# Nested Multiple Instance Learning with Attention Mechanisms

###### Abstract

Multiple instance learning (MIL) is a type of weakly supervised learning where
multiple instances of data with unknown labels are sorted into bags. Since
knowledge about the individual instances is incomplete, labels are assigned to
the bags containing the instances. While this method fits diverse applications
were labelled data is scarce, it lacks depth for solving more complex
scenarios where associations between sets of instances have to be made, like
finding relevant regions of interest in an image or detecting events in a set
of time-series signals. Nested MIL considers labelled bags within bags, where
only the outermost bag is labelled and inner-bags and instances are
represented as latent labels. In addition, we propose using an attention
mechanism to add interpretability, providing awareness into the impact of each
instance to the weak bag label. Experiments in classical image datasets show
that our proposed model provides high accuracy performance as well as spotting
relevant instances on image regions.

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Index Terms—  Weakly supervised learning, multiple instance learning,
attention mechanism, interpretability, feature learning

##  1 Introduction

In recent years deep neural networks (DNN) have provided state of the art
results in many applications. Most emphasis has been on fully-supervised
learning which requires a large amount of labelled training and validation
data for all machine learning techniques and even more so for DNN. It can be
challenging and time consuming, and sometimes infeasible, to produce such
large labelled datasets. For classification of images a single label states
the class for the entire image, for which ImageNet is an example of a large
image set with such labels [1]. But if the task is to localize a region within
the image, then a more detailed label is needed for learning in a straight
forward supervised manner.

In some applications it can also be that a detailed label is not possible to
produce, like a medical image where follow-up information of a patient is
known, i.e. true prognosis label, but it is unknown if it is possible to
predict that prognosis based on the particular image/data. Such labels can be
regarded as weak labels if the task is to find regions in the image. In
general, in the medical domain datasets, images thoroughly annotated with
regions of interest (ROIs) from medical experts are often not available, and
only clinical information for the patient is given. The research community is
working on ways of learning good machine learning and DNN models less
dependent on labelled data by unsupervised, semi-supervised learning, and
based on limited, lacking, noisy and weak labels.

Multiple instance learning (MIL) refers to a learning method where a number of
samples, called instances, are individually unlabelled [2]. However, there
exist a label for a group of such instances, called a bag. Usually in MIL a
bag-label is defined as positive if at least one of the instances are
positive, and negative if all instances are negative, we will refer to this as
the MIL assumption. An example of instances, bags and bag-labels can be to let
the diagnosis of a histopathological image be the bag-label, patches of the
image can be regarded as instances, and the whole image as a bag. Histological
whole slide images (WSI) are in general very large, and under the MIL
assumption, having one patch considered positive will see the whole image as
positive. Sometimes this is not a good model, and a more complex structure of
the instance-bag relationship is needed. One way to handle more complex
scenarios is extending MIL by assembling a bag-of-bags, which can be referred
to as nested MIL (NMIL).

Even if such a system can learn complex scenarios, the impact of individual
instances or inner-bags will be lost with a straightforward setup of MIL or
NMIL, resulting in a low degree of explainability and applicability for
interpreting a particular decision, which has been one of the major focus in
deep learning recently [3, 4]. Attention mechanisms can be integrated in the
DNN structure to provide a degree of explainability while showing comparable
or improved performance on their predictions. Attention scores generated
during a forward pass reveal how input features are weighted, and can be
visualized as a magnitude value of the influence on the final decision that
would not have been observable otherwise [5, 6, 7, 8].

In this paper we propose a method for handling learning with weak labels where
the relationship between the detailed instance labels and the weak label is
complex, by doing nested multiple instance learning, and where a degree of
interpretability in terms of understanding which instances are most relevant
for the weak label is offered by an attention mechanism, motivated by the task
of finding a region in an image with multiple positive instances.

