# Improving Cognitive Diagnosis Models with Adaptive Relational Graph Neural
Networks

Pengyang Shao  Hefei University of TechnologyAnhuiChina ,  Chen Gao  Tsinghua
UniversityBeijingChina ,  Lei Chen<EMAIL_ADDRESS>Tsinghua
UniversityBeijingChina ,  Yonghui yang  Hefei University of
TechnologyAnhuiChina ,  Kun Zhang  Hefei University of TechnologyAnhuiChina
and  Meng Wang  Hefei University of TechnologyAnhuiChina

###### Abstract.

Web-based online education has emerged as a powerful tool in realizing the
United Nations’ Sustainable Development Goals, particularly in ensuring
inclusive and equitable quality education and promoting lifelong learning
opportunities for all111https://sdgs.un.org/goals/goal4. Educational Cognitive
Diagnosis (CD) algorithms are an area of growing research interest in the
context of online education. Typically, these CD algorithms assist students by
inferring their abilities (i.e., their proficiency levels on various knowledge
concepts). The proficiency levels can enable further targeted skill training
and personalized exercise recommendations, thereby promoting students’
learning efficiency in online education. Recently, researchers have found that
building and incorporating a student-exercise bipartite graph is beneficial
for enhancing diagnostic performance. However, there are still limitations in
their studies. On one hand, researchers overlook the heterogeneity within
edges, where there can be both correct and incorrect answers. On the other
hand, they disregard the uncertainty within edges, e.g., a correct answer can
indicate true mastery or fortunate guessing. To address the limitations, we
propose Adaptive Semantic-aware Graph-based Cognitive Diagnosis model (ASG-
CD), which introduces a novel and effective way to leverage bipartite graph
information in CD. Specifically, we first map students, exercises, and
knowledge concepts into a latent representation space and combine these latent
representations to obtain student abilities and exercise difficulties. After
that, we propose a Semantic-aware Graph Neural Network Layer to address edge
heterogeneity. This layer splits the original bipartite graph into two
subgraphs according to edge semantics, and aggregates information based on
these two subgraphs separately. To mitigate the impact of edge uncertainties,
we propose an Adaptive Edge Differentiation Layer that dynamically
differentiates edges, followed by keeping reliable edges and filtering out
uncertain edges. Extensive experiments on three real-world datasets have
demonstrated the effectiveness of ASG-CD. As a bypass, we will release our
codes to support the community222The implementation will be publicly available
after paper acceptance.

Chen Gao and Pengyang Shao contributed equally to this work.

Lei Chen is the corresponding author.

††copyright: none

##  1\. introduction

The United Nations’ Sustainable Development Goals include ensuring inclusive
and equitable quality education, as well as promoting lifelong learning
opportunities for all. The emergence of intelligent education, particularly
online education, has provided a valuable opportunity for the majority of
people to access education. However, unlike traditional teaching methods,
online education lacks a role that can analyze students’ weaknesses in
learning, such as identifying which knowledge concepts students have not yet
mastered. This role is significant in education, as it serves as the
foundation for other learning tasks, e.g., providing learning recommendations
or conducting specific skill training (wu2022multi, ; kuh2011piecing, ).

Educational CD fulfills the aforementioned needs by collecting students’
historical behaviors, utilizing trainable parameters to represent student
abilities, and then optimizing student abilities to fit their behavioral data
(embretson2013item, ; wang2020neural, ). Classic CD models adopt free
embeddings to represent student abilities (embretson2013item, ;
reckase1997past, ; koren2009matrix, ), which only allows for response log
prediction without providing detailed analyses of students’ proficiency levels
on concepts. Recently, neural network-based CD models have garnered
significant attention in the field of intelligent education due to their
detailed diagnoses and easy-to-implementation (wang2023dynamic, ;
wang2020neural, ; gao2023leveraging, ; li2022hiercdf, ). As shown in Figure 1
(a), these models take students’ responses logs, and expert-labeled exercise-
concept relations as input, and output students’ abilities (i.e., their
proficiency levels on all concepts). The cornerstone of these models lies in a
diagnostic function that establishes connections among exercise-concept
relations, students’ proficiency levels on concepts, and response logs
(wang2022neuralcd, ).

Figure 1. (a) Process of educational CD. These models take students’ responses
logs, and exercise-concept relations as input, and output students’
proficiency levels on all concepts. (b) A tiny example of two students (Bob
and John). Their interactions with exercises have extensively overlapped,
leading to numerous second-order connections between them.

Recently, researchers have proposed integrating Graph Neural Networks (GNNs)
into CD models to improve diagnostic performance (gao2021rcd, ; wang2023self,
). Typically, these methods introduce a new bipartite graph perspective, with
students and exercises as nodes and student-exercise response logs as edges.
Based on the bipartite graph, GNN aggregates information from neighboring
nodes, effectively capturing the similarities among students and exercises for
representation enhancements. Figure 1 (b) visually illustrates underlying
reasons for the effectiveness of GNN in CD. Suppose that there are two
students (Bob and John), and their completed exercises have extensively
overlapped. That is, in the bipartite graph, there are numerous second-order
connections between Bob and John. As John has not encountered a specific
concept k5, we can not accurately estimate his proficiency level on that
concept without GNN. The presence of multiple connected edges between Bob and
John enables GNN to efficiently exchange information between them, resulting
in similarity between their abilities. In practical cases, each student will
establish second-order connections with a large number of other students. As
the number of overlapped response logs between two students increases, their
connection will become stronger. In sum, GNN can effectively enhance student
abilities by aggregating information based on the bipartite graph.

