# Online Continual Learning via the Knowledge Invariant and Spread-out
Properties

Ya-nan Han  Jian-wei Liu<EMAIL_ADDRESS>Department of Automation, College
of Information Science and Engineering, China University of Petroleum Beijing,
Beijing, China

###### Abstract

The goal of continual learning is to provide intelligent agents that are
capable of learning continually a sequence of tasks using the knowledge
obtained from previous tasks while performing well on prior tasks. However, a
key challenge in this continual learning paradigm is catastrophic forgetting,
namely adapting a model to new tasks often leads to severe performance
degradation on prior tasks. Current memory-based approaches show their success
in alleviating the catastrophic forgetting problem by replaying examples from
past tasks when a new task is learned. However, these methods are infeasible
to transfer the structural knowledge from previous tasks i.e., similarities or
dissimilarities between different instances. Furthermore, the learning bias
between the current task and prior tasks is also an urgent problem that should
be solved. In this work, we propose a new method, named Online Continual
Learning via the Knowledge Invariant and Spread-out Properties (OCLKISP), in
which we constraint the evolution of the embedding features via Knowledge
Invariant and Spread-out Properties (KISP). Thus, we can further transfer the
inter-instance structural knowledge of previous tasks while alleviating the
forgetting due to the learning bias. We empirically evaluate our proposed
method on four popular benchmarks for continual learning: Split CIFAR 100,
Split SVHN, Split CUB200, and Split Tiny-Image-Net. The experimental results
show the efficacy of our proposed method compared to the state-of-the-art
continual learning algorithms.

###### keywords:

continual learning; catastrophic forgetting; knowledge invariant and spread-
out properties; knowledge transfer.

††journal: arXiv

##  1 Introduction

With the development of deep learning, current deep models can perform well on
various computer vision problems[1, 2, 3, 4], such as object detection [5],
image classification [2, 6], semantic segmentation [7], and so on. The
standard learning process of current deep models is underpinned by the
assumption that the data is independently and identically distributed (i.i.d).
However, in our ever-changing world, the training data often be obtained from
an open environment [8], which is with stream format [9]. To solve this
challenge, the model should learn new tasks incrementally instead of
restarting the training process from scratch. A naïve method is to finetune
the model on the incoming new tasks, however, it suffers the forgetting of
past tasks when new ones are trained, a phenomenon known as catastrophic
forgetting [10]: due to the absence of previous data, the performance on prior
tasks drops significantly. Continual learning [11, 12, 13] aims to learn a
sequence of tasks by leveraging the knowledge obtained in the past while not
forgetting how to perform previous tasks. Fig.1 illustrates the setup of
continual learning. As is shown in Fig.1, when the first task arrives, the
model is required to classify dogs and spiders. After that, the model is
incrementally updated with another new task, i.e. , bag and box, and it needs
to correctly classify both the past task (dogs and spiders) and the current
new task (bag and box), simultaneously. That means, when the new task arrives
sequentially, the agent is required to learn a new task while should not
forget how to address the previous tasks.

Figure 1: Paradigm for continual learning. When the task comes sequentially,
we need to learn a classifier for all tasks incrementally. However, the
protocol only allows us to access to the data of current tasks. After
completing the learning stage of every task, we test the performance of the
current model among all seen tasks, i.e., it should perform well on the
current task, while effectively preventing catastrophic forgetting.

Recently, several works have attempted to resolve catastrophic forgetting for
a continual learning scene, which can be coarsely grouped into two categories
according to whether maintain a memory. The methods without memory [14, 15,
16, 17] constrain the updating of the model parameters to resist catastrophic
forgetting and the key difference for this category of approach lies in the
way of parameter importance measurement, such as knowledge distillation loss
[14], the Fisher information matrix [16kirkpatrick], and so on. Other works
selectively save some examples to prevent forgetting [18, 19, 20, 21].
Compared to other methods, the approaches based on episodic memory can
effectively prevent forgetting and achieve superior performance. Thus, in this
paper, we also maintain an episodic memory which is used to jointly train to
prevent forgetting. However, in this continual learning setting, it is worthy
to note that since the budget for storing the examples from the previous tasks
is limited for practical continual learning, which can create a bias towards
the current task. Such bias makes the model more prone to the current tasks by
which worsens forgetting. Recalling the existing researches on this issue,
some approaches [22, 23] attempt to apply an additional subset of examples to
alleviate this problem, however, this kind of method requires the whole data
of current task which is unavailable for the online continual learning setup.
Furthermore, the knowledge distillation strategy used in LWF [14] is to
minimize the KL divergence between the soft-max scores (input called logit) of
the old and the new tasks. Others [24, 25] distill knowledge from the
penultimate layer. Zhou et.al [26] directly leverage all previous model
snapshots which means that directly matches the outputs of current network
with those from the corresponding previous task models. However, we find that
they treat all logits of the intermediate feature levels as separation. That
means, these methods only require the current model to mimic the previous
model’s logit output values, thus neglecting the important structural
knowledge existing in the previous tasks, i.e., similarities or
dissimilarities between the logits. We anticipate that fully transferring the
important intrinsic structural knowledge underlying in the old tasks is
benefit to current task learning, thus further resisting forgetting. In
addition, in recent studies, existing works mainly focus on the situations
where the whole task’s training data arrives at each step and allow the agent
to learn the current tasks for many epochs. However, this assumption is not
suitable for many real-world scenarios where the training data is observed
sequentially in a stream form. Based on this fact, in this paper, we consider
a more realistic but difficult scenario, called online continual learning
where the agent only has one look at each instance. It is obvious that
continual learning is closer to animal learning[27].

In response to the above problems, motivated by positive concentrated and
negative separated in the self-supervised field [28], we propose a new
framework under the online continual learning setting, called Online Continual
Learning via the Knowledge Invariant and Spread-out Properties (OCLKISP).
Particularly, we propose a new method to transfer the important structural
knowledge of the previous tasks via knowledge invariant and spread-out
properties for the continual learning setting. In detail, facing the incoming
of new tasks, the model should not forget previous knowledge learned from past
tasks. That means, the inter-instance space location relationships should keep
invariant, in other words, the embedding features of episodic memory learned
from the previous model and current model respectively should maintain
invariant or the highest similarity as much as possible. However, the
embedding features of different examples are separated, which can be
beneficial to learn the similarities or dissimilarities of the output
distribution among different examples in episodic memory and then better
maintain the structure of the embedding features as the task arrives. Here, we
transform the multi-class classification problem to a binary classification
problem and apply maximum likelihood estimation (MLE) to optimize the
continual learning effect.

In summary, the major contributions of this work are as follows:

  * $\bullet$

We proposed a novel online continual learning method via the knowledge
invariant and spread-out properties (OCLKISP), which can transfer the
structural knowledge from previous tasks and then improve the overall
performance of the model.

  * $\bullet$

We analyze the learning bias of the replay-based method in online continual
learning from the angle of embedding features and the classification accuracy
respectively. Then, the forgetting problem due to the learning bias can be
alleviated via our OCLKISP method.

  * $\bullet$

We evaluate our proposed method on four popular datasets, and the experimental
results illustrate the effectiveness of our OCLKISP method in a continual
learning setting. The ablation studies are conducted to further understand the
contribution of KISP.

The remaining of this paper is organized as follows. In Section 2, we give a
survey of the related works. In Section 3, we illustrate the problem of
continual learning. In Section 4, we introduce the proposed approach of this
work. In Section 5, contrast experiments and ablation studies are performed to
demonstrate the effectiveness of our proposed method. Finally, conclusions and
future researches are given in Section 6.

##  2 Related Work

1) Continual learning

Continual learning is now a popular topic in the field of machine learning
[15], [29, 30, 31, 32]. The ultimate goal of continual learning is to achieve
good performance on the current task while not forgetting how to perform past
tasks. According to whether maintain a memory, the method of continual
learning can be coarsely grouped into two groups: Non-memory-based strategy
and Memory-based strategy.

Non-memory-based strategy This line of works avoids preserving raw inputs and
alleviating memory requirements. We can further divide these methods into
parameter-based, distillation-based, and architecture-based methods. Firstly,
the methods of parameter-based such as EWC, SI, et al, attempt to use fixed-
capacity models and add a regularization term on the loss function to control
the changes on significant parameters. The differences among these works lie
in the way to estimate the weight importance. For example, a representative
method called Elastic Weight Consolidation(EWC) [16] applies the Fisher
information matrix to estimate the important parameters and then prevent the
large change for the important parameters. In Synaptic Intelligence (SI) [15],
the importance of weight is computed by the cumulative change of distance
after learning new tasks, where the importance is estimated in an online
manner during train. Compared to EWC, SI is more intuitive and persuasive in
terms of the measure about the importance. In addition, Memory Aware Synapses
(MAS) [33] applies the sensitivity of the learned function instead of loss to
measure the importance of weights. Another line of work is distillation-based
methods and a typical model is Learning Without Forgetting (LWF) [14], which
constrains the change of the model predictions on past tasks via a
distillation loss [34]. Although the methods lessen the catastrophic
forgetting, they usually fail to solve the forgetting problem well and perform
non-favorably when the sequence of tasks is long [35]. Others focus on the
network structure which tries to train a separate model for each incoming task
to resist catastrophic forgetting. For example, Progressive Neural Network
(PNN) [36] alleviates catastrophic forgetting by fixing the parameters of the
network from past tasks while learning new tasks by expanding the network.
Other relevant works [37, 38, 39, 40] consider adaptively selecting the
network structure based on the correlation of knowledge and then improving the
efficiency of the network capacity.

Memory-based strategy This line of work saves prototype instances of former
tasks in episodic memory and then a rehearsal or pseudo-rehearsal process is
used to prevent catastrophic forgetting when learning a new task. Incremental
Classifier and Representation Learning (ICARL) [40] combines knowledge
distillation and feature learning to resist forgetting. Specifically, the
examples closest to the feature mean of every class are picked and stored in
fixed memory. During training, the distillation loss between targets obtained
from previous model predictions and current model predictions is applied to
alleviate forgetting. Gradient Episodic Memory (GEM) [18] and Average-GEM
(AGEM) [41] apply an inequality constraint via episodic memory to prevent the
forgetting for former tasks. Hou et al [24] propose to learn a unified
classifier via a cosine linear layer and then incrementally learn new
knowledge in a continual learning setting. Wu et al [23] and Pham et al [19]
utilize the samples to build an extra validation set and then further improve
the generalization of the model. Along with the development of the generative
model, such as auto-encoder [42], VAE [43], or GAN [44], another direction is
to use a generative model to mimic the samples of former tasks and then we
prevent forgetting by replaying the generated data of previous tasks [23, 45,
46]. Compared to the methods of Non-memory based, the major drawback of this
kind of method is to require additional computation and storage of raw input
samples.

2) Data imbalance

Recently, the method with experience replay has achieved significant
improvements in continual learning setting. However, in a replay approach, the
data from prior tasks are limited compared to current task which would cause
the data imbalance between the previous tasks and current task and then create
a learning bias. Many researchers have attempted to resolve this issue in
batch continual learning. Francisco et al. [22] attempt to apply an additional
balanced subset of examples to finetuning the model. Yue et al. [47] apply a
bias correction layer to further avoid the strong bias towards current task.
Bowen et al. [48] attempt to correct the bias via a weight alignment approach
in the fully-connected layer after each update. However, these methods require
that the data of current task is available and then train multiple epochs or
many learning stages is applied to alleviate the learning bias, which is not
appropriate for online continual learning where the data of each task arrives
in the data stream. We also note that some distillation-based methods [19, 26,
34] can be used to alleviate this problem, such as requiring the corresponding
layer in the previous model to be exactly the same as the current model.
However, these methods only require the current model to mimic the previous
output value of each layer in episodic memory, which is not beneficial to
capture the structure knowledge of previous tasks. Based on the above study,
in this work, we design a new loss function to deal with the learning bias.