##  2 Related work

Several applications have been developed that use MIL for overcoming the lack
of annotated regions of interest in an image [9, 10]. Chen et al [11] train
end-to-end feeding an entire WSI in a strong supervision manner, adopting a
variant of MIL. To provide further intuition into the composition of the bags
and the relevance that individual instances carry in the classification,
Chikontwe et al [12] propose a center loss that characterizes intra-class
variations by minimizing the distance among instances from the same class. Li
et al [13] propose a dual stream architecture to learn instance and bag
classifiers at once, where the first instance would be an instance classifier
and the second stream aggregates the instances into a bag embedding which is
in fact a bag classifier. Also, He et al [14] use a clustering-based strategy
for obtaining hidden structure information in the feature space in order to
discover positive instances. These methods work well under the MIL assumption,
but would not necessarily be able to understand more complex scenarios. The
concept of bag-of-bags is first seen in an application for prostate cancer
detection using magnetic resonance images [15], but the concept of a nested
architecture for MIL is later introduced by Tibo et al [16, 17], where they
present the use of bag layers to aggregate instance-level representation into
a bag-level representation.

Attention mechanisms were introduced into weakly supervised methods to give
insight into the model’s decision making and ability to pick out instances of
interest [18, 19, 20, 21, 22, 23]. These architectures use an attention
mechanism to distinguish subgroups of instances within a bag, for which the
individual instance representations are aggregated to obtain a single bag
representation and collapse into a supervised learning problem.

In this work we are further developing the NMIL for complex scenarios and
combining it with the attention mechanism for interpretability, motivated from
the application of finding a specific diagnostic region in a large WSI. In the
experiments carried out, we show that the attention mechanism correctly
identifies key instances also in complex scenarios where simple MIL
architectures working under the MIL assumption fails.

##  3 Methodology

###  3.1 Nested Multiple Instance Learning

Multiple instance learning (MIL) is a weakly supervised method trained in a
supervised manner considering outermost bags and their corresponding labels.
For the conventional MIL binary setting, a dataset
$\mathcal{X},\mathcal{Y}=\left\\{(X^{i},y^{i}),\forall i=1,...,N\right\\}$ is
formed of pairs of sample sets $X$ and their corresponding labels $y$, where
$i$ denotes the current sample for a total of $N$ samples. A sample $X$
consists of a bag of instances:

$X=\left\\{x_{l},\forall l=1,...,L\right\\}$ (1)

where $L$ is the number of instances in the bag, alongside with a label
$y\in\left\\{0,1\right\\}$ associated with the bag. Although each instance
might be associated with an unknown label $y_{l}$, they are in general unknown
hence only the bag label is used during training. At the inference stage, a
test set might be associated with labels both at bag and instance level, to be
able to provide performance metrics. Under the conventional binary
classification MIL assumption, a bag label is positive with the single
presence of a positive instance, expressed as:

$y=\underset{l}{max}\\{y_{l}\\}$ (2)

In contrast with conventional MIL, nested MIL setting consists of levels of
bags within bags where only the innermost bags contain instances. NMIL attains
an advantage from MIL when complex scenarios are presented and the arrangement
of subsets of instances within a bag are to be considered to compute a bag
prediction. In Figure 1 the idea of grouping instances in inner-bags, inner-
bags in larger bags, and finally in one outermost bag is illustrated.

Fig. 1: Nested bag-of-bags. Instances (crosses and circles) are drawn into the
innermost bags to form sets of them which are recursively grouped within the
outermost bag.

Let $J$ denote the number of nested levels, $K_{j}$ number of bags at level
$j$ and $L_{j,k}$ denote the number of instances or bags in a bag $k$. For
$J=1$, i.e. NMIL with one level of nesting, the NMIL notation corresponds to
the ordinary MIL notation described in Equation (1). A level set $X_{j}$
contains a set of bags of instances $X_{j,k}$:

$X_{j}=\left\\{X_{j,k},\forall k=1,...,K_{j}\right\\}$ (3)

for $j$ defining the current nesting level up to $J$ levels of nesting, and
the current bag at that level $j$, which maximum value $K_{j}$ is an arbitrary
value that can vary from level to level. For a given bag $k$ at level $j$, a
bag of instances $X_{j,k}$ is expressed as:

$X_{j,k}=\left\\{x_{j,k,l},\forall l=1,...,L_{j,k}\right\\}$ (4)

where $l$ defines the instance number up to $L_{j,k}$ for a bag $k$ at a level
$j$. Subsequently, a set of latent labels of instances $Y_{j,k}$ in a bag $k$
and level $j$ are defined as:

$Y_{j,k}=\left\\{y_{j,k,l},\forall l=1,...,L_{j,k}\right\\}$ (5)

For simplicity in the experiments section, we will refer to an instance latent
label $y_{j,k,l}$ as $y^{j}_{l}$ to describe the latent label of an instance
$l$ in a level $j$, omitting bag index $k$.