While graph-based CD models have demonstrated remarkable performance, we
contend that they still have inherent limitations. Previous works often
overlook certain unique characteristics of response logs, leading to a
misunderstanding of edge heterogeneity and uncertainty (gao2021rcd, ;
wang2023self, ). For illustration, we take John and Mark in Figure 2 as an
example. To display edge heterogeneity, we consider a crucial question:
”Should John’s proficiency levels on concepts be similar to Mark’s?” Current
models think that John’s and Mark’s completed exercises extensively overlap
(gao2021rcd, ; wang2023self, ), resulting in many second-order connections and
a high similarity between these two students (Figure 2 (a)). However, we
notice that response logs can be either correct or incorrect. Although Mark
and John have extensive overlap in completed exercises, they provide different
answers to all these exercises, indicating that their abilities are
fundamentally opposite. That is, there should be no second-order connections
between these two students. As shown in Figure 2 (b), we split the bipartite
graph into two subgraphs according to correct/incorrect answers. Obviously,
there are no connections between John and Mark in either of the subgraphs,
indicating that GNN will not share their information.

Additionally, we suspect that each edge may involve uncertainty, i.e.,
fortunate guessing or careless mistakes. In Figure 2 (c), John has answered
one exercise related to concept k2 incorrectly but answered the rest
correctly, indicating that John may have already mastered concept k2. We
further question whether John’s incorrect answer to $e_{5}$ is due to his lack
of concept understanding or a careless mistake. It is difficult to provide a
definitive answer. Therefore, we consider differentiating and filtering out
these uncertain edges in the graph structure (i.e., edges that cannot
accurately reflect students’ proficiency levels on concepts). It is worth
noting that although some studies have introduced heterogeneous edges in
student performance prediction, they have not considered how to diagnose
students’ proficiency levels on concepts or address uncertainty, leading to
suboptimal diagnostic performance (liu2021graph, ; wu2022multi, ).

Figure 2. Analyses about how to convert response logs to a bipartite graph. We
take student John and Mark as an example. (a) Current methods (gao2021rcd, ;
wang2023self, ); (b) Distinguishing edge heterogeneity; (c) Considering edge
uncertainties.

To explore edge heterogeneity and uncertainty in the bipartite graph, in this
paper, we propose a novel Adaptive Semantic-aware Graph-based Cognitive
Diagnosis model (ASG-CD). Specifically, we first adopt an embedding module
that maps students, exercises, and knowledge concepts to a latent
representation space (wang2022neuralcd, ). Then, we utilize a matrix
factorization layer with inputting students’ and concepts’ latent
representations to obtain students’ proficiency levels on concepts. Similarly,
exercise difficulties can be obtained in an analogous way. In the following,
we devise two layers for addressing edge heterogeneity and uncertainty in the
bipartite graph, respectively. One is the Semantic-aware GNN Layer, which is
designed to distinguish the different edge semantics. Inspired by graph
convolutional matrix completion (berg2017graph, ), we split the bipartite
graph into two subgraphs based on different edge semantics (correct answers
and incorrect answers), and then aggregate information based on these two
subgraphs separately. The other layer is the Adaptive Edge Differentiation
Layer, in which we dynamically differentiate and filter out uncertain edges
for a reliable graph structure. As CD can predict students’ proficiency levels
on concepts by analyzing their response logs, we calculate the deviations
between the predicted and real response logs to differentiate uncertain edges.
In the following, to filter out uncertain edges, we select edges with lower
deviations to construct the bipartite graph. Obviously, this layer can help us
differentiate student-exercise edges that are more consistent with model
predictions. Note that, we consider that those edges can better reflect the
students’ true abilities. Conversely, other uncertain edges contradict true
abilities. Inspired by adaptive learning in machine learning
(balachandran2018importance, ; liu2022agfa, ), during each epoch, we first
dynamically differentiate each edge, and then filter out uncertain edges.
Extensive experiments on three real-world datasets have clearly demonstrated
the effectiveness of our proposed ASG-CD. In summary, the major contributions
of this paper are summarized as follows:

  * •

We observe edge heterogeneity and uncertainty in the student-exercise
bipartite graph. We argue that previous works neglect their negative effects,
leading to suboptimal diagnostic performance of CD models.

  * •

We propose a novel ASG-CD model. Specifically, to address edge heterogeneity,
we propose Semantic-aware GNN Layer which models diverse semantics of edges.
To avoid effects of edge uncertainty, we devise Adaptive Edge Differentiation
Layer by adaptively differentiating and filtering out uncertain edges in the
bipartite graph.

  * •

Extensive experiments on three real-world datasets have demonstrated the
effectiveness of ASG-CD. We achieve an improvement of over 1% than best
baselines on accuracy metrics on the ASSIST dataset and Junyi dataset.
Morever, we achieve an improvement of over 17% on DOA metric on the Junyi
dataset.

##  2\. Relation Works

As we focus on how to apply graph information for better diagnostic
performance, in this section, we summarize prior works in educational CD, and
graph-based models for edge prediction.

###  2.1. Educational Cognitive Diagnosis

Researches on educational CD have primarily emerged from the field of
psychology. Two classic methods in this area are Item Response Theory (IRT)
(embretson2013item, ) and DINA (de2009dina, ). IRT utilizes continuous one-
dimensional parameters to represent student and exercise entities. Further, it
employs a logistic function to infer students’ performance on exercises.
Multidimensional item response theory (MIRT) has been proposed for better
ability estimations by extending parameter dimensions (reckase1997past, ;
bonifay2019multidimensional, ). DINA represents student and exercise entities
with binary discrete variables. Then, it introduces slip and guess parameters
to better fit real-world scenarios (de2009dina, ).

With the recent advancements in neural networks, researchers have turned their
attention on leveraging neural networks in educational CD (gao2023leveraging,
; tong2021item, ; huang2021group, ; wang2023self, ). NCDM framework first
utilizes high-dimensional continuous representations to model students’
abilities and exercise difficulties, and proposes a novel diagnostic functions
to incorporate all parameters to obtain final results (wang2020neural, ).
Building upon the NCDM framework, graph-based CD models integrate graph
structure information into CD models (gao2021rcd, ; wang2023self, ). Compared
to these CD models, our distinction lies in further exploring the semantic
heterogeneity and uncertainty within edges in the bipartite graph to further
enhance diagnostic performance.

