurons by examining the mean and
standard deviation on each neuron’s activation; 3. network sparsity:
investigating the network support of MCDropout in comparison with
deterministic neural networks. Our findings suggest that MCDropout further
sparsifies and regularizes the deterministic neural networks and thus provides
higher robustness against noisy labels.

##  1 Introduction

Neural networks exhibit state-of-the-art performance on many learning tasks,
such as classification and segmentation. However, training these networks
requires an abundance of carefully labeled data; networks tend to overfit
quickly to noise in training labels, which makes their application to noisy
real-world problems less effective. Expert-labeled data is expensive and time-
consuming to collect; label noise is common in less carefully crafted datasets
due to measurement inaccuracies, human error, etc.

Nonetheless the latter type of data, albeit noisy, is much more readily
available and in much larger quantities. One recent strategy shown to perform
well on datasets containing significant amounts of label noise is augmenting
the neural network with an uncertainty estimation method like Monte Carlo
Dropout [5]. These uncertainty estimation models display a delayed
memorization effect of noisy training labels, and can generalize better to
clean test data. Augmenting models with Monte Carlo Dropout shows a slower
degradation of classification performance, consistent on benchmark datasets
like MNIST and CIFAR-10[5]. In addition to its resilient performance against
noisy training labels, MCDropout also does not add training overhead and only
adds minimal cost to inference time.

The robustness property and low-computational cost of MCDropout indicate it as
an effective and practical solution against noisy labels. In this paper, our
goal is to not only determine whether MCDropout performs consistently better
in these noisy-label situations, but also provide an in-depth analysis for
_why_ it performs better. We present an investigation into the performance and
latent representation learned by a model augmented with MCDropout. We first
evaluate the accuracy of MCDropout models in comparison with deterministic
neural networks on datasets like MNIST and CIFAR-10 with artificially injected
noisy labels and Animal-10n with natural annotation noise. Second, we measure
neuron responsiveness in each layer, to better explore the differences between
latent representations learned by certainty and MCDropout models. Finally we
study network sparsity and find that the sparsity property offered by
MCDropout models contribute to robustness against noisy training labels. To
our knowledge, our work provides the first detailed analysis of MCDropout in
the setting of noisy labels.

The rest of the paper is organized as follows. In Section 2, we provide the
background information on the noisy label setting, label noise taxonomies,
Monte Carlo Dropout and related work. In Section 3, we describe our study
directions including measuring efficacy, neuron responsiveness via volatility
and network sparsity. In Section 4, we demonstrate the effectiveness of
MCDropout on empirical datasets such as MNIST, CIFAR10 with artificially
corrupted training labels and Animal-10n a real-world dataset containing
annotation noise. We further analyze the neuron responsiveness and network
sparsity by MCDropout in comparison with deterministic networks. Finally in
Section 5, we discuss optimal placement for MCDropout on a neural network and
conclude our paper.

##  2 Preliminaries

In this section, we present the problem statement, the preliminaries on label
noise, Monte Carlo Dropout and related work.

We consider a fully supervised learning problem in image classification, where
the images and its associated labels in the training set, denoted by
$\mathcal{T}_{train}:=\\{(X_{i},Y_{i})\\}_{i=1}^{n}$, with $n$ denoting the
total number of training samples and all the pairs
$\\{(X_{i},Y_{i})\\}_{i=1}^{n}$ sampled i.i.d from a joint distribution
$F_{X,Y}$. However instead of observing all the correctly annotated labels, we
observe the training data
$\mathcal{T}_{tr}:=\\{(X_{i},\hat{Y}_{i})\\}_{i=1}^{n}$, where given by a
probabilistic process, $\hat{Y}_{i}$ deviates from $Y_{i}$. Our exploitation
task is to learn a robust classifier on
$\mathcal{T}_{tr}=\\{(X_{i},\hat{Y}_{i})\\}_{i=1}^{n}$ containing noisy labels
such that the classification efficacy on incoming test image $X$ can best
predict the unknown label $Y$.