3) Contrastive Learning

Recently, contrastive learning [49, 50, 51] has shown great potential to learn
general representations by constructing positive and negative examples. There
are some prior works in this area. Typically, a random data augmentation, such
as scaling, crop, and flip, is employed to every image and then leverages the
augmented image as a positive instance. That means we aim to learn more
concentration features that should be invariant under different data
augmentations. Then, a small batch of randomly selected examples can be
considered as negative examples of each image. Under this assumption, some
works [51, 52, 53] attempt to separate each instance from other negative
instances which leads to a spread-out property. Xiao et al.[54] consider local
features of training inputs and different inputs as positive and negative
examples respectively. Tan et al. [55] proposed hyperspherical consistency
regularization (HCR) to regularize the classifier which would enforce data
points on hyperspheres to have similar structures.

Contrastive representation learning [56, 52] has also shown superior
downstream task performance via the positive concentrated and negative
separated properties. In the Multiview setting, Tian et al. [56] consider that
every view is noisy and incomplete, however, the important information should
be shared among all views. Based on this assumption, different views of input
are recognized as the positive instance and then the model can further capture
the shared information. Lin et al.[57] adopt this property to video
representation learning. Contrastive learning is now also widely applied to
other computer vision fields, e.g., objective detection [58], natural language
domain [59], and signature verification [60]. In this work, inspired by the
positive concentrated and negative separated properties, we mainly apply this
property to the continual learning setting based on our novel observation.

##  3 Problem Formulation

Continual Learning Before diving into the details of our approach proposed in
this work, we first introduce the definition of continual learning followed by
a classical memory-based continual learning scenario. Continual learning
studies the problem of learning from the streaming tasks, with the goal of
incrementally gaining knowledge from data of tasks. Assume there is a sequence
of task $\left\\{{1,2,\cdots,t,\cdots,{\rm T}}\right\\}$, where ${\rm T}$
denotes the total number of training tasks. Each task $t$ has its own dataset
$D^{t}{\rm{=}}\left\\{{\left({x_{i}^{t},\,y_{i}^{t}}\right)}\right\\}_{i=1}^{N_{t}}$,where
$x_{i}^{t}$ is the i-th training instance and $y_{i}^{t}$ is the corresponding
ground truth label. It is worth noting that there is no overlapping class
among different tasks. During the training process of task $t$, the previous
data $D^{1},D^{2},\cdots,D^{{\kern 1.0pt}t{\rm{-}}1}$ are no longer available.
Our goal at every step is not only to learn the knowledge from the current
task, but also to retain the knowledge learned from former tasks. After ending
every task, we evaluate the performance of the trained model on all currently
seen tasks.

The crux problem is that when a new task $t$ arrives, the model should
incrementally obtain the ability to correctly classify the new class and all
currently seen classes, simultaneously. Here, we split the trained model into
two parts: embedding function $\phi\left(\cdot\right)$ and a linear
classification head $W^{t}$. When the training instance $x_{i}^{t}$ is
observed, firstly, the embedding feature $f_{i}^{t}{\text{ =
}}\phi\left({x_{i}^{t}}\right)$ is generated by the embedding function. After
that, we compute its classification probability for class $y_{i}^{t}$ by
$\sigma\left({\left[{W^{t}}\right]^{T}f_{i}^{t}}\right)$ where
$\sigma\left(\cdot\right)$ is the soft-max operator. Similarly, the
classification probability about the all currently seen class label set is
calculated as $\sigma\left({\left[W\right]^{T}f_{i}^{t}}\right)$, where
$W{\text{ = }}\left[{W^{1},W^{2},\cdots,W^{t}}\right]$ denotes the
concatenation of all the current classification heads, i.e., the set of logits
for tasks $1,\cdots,t$. As aforementioned, normally, memory-based approaches
prevent catastrophic forgetting via a tiny episodic memory. A typical way is
to compute cross-entropy using these examples from episodic memory and current
task respectively:

$L_{CE}\left({x^{t},y^{t},M{\kern 1.0pt}^{t}}\right){\text{ = -
E}}_{\left({x\,^{t},\;y\,^{t}}\right)\cup
M\,^{t}}\sum\limits_{c=1}^{C}{{\mathbf{1}}_{\left[{c=y\,^{t}}\right]}}\sigma\left({\left[W\right]^{T}\phi\left({x\,^{t}}\right)}\right)$
(1)

where $M{\kern 1.0pt}^{t}$ is episodic memory. We optimize the above cross-
entropy loss about all previous replay examples in episodic memory and current
novel examples, thus learning new knowledge while alleviating the forgetting.

Eq.(1) describes a way to prevent forgetting for continual learning by memory
replay. Recently, a lot of literature has shown that compared with other
state-of-the-art methods, methods based on experience replay can achieve
competitive performance in the online continual learning setting. However,
there are some intractable problems which should be solved for these
approaches. Firstly, note that the training model only observes none or a few
examples of previous tasks due to the small episodic memory but substantially
more for the current task. Under this circumstance, as new tasks are learned
sequentially, the training process can create a bias toward the current task,
thus leading to the forgetting of knowledge learned from previous tasks.

Here, to more closely and clearly show what the problem is, we make a
quantitative analysis. Specifically, we measure the change in cosine distance
for embedding features of episodic memory in the buffer after each update when
a new task comes. As is illustrated in Fig. 2, due to unbalance data and
learning bias, as new knowledge is continuously learned, the change of the
embedding features in the buffer gradually increases. Moreover, in Fig. 3, we
plot the evolution of task classification accuracy accordingly. We observe
that the task classification accuracy performs poorly as new tasks are
learned.

From Figs. 2 and 3, we found that the knowledge learned from previous tasks
changes. Thus, the forgetting uninterruptedly and deterioratingly occurs as
new knowledge is learned sequentially. So, we must further take some actions
to maintain the knowledge invariant and then prevent forgetting due to the
learning bias.

Fortunately, we find that there is some side information neglected in this
setting. Specifically, maintaining the structured knowledge in episodic memory
(i.e., similarities or dissimilarities between the embedding features
$f_{i}^{\,t}$ and $f_{j}^{\,t}$) as the new task sequentially arrives is
useful means. By means of preserving the structured knowledge in the continual
learning setup it is promising to alleviate the learning bias.

(a) On the CIFAR100 dataset (b) On the Tiny-Image-Net dataset

Figure 2: The change curve of embedding features on consecutive tasks on the
CIFAR100 and Tiny-Image-Net datasets respectively. The difference is measured
by cosines distance in the buffer when new task is learned. We observe that
when the new task arrives, the difference increases as new knowledge is
learned and the red line indicates the trend of change.

(a) On the CIFAR100 dataset (b) On the Tiny-Image-Net dataset

Figure 3: Evolution curves of the task classification accuracies when new
tasks come on the CIFAR100 and Tiny-Image-Net datasets respectively.

##  4 Online Continual Learning via the Knowledge Invariant and Spread-out
Properties (OCLKISP)

In this section, we introduce our proposed OCLKISP for online continual
learning detailly. In subsection 4.1, we start the approach discussion with a
brief introduction to the framework of OCLKISP. Then, we present the details
of our proposed OCLKISP in subsection 4.2. Lastly, we further make the
rationale analysis for our proposed approach in subsection 4.3.

###  4.1 The Proposed OCLKISP Method

Different from the typic memory-based method, in our model we further apply
the knowledge invariant and spread-out properties to capture the important
structural knowledge of the previous tasks, i.e., similarities or
dissimilarities between the embedding features $f_{i}^{t}$ and $f_{j}^{t}$.
Specifically, we aim to maintain the following properties: i) _knowledge
invariant_ , the embedding features of episodic memory learned from the
previous tasks and current task respectively should keep invariant or the
highest similarity as much as possible, which can encourage the model to
maintain the prior knowledge learned from the old tasks; ii) _knowledge
separated,_ the embedding features of different examples are separated, which
can be beneficial to learn the similarities or dissimilarities of the
embedding features in episodic memory and then better maintain the structure
of the embedding features as the task arrives. In this way, our OCLKISP model
is better able to resist forgetting and improve it’s the performance of the
model. A visual explanation is shown in Fig. 4.

Figure 4:  Illustration of the flowchart of our proposed OCLKISP approach.

In Fig. 4, we use $M{\kern 1.0pt}^{t}$ and ${\mathbf{x}}{\kern 1.0pt}^{t}$ to
represent episodic memory and the current training data, respectively.
${\mathbf{f}}{\kern 1.0pt}_{{\text{cur}}}$ and ${\mathbf{f}}{\kern
1.0pt}_{{\text{cur}}}^{M^{t}}$ represent the low dimensional embedding
features of the current training data and episodic memory from the current
model, respectively. ${\mathbf{f}}{\kern 1.0pt}_{{\text{pre}}}^{M^{t}}$
represents the low dimensional embedding features of episodic memory from the
previous models. The input low dimensional features ${\mathbf{f}}{\kern
1.0pt}_{{\text{cur}}}^{M^{t}}$ and ${\mathbf{f}}{\kern
1.0pt}_{{\text{pre}}}^{M^{t}}$ are normalized via $l_{2}$ are normalized via
the L2 norm to acquire the embedding features. After that, the model builds
the classifier from these mixed training data where the episodic memory is
used to prevent forgetting. We apply the knowledge invariant and spread-out
properties to regulate the evolving process of representation. Finally,
$L_{CE}$ and $L_{{\text{KISP}}}$ represent cross-entropy loss and constraint
item based on knowledge invariant and spread-out properties, respectively.

###  4.2 OCLKISP

The OCLKISP approach proposed in this paper is described in detail below.
Specifically, at phase t, that means when the task $t$ comes, we perform
memory relay via a tiny episodic memory $M{\kern 1.0pt}^{t}{\text{ =
}}\left\\{{x_{1},x_{2},\cdots,x_{m}}\right\\}$ where $m$ denotes the total
number of episodic memory at phase t. Then, we record the output of episodic
memory as ${\mathbf{f}}_{{\text{cur}}}^{M^{t}}{\text{ =
}}\phi\left({M^{t}}\right)$ in the penultimate layer. To measure different
features more efficiently and fairly, we map the low dimensional features into
the normalized embedding features ${\mathbf{\bar{f}}}_{{\text{cur}}}^{M^{t}}$,
where ${\mathbf{\bar{f}}}_{{\text{cur}}}^{M^{t}}{\text{ =
}}{{{\mathbf{f}}_{{\text{cur}}}^{M^{t}}}\mathord{\left/{\vphantom{{{\mathbf{f}}_{{\text{cur}}}^{M^{t}}}{\left\|{{\mathbf{f}}_{{\text{cur}}}^{M^{t}}}\right\|}}}\right.\kern-1.2pt}{\left\|{{\mathbf{f}}_{{\text{cur}}}^{M^{t}}}\right\|}}$
denotes the $L_{2}$ normalized. We use
${\mathbf{\bar{f}}}_{{\text{pre}}}^{M^{t}}$ to represent the normalized
embedding features of the episodic memory from the previous model.

To further transfer the structural knowledge of episodic memory and prevent
forgetting, in this work, we attempt to directly solve it as a binary
classification problem via maximum likelihood estimation (MLE). In particular,
for instance $x_{i}\in M^{t}$, the embedding feature
$\bar{f}_{{\text{pre}}}^{i}$ and $\bar{f}_{{\text{cur}}}^{i}$ from the current
model and previous model respectively should be classified into label $i$, and
other instance $x_{j}\in M^{t},i\neq j$ should not be categorized into label
$i$. By this way, we can effectively preserve the geometric configuration of
the episodic memory to resist forgetting. Thus, the probability of the current
embedding feature $\bar{f}_{{\text{cur}}}^{i}$ being recognized as the label
$i$ is denoted by

$p\left({i\left|{\bar{f}_{{\text{cur}}}^{i}}\right.}\right)=\frac{{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}{{\sum\nolimits_{k=1}^{m}{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}}$
(2)

Here,$\tau$ is a scaling factor. Thus, the probability of
$\bar{f}_{{\text{cur}}}^{j}$ being considered as the label $i$ is given by,

$p\left({i\left|{\bar{f}_{{\text{cur}}}^{j}}\right.}\right)=\frac{{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}{{\sum\nolimits_{k=1}^{m}{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}},i\neq
j$ (3)

Correspondingly, the probability that $\bar{f}_{{\text{cur}}}^{j},i\neq j$ is
not classified into label $i$ is
$1\,-p\left({i\left|{\bar{f}_{{\text{cur}}}^{j}}\right.}\right)$.

