# Consistency and Monotonicity Regularization for Neural Knowledge Tracing

Seewoo Lee1 Youngduck Choi1 Juneyoung Park1111Contact Author Byungsoo
Kim1&Jinwoo Shin2
1Riiid! AI Research
2Graduate School of AI, KAIST
{seewoo.lee, youngduck.choi, juneyoung.park<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Knowledge Tracing (KT), tracking a human’s knowledge acquisition, is a central
component in online learning and AI in Education. In this paper, we present a
simple, yet effective strategy to improve the generalization ability of KT
models: we propose three types of novel data augmentation, coined replacement,
insertion, and deletion, along with corresponding regularization losses that
impose certain consistency or monotonicity biases on model’s predictions for
the original and augmented sequence. Extensive experiments on various KT
benchmarks show that our regularization scheme consistently improves the model
performances, under 3 widely-used neural networks and 4 public benchmarks,
e.g., it yields 6.3% improvement in AUC under the DKT model and the
ASSISTmentsChall dataset.

##  1 Introduction

In recent years, Artificial Intelligence in Education (AIEd) has gained much
attention as an emerging field to elevate educational technology. Especially
due to the circumstances from the COVID-19 pandemic, much of the education
industry was forcibly moved to an online environment which inevitably allowed
much opportunity to utilize educational data. The ability to diagnose students
through data and provide personalized learning paths have become a critical
edge to uplift online education. The most fundamental aspect of assessing
student’s current knowledge state has been the focus of AIED research, a task
commonly known as Knowledge Tracing(KT). Creating a more precise and a robust
KT model has become an essential path to develop a highly effective and AI-
based educational system.

(a) ASSISTments2015

(b) ASSISTmentsChall

(c) STATICS2011

(d) EdNet-KT1

Figure 1:  Distribution of the correctness rate of past interactions when the
response correctness of current interaction is fixed, for 4 knowledge tracing
benchmark datasets. Orange (resp. blue) represents the distribution of
correctness rate (of past interactions) where current interaction’s response
is correct (resp. incorrect). $x$ axis represents previous interactions’
correctness rates (values in $[0,1]$). The orange distribution lean more to
the right than the blue distribution, which shows the monotonicity nature of
the interaction datasets.

Since the work of Piech et al. (2015), deep neural networks have been widely
used for the KT modeling. Current research trends in the KT literature
concentrate on building more sophisticated, complex and large-scale models,
inspired by model architectures from Natural Language Processing (NLP), such
as LSTM Hochreiter and Schmidhuber (1997) or Transformer Vaswani et al. (2017)
architectures. Further investigations to introduce additional components based
on educational contexts such as textual information Huang et al. (2019).
However, not all educational data are sufficiently large and more often than
not, the larger model sizes lead to overfitting and ultimately hurt the
model’s generalizabiliy Gervet et al. (2020) (See Figure 1 of the Appendix).
To the best of our knowledge, only a handful of the literature addresses such
issues and even then, the scope is limited to regularization Yeung and Yeung
(2018); Sonkar et al. (2020).

Figure 2:  Augmentation strategies and corresponding bias on model’s
predictions. Each tuple represents question id and response of the student’s
interaction (1 means correct). Replacing interactions with similar questions
($Q_{1},Q_{3}$ to $Q_{1}^{\prime},Q_{3}^{\prime}$) does not change model’s
predictions drastically. Introducing new interactions with correct responses
$(Q^{\prime},Q^{\prime\prime})$ increases model’s predictions, but deleting
such interaction $(Q_{1},1)$ decreases them.

To address the issue, we propose simple, yet effective data augmentation
strategies for improving the generalization ability of KT models, along with
novel regularization losses for each strategy. In particular, we suggest three
types of data augmentation, coined (skill-based) replacement, insertion, and
deletion. Specifically, we generate augmented (training) samples by randomly
replacing questions that a student solved with similar questions or
inserting/deleting interactions with fixed responses. Furthermore, during
training, we impose certain consistency (for replacement) and monotonicity
(for insertion/deletion) bias on the model’s predictions by optimizing
corresponding regularization losses that compares the original and the
augmented interaction sequences. Such regularization strategies are motivated
from our observation that existing knowledge tracing models’ prediction often
fails to satisfy the consistency and monotonicity condition, e.g., see Figure
4 in Section 3. Here, our intuition behind the proposed consistency
regularization is that the model’s output for two interaction sequences with
same response logs for similar questions should be close. Next, the proposed
monotonicity regularization is designed to enforce the model’s prediction to
be monotone with respect to the number of questions that are correctly (or
incorrectly) answered, i.e., a student is more likely to be correct (or
incorrectly) in the next question if the student was more correct in the past.
By analyzing the distribution of previous correctness rate, we can observe
that the existing student interaction is indeed monotonic as shown in Figure
1. The overall augmentation and regularization strategies are sketched in
Figure 2.