###  2.2. Graph-based Models for Edge Prediction

Considering that student-exercise response logs can form a bipartite graph,
student-exercise interaction prediction can also be seen as an edge prediction
task in the bipartite graph (liu2021graph, ). GNN leverages graph structures
to propagate information, enabling effective analysis in various domains
(hu2022learning, ; xia2022simgrace, ; yang2022graph, ). NGCF first consider
introducing attention-based graph neural networks to better handle interaction
data (wang2019neural, ). Further researches have demonstrated that simplifying
GNN is beneficial for effectively capturing graph structural information from
interaction data (he2020lightgcn, ; chen2020revisiting, ). For edges with
multiple semantic information, it has been proven effective to constructing
sub-graphs based on edge semantics (berg2017graph, ; shuai2022review, ;
schlichtkrull2018modeling, ).

To further exploit the graph information, researchers have studied adaptive
learning, a classic method in machine learning for feature completion or
selection (balachandran2018importance, ; liu2022agfa, ; wen2022trend, ). This
idea has been widely applied in graphs (wang2018edge, ; chen2023adap, ), such
as completing node features (wu2020joint, ), exploring potential connected
edges (yang2021enhanced, ), or design specific activation functions
(zhang2022graph, ). Researchers have proposed adaptively modifying and
fulfilling node features for better graph representations (wu2020joint, ). In
this paper, inspired by adaptive learning, we propose dynamically
distinguishing and filtering out uncertain edges during the training process.
This operation can ensure that the bipartite graph is reliable and stable,
thereby improving diagnostic performance.

Figure 3. The overall structure of our proposed ASG-CD.

##  3\. The Proposed Model

In this section, we first present important notations and task formulation,
followed by the overall structure of our proposed ASG-CD. After that, we
introduce each component of ASG-CD in detail.

###  3.1. Important Notations and Task Formulation

In CD, there are students $S(|S|=M)$, exercises $E(|E|=N)$, and knowledge
concepts $K(|K|=T)$. Two types of relationships are utilized among these
entities. First, students practice some exercises, forming response logs,
denoted as a set of triplets $\\{(s,e,r_{se})\\}$. If student $s$ answers
exercise $e$ correctly, $r_{se}=1$. Otherwise, $r_{se}=0$. $R_{train}$ and
$R_{test}$ denote the training and testing triplets, respectively. Second, the
relations between exercises and concepts are denoted by
$\mathbf{Q}=\\{q_{ek}\\}_{N\times T}$. Usually, $\mathbf{Q}$ is pre-defined by
experts. If exercise $e$ is related to concept $k$, $q_{ek}=1$; otherwise,
$q_{ek}=0$. Another perspective for the input data ${R}_{train}$ is a student-
exercise bipartite graph with the adjacency matrix $\mathbf{X}$. If student
$s$ answers exercise $e$ correctly, $x_{se}=1$. If student $s$ answers
exercise $e$ incorrectly, $x_{se}=0$. Otherwise, $x_{se}$ is missing. Current
neural network-based CD models mainly adopt the NCDM framework
(gao2023leveraging, ; wang2020neural, ; wang2023dynamic, ; gao2021rcd, ;
li2022hiercdf, ). Specifically, NCDM framework uses
$\mathbf{A}=[\mathbf{a}_{1},...,\mathbf{a}_{s},...,\mathbf{a}_{M}]^{\top}\in\mathbb{R}^{M\times
T}$ to represent all students’ abilities (i.e., proficiency levels on all
concepts) $\mathbf{a}_{s}=[{a}_{s1},...,{a}_{sk},...,{a}_{sT}]$. Note that,
each dimension of $\mathbf{a}_{s}$ has independent meaning, e.g., ${a}_{sk}$
denotes student $s$’s proficiency level on concept $k$. NCDM framework also
introduces exercise difficulties
$\mathbf{D}=[\mathbf{d}_{1},...,\mathbf{d}_{e},...,\mathbf{d}_{N}]^{\top}\in\mathbb{R}^{N\times
C}$ and discriminations
$\mathbf{h}^{disc}=[{h}_{1}^{disc},...,{h}_{e}^{disc},...,{h}_{N}^{disc}]^{\top}\in\mathbb{R}^{N\times
1}$.

Based on above notations, we formulate the CD task as follows,
Input: The response logs ${R}_{train}$ and exercise-concept relations
$\mathbf{Q}$.
Output: A CD model to infer students’ abilities $\mathbf{A}$ (i.e.,
proficiency levels on all concepts) through response prediction.

###  3.2. Overall Structure

Figure 3 illustrates the overall structure of our proposed ASG-CD. Note that,
apart from treating student-exercise response logs as input, we further focus
on building and utilizing a reliable student-exercise bipartite graph to
enhance student abilities $\mathbf{A}$ in CD. The following parts in this
section provide detailed discussions about each component of ASG-CD, as
follows:

  * •

Embedding Module and Matrix Factorization Layer. We first embed entities
(i.e., students, exercises, and concepts) into latent embeddings, and utilize
matrix factorization techniques to obtain student abilities and exercise
difficulties by combining these latent embeddings (he2017neural, ;
koren2009matrix, ).

  * •

Semantic-aware GNN Layer. Inspired by previous works (berg2017graph, ), we
devise a semantic-aware GNN layer that respectively aggregates information
based on edges with different semantics. This layer can capture edge
heterogeneity to better enhance student abilities and exercise difficulties.

  * •

Prediction Layer. We adopt a widely-used diagnostic function which
incorporates student abilities and exercise parameters into student-exercise
hidden representations, followed by mapping hidden representations to
predicted responses (wang2020neural, ; wang2022neuralcd, ).

  * •

Adaptive Edge Differentiation Layer. This layer is designed to differentiate
and filter out uncertain edges, mitigating adverse impacts in GNN. With the
aid of this layer, we can adaptively build a reliable graph to enhance
diagnostic performance.