Across this paper, we refer to the deterministic neural network without
uncertainty estimation as certainty model or deterministic model”
interchangeably. We refer to the neural network augmented with MCDropout
layers as the MCDropout model.

###  2.1 Label Noise Taxonomies

There are several categorizations of noise labels. One commonly used
categorization depends on whether or not the noisy label depends on the
features. If the noisy label generation process is conditionally independent
of the features, then a noise transition matrix $T_{c\times c}$, where $c$ is
the number of classes, is sufficient to describe the label noise generation
process. Each entry in $T_{ij}=p_{ij}$ is a probability such that the true
label will be changed into a noisy label with probability $p_{ij}$. If the
observed label is different from the true label with a uniform probability,
then the noise is considered to be label-independent and this noise is called
considered symmetric or uniform noise. If the observed label is changed from
the true label with probabilities depending on the original ground truth, then
the noise is label-dependent and called asymmetric noise. On the other hand,
if the corruption process depends on the features and labels, the label noise
is called instance-dependent. A more recent study proposes a new but practical
assumption within instance-dependent label noise, defined as part-dependent
label noise, where the noise depends partially on an instance [26].

Another perspective on label noise is via uncertainty characterization [5].
The noisy label generation process is probabilistic and random. Naturally
uncertainty characterization comes into play. From the notion of deep learning
uncertainty, the noise in the labels can be considered a type of aleatoric
uncertainty, a measurement of the intrinsic and irreducible uncertainty within
the data. Within aleatoric uncertainty, homoscedastic uncertainty is constant
across the input while heteroscedastic uncertainty is dependent on the input.
Hence if the noise transition matrix is a uniform or symmetric one, then the
label noise can be considered homoscedastic; if it is label-dependent, then
the label noise can be considered heteroscedastic. In recent noise simulation
schemes, label noise is applied on samples that are more likely to be
mislabeled given by pre-learned model [1]. We consider such type of noise as
epistemic uncertainty, a term that describes uncertainty induced by models.

###  2.2 Monte Carlo Dropout

The deep learning uncertainty perspective to characterize label noise inspires
us to study label noise via deep learning uncertainty estimation techniques.
Chen et al [5] proposed using epistemic uncertainty estimation methods when
learning with noisy labels. Comparing Monte Carlo Dropout, Bootstrap [15],
Bayesian CNN upon Bayes by Backprop [4] and certainty neural networks trained
in noisy label settings, the authors discovered that Monte Carlo Dropout
(MCDropout) had a prolonged memorization effect and possessed the best
classification performance on test set.

We also included Figure 1 as our motivational example here. Hence in this
paper, we laser-focus on the study of why MCDropout possesses robustness
against noisy labels in comparison with certainty models. In this section, we
provide the background information on MCDropout.

The core idea of MCDropout is to enable dropout regularization at both
training and test time. With multiple forward passes at inference time, the
prediction is not deterministic and can be used to estimate the posterior
distribution. As a result, MCDropout offers Bayesian interpretation. First
proposed in [8], the authors established the theoretical framework of
MCDropout as approximate Bayesian inference and proved MCDropout minimises the
Kullback–Leibler divergence between an approximate distribution and the
posterior of a deep Gaussian process. More formally, let $d_{l}$ denotes
dropout at the $l$-th layer of a neural network, where $d_{l}\sim
Bernoulli(p)$. Then at inference time, with $K$ forward passes, we obtained a
distribution of $K$ logits and predictions per test data, where we can compute
the expected value, standard deviation, variation ratio and entropy to assess
uncertainty.