We demonstrate the effectiveness of the proposed method with 3 widely used
neural knowledge tracing models - DKT Piech et al. (2015), DKVMN Zhang et al.
(2017), and SAINT Choi et al. (2020a) \- on 4 public benchmark datasets -
ASSISTments2015, ASSISTmentsChall, STATICS2011, and EdNet-KT1. Extensive
experiments show that, regardless of dataset or model architecture, our scheme
remarkably increases the prediction performance - 6.3% gain in Area Under
Curve (AUC) for DKT on the ASSISTmentsChall dataset. In particular, ours is
much more effective under smaller datasets: by using only $25\%$ of the
ASSISTmentsChall dataset, we improve AUC of the DKT model from 69.68% to
75.44%, which even surpasses the baseline performance 74.4% with the full
training set. We further provide various ablation studies for the selected
design choices, e.g., AUC of the DKT model on the ASSISTments2015 dataset
dropped from 72.44% to 66.48% when we impose ‘reversed’ (wrong) monotonicity
regularization. The findings from the current study contribute to existing KT
literature by providing a novel generalization mechanism which provide a
strong baseline for future augmentation and regularization research.

###  1.1 Related Works and Preliminaries

Data augmentation is arguably the most trustworthy technique to prevent
overfitting or improve the generalizability of machine learning models. In
particular, it has been developed as an effective way to impose a domain-
specific, inductive bias to a model. For example, for computer vision models,
simple image warpings such as flip, rotation, distortion, color shifting,
blur, and random erasing are the most popular data augmentation methods
Shorten and Khoshgoftaar (2019). For NLP models, it is popular to augment
texts by replacing words with synonyms Zhang et al. (2015) or words with
similar (contextualized) embeddings Kobayashi (2018). Recently, Wei and Zou
(2019) show that even simple methods like random insertion/swap/deletion could
improve text classification performances.

The aforementioned data augmentation techniques have been used not only for
standard supervised learning setups, but also for various unsupervised and
semi-supervised learning frameworks, by imposing certain inductive biases to
models. For example, consistency learning Berthelot et al. (2019) impose a
consistency bias to a model so that the model’s output is invariant under the
augmentations, by means of training the model with consistency regularization
loss (e.g. $L^{2}$-loss between outputs). Abu-Mostafa (1992) suggested general
theory for imposing such inductive biases (which are stated as _hints_) via
additional regularization losses. Their successes highlight the importance of
domain specific knowledge for designing appropriate data augmentation
strategies, but such results are rare in the domain of AIEd, especially for
Knowledge Tracing.

Knowledge tracing (KT) is the task of modeling student knowledge over time
based on the student’s learning history. Formally, for a given student
interaction sequence $(I_{1},\dots,I_{T})$, where each $I_{t}=(Q_{t},R_{t})$
is a pair of question id $Q_{t}$ and the student’s response correctness
$R_{t}\in\\{0,1\\}$ ($1$ means correct), KT aims to estimate the following
probability

$\displaystyle\mathbb{P}[R_{t}=1|I_{1},I_{2},\dots,I_{t-1},Q_{t}],$ (1)

i.e., the probability that the student answers correctly to the question
$Q_{t}$ at $t$-th step. Corbett and Anderson (1994) proposed Bayesian
Knowledge Tracing (BKT) that models a student’s knowledge as a latent variable
in a Hidden Markov Model. Also, various seq2seq architectures including LSTM
Hochreiter and Schmidhuber (1997), MANN Graves et al. (2016), and Transformer
Vaswani et al. (2017) are used in the context of KT and showed their efficacy.
Deep Knowledge Tracing (DKT) is the first deep learning based model that
models student’s knowledge states as LSTM’s hidden state vectors Piech et al.
(2015). Dynamic Key-Value Memory Network and its variation can exploit
relationships between questions/skills with concept vectors and concept-state
vectors with key and value matrices, which is more interpretable than DKT
Zhang et al. (2017). Transformer based models Pandey and Karypis (2019); Choi
et al. (2020a) are able to learn long-range dependencies with their self-
attention mechanisms and be trained in parallel. Utilizing additional features
of interactions, such as question texts Huang et al. (2019) and time
information Nagatani et al. (2019), is another way to improve performances.