###  3.3. Embedding Module and Matrix Factorization Layer

Traditional methods directly model student abilities with variables whose
dimension equal to the number of concepts (gao2021rcd, ; wang2020neural, ;
de2009dina, ). They can successfully represent proficiency levels on all
concepts, however, they have an obvious drawback: dimensions of abilities
$\mathbf{A}$ are independent in the process of optimization. This is not the
ideal situation. Obviously, knowledge concepts are related, therefore, some
dimensions of $\mathbf{A}$ are correlated. E.g., persons who understand
refraction of light may already understand reflection of light. Directly
extracting such correlations from the data is complex. Alternatively, we
borrow the success of capturing student-concept collaborative information
(wang2022neuralcd, ; ma2022knowledge, ), implicitly modeling student and
concept similarities for CD. Specifically, we apply choose Matrix
Factorization techniques in CD (koren2009matrix, ; he2017neural, ).

The process can be divided into two parts. First, we adopt an embedding
module, which embeds entities into: student latent embeddings
$\mathbf{U}=[\mathbf{u}_{1},...,\mathbf{u}_{s},...,\mathbf{u}_{M}]^{\top}\in\mathbb{R}^{M\times
Z}$, exercise latent embeddings
$\mathbf{V}=[\mathbf{v}_{1},...,\mathbf{v}_{e},...,\mathbf{v}_{N}]^{\top}\in\mathbb{R}^{N\times
Z}$ and concept latent embeddings
$\mathbf{O}=[\mathbf{o}_{1},...,\mathbf{o}_{k},...,\mathbf{o}_{T}]^{\top}\in\mathbb{R}^{T\times
Z}$. $Z$ denotes the dimension of free embeddings. Second, we utilize a simple
yet effective PMF model as the matrix factorization layer (koren2009matrix, ),
formulated as:

(1)
$a_{sk}^{(0)}=\sigma(<\mathbf{u}_{s},\mathbf{o}_{k}>),~{}{d}_{ek}^{(0)}=\sigma(<\mathbf{v}_{e},\mathbf{o}_{k}>),$

where $<,>$ denotes the inner dot of corresponding representations.
$a_{sk}^{(0)}$ denotes the student $s$’s ability on concept $k$. The original
ability $\mathbf{a}_{s}^{(0)}$ of student $s$ can be formulated as
$\mathbf{a}_{s}^{(0)}=[{a}_{s1}^{(0)},...,{a}_{sk}^{(0)},...,{a}_{sT}^{(0)}]$.
Similarly, $\mathbf{d}_{e}^{(0)}$ denotes exercise $e$’s original difficulty,
and
$\mathbf{d}_{e}^{(0)}=[{d}_{e1}^{(0)},...,{d}_{ek}^{(0)},...,{d}_{eT}^{(0)}]$.

###  3.4. Semantic-aware GNN Layer

Recently, researchers have found that response logs naturally form a student-
exercise bipartite graph, and the graph structure has been proven beneficial
for diagnosis (gao2021rcd, ; wang2023self, ; liu2021graph, ). In the student-
exercise bipartite graph, there are two types of edges with fundamentally
different semantic meanings: correct answers and incorrect answers. The key to
the success of GNN lies in information propagation among connected nodes in
the graph structure. For a student node, its subgraph structure can help
capture similar students’ information from its second-order connected
students, thereby enhancing its representation capability. When considering
second-order connections between student nodes, we argue that there should
only be a second-order edge if two students have the same response on an
exercise. On the contrary, if two students have different responses, it is
inappropriate to establish a second-order edge between them. Ignoring this
edge heterogeneity poses a risk of confounding semantic information in GNN. We
have observed that attention modules do not always accurately capture the
semantics of these distinct edges, leading to suboptimal diagnostic
performance of graph-based CD models (gao2021rcd, ; wang2023self, ).

To capture the heterogeneity, we follow a classic GCMC model by splitting the
bipartite graph into subgraphs based on edge semantics (berg2017graph, ), and
devise a Semantic-aware GNN Layer. In the following, we introduce the layer
from the perspective of message passing and aggregation.

####  3.4.1. Message Passing.

The student-exercise adjacency matrix $\mathbf{X}$ consists of three states:
correct answers (forming set $X_{1}$), incorrect answers (forming set
$X_{0}$), and non-interactions (missing values). To capture edge heterogeneity
in message passing, we can allocate a unique transformation to each certain
type of edge semantics, resulting in edge-specific messages $\mu$ from
exercise $e$ to student $s$ of the following form:

(2) $\displaystyle\mu_{e\rightarrow
s,1}^{(l)}=\frac{1}{\sqrt{|\mathcal{N}_{s,1}||\mathcal{N}_{e,1}|}}W_{1}\mathbf{d}_{e}^{(l-1)},$
$\displaystyle\mu_{e\rightarrow
s,0}^{(l)}=\frac{1}{\sqrt{|\mathcal{N}_{s,0}||\mathcal{N}_{e,0}|}}W_{0}\mathbf{d}_{e}^{(l-1)},$

where $\mu_{e\rightarrow s,1}^{(l)}$ and $\mu_{e\rightarrow s,0}^{(l)}$ denote
message passed by 1 edges (correct answers) and message passed by 0 edges
(incorrect answers), respectively.
$\sqrt{|\mathcal{N}_{s,1}||\mathcal{N}_{e,1}|}$ and
$\sqrt{|\mathcal{N}_{s,0}||\mathcal{N}_{e,0}|}$ denotes the normalization
constant called symmetric normalization. $\mathcal{N}_{s,1},\mathcal{N}_{e,1}$
denote the neighbor sets of student $s$ and exercise $e$ with 1 edges.
Similarly, $\mathcal{N}_{s,1},\mathcal{N}_{e,1}$ denote the neighbor sets with
0 edges. Obviously, $\mathcal{N}_{s,1},\mathcal{N}_{e,1}$ and
$\mathcal{N}_{s,0},\mathcal{N}_{e,0}$ can be obtained from $X_{1}$ and
$X_{0}$, respectively. $W_{1}$ and $W_{0}$ are edge-type specific parameter
matrices for edge 1 and edge 0, respectively. $\mathbf{d}_{e}^{(l-1)}$ denotes
exercise $e$’s difficulty at layer $(l-1)$, and $\mathbf{d}_{e}^{0}$ is the
exercise $e$’s original difficulty. Messages from student $s$ to exercise $e$
can be processed in an analogous way, which can be formulated as:

(3) $\displaystyle\mu_{s\rightarrow
e,1}^{(l)}=\frac{1}{\sqrt{|\mathcal{N}_{s,1}||\mathcal{N}_{e,1}|}}W_{1}\mathbf{a}_{s}^{(l-1)},$
$\displaystyle\mu_{s\rightarrow
e,0}^{(l)}=\frac{1}{\sqrt{|\mathcal{N}_{s,0}||\mathcal{N}_{e,0}|}}W_{0}\mathbf{a}_{s}^{(l-1)},$

where $\mathbf{a}_{s}^{(l-1)}$ denotes student $s$’s ability at layer $(l-1)$.
Eq.(2) and Eq.(3) can be understood as splitting the original graph into
different subgraphs according to edge types, and then propagating information
based on each subgraph.

####  3.4.2. Message Aggregation.

After message passing, we can obtain incoming messages at every node from two
types of edges. In message aggregation, for each node, we first sum incoming
messages over all neighbors under a specific edge-type. Subsequently, we
accumulate the messages from all edge-types into vector representations at the
$l$-th layer, formulated as:

(4)
$\displaystyle\mathbf{a}_{s}^{(l)}=\mathbf{W}^{(l)}(\sum_{e\in\mathcal{N}_{e,1}}\mu_{e\rightarrow
s,1}+\sum_{e\in\mathcal{N}_{e,0}}\mu_{e\rightarrow s,0}),$
$\displaystyle\mathbf{d}_{e}^{(l)}=\mathbf{W}^{(l)}(\sum_{s\in\mathcal{N}_{s,1}}\mu_{s\rightarrow
e,1}+\sum_{s\in\mathcal{N}_{s,0}}\mu_{s\rightarrow e,0}).$

After stacking $L$ layers of message passing and aggregation, we treat the
vector representations at the $L$-th layer as the final student ability and
exercise difficulty:

(5)
$\displaystyle\mathbf{a}_{s}=\mathbf{a}_{s}^{(L)},\mathbf{d}_{e}=\mathbf{d}_{e}^{(L)}.$

After obtaining the final student abilities and exercise difficulties, the
next step is to establish connections between these representations and
student-exercise response logs for optimization.

###  3.5. Prediction Layer

In this layer, we adopt the NCDM framework to map parameters into predicted
response logs (wang2020neural, ). For a student-exercise response log, we tend
to believe that a student is more likely to answer an exercise correctly when
the student’s ability exceeds the exercise difficulty on corresponding
knowledge concepts. Additionally, the exercise discriminations are taken into
consideration. Finally, the diagnostic function can be formulated as:

(6) $\displaystyle\mathbf{p}_{se}$
$\displaystyle=\mathbf{Q}_{e}\odot(\mathbf{a}_{s}-\mathbf{d}_{e})\times{h}_{e}^{disc},$
$\displaystyle\hat{r}_{se}$ $\displaystyle=\sigma(MLPs(\mathbf{p}_{se})),$

where $\mathbf{p}_{se}$ denotes the hidden representation between student $s$
and exercise $e$. The framework further utilizes $\mathbf{p}_{se}$ to infer
response logs. $\mathbf{Q}_{e}$ denotes the relations between concepts and
exercise $e$, and $\odot$ denotes the element-wise product. $\hat{r}_{se}$
denotes the predicted student $s$’s response log on exercise $e$. Then, we
have opted to utilize BCE loss function for model optimization, formulated as:

(7) $Loss=-\sum_{s,e,r_{se}\in
R_{train}}(r_{se}log(\hat{r}_{se})+(1-r_{se})log(1-\hat{r}_{se})).$

With the aids of Eq.(6) and Eq.(7), we can successfully optimize all the
parameters of students and exercises.

###  3.6. Adaptive Edge Differentiation Layer

How to distinguish fortunate guesses (careless mistakes) from mastery (no
mastery) is a remaining question. For example, a student answers multiple
different exercises related to the same knowledge concept. Only one of the
exercises is answered incorrectly but the others are answered correctly. In
this case, the incorrect answer does not accurately reflect the student’s
proficiency level on that concept. This is crucial for GNNs because these
uncertain edges may propagate incorrect information during aggregation,
affecting diagnostic performance. One intuitive approach is to first train a
model to identify which edges are uncertain, and then construct a reliable
graph to retrain CD models. However, this two-stage design is too time-
consuming. Inspired by adaptive learning in machine learning (wang2018edge, ;
botu2015adaptive, ), we propose dynamically selecting to re-construct the
bipartite graph structure. The graph structure in each epoch consists of two
edge sets:

(8) $\displaystyle X_{1}=\\{(s,e)|{\hat{r}_{se}>\epsilon\land
x_{se}=1}\\},X_{0}=\\{(s,e)|{\hat{r}_{se}<1-\epsilon\land x_{se}=0}\\},$

where hyperparameter $\epsilon$ denotes the threshold for adaptively
distinguishing uncertain edges. Eq.(8) aligns with our example: for a specific
concept, if a student answers corresponding exercises correctly multiple times
and only answers incorrectly once, CD models will infer that the student has
already mastered the concept. Here, the incorrect answer will be considered as
an uncertain edge. Considering that we can not fully determine or figure out
the true semantics of these uncertain edges, we choose not to utilize them in
GNN rather than assigning them specific semantics. Note that, modifications of
edge sets ($X_{0}$, $X_{1}$) would affect GNN process by updating neighbor
sets
($\mathcal{N}_{s,1}$,$\mathcal{N}_{s,0}$,$\mathcal{N}_{e,1}$,$\mathcal{N}_{e,0}$)
in Eq.(2-3).