Assuming different embedding features being recognized as label $i$ are
independent, the joint distribution of $\bar{f}_{{\text{cur}}}^{i}$ being
categorized into label $i$ and $\bar{f}_{{\text{cur}}}^{j},i\neq j$ not being
viewed as label $i$ is given by

$P_{i}=p\left({i\left|{\bar{f}_{{\text{cur}}}^{i}}\right.}\right)\prod\limits_{j\neq
i}{\left({1\,-\,p\left({i\left|{\bar{f}_{{\text{cur}}}^{j}}\right.}\right)}\right)}$
(4)

The negative log likelihood is defined by,

$J_{i}=-\log
p\left({i\left|{\bar{f}_{{\text{cur}}}^{i}}\right.}\right)-\sum\limits_{j\neq
i}{\log\left({1\,-\,p\left({i\left|{\bar{f}_{{\text{cur}}}^{j}}\right.}\right)}\right)}$
(5)

We minimize the sum of the above Eq.(4) over all the embedding features for
current episodic memory, which is given by,

$J_{i}=-\sum\limits_{i}{\log
p\left({i\left|{\bar{f}_{{\text{cur}}}^{i}}\right.}\right)}-\sum\limits_{i}{\sum\limits_{j\neq
i}{\log\left({1\,-\,p\left({i\left|{\bar{f}_{{\text{cur}}}^{j}}\right.}\right)}\right)}}$
(6)

Then, the final loss function is formulated as follows,

$L_{{\text{total}}}{\text{ = }}L_{CE}\left({D^{t},M{\kern
1.0pt}^{t}}\right)+\lambda
J_{i}\left({{\mathbf{\bar{f}}}_{\,{\text{pre}}}^{M_{t}},{\mathbf{\bar{f}}}_{\,{\text{cur}}}^{M_{t}}}\right)$
(7)

where $L_{CE}$ is the cross-entropy loss and $J_{i}$ is the regular term based
on the knowledge invariant and spread-out properties. We use a parameter
$\lambda$ to balance $L_{CE}$ and $J_{i}$.

###  4.3 Rationale Analysis

Here, we also make a detailed rationale analysis about why minimizing Eq.(5)
can realize the knowledge invariant and spread-out properties. Minimizing the
above Eq.(5) can be recognized as maximizing Eq.(2) and minimizing Eq.(3)
respectively. Noting that for Eq.(3), it can be rewritten as,

$p\left({i\left|{\bar{f}_{{\text{cur}}}^{i}}\right.}\right)=\frac{{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}{{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right)+\sum\nolimits_{k\neq
i}{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}}$
(8)

maximizing Eq.(2) means maximizing
$\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right)$
and minimizing
$\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right),k\neq
i$. Maximizing
$\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right)$
can increase the cosine similarity between $\bar{f}_{{\text{pre}}}^{i}$ and
$\bar{f}_{{\text{cur}}}^{i}$ where $\bar{f}_{{\text{pre}}}^{i}$ and
$\bar{f}_{{\text{cur}}}^{i}$ are normalized features extracted by the previous
model and those by the current one, respectively, which can encourage the
knowledge learned by the current model to be consistent to those by the
previous model. On the other hand, minimizing
$\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{i}}\tau}}\right.\kern-1.2pt}\tau}}\right),k\neq
i$ requires $\bar{f}_{{\text{cur}}}^{i}$ and other features of the samples are
separated. Considering all the instances within the current episodic memory,
the embedding features are compelled to be separated from each other, which
can lead to the spread-out property.

In a similar manner, Eq.(3) can be rewritten as,

$p\left({i\left|{\bar{f}_{{\text{cur}}}^{j}}\right.}\right)=\frac{{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}{{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{j}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{j}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\tau}}\right.\kern-1.2pt}\tau}}\right)+\sum\nolimits_{k\neq
j}{\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{k}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\tau}}\right.\kern-1.2pt}\tau}}\right)}}},i\neq
j$ (9)

Note that the inner product
$\left({\bar{f}_{{\text{pre}}}^{j}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}$
tends to 1 and the scale factor $\tau$ generally is set to a small value (set
to 0.1 in our experiment). Thus, the term
$\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{j}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{j}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\tau}}\right.\kern-1.2pt}\tau}}\right)$
generally dominates the value of the whole denominator. Minimizing Eq.(3)
encourages the value
of$\exp\left({{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\mathord{\left/{\vphantom{{\left({\bar{f}_{{\text{pre}}}^{i}}\right)^{T}\bar{f}_{{\text{cur}}}^{j}}\tau}}\right.\kern-1.2pt}\tau}}\right)$
should be minimized, which aims at increasing the dissimilarity between
$\bar{f}_{{\text{pre}}}^{i}$ and $\bar{f}_{{\text{cur}}}^{j}$. Therefore, it
would further encourage the spread-out property and then be beneficial to
transfer the structural knowledge underlying in embedding features for
previous tasks to current task, such as the surrounding of instance embedding
features, inter-instance location relationships and inter-instance geometric
structure information.

##  5 Experimental evaluations

In this section, we compare our OCLKISP with other baselines on several
datasets used in continual learning to verify the effectiveness of our
proposed method. Then, we introduce the metrics and implementation details
used in our experiments. Finally, we also perform ablation studies to validate
the improvement of my proposed method.

###  5.1 Experiment Setting

Datasets Following the protocol defined in [18], we evaluate the performance
of my proposed method and baselines on Split SVHN, Split CIFAR100, Split
CUB200, Split TinyImageNet200.

  * 1)

Split SVHN [61]. The SVHN [62] consists of 600,000 images with the size of
$32\times 32$ pixels with 10 classes. In our experiments, we part this dataset
into 5 disjoint tasks and every task has no overlap classes.

  * 2)

Split CIFAR100 [18]. CIFAR100 [63] contains 60000 images with the size of
$32\times 32$ pixel in 100 object categories, where 50000 images are applied
to train a model and others are used to test. In this work, we divide this
dataset into 20 disjoint tasks and each task contains 5 object categories.

  * 3)

Split CUB200 [64]. This benchmark is a fine-grained image dataset with 200
bird species. This dataset includes 11788 images in total. Among 11788, 5994
images are the training data and 5794 images are the testing data. Here, we
divide CUB200 into 20 disjoint subsets and every subset is recognized as a
task.

  * 4)

Split Tiny-Image-Net200 [65]. This dataset [66] contains 120,000 images with
the size of pixels in 200 classes and we divide 200 object categories into 40
5-way classification tasks in this paper.

Baselines We extensively compare our OCLKISP with following several state-of-
the-art continual learning approaches, including Finetune: we learn this model
without any regularization and episodic memory; LwF(Learning without
Forgetting) [14]: this method apply knowledge distillation to resist
catastrophic forgetting; EWC(Elastic Weight Consolidation) [16]: this method
constrain the change of the important parameters via Fisher Information Matrix
to learn continually a sequence of tasks; ICARL(Incremental Classifier and
Representation Learning) [40]: this model apply nearest center mean as
classifier at inference and knowledge distillation is used to avoid
forgetting; GEM(Gradient Episodic Memory) [18]:an episodic memory is used to
avoid forgetting and positive backward transfer is considered in this model;
AGEM(Average Gradient Episodic Memory) [41]: this model apply the average
gradient information of examples in the episodic memory to control the change
of the model; ER(Experience Replay) [67]: In this method, experience replay is
used to alleviate forgetting; DER and DER++ (Dark Experience Replay) [68]:
These two approaches address forgetting through mixing rehearsal with
knowledge distillation, thus promoting consistency with its past while
alleviating the negative effects due to data imbalance. Note that DER++ equips
the objective function of DER with an additional term on buffer datapoints.

Metrics For comprehensive evaluations, we adopt three standard metrics: The
Final Accuracy (FA) [18], The Global Average Accuracy (GA) [69], Forgetting
Measure (FM) [41], and Learning Accuracy (LA) [21]. FA denotes the final
average classification accuracy of all observed tasks
$\left\\{{1,2,\cdots,t,\cdots,T}\right\\}$ after training the whole sequence;
Different from FA, GA is an accuracy metric that takes into account the
performance of a model at each task phase $t$ better characterizes the dynamic
aspects of continual learning. GA and FA are used to measure the performance
of the final model from different perspectives. FM indicates the influence
that learning a new task has the performance on prior tasks, in other words,
the average forgetting of all previous tasks after learning a new task; LA
reflects the performance of a model on a task right after it finishes learning
that task, which can indicate the ability to learn new knowledge to some
extent.

Here, given the train-test accuracy matrix $a_{i,j}$, which denotes the test
classification accuracy of the model on task $t_{j}$ after observing the last
instance about the current task $t_{i}$;$T$ is the total number of tasks in
the current task sequence. Thus, the FA, GA, FM, and LA are defined as
follows:

${\text{The Final Accuracy (FA) :
ACC}}=\frac{1}{T}\sum\limits_{i=1}^{T}{a_{T,i}}{\text{ }}$ (10)

${\text{The Global Accuracy (GA) : GA}}=\frac{{\sum\limits_{i\geqslant
j}^{T}{a_{i,j}}{\text{
}}}}{{{{T(T+1)}\mathord{\left/{\vphantom{{T(T+1)}2}}\right.\kern-1.2pt}2}}}{\text{
}}$ (11)

${\text{Forgetting Measure (FM) :
FM}}=\frac{1}{{T-1}}\sum\limits_{j=1}^{T-1}{\mathop{\max}\limits_{l<T}a_{l,j}-a_{T,j}}$
(12)

${\text{Learning Accuracy (LA) :
LA}}=\frac{1}{T}\sum\limits_{i=1}^{T}{a_{i,i}}{\text{ }}$ (13)

Implementation Details In this work, we follow the implementation details
proposed in the literature [41]. In particular, during the training process,
we also use the cross-validation principle via bilevel optimization [70] to
cross-validate the parameter of our model and perform online continual
learning [71, 72]. In addition, because we insist that the learning process is
a “single pass through data” which means that the model is only trained for
one epoch for each task in all of our experiments. For each method, we use the
same hyper-parameter configuration and protocol as provided in the
corresponding original paper. We set each data batch size to 10 for all
methods. Here, for comparisons of memory-based methods, such as GEM, A-GEM,
ER, follow [41, 71], the size of episodic memory for every task is set to 65,
25, 65 and 128 on Split CIFAR10, Split SVHN, Split CIFAR100 and Split Tiny-
ImageNet respectively.

Specifically, for SVHN, CIFAR, Tiny-Image-Net, we use a reduced ResNet18 as
backbone, and details of network architecture are shown in the following
Fig.4. For CUB200, we use a pretrained full ResNet18 [73] as the baseline
architecture. We apply SGD to train our model with the minibatch size of 10 in
an online learning manner. For SVHN and CUB200 datasets, one gradient update
is sufficient. Conversely, for CIFAR100 and Tiny-ImageNet, we perform a few
iterations over a batch and the number of iterations is set to 2 and 3
respectively, as done by works [74, 75] Moreover, we select the ring buffer
proposed in [18] as our memory writing strategy. Lastly, for each experimental
result, the experiments are conducted five times, and the average FA, GA, FM,
and LA are reported in this work.