###  2.3 Related Work on Deep Learning with Noisy Labels

While there has not been much work on applying epistemic uncertainty methods
to address noisy labels, an abundant of research has been done in deep
learning noisy labels ranging from loss function adjustment, robust
architecture design, data processing, data filtering and so on. Authors in
[10, 28, 25, 18] devised robust loss function to achieve a smaller risk for
unseen clean data when learning with noisy labels. Sample selection techniques
as to filter the clean labels for training and removing the noisy labels have
been proposed in [14, 11, 27, 20]. Sample selection and label correction for
spatial computing is studied in [6]. Devising loss to estimate noise
transition matrix and correct the labels are studied in [22, 12, 2]. Semi-
supervised learning is another field of techniques on noisy labels, where the
noisy labelled data are treated as unlabeled and clean labelled data are as
labeled [21, 7, 17].

Figure 1: (Left): MNIST test accuracy when training labels contain 15% noise. (Right): MNIST test accuracy when training labels contain 35% noise. Our previous study suggests that MCDropout has the best classification performance among a few other uncertainty estimation methods. Further MCDropout does not increase training time per epoch and has relatively cheap inference cost. Hence in this paper, we focus on investigating the robustness of MCDropout when training with noisy labels. Certain |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
MCDropout |  |  |  |  |  |  |  |  |  |  |  |  |
Figure 2:  Activation maps for our MNIST experiment. We show the activations from 10 neurons on a random image from MNIST’s test set, from the certain LeNet5’s second convolution layer (top) and the MCDropout model’s second convolution layer. Brighter values indicate higher activation values. The chosen layer contains 16 neurons total; for each model, we choose the 10 neurons with the highest mean activation value. The certain model activates much more strongly, while the MCDropout model’s activations are more muted, and spatially sparse.  Certain |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
MCDropout |  |  |  |  |  |  |  |  |  |  |  |  |
Figure 3: Activation maps for our CIFAR10 experiment. We show the activations from 10 neurons on a random image from CIFAR10’s test set, from the certain ConvNet’s first convolution layer (top) and the MCDropout model’s first convolution layer. The chosen layer contains 48 neurons total; for each model, we choose the 10 neurons with the highest mean activation value. The brighter the value at a pixel, the higher the activation at that point. We see on visual inspection that MCDropout has smaller overall activations: there are no bright yellow areas in the bottom row’s feature maps, while we do for several maps in the top row. In addition, the activations are more spatially sparse: many of the displayed MCDropout model’s featuremaps have entire regions with no activations. This is not as noticeable in the certain model.  Certain |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
MCDropout |  |  |  |  |  |  |  |  |  |  |  |  |
Figure 4:  Activation maps for our Animal-10n experiment. We show the
activations from 10 neurons on a random image from Animal-10n’s test set, from
the certain ConvNet’s third convolution layer (top) and the MCDropout model’s
third convolution layer. Brighter values indicate higher activation values.
The chosen layer contains 196 neurons total; for each model, we choose the 10
neurons with the highest mean activation value. While less obvious than the
comparison in Figure 3 and Figure 2, the certain model still shows higher and
less spatially sparse activations than the MCDropout model.

##  3 Investigation

Our goal is to analyze the latent representations learned by MCDropout models,
particularly in comparison with certainty models, trained in the presence of
noisy labels. Similar to the definitions presented in Bau et al [3], we use
the term _representation_ to describe the outputs of a particular layer in a
model. More specifically: which channels of the layer have been activated for
various data inputs? How strongly have these channels been activated? What is
the variation in a specific channel’s possible activations? Comparing the
representations lends insight and intuition as to why one model may perform
better than another one. Essentially, we investigate why MCDropout performs
better than a certainty model by comparing the different latent
representations learned by the two models respectively.

Again following the vocabulary used in Bau et al [3], we refer to feature maps
as the output of every layer in the network–the aggregate of the feature maps
makes up the network’s learned representation. We refer to a neuron as a
specific channel of the feature map. In this paper, we use the term
_activation gamut_ to refer to all the possible values that a particular
neuron can produce. We can approximate the activation gamut as the set of a
neuron’s activation values for each image in a dataset.