##  4\. Experiments

Table 1. The detailed statistics of three datasets.

Dataset | ASSIST | Junyi | MOOC-Radar
---|---|---|---
#Students | 2,493 | 10,000 | 14,224
#Exercises | 17,746 | 734 | 2,513
#Knowledge concepts | 123 | 734 | 580
#Response logs | 267,415 | 408,057 | 898,933
#Response logs per student | 107.266 | 40.8 | 63.198
#Knowledge concepts per exercise | 1.192 | 1 | 1
#Sparsity in student-concept interactions | 92.212% | 94.441% | 98.985%
#Sparsity in student-exercise logs | 99.396% | 94.441% | 97.485%

In the section, we try to answer these Research Questions (RQ):

  * RQ1:

Does ASG-CD have consistently superior performance on three datasets? (Section
4.2)

  * RQ2:

Are all components in ASG-CD important? (Section 4.3)

  * RQ3:

What is the impact of different hyperparameters on the model? Can ASG-CD
consistently achieve better diagnostic performance than the second-best CD
model? (Section 4.4)

###  4.1. Experimental Settings

####  4.1.1. Datasets.

In the experimental part, we choose three real-world datasets, i.e., ASSIST
dataset (ASSISTments 2009-2010 ”skill
builder”)333https://sites.google.com/site/assistmentsdata/feng2009, Junyi
dataset (Junyi Academy Math Practicing
Log)444https://pslcdatashop.web.cmu.edu/DatasetInfo?datasetId=1198, and MOOC-
Radar dataset555https://github.com/THU-KEG/MOOC-Radar. ASSIST is a public
available dataset collected by the ASSISTments online tutoring systems
(feng2009addressing, ). It contains a wealth of student-exercise response logs
and expert-labeled exercise-concept relations $\mathbf{Q}$. In the pre-
processing step, we remove students with fewer than 15 logs to ensure an
adequate amount of data for effective cognitive diagnosis. Junyi dataset is an
open dataset collected by an e-learning website, Junyi Academy. The unique
characteristic of Junyi dataset lies in one-to-one correspondence between
knowledge concepts and exercises. Similar to previous work (li2022hiercdf, ),
we remove students that has less than 15 response logs, and then randomly
choose 10,000 students from the remaining students. MOOC-Radar is a recently
collected dataset from students’ learning records in MOOCs (yu2023moocradar,
), which contains abundant response logs. After data pre-processing, we
randomly split historical student-exercise response logs into training, and
testing parts with a certain ratio of 8:2 on all three datasets. We record the
detailed statistics in Table 1.

####  4.1.2. Metrics.

A challenge in CD lies in the lack of ground truth for students’ proficiency
levels on various concepts. Similar to previous studies (gao2021rcd, ;
li2022hiercdf, ; wang2020neural, ; wang2023self, ), we evaluate the prediction
performance of response logs as an alternative approach. Three commonly-used
accuracy metrics have been adopted for performance evaluation: Accuracy (ACC),
Root Mean Square Error (RMSE) (willmott2005advantages, ), and Area Under the
Curve (AUC) (myerson2001area, ). Additionally, we employ a widely-used metric
in educational CD called Degree of Agreement (DOA) (fouss2007random, ;
wang2020neural, ; wang2022neuralcd, ). Note that, DOA measures the consistency
between the predicted proficiency levels on concepts and the observed patterns
in the student-exercise response logs. By employing these evaluation metrics,
we can assess accuracy performance and consistency performance of our proposed
ASG-CD in predicting student proficiency levels based on response logs. These
above metrics provide a comprehensive evaluation of model utilities.

####  4.1.3. Baselines.

We select the following models as baselines:

  * •

IRT (embretson2013item, ). Item Response Theory (IRT) utilizes one-dimensional
parameters to represent students and exercises, respectively.

  * •

MIRT (reckase1997past, ). Compared to IRT, Multidimensional Item Response
Theory (MIRT) further utilizes high-dimensional parameters to represent
students and exercises.

  * •

PMF (mnih2007probabilistic, ). It treats response log prediction as
interactive behavior prediction by utilizing matrix factorization techniques.

  * •

EKLN (liu2021graph, ). Compared to PMF, it incorporates the student-exercise
bipartite graph to enhance diagnostic performance.

  * •

DINA (de2009dina, ). As a classic CD model, it adopts binary variables to
represent students and exercises. Guess and slip parameters are also
introduced to fit the real-world situation.

  * •

NCDM (wang2020neural, ). It applies neural networks into CD, and uses high-
dimensional representations to represent students’ abilities.

  * •

RCD (gao2021rcd, ). Compared to NCDM, it further incorporates complex student-
exercise bipartite graph structure information and concept relationships into
CD models.

  * •

CWNCD (wang2022continuous, ). Compared to NCDM, it assign learnable weights as
importance degrees of different concepts in each exercise.

  * •

KaNCD (wang2022neuralcd, ). It adopts matrix factorization techniques on
representations of student abilities and exercise difficulties.

Note that, IRT, MIRT, PMF and EKLN only adopt inner dot or subtraction among
student and exercise embeddings to fit student-exercise response logs. These
models cannot infer students’ proficiency levels on all concepts.

####  4.1.4. Hyperparameter Settings.

We set all models to randomly sample 8,192 student-exercise response logs per
batch. Additionally, we use the Adam optimizer and search the learning rate
from {0.0001, 0.0005, 0.001, 0.005, 0.01} for each model. For the number of
GNN layers, we search from {1, 2, 3, 4, 5} and select the best result. For the
thresholds of reliable and uncertain answers, we search $\epsilon$ from {0.01,
0.05, 0.1, 0.25, 0.5}. Finally, all experiments are conducted on a single
NVIDIA 3090 graphics card with 24G memory.