Figure 5: Details of model architectures used in this work on CIFAR dataset

###  5.2 Results and Analysis

In this subsection, we evaluate our OCLKISP method’s performance by comparing
it with other state-of-the-art models in terms of FA, GA, FM, and LA with
quantitative and qualitative results. Furthermore, in order to verify the
effectiveness of our proposed method, we also conduct ablation studies. The
specific experimental results and analysis are shown below.

####  5.2.1 OCLKISP Performance on different datasets

1) Different Evaluation Metrics on various datasets

The numerical results of different baselines are reported in Table 2 on four
popular benchmarks including Split Cifar100(20T), Split SVHN(5T), Split
CUB200(20T), and Split Tiny-Image-Net (40T). Across all approaches, the higher
FA and GA indicate the better performance of the model; The value of LA
reflects the ability to learn new knowledge, higher is better; Then, FM is
used to measure the forgetting of previous knowledge, and this value is
higher, catastrophic forgetting is more prominent.

All numerical results are reported in table 2. From the experimental results,
we make several observations. Firstly, we can see that our OCLKISP method
outperforms all the baselines by a large margin across all the benchmarks
consistently, for example, our proposed method achieves the best FA, LA and GA
on all dataset. Besides, consistent with the finding of previous studies, the
regularization methods, such as EWC, LWF, exhibit relatively poor performance
and catastrophic forgetting of the previously learned knowledge. Compared with
regularization-based methods, the ICARL algorithm makes a further improvement
via the nearest-example algorithm. Then, we also note that all of the
considered memory-based approaches, such as ER, GEM, AGEM, were able to obtain
competitive performance on all datasets.

Table 1: The results of different baselines on different datasets, i.e., 95% confidence intervals reported after the symbol ±. Methods | Dataset | FA | GA | LA | FM
---|---|---|---|---|---
Finetune | Split Cifar100 | 34.93±0.99 | 41.90±2.02 | 66.01±0.32 | 32.74±0.96
EWC | 38.24±3.62 | 42.75±3.20 | 64.38±0.73 | 26.86±3.74
LWF | 42.24±1.19 | 44.61±1.60 | 64.25±0.61 | 22.40±1.00
ICARL | 46.48±0.47 | 49.69±1.03 | 66.01±1.56 | 18.77±1.66
GEM | 62.63±0.58 | 61.18±1.04 | 67.99±0.55 | 7.08±0.41
AGEM | 55.59±1.43 | 57.08±0.61 | 67.56±0.84 | 13.27±0.81
ER | 63.04±1.56 | 62.29±0.76 | 68.73±0.84 | 5.83±2.17
DER | 65.10±0.65 | 63.92±0.67 | 69.53±0.99 | 6.11±0.31
DER++ | 67.84±1.14 | 64.59±3.07 | 69.33±0.49 | 3.47±0.46
OCLKISP | 68.70±1.00 | 64.81±2.23 | 70.47±0.44 | 2.81±0.42
Methods | Dataset | FA | GA | LA | FM
---|---|---|---|---|---
Finetune | Split CUB200 | 34.93±1.27 | 41.90±1.06 | 72.30±1.60 | 12.69±2.23
EWC | 65.45±1.46 | 67.87±1.84 | 74.52±0.71 | 11.82±1.48
LWF | 66.43±2.31 | 67.59±2.46 | 75.64±1.53 | 11.19±2.50
ICARL | 67.59±1.55 | 73.45±3.90 | 76.55±0.39 | 9.81±1.62
GEM | 81.69±1.57 | 79.39±1.68 | 77.01±1.81 | 2.23±1.57
AGEM | 73.77±0.25 | 73.45±1.12 | 75.96±0.57 | 5.16±0.37
ER | 81.42±1.07 | 81.05±0.51 | 78.67±0.74 | 3.59±0.93
DER | 83.17±0.43 | 80.46±0.82 | 76.59±1.30 | 1.15±0.79
DER++ | 83.32±0.46 | 81.02±0.51 | 77.56±1.41 | 1.22±0.56
OCLKISP | 85.08±0.51 | 83.45±0.73 | 83.22±0.69 | 1.62±0.49
Finetune | Split Tiny-Image-Net | 34.10±2.80 | 35.52±3.98 | 64.82±0.49 | 31.54±3.01
EWC | 35.09±1.57 | 35.49±4.15 | 65.13±1.16 | 30.88±1.11
LWF | 36.41±0.30 | 35.73±4.38 | 64.77±0.23 | 29.12±0.36
ICARL | 43.87±0.31 | 45.89±3.08 | 66.19±0.54 | 23.04±0.57
GEM | 63.24±0.78 | 61.76±2.27 | 67.00±0.63 | 6.46±0.43
AGEM | 60.37±0.79 | 57.17±1.98 | 66.75±0.38 | 7.92±0.84
ER | 67.44±0.19 | 63.36±2.86 | 68.48±0.33 | 3.65±0.36
DER | 67.50±0.43 | 64.05±3.08 | 69.29±0.78 | 4.58±0.17
DER++ | 69.43±0.37 | 66.78±3.04 | 71.31±0.41 | 4.79±0.19
OCLKISP | 69.96±0.54 | 67.55±2.19 | 71.91±0.35 | 3.49±0.39
Finetune | Split SVHN | 75.87±7.18 | 79.34±8.15 | 96.07±0.10 | 25.25±7.06
EWC | 82.65±1.52 | 81.42±6.86 | 94.39±1.36 | 14.66±3.48
LWF | 83.27±1.53 | 79.50±7.06 | 93.97±0.87 | 13.37±2.86
ICARL | 86.89±1.59 | 85.41±4.74 | 95.39±0.81 | 10.62±2.77
GEM | 90.00±1.25 | 89.76±2.68 | 95.72±0.79 | 7.14±2.41
AGEM | 90.84±1.43 | 89.81±2.15 | 95.26±0.52 | 5.54±1.67
ER | 91.57±1.07 | 91.72±2.05 | 96.43±0.32 | 6.08±1.43
DER | 93.02±0.71 | 92.42±1.81 | 95.88±0.47 | 3.67±1.17
DER++ | 93.44±1.16 | 93.29±1.31 | 96.26±0.15 | 3.40±1.84
OCLKISP | 93.55±0.28 | 93.66±1.51 | 96.60±0.25 | 3.81±0.32

2) The evolution of the final average accuracy as the number of tasks
increases

As new tasks are learned, the evolution of the final accuracies and global
accuracies in different datasets are shown in Figs. 6 and 7 respectively. Each
curve illustrates the different methods’ final accuracies or global accuracies
for all the past tasks in the current training phase. Firstly, we can find
that the OCLKISP method proposed in this work outperforms all the baselines by
a significant margin across the various benchmarks consistently, which would
further verify the above observations. Then, we also observe that the curves
of traditional methods, such as finetune, LWF, and EWC, declines dramatically
as more tasks come, which means, these methods have limited ability to resist
catastrophic forgetting. Moreover, the downward trend of those memory-based
approaches is relatively slow, which indicates that these methods with the
experience replay strategy can more effectively alleviate forgetting than
regularization-based methods. It is also worth to be highlighted that as more
tasks are learned, our OCLKISP method enjoys superior performance than other
baselines, which illustrates that the OCLKISP method exhibits a superior
ability to prevent forgetting in a continual learning setting.

(a) Split SVHN

(b) Split CUB200

(c) Split Cifar100

(d) Split Tiny-ImageNet

Figure 6: Evolution curves of the final accuracies as new tasks arrive on
different datasets

(a) Split SVHN

(b) Split CUB200

(c) Split Cifar100

(d) Split Tiny-ImageNet

Figure 7: Evolution curves of the global accuracies as new tasks arrive on
different datasets

3) The impact of KISP against other baselines

In this section, we compare our OCLKISP method with two variations of
baselines and further analyze the impact of the KISP. The different methods
are explained as follows: w/o KISP: our model is trained without KISP; OCL-
LFC: Compared to w/o KISP, the Less-Forget Constraint(LFC) [24] is employed
which can encourage the orientation of features extracted via current model to
be consistent to those by previous model, and then we can prevent the
influence due to the learning bias; OCL-RLD [25]: this method, called
representation-level distillation(RLD) prevent the learning bias by guiding
the representation of the episodic memory on current model close to the
previous model.

The experimental results are shown in Table 2. With the help of the LFC and
RLD, OCL-LFC and OCL-RLD are better than w/o KISP which illustrates those can
alleviate the influence of the bias to some extent, such as the final accuracy
on Split CIFAR100 is boosted from 66.03% to 67.60% and 68.59% respectively.
However, those methods neglect the important structural knowledge of previous
tasks, i.e., similarities or dissimilarities between the different embedding
features. Based on this, in this work, we design a novel loss via the
knowledge invariant and spread-out properties. In our OCLKISP method, the
knowledge invariant is used to alleviate the forgetting due to the learning
bias. Compared to LFC and RLD, our proposed method can further transfer the
structure knowledge from prior tasks and then achieve slightly better
accuracies than other models, such as the higher FA on all datasets. Note that
OCLKISP obtains worse global accuracy than OCL-LFC on the Split CUB200
dataset. This result means the final accuracies of our OCLKISP are not always
better than OCL-LFC at each task phase when the task comes sequentially.
However, our proposed model has a higher FA than OCL-LFC which means that when
all tasks are learned, our OCLKISP enjoys a better final performance than
other models. The following results validate the effectiveness of our proposed
method.

Table 2: The numerical results on different datasets, i.e., 95% confidence intervals reported after the symbol ± Methods | Datasets | FA | GA | LA | FM
---|---|---|---|---|---
w/o KISP | Split Cifar100 | 66.03±1.34 | 63.30±1.70 | 69.20±0.62 | 4.56±0.94
OCL-LFC | 67.60±0.49 | 64.07±2.85 | 69.36±0.52 | 2.33±1.68
OCL-RLD | 68.59±0.19 | 64.59±2.36 | 69.77±0.64 | 2.99±0.45
OCLKISP | 68.70±1.00 | 64.81±2.23 | 70.47±0.44 | 2.81±0.42
w/o KISP | Split CUB200 | 84.31±0.48 | 83.14±0.24 | 82.58±0.76 | 1.91±0.59
OCL-LFC | 84.48±0.24 | 84.00±0.45 | 83.21±0.54 | 2.45±0.49
OCL-RLD | 84.45±0.52 | 83.05±0.57 | 82.33±0.36 | 1.84±0.43
OCLKISP | 85.08±0.51 | 83.45±0.73 | 83.22±0.69 | 1.62±0.49
w/o KISP | Split SVHN | 92.44±1.32 | 93.11±1.55 | 96.42±0.22 | 5.06±1.77
OCL-LFC | 92.88±0.74 | 93.35±1.65 | 96.58±0.10 | 4.62±0.91
OCL-RLD | 92.24±0.68 | 92.60±2.12 | 96.63±0.19 | 5.51±0.87
OCLKISP | 93.55±0.28 | 93.66±1.51 | 96.60±0.25 | 3.81±0.32
w/o KISP | Split Tiny-Image-Net | 68.89±0.53 | 66.99±1.75 | 71.75±0.61 | 4.28±0.62
OCL-LFC | 69.01±0.30 | 66.77±2.32 | 71.49±0.20 | 3.94±0.11
OCL-RLD | 69.05±0.49 | 66.89±1.74 | 71.96±0.88 | 3.90±0.22
OCLKISP | 69.96±0.54 | 67.55±2.19 | 71.91±0.35 | 3.49±0.39

####  5.2.2 Ablation study

In this work, we propose a novel OCLKISP method and during the training
process in continual learning, OCLKISP can capture the beneficial structure
knowledge of the embedding features in episodic memory via the knowledge
invariant and spread-out properties and then enhance the transfer of knowledge
from old tasks. To provide more in-depth insight into the working mechanism of
our proposed method, we conduct additional ablative experiments in which we
further discuss the importance of knowledge invariant and spread-out
properties (KISP) for our proposed method.