We compare the classification efficacy, neuron responsiveness, and network
sparsity by the two models respectively. To understand how two models have
learned and encoded information differently, we evaluate trained models on
test set and cache neuron activations from each layer, where we derive
statistics such as mean and standard deviations on each neuron with respect to
data samples from the test set.

|  |
---|---|---
Figure 5: We show the train and test accuracies per epoch for (Left) the
certain and MCDropout LeNet5 models on MNIST; (middle) the certain and
MCDropout ConvNet models on CIFAR10; (right) the certain and MCDropout ConvNet
models on Animal-10n. All of these models are trained in a noisy-label setting
with 35 percent noise. In all these cases, the MCDropout model has a better
validation accuracy than the certain model; the certain model generally
overfits to training set noise.

###  3.1 Measuring Efficacy

We first train MCDropout and certainty models on training data with noisy
labels and evaluate their accuracy on a cleanly labeled test set. We present
the learning behaviors during training and testing over epochs.

###  3.2 Measuring Responsiveness

Next we compare neuron responsiveness measured by volatility in the two
models. We define volatility as the standard deviation of a neuron’s
activations over a dataset; if a neuron is capable of producing vastly
different activation values for different input images, the neuron’s
activation gamut would possess high standard deviation, indicating a highly
responsive neuron.

To compute the activation gamut of a neuron, we first cache the feature maps
$u_{j}^{i}$ , post-ReLu, produced by the $i$-th neuron on the $j$-th test set
image. We find the mean activation value, $a_{i}^{j}$ for the feature map
$u_{j}^{i}$. In other words, for a feature map with $n$ rows and $m$ columns,

$a_{i}^{j}=\frac{\sum_{r=0}^{n}\sum_{c=0}^{m}u_{i}^{j}(r,c)}{nm}.$

Per neuron, this results in $j$ values which compose its activation gamut
$A_{i}$

$A_{i}=\\{a_{i}^{0},a_{i}^{1}...a_{i}^{j}\\}.$

We can perform statistical analysis on these gamuts and aggregate them per-
layer, such as finding the mean activation value $V_{l}$ of all $I$ neurons in
the $l$-th network layer:

$V_{l}=\frac{\sum_{i=0}^{I}mean(A_{i})}{I}.$

We also find the average gamut standard deviation $S_{l}$ for all $I$ neurons
in the $l$-th network layer

$S_{l}=\frac{\sum_{i=0}^{I}std(A_{i})}{I}.$

We would observe the activation gamut of a volatile neuron to possess a higher
standard deviation than that of a nonvolatile neuron. The activation gamut of
a volatile neuron may also include extremes, showing a higher maximum
activation than a non-volatile neuron.

###  3.3 Measuring Sparsity

Along with research directions into network uncertainty and robustness, neural
network sparsity has become a subject of interest for many machine learning
researchers [9, 13, 24]. Sparse neural networks are desirable because they
require less computation at test time, demand less memory [9], and are less
likely to overfit to training data [19]. In the context of our investigation,
the tendency for sparse neural networks to overfit more slowly to training
data can allow them to avoid memorizing noisy training labels. We can evaluate
network sparsity on a per-neuron level: which neurons never or rarely
activate, for any and all test samples, and how common are these neurons
throughout the entire model? Network sparsity can be defined as the subset of
neurons output a value that is always zero [9], or very close: these neurons
do not affect the final predictions in any significant manner. The larger the
subset of neurons with this property, the more sparse the network’s learned
representation is. Because neural networks can easily overfit to noise in
training labels [5], we are interested in the observed property of sparse
models to overfit more slowly. With fewer tunable parameters available, sparse
models have fewer degrees of freedom to overfit to noise.