###  3.2 Attention Mechanism

Following the lines of Ilse et al [21], a multiple instance block is
constructed using an embedding-level approach to obtain a bag-level
representation from the instances within, as depicted in Figure 2. The
proposed attention module’s input corresponds to low-dimensional embeddings
generated by a feature extractor or by bag embedding representations of
previous levels. In order to obtain a bag representation, this module observes
those embeddings and compute an attention score that leverages the
meaningfulness of the features extracted for the given task. The sum of the
scores must be equal to 1 to work even with arbitrary bag sizes.

Fig. 2: Multiple instance block. Input embeddings are fed into the attention
module to compute an attention score. Those scores are used to compute a
weighted representation of the themselves based on the attention score given.
Then, weighted embeddings are aggregated to create a final bag embedding.

For ease of exposition, we define indexes $\delta=(j,k,l)$,
${{\delta}^{\prime}}=(j+1,{k}^{\prime},{l}^{\prime})$, $\gamma=(j,k)$ and omit
the use of training sample index $i$. An attention score $a_{\delta}$ for a
given input embedding $x_{\delta}$ is calculated as:

$a_{\delta}=\frac{\mathrm{exp}\\{\mathbf{w}^{\top}(\mathrm{tanh}(\mathbf{Vx_{\delta}^{\top}})\odot\mathrm{sigm}(\mathbf{Ux_{\delta}^{\top})})\\}}{\sum_{l=1}^{M_{\gamma}}\mathrm{exp}\\{\mathbf{w}^{\top}(\mathrm{tanh}(\mathbf{Vx_{\delta}^{\top}})\odot\mathrm{sigm}(\mathbf{Ux_{\delta}^{\top})})\\}}$
(6)

where $\mathbf{w}\in\mathbb{R}^{L\times 1}$, $\mathbf{V}\in\mathbb{R}^{L\times
M}$ and $\mathbf{U}\in\mathbb{R}^{L\times M}$ are trainable parameters and
$\odot$ is an element-wise multiplication. Furthermore, the hyperbolic tangent
$\mathrm{tanh}(\cdot)$ and sigmoid $\mathrm{sigm}(\cdot)$ are included to
introduce non-linearity for learning complex applications. Finally, an
aggregation function $\Xi$ will transform a leveraged bag of embeddings
$\mathbf{\tilde{X}_{\gamma}}$ to obtain a bag representation
$\mathbf{x_{{\delta}^{\prime}}}$ as:

$\mathbf{x_{{\delta}^{\prime}}}=\Xi(\mathbf{\tilde{X}_{\gamma}})=\Xi(a_{\delta}\cdot\mathbf{x_{\delta}})$
(7)

where $\Xi$ can be replaced by either the $mean$ or $max$ operators.

Fig. 3: Nested MIL model. The feature extractor (light blue) projects all
instances into low-dimensional embeddings and consecutive multiple instance
blocks (purple) aggregate deeper levels into more superficial representations
until a bag-of-bags embedding is fed to the classifier (light red) for
obtaining a bag prediction, $\hat{y}$. Several MI blocks and feature
extractors are depicted per level, yet there is just one. To extend MIL into
NMIL with J levels of nesting J multiple instance blocks are required.

##  4 NMIL ARCHITECTURE

The proposed neural network architecture combines a recursive bag processing
of low-dimensional embeddings along with an attention mechanism for which a
final bag representation is formed and a prediction is given by a classifier.
The proposed multiple instance block computes attention scores for each of the
input embeddings and aggregates them to create an embedding that represents
the contents of the bag.