Table 3 and Fig. 8 show the experimental results of the ablation studies,
where w/o KISP denotes the method removing the constraint of knowledge
invariant and spread-out properties on our OCLKISP method. By comparing the
results of Table 3, we observe that our OCLKISP method without KISP exhibits a
worse performance than OCLKISP on all datasets, for example, there is a 2.67%
decline on the Split CIFAR100 dataset in terms of FA. Thus, we argue that the
idea of KISP is indeed beneficial to enhancing the transfer of knowledge
learned in past tasks and then alleviating forgetting in the continual
learning environment. Furthermore, in order to further analyze the performance
variation from a global accuracy view, Fig. 8 shows the performance variation
of GA as new tasks arrive. We can observe that on SVHN and CUB200 datasets,
our OCLKISP method achieves superior performance than the w/o KISP method
consistently. Furthermore, it is noted that in all cases, compared with the
w/o KISP method, our OCLKISP method offers a more significant improvement when
the number of tasks increases. These results validate the effectiveness of our
proposed method, which also shows that our proposed OCLKISP is a promised
approach in the context of online continual learning.

Table 3: The numerical results of different ablation studies, i.e., 95% confidence intervals reported after the symbol ±. Methods | Datasets | FA | GA | LA | FM
---|---|---|---|---|---
w/o KISP | Split Cifar100 | 66.03±1.34 | 63.30±1.70 | 69.20±0.62 | 4.56±0.94
OCLKISP | 68.70±1.00 | 64.81±2.23 | 70.47±0.44 | 2.81±0.42
w/o KISP | Split CUB200 | 84.31±0.48 | 83.14±0.24 | 82.58±0.76 | 1.91±0.59
OCLKISP | 85.08±0.51 | 83.45±0.73 | 83.22±0.69 | 1.62±0.49
w/o KISP | Split SVHN | 92.44±1.32 | 93.11±1.55 | 96.42±0.22 | 5.06±1.77
OCLKISP | 93.55±0.28 | 93.66±1.51 | 96.60±0.25 | 3.81±0.32
w/o KISP | Split Tiny-Image-Net | 68.89±0.53 | 66.99±1.75 | 71.75±0.61 | 4.28±0.62
OCLKISP | 69.96±0.54 | 67.55±2.19 | 71.91±0.35 | 3.49±0.39

(a) Split SVHN

(b) Split CUB200

(c) Split Cifar100

(d) Split Tiny-ImageNet

Figure 8: Performance variation of the global accuracy on various datasets

####  5.2.3 Comparison of the effects of the different episodic memory sizes

In this subsection, we study the effects of the different episodic memory
sizes on different baselines. We evaluate five competing approaches GEM, ER,
DER++, w/o KISP and OCLKISP on Split CIFAR and Split CUB datasets. The
experimental results of every benchmark against the different episodic memory
sizes are shown in Tables 4 and 5. Furthermore, Fig. 9 plots the final and
global accuracy curves of each method. Firstly, we observe that all methods
exhibit better overall performance when the episodic memory size increases.
That means a larger memory is able to result in a better representation of the
original training data, which can help remove catastrophic forgetting. Then,
compared to w/o KISP, we found that the performance of OCLKISP consistently
improves, e.g., offers 0.5% to 2% improvements on Split CIFAR100. These
results illustrate that the idea of KISP indeed helps improve the performance
of the model across different memory sizes. Moreover, the experimental results
show that OCLKISP can outperform existing approaches considered in this work
on almost all metrics across different memory sizes, which further validates
the effectiveness of OCLKISP.

Table 4: Numerical results for Split Cifar100 dataset with memory size M per task and 95% confidence interval reported after symbol ±. Methods | M=25 | M=45
---|---|---
FA | GA | FM | FA | GA | FM
GEM | 57.62±1.39 | 57.46±0.83 | 10.91±1.25 | 60.97±1.01 | 60.27±1.38 | 8.19±0.83
ER | 57.78±1.65 | 58.12±0.63 | 12.65±1.65 | 61.83±1.17 | 60.05±1.21 | 8.19±0.96
DER++ | 61.93±1.30 | 58.64±1.76 | 7.37±0.92 | 66.39±0.87 | 62.05±3.09 | 4.90±0.84
w/o KISP | 59.81±0.59 | 58.07±1.92 | 9.92±1.06 | 65.50±0.64 | 62.06±2.26 | 4.64±0.45
OCLKISP | 60.65±1.48 | 58.42±1.55 | 8.93±1.42 | 66.32±0.49 | 62.52±2.53 | 4.14±0.24
Methods | M=65 | M=85
GEM | 62.63±0.58 | 61.18±1.04 | 7.08±0.41 | 64.24±1.03 | 62.30±2.20 | 5.29±0.82
ER | 63.04±1.56 | 62.29±0.76 | 5.83±2.17 | 65.28±0.32 | 63.62±0.82 | 5.94±0.37
DER++ | 67.84±1.14 | 64.59±3.07 | 3.47±0.46 | 68.53±0.94 | 65.41±3.75 | 3.06±0.50
w/o KISP | 66.03±1.34 | 63.30±1.70 | 4.56±0.94 | 68.83±0.59 | 65.81±2.65 | 3.21±0.38
OCLKISP | 68.70±1.00 | 64.81±2.23 | 2.81±0.42 | 69.38±0.49 | 66.36±2.24 | 2.44±0.31
Table 5: Numerical results for Split CUB dataset with memory size M per task and 95% confidence interval reported after symbol ±. Methods | M=25 | M=45
---|---|---
FA | GA | FM | FA | GA | FM
GEM | 74.40±2.03 | 75.59±0.94 | 7.03±1.76 | 80.27±0.80 | 78.54±0.59 | 2.88±1.00
ER | 76.31±1.36 | 75.95±0.39 | 7.03±1.12 | 80.26±1.30 | 78.84±2.38 | 3.09±0.97
DER++ | 78.45±1.04 | 76.63±1.33 | 3.48±1.11 | 81.56±1.68 | 79.83±0.83 | 2.18±0.75
w/o KISP | 79.73±0.76 | 80.17±0.86 | 4.97±0.54 | 83.70±0.22 | 82.31±1.00 | 1.84±0.23
OCLKISP | 80.90±0.67 | 81.05±0.74 | 4.42±0.69 | 83.82±0.61 | 82.59±0.32 | 2.39±0.61
Methods | M=65 | M=85
GEM | 81.69±1.57 | 79.39±1.68 | 2.23±1.57 | 82.18±2.45 | 81.61±1.73 | 2.70±2.09
ER | 81.42±1.07 | 81.05±0.51 | 3.59±0.93 | 82.35±1.11 | 81.67±0.89 | 2.21±1.45
DER++ | 83.32±0.46 | 81.02±0.51 | 1.22±0.56 | 83.91±0.59 | 82.16±2.57 | 1.15±0.76
w/o KISP | 84.31±0.48 | 83.14±0.24 | 1.91±0.59 | 85.49±0.25 | 84.63±0.25 | 1.67±0.57
OCLKISP | 85.08±0.51 | 83.45±0.73 | 1.62±0.49 | 85.72±0.85 | 84.84±0.40 | 1.79±0.03

(a) Split SVHN

(b) Split CUB200

(c) Split Cifar100

(d) Split Tiny-ImageNet

Figure 9: Comparison of the effects of the different episodic memory sizes on
Split Cifar100 and Split CUB200 datasets. a) and b) plot the final accuracies
with different memory sizes. c) and d) plot the global accuracies with
different memory sizes.

##  6 Conclusions and future work

Continual learning aims to prevent catastrophic forgetting when new tasks are
added. In this paper, we have studied the potentials and limitations of
current popular continual learning paradigms. The experimental results
illustrate the effectiveness of some state-of-the-art approaches, especially
memory-based approaches. However, the learning bias between the current and
previous tasks during experience replay is also a key problem, which can
enhance forgetting. Furthermore, existing methods also neglect the transfer of
structural knowledge in episodic memory which would be harmful to learning
tasks sequentially. Inspired by this, this work introduces a novel continual
learning method called Online Continual Learning via the Knowledge Invariant
and Spread-out Properties (OCLKISP). The proposed approach can further
transfer the structural characteristics of instances via knowledge invariant
and spread-out properties when continually learning a sequence of tasks, which
can also be beneficial to overcome forgetting caused by the learning bias.
Finally, comprehensive experiments are conducted on various benchmark datasets
to validate the effectiveness of our proposed paradigm.

In the future, there are several directions worth considering. Firstly, in
this paper, we have illustrated the feasibility of our OCLKISP approach in
classification settings and we can consider further extending our proposed
method to more avenues of computer vision, such as continual face recognition
and continual semantic segmentation. Besides, we can also consider some more
realistic problems during the practical application, such as concept drift and
abnormal detection. Furthermore, note that experience replay is an important
means to overcome catastrophic forgetting, thus it would be interesting to
study how to select the more representative samples which can better preserve
latent decision boundaries for previously observed tasks. Another interesting
extension to this work would be to use generative models to generate new
examples which are used instead of episodic memory.