Table 2. Overall performance on the ASSIST dataset. We use bold font to
emphasize the best results and underline to indicate the second-best results.
The improvement is derived from the second-best results on each metric.

Metric Model | IRT | MIRT | PMF | EKLN | DINA | NCDM | RCD | CWNCD | KaNCD | ASG-CD | Impro
---|---|---|---|---|---|---|---|---|---|---|---
ACC $\uparrow$ | 0.7160 | 0.7170 | 0.7065 | 0.7288 | 0.6737 | 0.7232 | 0.7279 | 0.7264 | 0.7304 | 0.7392 | 1.205%
RMSE $\downarrow$ | 0.4366 | 0.4352 | 0.4429 | 0.4324 | 0.4672 | 0.4350 | 0.4333 | 0.4341 | 0.4319 | 0.4212 | 2.477%
AUC $\uparrow$ | 0.7445 | 0.7458 | 0.7469 | 0.7611 | 0.7294 | 0.7559 | 0.7602 | 0.7583 | 0.7650 | 0.7765 | 1.503%
DOA $\uparrow$ | - | - | - | - | 0.6235 | 0.6422 | 0.6525 | 0.6486 | 0.6563 | 0.6657 | 1.432%

Table 3. Overall performance on the Junyi dataset. We use bold font to
emphasize the best results and underline to indicate the second-best results.
The improvement is derived from the second-best results on each metric.

Metric Model | IRT | MIRT | PMF | EKLN | DINA | NCDM | RCD | CWNCD | KaNCD | ASG-CD | Impro
---|---|---|---|---|---|---|---|---|---|---|---
ACC $\uparrow$ | 0.7511 | 0.7576 | 0.7650 | 0.7668 | 0.4993 | 0.7426 | 0.7496 | 0.7462 | 0.7550 | 0.7768 | 1.304%
RMSE $\downarrow$ | 0.4071 | 0.4062 | 0.4017 | 0.3997 | 0.5246 | 0.4174 | 0.4126 | 0.4255 | 0.4100 | 0.3918 | 1.976%
AUC $\uparrow$ | 0.7878 | 0.7904 | 0.8019 | 0.8042 | 0.6314 | 0.7806 | 0.7824 | 0.7861 | 0.7898 | 0.8192 | 1.865%
DOA $\uparrow$ | - | - | - | - | 0.4896 | 0.5075 | 0.5332 | 0.5419 | 0.6030 | 0.7115 | 17.993%

Table 4. Overall performance on the MOOC-Radar dataset. We use bold font to
emphasize the best results and underline to indicate the second-best results.
The improvement is derived from the second-best results on each metric.

Metric Model | IRT | MIRT | PMF | EKLN | DINA | NCDM | RCD | CWNCD | KaNCD | ASG-CD | Impro
---|---|---|---|---|---|---|---|---|---|---|---
ACC $\uparrow$ | 0.8486 | 0.8576 | 0.8504 | 0.8586 | 0.7798 | 0.8541 | 0.8590 | 0.8604 | 0.8613 | 0.8686 | 0.848%
RMSE $\downarrow$ | 0.3263 | 0.3198 | 0.3247 | 0.3179 | 0.3752 | 0.3231 | 0.3171 | 0.3151 | 0.3144 | 0.3079 | 2.067%
AUC $\uparrow$ | 0.8573 | 0.8663 | 0.8634 | 0.8723 | 0.8127 | 0.8700 | 0.8751 | 0.8776 | 0.8785 | 0.8862 | 0.876%
DOA $\uparrow$ | - | - | - | - | 0.6080 | 0.6044 | 0.6724 | 0.7062 | 0.7135 | 0.7189 | 0.757%

###  4.2. Overall Performance (RQ1)

We report overall performance in Table 2, Table 3 and Table 4. There are
several observations from these three tables.

  * •

First of all, ASG-CD has the best performance on three datasets. On three
datasets, ASG-CD achieves an improvement of nearly 1% on accuracy performance
(ACC, RMSE, AUC). Furthermore, ASG-CD has an improvement of over 10% on the
DOA metric on Junyi dataset. This clearly demonstrates the stable and strong
modeling ability of our proposed ASG-CD.

  * •

Second, compared to another graph-based CD model (RCD), ASG-CD achieves
significant improvements on all three datasets. Specifically, compared to RCD,
ASG-CD has an improvement of over 3% on accuracy performance and over 30% on
the DOA metric on the Junyi dataset. This indicates that ASG-CD can better
make use of graph structure for diagnosis.

  * •

Third, compared to state-of-the-art KaNCD model, our proposed ASG-CD achieves
the most accuracy improvement on the Junyi dataset. The reason is the unique
characteristic of Junyi dataset, i.e., one-to-one correspondence between
concepts and exercises. That is to say, concepts that we encountered in the
testing set will be totally different from those in the training set. This
indicates that our proposed ASG-CD can better handle some extreme data
situations.

  * •

Last but not least, we find that PMF and EKLN achieve better performance than
current NN-based CD models on the Junyi dataset. This indicates a promising
future direction: when faced with diagnosing new knowledge concepts,
researchers can leverage the collaborative information between students and
exercises.

Due to page limitations, in remaining parts of this section, we present the
results of the ablation studies and hyperparameter analyses on two smaller
datasets, ASSIST and Junyi.

###  4.3. Ablation Studies (RQ2)

We have conducted ablation experiments about three layers of ASG-CD, and
results are presented in Table 5. The first row does not take any additional
layers into consideration, therefore, it corresponds to the NCDM model
(wang2020neural, ). The last row considers all layers, therefore, it
corresponds to our proposed ASG-CD.