Figures 2 and 3 show the building block and the model architecture for an
independent NMIL framework, respectively. As in the conventional MIL setting,
all instances are projected into embeddings, which later are weighted and
aggregated into a bag-of-bags embedding after passing through the MI blocks.
Finally, a classifier predicts the label $\hat{y}$ for the sample $X$.

##  5 EXPERIMENTAL SETUP

Several experiments were carried out to show the usefulness of NMIL against
MIL under different types of data and tasks. Also, we show the importance of
attention-based models to obtain further interpretability from meaningful
instances and comparing the performance to regular aggregation techniques.
MNIST [24] and PCAM [25, 26] image datasets were used in the experiments. All
models were trained using stochastic gradient descent (SGD) optimizer, binary
cross-entropy loss function and early stopping. The models are implemented in
Python 3.6 using Tensorflow machine learning library.

We have conducted three experiments as follows. In the two first, we have
considered as the positive instance class $y^{1}_{l^{+}}$ the digit 9 for
MNIST and tiles with metastatic tissue for PCAM. Dataset samples were arranged
to form a 2-level setting. The third experiment was carried out exclusively
for MNIST on a 3-level setting. Latent labels introduced in intermediate
levels are formulated to obtain the resulting bag-of-bags label exclusively
for the train and test set, and therefore are unknown to the model, which is
trained entirely on weak bag-of-bags labels. The bag-of-bags embedding
classifier is trained in a fully-supervised manner due to the nature of the
model’s architecture, which encapsulates all information from several
instances into a single representation, hence matching one bag-of-bags
embedding to a weak label. Then, we evaluate if the model might learn these
inherent arrangements and reflect it on the attention scores.

Experiment 1: A dataset is constructed following the MIL assumption described
in Equation (2). In this assumption there is nothing to gain in using NMIL,
but we want to show that NMIL is flexible and will perform comparably to MIL.
Conventional MIL with and without attention is compared to a 2-level NMIL
model with random grouping of the bags at the first level.

Experiment 2: A dataset is constructed such that at least 2 instances from
from the same inner-bag have to be positive for the weak bag-of-bags label to
be positive. We do not expect a model trained as ordinary MIL to be able to
solve this task. A 2-level NMIL model, however, can solve this if the inner-
bag grouping is meaningful. For example, in region-based analysis of medical
images, an object belonging to a positive class is located in a certain region
and not scattered across the entire image. Therefore, a certain positive
region of the image will contain several positive instances, but not others.
Regions containing few positives are regarded as noise or missclassified
instances and they should not be reflected in the overall prediction.

$\begin{matrix}\mathbf{1^{st}}&\mathbf{2^{nd}}\\\
y^{2}_{l}=\left\\{\begin{matrix}0,\;\;\\#y^{1}_{l^{+}}\leq\,1\\\
1,\;\;\\#y^{1}_{l^{+}}\,>1\end{matrix}\right.&y=\left\\{\begin{matrix}0,\;\;y^{2}_{l}\notin
1\\\ 1,\;\;y^{2}_{l}\in 1\end{matrix}\right.\par\end{matrix}$ (8)

Fig. 4: From left to right, 1, 2 and 3-level partitioning. Red dotted lines
separate bags-of-instances at a second level, while greens at a third level.
The task of experiment 3 is to find out if there exist a second level region
that contains at least one first level region with odd numbers but no first
level regions containing only even numbers.

Experiment 3: The goal of this experiment is to prove that a nested
implementation performs in complex scenarios and it is easily adaptable to the
data arrangement required to solve the task at hand. As depicted in Figure 4,
we can observe how the partitioning of a set of instances can influence a
final weak label. In order to do that, a dataset consisting of MNIST images is
constructed following the latent labels arrangement described in Equation (9).
If we want to find a ROI that contains a bag of odd numbers and not one of
even numbers, a 2-level solution is not enough to overcome this task, hence 3
levels of nesting are required. Such a scenario is not possible to learn using
MIL as proven in experiment 2, thus corresponding MIL tests were never carried
out.