## References

  * [1] R. Girshick, Fast r-cnn, in: Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2015, pp. 1440–1448.
  * [2] A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, Advances in neural information processing systems 25 (2012) 1097–1105.
  * [3] J. Long, E. Shelhamer, T. Darrell, Fully convolutional networks for semantic segmentation, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 3431–3440.
  * [4] Y. Zhang, Z. Qiu, T. Yao, D. Liu, T. Mei, Fully convolutional adaptation networks for semantic segmentation, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 6810–6818.
  * [5] M. Z. Alom, M. Hasan, C. Yakopcic, T. M. Taha, V. K. Asari, Inception recurrent convolutional neural network for object recognition, Machine Vision and Applications 32 (1) (2021) 1–14.
  * [6] G. Chandra, M. R. Challa, Ae-cnn based supervised image classification, in: International Conference on Computer Vision and Image Processing, Springer, 2020, pp. 434–442.
  * [7] S. Pereira, A. Pinto, J. Amorim, A. Ribeiro, V. Alves, C. A. Silva, Adaptive feature recombination and recalibration for semantic segmentation with fully convolutional networks, IEEE transactions on medical imaging 38 (12) (2019) 2914–2925.
  * [8] D.-W. Zhou, H.-J. Ye, D.-C. Zhan, Co-transport for class-incremental learning, arXiv preprint arXiv:2107.12654 (2021).
  * [9] L. Golab, M. T. Ozsu, Data stream management issues–a survey, Tech. rep., Technical Report, Apr. 2003. db. uwaterloo. ca/~ ddbms/publications/stream … (2003).
  * [10] A. Robins, Catastrophic forgetting in neural networks: the role of rehearsal mechanisms, in: Proceedings 1993 The First New Zealand International Two-Stream Conference on Artificial Neural Networks and Expert Systems, IEEE, 1993, pp. 65–68.
  * [11] M. B. Ring, et al., Continual learning in reinforcement environments (1995).
  * [12] S. Thrun, L. Pratt, Learning to learn: Introduction and overview, in: Learning to learn, Springer, 1998, pp. 3–17.
  * [13] S. Thrun, A lifelong learning perspective for mobile robot control, in: Intelligent robots and systems, Elsevier, 1994, pp. 201–214.
  * [14] Z. Li, D. Hoiem, Learning without forgetting, in: Computer Vision-14th European Conference, ECCV 2016, Proceedings, Springer-Verlag Berlin Heidelberg, 2016, pp. 614–629.
  * [15] F. Zenke, B. Poole, S. Ganguli, Continual learning through synaptic intelligence, in: International Conference on Machine Learning, PMLR, 2017, pp. 3987–3995.
  * [16] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan, J. Quan, T. Ramalho, A. Grabska-Barwinska, et al., Overcoming catastrophic forgetting in neural networks, Proceedings of the national academy of sciences 114 (13) (2017) 3521–3526.
  * [17] A. Chaudhry, P. K. Dokania, T. Ajanthan, P. H. Torr, Riemannian walk for incremental learning: Understanding forgetting and intransigence, in: European Conference on Computer Vision, Springer, 2018, pp. 556–572.
  * [18] D. Lopez-Paz, M. Ranzato, Gradient episodic memory for continual learning, Advances in neural information processing systems 30 (2017) 6467–6476.
  * [19] Q. Pham, D. Sahoo, C. Liu, S. C. Hoi, Bilevel continual learning, arXiv preprint arXiv:2007.15553 (2020).
  * [20] A. Chaudhry, M. Rohrbach, M. Elhoseiny, T. Ajanthan, P. K. Dokania, P. H. Torr, M. Ranzato, Continual learning with tiny episodic memories (2019).
  * [21] M. Riemer, I. Cases, R. Ajemian, M. Liu, I. Rish, Y. Tu, G. Tesauro, Learning to learn without forgetting by maximizing transfer and minimizing interference, arXiv preprint arXiv:1810.11910 (2018).
  * [22] F. M. Castro, M. J. Marín-Jiménez, N. Guil, C. Schmid, K. Alahari, End-to-end incremental learning, in: Proceedings of the European conference on computer vision (ECCV), 2018, pp. 233–248.
  * [23] Y. Wu, Y. Chen, L. Wang, Y. Ye, Z. Liu, Y. Guo, Y. Fu, Large scale incremental learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 374–382.
  * [24] S. Hou, X. Pan, C. C. Loy, Z. Wang, D. Lin, Learning a unified classifier incrementally via rebalancing, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 831–839.
  * [25] Z. Ni, H. Shi, S. Tang, Y. Zhuang, Alleviate representation overlapping in class incremental learning by contrastive class concentration, arXiv preprint arXiv:2107.12308 (2021).
  * [26] P. Zhou, L. Mai, J. Zhang, N. Xu, Z. Wu, L. S. Davis, M2kd: Multi-model and multi-level knowledge distillation for incremental learning, arXiv preprint arXiv:1904.01769 (2019).
  * [27] J. T. Wixted, et al., The psychology and neuroscience of forgetting, Annual review of psychology 55 (1) (2004) 235–269.
  * [28] M. Ye, X. Zhang, P. C. Yuen, S.-F. Chang, Unsupervised embedding learning via invariant and spreading instance feature, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 6210–6219.
  * [29] S. Ebrahimi, F. Meier, R. Calandra, T. Darrell, M. Rohrbach, Adversarial continual learning, arXiv (2020).
  * [30] J. Xu, J. Ma, X. Gao, Z. Zhu, Adaptive progressive continual learning, IEEE Transactions on Pattern Analysis and Machine Intelligence (2021).
  * [31] M. Delange, R. Aljundi, M. Masana, S. Parisot, X. Jia, A. Leonardis, G. Slabaugh, T. Tuytelaars, A continual learning survey: Defying forgetting in classification tasks, IEEE Transactions on Pattern Analysis and Machine Intelligence (2021).
  * [32] A. Daruna, M. Gupta, M. Sridharan, S. Chernova, Continual learning of knowledge graph embeddings, IEEE Robotics and Automation Letters 6 (2) (2021) 1128–1135.
  * [33] R. Aljundi, F. Babiloni, M. Elhoseiny, M. Rohrbach, T. Tuytelaars, Memory aware synapses: Learning what (not) to forget, in: Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 139–154.
  * [34] G. Hinton, O. Vinyals, J. Dean, Distilling the knowledge in a neural network, arXiv preprint arXiv:1503.02531 (2015).
  * [35] C.-H. Tu, C.-E. Wu, C.-S. Chen, Extending conditional convolution structures for enhancing multitasking continual learning, in: 2020 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC), IEEE, 2020, pp. 1605–1610.
  * [36] A. A. Rusu, N. C. Rabinowitz, G. Desjardins, H. Soyer, J. Kirkpatrick, K. Kavukcuoglu, R. Pascanu, R. Hadsell, Progressive neural networks, arXiv preprint arXiv:1606.04671 (2016).
  * [37] J. Yoon, J. Lee, E. Yang, S. J. Hwang, Lifelong learning with dynamically expandable network, in: International Conference on Learning Representations, International Conference on Learning Representations, 2018.
  * [38] A. Mallya, D. Davis, S. Lazebnik, Piggyback: Adapting a single network to multiple tasks by learning to mask weights, in: Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 67–82.
  * [39] A. Mallya, S. Lazebnik, Packnet: Adding multiple tasks to a single network by iterative pruning, in: Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, 2018, pp. 7765–7773.
  * [40] S.-A. Rebuffi, A. Kolesnikov, G. Sperl, C. H. Lampert, icarl: Incremental classifier and representation learning, in: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE Computer Society, 2017, pp. 5533–5542.
  * [41] A. Chaudhry, R. Marc’Aurelio, M. Rohrbach, M. Elhoseiny, Efficient lifelong learning with a-gem, in: 7th International Conference on Learning Representations, ICLR 2019, International Conference on Learning Representations, ICLR, 2019.
  * [42] Y. Bengio, A. Courville, P. Vincent, Representation learning: A review and new perspectives, IEEE transactions on pattern analysis and machine intelligence 35 (8) (2013) 1798–1828.
  * [43] D. P. Kingma, M. Welling, Auto-encoding variational bayes, arXiv preprint arXiv:1312.6114 (2013).
  * [44] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, Y. Bengio, Generative adversarial networks, Communications of the ACM 63 (11) (2020) 139–144.
  * [45] H. Shin, J. K. Lee, J. Kim, J. Kim, Continual learning with deep generative replay, arXiv preprint arXiv:1705.08690 (2017).
  * [46] O. Ostapenko, M. Puscas, T. Klein, P. Jahnichen, M. Nabi, Learning to remember: A synaptic plasticity driven framework for continual learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 11321–11329.
  * [47] Y. Wu, Y. Chen, L. Wang, Y. Ye, Z. Liu, Y. Guo, Y. Fu, Large scale incremental learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 374–382.
  * [48] B. Zhao, X. Xiao, G. Gan, B. Zhang, S.-T. Xia, Maintaining discrimination and fairness in class incremental learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 13208–13217.
  * [49] R. Hadsell, S. Chopra, Y. LeCun, Dimensionality reduction by learning an invariant mapping, in: 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), Vol. 2, IEEE, 2006, pp. 1735–1742.
  * [50] K. Q. Weinberger, L. K. Saul, Distance metric learning for large margin nearest neighbor classification., Journal of machine learning research 10 (2) (2009).
  * [51] T. Chen, S. Kornblith, M. Norouzi, G. Hinton, A simple framework for contrastive learning of visual representations, in: International conference on machine learning, PMLR, 2020, pp. 1597–1607.
  * [52] M. Ye, X. Zhang, P. C. Yuen, S.-F. Chang, Unsupervised embedding learning via invariant and spreading instance feature, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 6210–6219.
  * [53] K. He, H. Fan, Y. Wu, S. Xie, R. Girshick, Momentum contrast for unsupervised visual representation learning, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp. 9729–9738.
  * [54] T. Xiao, C. J. Reed, X. Wang, K. Keutzer, T. Darrell, Region similarity representation learning, in: Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp. 10539–10548.
  * [55] C. Tan, Z. Gao, L. Wu, S. Li, S. Z. Li, Hyperspherical consistency regularization, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 7244–7255.
  * [56] Y. Tian, D. Krishnan, P. Isola, Contrastive multiview coding, in: European conference on computer vision, Springer, 2020, pp. 776–794.
  * [57] Y. Lin, X. Guo, Y. Lu, Self-supervised video representation learning with meta-contrastive network, in: Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp. 8239–8249.
  * [58] S. R. Silva, T. Vieira, D. Martínez, A. Paiva, On novelty detection for multi-class classification using non-linear metric learning, Expert Systems with Applications 167 (2021) 114193.
  * [59] H. Fu, S. Zhou, Q. Yang, J. Tang, G. Liu, K. Liu, X. Li, Lrc-bert: latent-representation contrastive knowledge distillation for natural language understanding, in: Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 35, 2021, pp. 12830–12838.
  * [60] D. Tsourounis, I. Theodorakopoulos, E. N. Zois, G. Economou, From text to signatures: Knowledge transfer for efficient deep feature learning in offline signature verification, Expert Systems with Applications 189 (2022) 116136.
  * [61] J. Rajasegaran, S. Khan, M. Hayat, F. S. Khan, M. Shah, itaml: An incremental task-agnostic meta-learning approach, in: 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, 2020, pp. 13585–13594.
  * [62] Y. Netzer, T. Wang, A. Coates, A. Bissacco, B. Wu, A. Y. Ng, Reading digits in natural images with unsupervised feature learning (2011).
  * [63] A. Krizhevsky, G. Hinton, et al., Learning multiple layers of features from tiny images (2009).
  * [64] C. Wah, S. Branson, P. Welinder, P. Perona, S. Belongie, The caltech-ucsd birds-200-2011 dataset (2011).
  * [65] G. Gupta, K. Yadav, L. Paull, La-maml: Look-ahead meta learning for continual learning, arXiv preprint arXiv:2007.13904 (2020).
  * [66] Y. Le, X. Yang, Tiny imagenet visual recognition challenge, CS 231N 7 (7) (2015) 3.
  * [67] A. Chaudhry, M. Rohrbach, M. Elhoseiny, T. Ajanthan, P. K. Dokania, P. H. Torr, M. Ranzato, Continual learning with tiny episodic memories. (2019).
  * [68] P. Buzzega, M. Boschini, A. Porrello, D. Abati, S. Calderara, Dark experience for general continual learning: a strong, simple baseline, arXiv preprint arXiv:2004.07211 (2020).
  * [69] N. Díaz-Rodríguez, V. Lomonaco, D. Filliat, D. Maltoni, Don’t forget, there is more than forgetting: new metrics for continual learning, arXiv preprint arXiv:1810.13166 (2018).
  * [70] B. Colson, P. Marcotte, G. Savard, An overview of bilevel optimization, Annals of operations research 153 (1) (2007) 235–256.
  * [71] Q. Pham, D. Sahoo, C. Liu, S. C. Hoi, Bilevel continual learning, arXiv preprint arXiv:2007.15553 (2020).
  * [72] A. Nichol, J. Achiam, J. Schulman, On first-order meta-learning algorithms, arXiv preprint arXiv:1803.02999 (2018).
  * [73] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 770–778.
  * [74] Z. Mai, R. Li, J. Jeong, D. Quispe, H. Kim, S. Sanner, Online continual learning in image classification: An empirical survey, Neurocomputing 469 (2022) 28–51.
  * [75] R. Aljundi, E. Belilovsky, T. Tuytelaars, L. Charlin, M. Caccia, M. Lin, L. Page-Caccia, Online continual learning with maximal interfered retrieval, in: H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, R. Garnett (Eds.), Advances in Neural Information Processing Systems, Vol. 32, Curran Associates, Inc., 2019.
URL
https://proceedings.neurips.cc/paper/2019/file/15825aee15eb335cc13f9b559f166ee8-Paper.pdf


# Discovering Process Models that Support Desired Behavior and Avoid Undesired
Behavior

Ali Norouzifar  1234-5678-9012 RWTH Aachen University
<EMAIL_ADDRESS>and  Wil van der Aalst  1234-5678-9012 RWTH
Aachen University<EMAIL_ADDRESS>
(2023)

###### Abstract.

Process discovery is one of the primary process mining tasks and starting
point for process improvements using event data. Existing process discovery
techniques aim to find process models that best describe the observed
behavior. The focus can be on recall (i.e., replay fitness) or precision.
Here, we take a different perspective. We aim to discover a process model that
allows for the good behavior observed, and does not allow for the bad
behavior. In order to do this, we assume that we have a desirable event log
($L^{+}$) and an undesirable event log ($L^{-}$). For example, the desirable
event log consists of the cases that were handled within two weeks, and the
undesirable event log consists of the cases that took longer. Our discovery
approach explores the tradeoff between supporting the cases in the desirable
event log and avoiding the cases in the undesirable event log. The proposed
framework uses a new inductive mining approach that has been implemented and
tested on several real-life event logs. Experimental results show that our
approach outperforms other approaches that use only the desirable event log
($L^{+}$). This supports the intuitive understanding that problematic cases
can and should be used to improve processes.