There are several observations derived from Table 5. First, compared to NCDM,
Matrix Factorization Layer serves as a strong foundation, leading to an
accuracy improvement of over 1% across both datasets. The reason for this
improvement is that students would not cover all concepts in the training data
(wang2022neuralcd, ). For students, their proficiency levels on their
uncovered knowledge concepts are close to the initial value, therefore,
predicted students’ performance on relevant exercises would be inaccurate.
Factorization can solve this problem because it can implicitly model the
collaborative signals to fill students’ proficiency levels on their uncovered
concepts. Second, Semantic-aware GNN Layer demonstrates remarkable
adaptability to CD by combining state-of-the-art model paradigms
(wang2020neural, ; wang2022neuralcd, ). Compared to some models that sacrifice
diagnostic capabilities for accuracy improvements (wang2023self, ;
liu2021graph, ), it not only achieves a remarkable over 2.5% accuracy
improvement but also retains its diagnostic capabilities intact. Third,
Adaptive Edge Differentiation Layer makes precise adjustments to the graph
structure, resulting in consistent accuracy improvements on both datasets.
This layer builds on the aforementioned foundations, showcasing its ability to
further enhance diagnostic performance.

Table 5. Ablation studies on ASSIST and Junyi datasets. The last row
corresponds to our proposed ASG-CD.

Factorization | GNN | Differentiation | ASSIST dataset | Junyi dataset
---|---|---|---|---
Eq.(1) | Eq.(2-4) | Eq.(8) |  ACC $\uparrow$ |  RMSE $\downarrow$ |  AUC $\uparrow$ |  DOA $\uparrow$ |  ACC $\uparrow$ |  RMSE $\downarrow$ |  AUC $\uparrow$ |  DOA $\uparrow$
✗ | ✗ | ✗ | 0.7232 | 0.4350 | 0.7559 | 0.6422 | 0.7426 | 0.4174 | 0.7806 | 0.5075
✓ | ✗ | ✗ | 0.7320 | 0.4296 | 0.7647 | 0.6569 | 0.7606 | 0.4067 | 0.7937 | 0.6426
✗ | ✓ | ✗ | 0.7370 | 0.4249 | 0.7707 | 0.6638 | 0.7717 | 0.3969 | 0.8112 | 0.6845
✓ | ✓ | ✗ | 0.7388 | 0.4230 | 0.7758 | 0.6662 | 0.7741 | 0.3927 | 0.8185 | 0.7087
✓ | ✓ | ✓ | 0.7392 | 0.4212 | 0.7765 | 0.6657 | 0.7768 | 0.3918 | 0.8192 | 0.7115

###  4.4. Hyperparameters Analyses (RQ3)

####  4.4.1. Number of GNN Layers.

Figure 4. Diagnostic performance with varying number of GNN layers $L$ on the
ASSIST and Junyi datasets.

In this part, we focus on studying the impacts of multi-layer GNN on
diagnostic performance. The results are in Figure 4. Firstly, we find that
smaller number of GNN layers ($L<4$) achieves better performance than the
bigger number ($L=4~{}or~{}5$). Specifically, on the ASSIST dataset, when
number of GNN layers is bigger than three, diagnostic performance is slightly
worse than the second-best model (KaNCD). The phenomenon suggests that we need
to be cautious about the over-smoothing caused by multiple GNN layers
(keriven2022not, ). Secondly, comparing two datasets, we observe that accuracy
performance (AUC, ACC) on the ASSIST dataset occurs when $L=1$, while the
optimal results on the Junyi dataset occur when $L=3$. Note that, Junyi
dataset has a one-to-one correspondence between concepts and exercises. This
indicates that graph algorithms have a clear advantage in extreme data
scenarios where CD models need to assess a student’s procifiency levels on
concepts they haven’t encountered.

####  4.4.2. Thresholds for Adaptive Learning.

Figure 5. Diagnostic performance with varying thresold $\epsilon$ on the
ASSIST and Junyi datasets.

We investigate the impact of removing uncertain edges by adjusting the
thresholds for determining uncertainty, and the corresponding results are
shown in Figure 3. Specifically, we test results for
$\epsilon\in\\{0,0.01,0.05,0.1,0.25,0.5\\}$, where $\epsilon=0$ means no edges
are removed.

Firstly, we find that thresholds $\epsilon\in\\{0,0.01,0.05,0.1,0.25\\}$ lead
to an improvement in diagnostic performance. This confirms that there are
instances where students make careless mistakes or fortunate guesses, and
removing these uncertain edges is beneficial for diagnosis. Secondly, we
observe a decrease in performance when $\epsilon=0.5$. This is because we
remove a wide range of edges, and some removed edges may carry correct
semantic information. In summary, our findings suggest that adjusting the
thresholds for removing uncertain edges can have a positive impact, but the
selection of thresholds should be done carefully to avoid removing
semantically important edges. Further research is needed to explore more
methods for identifying uncertain edges in the bipartite graph.

##  5\. Conclusion

As one of the United Nations’ Sustainable Development Goals, online education
has received widespread attention. Educational Cognitive diagnosis is a
critical task in online education. In this paper, we proposed ASG-CD to
capture the edge heterogeneity and uncertainty in the bipartite graph.
Firstly, we designed an embedding module and a matrix factorization layer to
obtain student abilities and exercise difficulties. These components enabled
us to represent students and exercises, with each dimension corresponding to a
specific concept. Then, we introduced a Semantic-aware GNN Layer to aggregate
information separately to capture edge heterogeneity. This layer allowed us to
utilize different semantic relationships among different nodes, enhancing the
diagnostic capability. Furthermore, we observed that some response logs would
contradict the model predictions, indicating uncertainties of corresponding
edges. Therefore, we proposed an Adaptive Edge Differentiation Layer to
dynamically filter out uncertain edges and construct a reliable graph during
model training. This layer helped improve the robustness and accuracy of the
diagnostic process. Extensive experimental results on three datasets have
demonstrated the superiority of ASG-CD.

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