##  4 Results

Metric | Model | conv0 | conv1 | fc1 | fc2 | fc3
---|---|---|---|---|---|---
Activation STD | Certain MNIST | 0.215 | 0.5367 | 2.386 | 1.335 | 1.5733
| MCDropout MNIST | 0.0646 | 0.1085 | 0.4207 | 0.2144 | 0.9182
Activation Mean | Certain MNIST | 1.009 | 1.3936 | 1.4381 | 0.8383 | -0.0324
| MCDropout MNIST | 0.2443 | 0.2041 | 0.1567 | 0.1208 | -0.498
Unresponsive neurons | Certain MNIST | 0.0 | 0.0 | 0.0916 | 0.0 | 0.0
| MCDropout MNIST | 0.1666 | 0.25 | 0.5083 | 0.2023 | 0.0
Table 1:  Quantitative metrics for our MNIST experiment, with the same table layout as described in Table 2. For all layers, the certain model’s activations posses higher means and standard deviations, while the uncertain model has more relatively unresponsive neurons and lower activation values. Notice how several of the layers in the certain model have _no_ relatively unresponsive neurons. Metric | Model | conv0 | conv1 | conv2 | conv3 | fc1 | fc2 | fc3
---|---|---|---|---|---|---|---|---
Activation STD | Certain ConvNet | 0.0602 | 0.0343 | 0.1715 | 0.1279 | 7.3578 | 11.8364 | 6.5248
| MCDropout ConvNet | 0.047 | 0.0123 | 0.0708 | 0.0871 | 4.3155 | 9.2449 | 4.480
Activation Mean | Certain ConvNet | 0.0818 | 0.04378 | 0.238 | 0.106 | 1.6077 | 5.038 | 0.075
| MCDropout ConvNet | 0.060 | 0.0149 | 0.091 | 0.0616 | 0.4894 | 3.340 | -2.424
Unresponsive neurons | Certain ConvNet | 0.4166 | 0.4583 | 0.4323 | 0.4414 | 0.5449 | 0.2031 | 0.0
| MCDropout ConvNet | 0.4583 | 0.71875 | 0.7083 | 0.6172 | 0.7851 | 0.0781 | 0.0
Table 2: Quantitative metrics for our CIFAR10 experiment. We show the standard deviation of the mean activation value per neuron (top), the average of the mean activation value per neuron (middle), and a ratio of the ConvNet layer’s neurons that are relatively unresponsive (bottom). We accumulate these metrics across all test samples. In general, the certain model’s layer activations have a higher standard deviation and higher mean (suggesting greater volatility). The uncertain model is less volatile and has more unresponsive neurons (suggesting sparsity). Metric | Model | conv0 | conv1 | conv2 | conv3 | fc1 | fc2 | fc3
---|---|---|---|---|---|---|---|---
Activation STD | Certain ConvNet | 0.2037 | 0.0202 | 0.0596 | 0.0304 | 2.077 | 5.071 | 5.695
| MCDropout ConvNet | 0.0191 | 0.0132 | 0.0277 | 0.0301 | 1.6326 | 4.032 | 3.8372
Activation Mean | Certain ConvNet | 0.0217 | 0.0146 | 0.0599 | 0.0191 | 0.4833 | 3.017 | -2.256
| MCDropout ConvNet | 0.0172 | 0.0086 | 0.0265 | 0.0148 | 0.210 | 1.534 | -1.9296
Unresponsive neurons | Certain ConvNet | 0.625 | 0.6354 | 0.4583 | 0.6367 | 0.4804 | 0.1094 | 0.0
| MCDropout ConvNet | 0.6666 | 0.7708 | 0.7448 | 0.4687 | 0.6679 | 0.5 | 0.0
Table 3: Quantitative metrics for our Animal-10n experiment, with the same
table layout as described in Table 2. While less exaggerated than results
shown on MNIST in Table 1 and CIFAR in Table 2, we see a similar trend. For
the majority of layers, the certain model is more volatile, with a larger
gamut of possible activation values and higher overall magnitude of
activations, while the uncertain model has more relatively unresponsive
neurons and a sparser representation.

We study the classification efficacy, neuron responsiveness, and network
sparsity of MCDropout and certainty model on three benchmark classification
datasets: MNIST, CIFAR10, and Animal-10n [23]. We use two different
architectures: LeNet5 [16] and ConvNet, a convolutional neural network
architecture with 4 convolutional layers followed by 3 fully connected layers.
To maximize the effect of MCDropout, we use an all-layer MCDropout
architecture where each layer in the certainty model is augmented with
MCDropout. Our investigation compares the findings for the original certainty
model and its augmented all-layer MCDropout model.