Process mining, Process discovery, Desirable and undesirable behavior

††copyright: acmcopyright††journalyear: 2023††copyright:
rightsretained††conference: The 38th ACM/SIGAPP Symposium on Applied
Computing; March 27-March 31, 2023; Tallinn, Estonia††booktitle: The 38th
ACM/SIGAPP Symposium on Applied Computing (SAC ’23), March 27-March 31, 2023,
Tallinn, Estonia††doi: 10.1145/3555776.3577818††isbn:
978-1-4503-9517-5/23/03††ccs: Applied computing Business process modeling

##  1\. Introduction

Process discovery is the problem of discovering a process model from an event
log that aims to represent the real process. In many applications, we have
valuable information about both desirable and undesirable behavior of
processes. Some cases might be handled differently in a process due to their
characteristics (De Leoni et al., 2016). For example, cases with a long
duration might have some undesirable characteristics that cause the delay.
Similarly, the outcome of the traces, the existence of a specific activity, or
a particular trace or event attribute could categorize the traces into
desirable and undesirable traces. The focus of current discovery techniques is
on discovering a process model that replays a single event log and avoids the
non-observed behavior of the process. Ignoring the undesirable behavior may
lead to process models that allow for this and generalize in an undesirable
direction. In addition, this type of information helps to support the
desirable behavior since the discovery algorithm can prioritize representing
the specifically desirable behavior.

The Inductive Miner–bi (IMbi) algorithm introduced in this paper uses the
problematic traces to discover better process models that are more specific to
desirable traces and avoid undesirable traces. The inductive miner algorithm
is one of the state-of-the-art discovery algorithms in the process mining
field (Leemans et al., 2013a). There are several extensions that are proposed
based on this algorithm (Leemans et al., 2013b; Brons et al., 2021). It is not
possible to directly adapt the inductive miner algorithm for the purpose of
this paper. In each recursion, the inductive miner algorithm tries to find a
cut that perfectly satisfies the cut type definitions. Considering a desirable
event log and an undesirable event log, we need an algorithm to compare and
rank different cuts even if they have some deviations or some important
relations are missing.

##  2\. Motivating Example

Let ${\mathcal{U}}_{A}$ be the universe of activities. A trace $\sigma=\langle
a_{1},a_{2},\ldots,a_{n}\rangle{\in}{\mathcal{U}}_{A}^{*}$ is a finite
sequence of activities. An event log $L\in\mathcal{B}({\mathcal{U}}_{A}^{*})$
is a multiset of traces. Consider $L^{+}=[\langle a,b,c\rangle^{40},\langle
b,d\rangle^{40},\langle c,b,d\rangle^{5},\langle b,d,c\rangle^{5},$ $\langle
a,d,c,b\rangle^{5},\langle a,c,d,b\rangle^{5}]$ as a multiset of desirable
traces and $L^{-}=[\\!\langle b,c,d\rangle^{20}\\!,$ $\\!\langle
b,d,c\rangle^{20},\\!$ $\langle a,c,b,d\rangle^{20},\langle
d,b,c\rangle^{20},\langle d,c,b\rangle^{20}]$ as a multiset of undesirable
traces. The goal is to discover a Petri net model that supports $L^{+}$ and
avoids $L^{-}$. Figure 1(a) shows a process model that is discovered
considering only $L^{+}$ using the Inductive Miner infrequent (IMf) (Leemans
et al., 2013b) algorithm with 20% filtering. Ignoring the undesirable event
log leads to this process model that perfectly fits both $L^{+}$ and $L^{-}$.
Figure 1(b) shows a good discovered model considering both $L^{+}$ and
$L^{-}$. This model avoids the behavior that is more specific to the
undesirable event log and supports the behavior that is more specific to the
desirable event log. In this paper, we also define novel evaluation metrics to
measure such improvements.

(a) A discovered model just using $L^{+}$

(b) A discovered model considering both $L^{+}$ and $L^{-}$

Figure 1. A motivating example

##  3\. How to evaluate the discovered models?

Conformance checking is a type of process mining that aims to check the
quality of discovered models from different perspectives. Fitness is designed
to ensure the model can replay the event log properly. Alignment fitness is
widely accepted to measure the deviations between an event log and a process
model (Carmona et al., 2018). Let ${\mathcal{U}}_{L}$ be the universe of event
logs and ${\mathcal{U}}_{M}$ be the universe of Petri net models. Consider
$fit_{align}{:}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{M}{\rightarrow}[0,1]$
as a function that calculates the alignment fitness. Another measurement to
evaluate the fitness criterion is calculating the percentage of fitting
traces. We define
$fit_{trace}{:}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{M}{\rightarrow}[0,1]$
as a function that calculates this value. Precision is designed to ensure that
the model does not allow for many behaviors that are not observed in the event
log. $prc{:}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{M}{\rightarrow}[0,1]$ is
the function that calculates escaping edges precision (ETC precision) (Carmona
et al., 2018). The goal of this paper is to discover a process model that
supports $L^{+}$ and avoids $L^{-}$.

The goal of this paper is to discover a process model that supports $L^{+}$
and avoids $L^{-}$. We measure this goal using the following evaluation
metrics. $1{-}fit_{align}(L{,}M)=\overline{fit_{align}}(L{,}M)$ and
$1{-}fit_{trace}(L{,}M)=\overline{fit_{trace}}(L{,}M)$.

  * •

alignment accuracy:
$acc_{align}{:}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{M}{\rightarrow}[-1{,}1]$
such that

$acc_{align}(L^{+},L^{-},M){=}fit_{align}(L^{+},M)-fit_{align}(L^{-},M)$

  * •

trace accuracy:
$acc_{trace}{:}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{M}{\rightarrow}[-1,1]$
such that

$acc_{trace}(L^{+},L^{-},M){=}fit_{trace}(L^{+},M)-fit_{trace}(L^{-},M)$

  * •

alignment F1-score:
$F1_{align}{:}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{M}{\rightarrow}[0,1]$
such that

$F1_{align}(L^{+},L^{-},M){=}\frac{2{\times}fit_{align}(L^{+},M){\times}\overline{fit_{align}}(L^{-},M)}{fit_{align}(L^{+},M){+}\overline{fit_{align}}(L^{-},M)}$

  * •

trace F1-score:
$F1_{trace}{:}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{L}{\times}{\mathcal{U}}_{M}{\rightarrow}[0,1]$
such that

$F1_{trace}(L^{+},L^{-},M){=}\frac{2{\times}fit_{trace}(L^{+},M){\times}\overline{fit_{trace}}(L^{-},M)}{fit_{trace}(L^{+},M)+\overline{fit_{trace}}(L^{-},M)}$

The four introduced evaluation metrics measure the same goal from different
perspectives. Increasing alignment accuracy indicates that the discovered
model can differentiate better between $L^{+}$ and $L^{-}$. It is not
acceptable to discover a model with a very low $fit_{align}(L^{+},M)$ but a
high $acc_{align}(L^{+},L^{-},M)$. To avoid sacrificing
$fit_{align}(L^{+},M)$, we designed alignment F1-score to maintain a balance
between increasing $fit_{align}(L^{+},M)$ and increasing
$\overline{fit_{align}}(L^{-},M)$. Alignment accuracy takes only the average
alignment fitness into account. Increasing alignment accuracy might result in
discovering a process model that enforces a subprocess that is observed in
neither $L^{+}$ nor $L^{-}$. This subprocess might punish
$fit_{align}(L^{-},M)$ more than $fit_{align}(L^{+},M)$ but block the token
flow path. In order to avoid such models, we propose to consider
$fit_{trace}(L^{+},M)$ and $fit_{trace}(L^{-},M)$ to define similar evaluation
metrics.

##  4\. Inductive Miner–bi

Consider $L^{+}$ as the event log contains desirable behavior and $L^{-}$ as
the event log contains undesirable behavior. The Inductive Miner–bi (IMbi)
algorithm constructs a process model recursively. The main idea of IMbi is to
consider both the desirable and the undesirable event logs in each recursion.
First, it checks whether there exists a base case. If it fails to find a base
case, then the algorithm tries to find an optimal cut. In Fig. 2, the general
approach in each recursion is illustrated.

Figure 2. Finding the optimal cut in one recursion of IMbi

$\mathcal{G}$ is a function that extracts a directly-follows graph from an
event log, such that $\mathcal{G}(L){=}(\Sigma,E)$, where

  * •

$\Sigma{=}\\{a{\in}\sigma|\sigma\in L\\}$ is the set of nodes.

  * •

$E{=}[(\sigma_{i},\sigma_{i+1})|\sigma{\in}L^{{}^{\prime}},1{\leq}i{<}|\sigma|]$
with
$L^{{}^{\prime}}=[\langle\triangleright\rangle.\sigma.\langle\square\rangle|\sigma\in
L]$ is the multiset of edges. $\triangleright$ is the special start activity
and $\square$ is the special end activity.

$\mathcal{G}(L^{+})$ and $\mathcal{G}(L^{-})$ in Fig. 2 represent the
extracted directly-follows graphs from the desirable and undesirable event
logs. The algorithm explores $\mathcal{G}(L^{+})$ and constructs the set of
possible cuts $C$. Each cut $c{=}(\oplus,\Sigma_{1},\Sigma_{2}){\in}C$ divides
the set of nodes into two disjoint sets, i.e., $\Sigma_{1}$ and $\Sigma_{2}$,
using a cut operator
$\oplus{\in}\\{\rightarrow,\times,\wedge,\circlearrowleft\\}$. $\rightarrow$
denotes the sequential cut, $\times$ represents the exclusive cut, $\wedge$
denotes the concurrent cut and $\circlearrowleft$ represents the loop cut.

Finding an optimal cut is not trivial. Figure 3 shows different possible cut
types and their specifications. The dashed lines are optional relations. The
red lines are the relations that are not allowed. The solid black lines are
the mandatory relations. Based on the definitions (Leemans et al., 2013a), if
one of the specifications of a cut type does not hold, then it should be
rejected. However, this decision does not take the severity of the deviating
or missing behaviors into account. In addition, we cannot compare the quality
of a cut based on the desirable and undesirable event logs. To address this
problem, we need to define some cost functions to quantify and compare the
quality of the cuts. Considering a possible cut $c{\in}C$,
$dev^{\oplus}_{G}(c)$ counts the number of deviating edges and
$mis^{\oplus}_{G}(c,sup)$ assigns a cost if some edges are missing.
$sup{\in}[0,1]$ is a user-defined parameter to specify how strict these cost
values should be with respect to missing edges. $mis^{\oplus}_{G}(c,sup)$
considers more cost with higher values of $sup$ if some edges are missing.
$cost^{\oplus}_{G}(c,sup){=}dev^{\oplus}_{G}(c){+}mis^{\oplus}_{G}(c,sup)$
assigns a cost to each cut $c{\in}C$.

Figure 3. Specifications of different cut types (dashed lines are optional,
black lines are mandatory and red lines are deviating edges)

$ov\\_cost_{G^{+},G^{-}}{:}C{\times}[0,1]{\times}[0,1]{\rightarrow}\mathbb{R}$
is a function that assigns an overall cost to each binary cut $c\in C$
considering both $cost^{\oplus}_{G^{+}}$ and $cost^{\oplus}_{G^{-}}$.
$ratio{\in}[0,1]$ is a process discovery parameter that specifies the
importance of the undesirable event log.