$\begin{matrix}\begin{matrix}\mathbf{1^{st}}&\mathbf{2^{nd}}\\\
y^{2}_{l}=\left\\{\begin{matrix}0,\;\;y^{1}_{l}\in\left\\{0,2,4,6,8\right\\}\\\
1,\;\;y^{1}_{l}\in\left\\{1,3,5,7,9\right\\}\\\
2,\;\;otherwise\;\;\;\;\;\;\;\;\;\;\;\end{matrix}\right.&y^{3}_{l}=\left\\{\begin{matrix}0,\;\;y^{2}_{l}\in\\{0\cap\bar{1}\\}\\\
1,\;\;y^{2}_{l}\in\\{\bar{0}\cap 1\\}\\\
2,\;\;otherwise\;\;\;\end{matrix}\right.\end{matrix}\\\
\begin{matrix}\;\;\;\;\;\;\;\mathbf{3^{rd}}\\\
\;\;\;\;\;\;\;y=\left\\{\begin{matrix}0,\;\;y^{3}_{l}\notin 1\\\
1,\;\;y^{3}_{l}\in 1\end{matrix}\right.\end{matrix}\end{matrix}$ (9)

##  6 RESULTS AND DISCUSSION

Result metrics for the experiments are listed in Table 1. For experiment 1, it
is observed that the choice of architecture does not have an effect on the
predictive power of the classifier. This is expected as nesting becomes
irrelevant when only the individual labels of the instances are meaningful but
not the arrangement of them across inner-bags. However, for experiment 2, MIL
is not able to comprehend that within the bag there are requirements to be met
between inner sets of instances. From the MIL model perspective, all instances
are at the same level and belong to the same set. NMIL is able to process
independently these subsets, hence grasping the true nature of the task, which
is shown in the computed metrics.

Table 1: F1 scores for experiments on MNIST and PCAM datasets. |  | MNIST | PCAM
---|---|---|---
|  | Exp1 | Exp2 | Exp3 | Exp1 | Exp2
MIL | w/o Att | 0.929 | 0.345 | N/A | 0.957 | 0.290
w/ Att | 0.957 | 0.472 | N/A | 0.973 | 0.286
NMIL | w/o Att | 0.923 | 0.855 | 0.556 | 0.964 | 0.700
w/ Att | 0.959 | 0.921 | 0.836 | 0.978 | 0.734

Moreover, implementing an attention mechanism provides an edge over the model
that does not, both on performance and interpretability, as shown in Figure 5.
Attention scores provide insight into the model’s decision-making and reflect
on the importance given on each of the processed elements, thus strengthening
positive features while diminishing the impact of negative ones. We can
observe that the attention mechanism is not only able to correctly identify
the positive instances at instance level, but also recognizes the positive
inner-bags, distinguishing them from the ones that contain noisy instances.
For medical imaging, this is particularly interesting for finding ROIs within
a cohort of patients of scarcely annotated images.

(a) MNIST

(b) PCAM

Fig. 5: Attention scores in noise experiments with 2-level NMIL for (a) MNIST
and (b) PCAM test examples. Instances are categorized by colors indicating
inner-bag belonging, while digits represent the true label of the instance.
Bar plots on the left column show the attention at level 1, while on the right
at level 2. Positive instances obtain the highest attention scores.

As for experiment 3, the model reaches a F1 score of 0.8358 with the use of
NMIL plus attention, thus proving that a nested implementation handles complex
scenarios in an efficient manner. The processing and aggregation nature of MI
blocks provides the flexibility needed to implement any desired data
arrangement regardless of the level of nesting and number of bags and
instances per level.

##  7 CONCLUSIONS

In this paper, we have shown how NMIL with attention mechanisms is
advantageous for solving more complex scenarios that MIL cannot, for when
dependencies among sets of bags are to be considered. Experiments for
processing interdependent subsets from images were carried out that strengthen
this belief. Moreover, NMIL networks can be used in a wide range of
applications due to its simplicity and flexibility. Finally, the
implementation of an attention mechanism helps to clearly identify key
instances and inner-bags contained in a set of bags, giving insight into the
practical relationship between latent labels and the attention given among
different levels. Future research should consider the potential effects of
NMIL with attention on a full-scale medical imaging dataset more carefully.

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