We train both models on noisy training labels. Because we are evaluating these
models on a classification task, mislabeled data simply means that training
samples labeled with the incorrect class. For MNIST and CIFAR10, we use a
uniform noise simulation scheme to add noise to 35% of our training labels: in
this scheme, each corrupted label has an equal chance of 35% being mislabeled
as any of the other classes. Once training is complete, we run our trained
models on clean test data and compute all the neurons’ individual activation
maps after the application of an activation function. All of our chosen
architectures use ReLu as their activation.

###  4.1 Classification Efficacy

We compare the performance of the certainty model and MCDropout models trained
on noisy data and plot their training and accuracy curves over time 5. We see
an emerging trend consistent across all models and datasets. The certain model
overfits to the noise in the training data and results in a similar or higher
final training accuracy than the MCDropout model. However, the MCDropout model
consistently produces a higher validation accuracy.

Next we investigate why MCDropout outperforms certainty by analyzing the
representations learned by both models. Consider the results on MNIST shown in
5, left. Given that the training accuracies of the certainty and MCDropout
model are quite similar after 100 epochs, both models are clearly learning
_something_. However given the vastly different test accuracies between the
two models–the uncertain models undoubtedly generalize better to the test
dataset–the models are representing information _differently_.

###  4.2 Neuron Responsiveness Measured by Volatility

Next, we compare the volatility of the neurons in the uncertain and certain
models. We do so with two strategies. As mentioned earlier, we cache each
neuron’s activation map for every image in the test set. We find the mean
activation value for each feature map. To measure volatility, we compare two
statistics per layer: the standard deviation of the layer’s mean activation
values and the mean of the layer’s mean activation values.

We show the results of this investigation for each dataset in Table 1 for
MNIST, Table 2 for CIFAR10 and Table 3 for Animal-10n. With the exception of a
few layers, the neurons in the certainty models activate more strongly: the
mean activation is higher for each layer and with greater variation: the
standard deviation of the activations is higher for each layer.

|  |  |
---|---|---|---
|  |  |
Figure 6: Histograms for our MNIST experiment, with the same set-up described in Figure 7. We show 8 neurons from the second convolution layer. Several neurons from the uncertain model (orange) are unresponsive to all inputs: notice how many of the uncertain activation distributions are centered around 0, with shorter tails. This suggests that the uncertain model is sparser. We include a quantitative summary of activation values over the entire test dataset in Table 1 |  |  |
---|---|---|---
|  |  |
Figure 7: Histograms for our CIFAR10 experiment. We gather the mean activation
values per neuron for each image in the CIFAR10 test set, for both certain and
MCDropout models. We show 8 neurons from the 1st convolution layer; for each
neuron, we plot a histogram of the mean activation values. Mean activations
from the certain model are in blue; uncertain model are in orange. Notice how
the range of the uncertain model’s distributions are smaller: uncertain
neurons produce a smaller gamut of possible mean activations. The uncertain
model’s distributions tend to be centered on or near 0. This suggests several
of the uncertain model’s neurons produce no activation no matter the input
image. We choose 8 neurons to fit in page requirements; we include a
quantitative summary of these metrics in Table 2.

|  |  |
---|---|---|---
|  |  |

Figure 8: Histograms for our Animal-10n experiment, with the same set-up
described in Figure 7. We show neurons from the third convolution layer.
Similar to Figure 7, several neurons from the uncertain model are unresponsive
(result in mean activations near 0) to all inputs. This suggests that the
uncertain model is sparser. We include a quantitative summary of activation
values over the entire dataset in Table 3

.

###  4.3 Network Sparsity

We can compare the sparsity via analysis of the neurons’ individual feature
maps. Sparser models contain more neurons whose feature maps have values close
to some constant c, usually 0, no matter the input sample from the test set.