$ov\\_cost_{G^{+}{,}G^{-}}(c,sup,ratio){=}cost_{G^{+}}(c,sup)-ratio{\cdot}cost_{G^{-}}(c,sup)$

The best cut in each recursion is the cut with the minimum $ov\\_cost$, i.e.,

$optimal\\_cut=\underset{c\in
C}{\arg\min}\\{ov\\_cost_{G^{+},G^{-}}(c,sup,ratio)\\}$

The algorithm continues with splitting the event logs based on the selected
optimal cut and then runs the next recursion.

##  5\. Experiment

The IMbi framework implements the approach and the source code is publicly
available111https://github.com/aliNorouzifar/InductiveMiner_bi.git. The BPIC
2017 event log is extracted from a loan application process. There are three
types of events in this process: Application, Offer, and Workflow events. We
considered the event log consisting of only Application and Offer events. The
outcome of an application could be A_pending, A_cancelled, or A_denied. Based
on the observations, if activity “W_call incomplete files” occurs in a trace,
the chance of having A_pending state at the end is higher. We divided the
event log into $L^{+}_{17}$ and $L^{-}_{17}$ based on the existence of
activity “W_call incomplete files” in the traces. Among 31509 traces, 15003
traces are in $L^{+}_{17}$, and 16506 traces are in $L^{-}_{17}$. In
$L^{+}_{17}$, 84% of the traces have A_pending, 6% of the traces have
A_cancelled , and 9% of the traces have A_denied as the final state (few
traces are incomplete). In $L^{-}_{17}$, 28% of the traces have A_pending, 57%
of the traces have A_cancelled, and 15% of the traces have A_denied as the
final state. The goal is to discover a process model that supports traces in
$L^{+}_{17}$ and avoids traces in $L^{-}_{17}$.

###  5.1. IMbi using Single Event Log

In this section, we show that the IMbi algorithm discovers acceptable process
models from a single event log ($ratio=0$). Otherwise, there is no guarantee
that we can use this algorithm for our final goal. We use the IMf algorithm as
the baseline. In this section, the infrequency filtering parameter of this
algorithm is referred to as $f$. We change $f$ in the range of $[0,1]$ and
calculate the alignment fitness and the precision for different discovered
models. If we set $ratio=0$ in the IMbi algorithm, it means the undesirable
event log is ignored. We can discover different process models from only the
desirable event log by changing the $sup$ parameter in the range of $[0,1]$.

The alignment fitness and the precision of the discovered models using the IMf
algorithm are illustrated in Fig. 4(a). The IMf algorithm, after searching for
the base case in each recursion, tries to find a cut that satisfies the cut
type definitions. If this cut does not exist, then it filters
$\mathcal{G}(L^{+})$ considering $f$. The algorithm removes infrequent
directly-follows relations. Then, it tries to find a cut again. Increasing $f$
in the IMf algorithm removes more infrequent directly-follows relations from
$G(L^{+})$. The discovered model using a high $f$ value, considers only the
most frequent directly-follows relations. This leads to a decrease in the
alignment fitness value and an increase in the precision value.

(a) Discovered models from $L^{+}_{17}$ using IMf

(b) Discovered models from $L^{+}_{17}$ using IMbi

Figure 4. The fitness and precision of the discovered models from $L^{+}_{17}$
using IMf and IMbi (with $ratio=0$)

The alignment fitness and precision value of the discovered models using the
IMbi algorithm are illustrated in Fig. 4(b). $sup$ parameter is used in the
process discovery algorithm to control the strictness of the algorithm to
assign some cost if a missing behavior exists. If $sup{=}0$, it means that the
algorithm only counts the deviating edges in $\mathcal{G}(L^{+})$ (red edges
in Fig. 3). The discovered model may allow for more behavior than the observed
behavior. The algorithm does not insist on the behavior that should be in
$\mathcal{G}(L^{+})$, but it is missing. By increasing the $sup$ parameter,
the process discovery algorithm considers some cost for the part of the
behavior that is missing in $\mathcal{G}(L^{+})$. This leads to discovering
process models that more specifically represent the observed behaviors.
Generally, as we increase the $sup$ parameter, the alignment fitness value
decreases, and the precision value increases. Comparing the IMf and IMbi
models shows that the $sup$ parameter in IMbi and $f$ parameter in IMbi have a
similar effect on the fitness and precision of the discovered models. The
fitness and precision value of the discovered models using the IMbi algorithm
are comparable to the discovered models using the IMf algorithm.

###  5.2. IMbi using Desirable and Undesirable Event Logs

The parameter $ratio$ is designed to control the involvement of the
undesirable event log in the model discovery. If $ratio{=}0$, it means that
the algorithm only considers the desirable event log, and if $ratio{=}1$, it
means that in each recursion, the cost of a cut in $\mathcal{G}(L^{+})$ and
$\mathcal{G}(L^{-})$ has an equal weight. Increasing $ratio$ might lead to
selecting the cuts that are not the best considering only
$\mathcal{G}(L^{+})$, but helps to support $L^{+}$ and avoid $L^{-}$. In this
section, we select a $sup$ parameter value based on the harmonic mean of the
alignment fitness and precision. Then, we change the $ratio$ parameter in the
range of $[0,1]$ to show the effect on the evaluation metrics. In addition to
the introduced evaluation metrics, we show the alignment fitness and precision
corresponding to only the desirable event log.

For BPIC 2017, we select $sup{=}0.3$, since the harmonic mean of fitness and
precision is the best based on Fig. 4(b). In Fig. 5, the calculated evaluation
metrics are depicted. In this figure, the dashed lines are calculated for the
model discovered from $L^{+}_{17}$ using the IMf algorithm with $f{=}0.3$ as
the baseline. The alignment accuracy of the baseline model is around 0, and
the trace accuracy is -0.07. The discovered model cannot distinguish well
between the desirable event log and the undesirable event log. The evaluation
metrics for the discovered model using the IMbi algorithm with $ratio=0$ and
$sup=0.3$ is slightly better since it has a better trace accuracy, but still
has difficulties differentiating between the desirable and undesirable event
logs. Increasing the $ratio$ parameter shows that the evaluation metrics
improve when we consider the undesirable event log in the process discovery.
The main improvement in the evaluation metrics is achieved after setting the
$ratio$ parameter to 0.6. At $ratio{=}0.6$, all evaluation metrics except
precision increase. After $ratio{=}0.8$, the evaluation metrics decrease since
the selected cuts in the recursions are influenced by the undesirable event
log, and the low precision leads to the allowance for some undesirable
behaviors in the discovered models.

Figure 5. Evaluation metrics of the discovered models using IMbi with
$sup=0.3$ (dashed lines are calculated for the baseline)

##  6\. Related Work

The focus of this paper is on discovering a process model that allows for the
desirable behavior and avoids the undesirable behavior of a given process.
This fresh look is different from the existing works in the literature. In
(Goedertier et al., 2009) and (De León et al., 2018), the authors use
artificial negative events to improve the process discovery. The goal is to
maintain a balance between the allowed behavior by the model and the observed
behavior in the event log. Improving the precision, generalization, and
complexity of the process models are the outcomes of their research. In
(Slaats et al., 2021), the discovery phase is considered as a binary
classification and the information about undesired behavior is used to improve
the discovered models. Machine learning evaluation metrics are used to
evaluate these models. The focus of (De Oliveira et al., 2020) and (Slaats et
al., 2021) is on discovering models to classify the traces into desirable and
undesirable. In our paper, we formalized our approach as a method to discover
Petri net models that represent the process and the designed evaluation
metrics reflect the descriptive aspects of the discovered process models. In
(Dees et al., 2017), a methodology is proposed to repair discovered models to
allow for a part of the behavior that has a positive impact on process KPIs.

##  7\. Conclusion

The experiments show that the discovered models using our approach outperform
other approaches that use only desirable event logs. Based on the designed
evaluation metrics, changing $sup$ and $ratio$ parameters lead to good
candidate models. For some parameter configurations, the evaluation metrics
could have some conflicts. Selecting the best process model depends on the
application. For example, if we improve the alignment accuracy, it may have an
inverse effect on the trace accuracy. The discovery algorithm may choose some
cuts that are not good enough based on neither $L^{+}$ nor $L^{-}$, but
punishing $L^{-}$ behavior may lead to this decision. Adding some process
knowledge to the discovery algorithm helps to avoid bad decisions. For
example, a set of rules extracted from process knowledge can be used to guide
the algorithm. IMbi has a high computational cost due to the many possible
candidate cuts that needs to be improved in the next steps. Some differences
between $L^{+}$ and $L^{-}$ might not be in the process structures. Our
algorithm only takes the control flow into account and discovers a Petri net
that has a different structure in order to support the desirable event log and
avoid the undesirable event log.

###### Acknowledgements.

This research was supported by the research training group “Dataninja”
(Trustworthy AI for Seamless Problem Solving: Next Generation Intelligence
Joins Robust Data Analysis) funded by the German federal state of North Rhine-
Westphalia.

## References

  * (1)
  * Brons et al. (2021) Dennis Brons, Roeland Scheepens, and Dirk Fahland. 2021\.  Striking a new balance in accuracy and simplicity with the probabilistic inductive miner. In _2021 3rd International Conference on Process Mining (ICPM)_. IEEE, 32–39.
  * Carmona et al. (2018) Josep Carmona, Boudewijn F. van Dongen, Andreas Solti, and Matthias Weidlich. 2018.  _Conformance Checking - Relating Processes and Models_.  Springer.
  * De León et al. (2018) Hernán Ponce De León, Lucio Nardelli, Josep Carmona, and Seppe KLM vanden Broucke. 2018.  Incorporating negative information to process discovery of complex systems.  _Information Sciences_ 422 (2018), 480–496.
  * De Leoni et al. (2016) Massimiliano De Leoni, Wil M. P. van der Aalst, and Marcus Dees. 2016\.  A general process mining framework for correlating, predicting and clustering dynamic behavior based on event logs.  _Information Systems_ 56 (2016), 235–257.
  * De Oliveira et al. (2020) Hugo De Oliveira, Vincent Augusto, Baptiste Jouaneton, Ludovic Lamarsalle, Martin Prodel, and Xiaolan Xie. 2020.  An optimization-based process mining approach for explainable classification of timed event logs. In _2020 IEEE 16th International Conference on Automation Science and Engineering (CASE)_. IEEE, 43–48.
  * Dees et al. (2017) Marcus Dees, Massimiliano de Leoni, and Felix Mannhardt. 2017\.  Enhancing process models to improve business performance: A methodology and case studies. In _OTM Confederated International Conferences” On the Move to Meaningful Internet Systems”_. Springer, 232–251.
  * Goedertier et al. (2009) Stijn Goedertier, David Martens, Jan Vanthienen, and Bat Baesens. 2009.  Robust process discovery with artificial negative events.  _Journal of Machine Learning Research_ 10 (2009), 1305–1340.
  * Leemans et al. (2013a) Sander JJ Leemans, Dirk Fahland, and Wil M. P. van der Aalst. 2013a.  Discovering block-structured process models from event logs-a constructive approach. In _International conference on applications and theory of Petri nets and concurrency_. Springer, 311–329.
  * Leemans et al. (2013b) Sander JJ Leemans, Dirk Fahland, and Wil M. P. van der Aalst. 2013b.  Discovering block-structured process models from event logs containing infrequent behaviour. In _International conference on business process management_. Springer, 66–78.
  * Slaats et al. (2021) Tijs Slaats, Søren Debois, and Christoffer Olling Back. 2021\.  Weighing the Pros and Cons: Process Discovery with Negative Examples. In _International Conference on Business Process Management_. Springer, 47–64.