For qualitative evaluation, we visualize the post-activation feature map of
individual neuron for a given sample of testing data. We can visually compare
how many neurons seem near constant or activate in only small patches of the
map. We show heatmaps from various layers in Figure 2 (MNIST), Figure 3
(CIFAR10), and Figure 4 (Animal-10n). In all cases, activation maps from
MCDropout models have spatially sparse activations: when they do activate, it
is in tight, localized regions, and large patches of each activation map
remain inactivated. In addition, several of the MCDropout activation maps show
very little activation at all. We also calculate the mean activation value per
neuron per image in each experiment’s test dataset. We plot per-neuron
histograms of these mean activation values for the certain and MCDropout
models in Figures 6 (MNIST), 7 (CIFAR10), and 8 (Animal-10n). We can then
compare the mean and support of the resulting activation distributions: in
many cases, the distributions from MCDropout models possess smaller supports
and are centered more closely around a mean activation value of 0.0. This
provides an intuitive understanding of why MCDropout is more robust against
noisy labels: the neurons that may be influenced by noisy labels in the
certainty model are not activated in MCDropout models. MCDropout layers
provide regularization against these “corrupted” neurons.

These qualitative traits show that the MCDropout model’s learned
representation is more sparse. For a quantitative analysis, we can count how
many neurons are “relatively unresponsive” based on their gamut of possible
activations for all the test images. Neurons that rarely activate–that is, the
mean of their activations for all images on the test set falls below some
epsilon threshold–are tallied in the final row of Tables 1 (MNIST), 2
(CIFAR10), and 3 (Animal-10n). We report these numbers as the ratio of
“relatively unresponsive neurons” to the total number of neurons in the layer.
The results show that the major of the MCDropout models’ layers have more dead
neurons than corresponding layers in the certain model does. This indicates
that the uncertain model has learned a more sparse representation.

##  5 Discussion

We have compared the representations, on a per-neuron level, learned by
MCDropout models and certainty models when trained with noisy labels. The
representation learned by MCDropout representation is less volatile but more
sparse, an apt justification for its greater effectiveness and generalization
in noisy-label scenarios. MCDropout provides regularization so that neurons
are not overly influenced by the noisy labels; as a result, these neurons are
not activated at test time, thus contributing to the robustness against noisy
label training. With fewer free parameters to over-explain training label
noise, MCDropout models forge representations that are less capable of
overfitting to noisy labels.

Our larger goal in this investigation is not to build state-of-the-art models
on any of the presented datasets or find the best network to deal with noisy
labels. Rather, we investigate interpretable metrics and observations from the
learned representations of models that identify why MCDropout model
outperforms certainty model.

In our experiment, we primarily analyzed all-layered MCDropout for the purpose
of maximizing the MCDropout effect. However we acknowledge there are other
different configurations of uncertainty placement. As seen in Figure 9, we
further analyze different MCDropout placement configurations on MNIST dataset
and discover that MCDropout on all layers possesses the best test
classification accuracy when training with noisy labels. Such behavior is
consistent with the theoretical establishment that all-layered MCDropout best
approximates Bayesian neural network. While other configurations such as
converting only convolutional layers, internal layers, final layers, etc., to
MCDropout layers still outperform the certainty model, the best-performing
model benefits from the most number of MCDropout layers. We believe research
directions on an optimal trade off between classification performance and
MCDropout layer placement is critical for noisy-label training with
constraints on memory or inference time.

Figure 9: We further analyze different MCDropout placement configurations and
discover that MCDropout on all layers (black line) possesses the optimal
results against noisy labels, particularly compared to the certain model
(yellow line).

We hope this investigation helps us ask and answer more questions. It would be
interesting to see if our observations about volatility and sparsity hold for
other uncertainty estimation or ensemble methods like Bootstrap [15] or Bayes
by Backprop [4]. These observations may also help us tune MCDropout-related
hyperparameters, such as the best locations to place MCDropout layer–or layers
capable of uncertainty estimation in general–in a model architecture.

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