##  2 Consistency and Monotonicity Regularization for KT

For a given set of data augmentations $\mathcal{A}$, we train KT models with
the following loss:

$\displaystyle\mathcal{L}_{\mathrm{tot}}=\mathcal{L}_{\mathrm{ori}}+\sum_{\mathrm{aug}\in\mathcal{A}}(\lambda_{\mathrm{aug}}\mathcal{L}_{\mathrm{aug}}+\lambda_{\mathrm{reg}\mathchar
45\relax\mathrm{aug}}\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{aug}}),$ (2)

where $\mathcal{L}_{\mathrm{ori}}$ is the commonly used binary cross-entropy
(BCE) loss for original training sequences and $\mathcal{L}_{\mathrm{aug}}$
are the same BCE losses for augmented sequences generated by applying
augmentation strategies $\mathrm{aug}\in\mathcal{A}$.222For replacement and
insertion, we do not include outputs for augmented interactions in
$\mathcal{L}_{\mathrm{aug}}$.  $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{aug}}$ are the regularization losses that impose consistency
and monotonicity bias on the model’s predictions for the original and
augmented sequence, which will be defined in the following sections. Finally,
$\lambda_{\mathrm{aug}},\lambda_{\mathrm{reg}\mathchar
45\relax\mathrm{aug}}>0$ are hyperparameters to control the trade-off among
$\mathcal{L}_{\mathrm{ori}}$, $\mathcal{L}_{\mathrm{aug}}$, and
$\mathcal{L}_{\mathrm{reg}\mathchar 45\relax\mathrm{aug}}$.

In the following sections, we introduce the three augmentation strategies,
replacement, insertion and deletion with corresponding consistency and
monotonicity regularization losses, $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{rep}}$, $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{cor}\\_\mathrm{ins}}$ (or $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{incor}\\_\mathrm{ins}}$) and
$\mathcal{L}_{\mathrm{reg}\mathchar 45\relax\mathrm{cor}\\_\mathrm{del}}$ (or
$\mathcal{L}_{\mathrm{reg}\mathchar 45\relax\mathrm{incor}\\_\mathrm{del}}$).

###  2.1 Replacement

Replacement, similar to the synonym replacement strategy in NLP, is an
augmentation strategy that replaces questions in the original interaction
sequence with other similar questions _without changing their responses_ ,
where _similar questions_ are defined as the questions that share the same
skills as the original question. Our hypothesis is that the predicted
correctness probabilities for questions in an augmented interaction sequence
should not change drastically compared to the original interaction sequence.
Formally, for each interaction in the original interaction sequence
$(I_{1},\dots,I_{T})$, we randomly decide whether the interaction will be
replaced or not, following the Bernoulli distribution with the probability
$\alpha_{\mathrm{rep}}$. If an interaction $I_{t}=(Q_{t},R_{t})$ with a set of
skills $S_{t}$ associated with the question $Q_{t}$ is set to be replaced, we
determine $I_{t}^{\mathrm{rep}}=(Q_{t}^{\mathrm{rep}},R_{t})$ by selecting a
question $Q_{t}^{\mathrm{rep}}$ with its associated set of skills
$S_{t}^{\mathrm{rep}}$ that satisfies $S_{t}\cap
S_{t}^{\mathrm{rep}}\neq\emptyset$. The resulting augmented sequence
$(I_{1}^{\mathrm{rep}},\dots,I_{T}^{\mathrm{rep}})$ is generated by replacing
$I_{t}$ with $I_{t}^{\mathrm{rep}}$ for
$t\in\mathbf{R}\subset[T]=\\{1,2,\dots,T\\}$, where $\mathbf{R}$ is a set of
indices to replace. Then we consider the following consistency regularization
loss:

$\displaystyle\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{rep}}=\mathbb{E}_{t\not\in\mathbf{R}}[(p_{t}-p_{t}^{\mathrm{rep}})^{2}],$
(3)

where $p_{t}$ and $p_{t}^{\mathrm{rep}}$ are model’s predicted correctness
probabilities for $t$-th question of the original and augmented sequences. The
output for the replaced interactions are not included in the loss computation.

Also, the replacement strategy has several variants depending on the dataset.
For instance, randomly selecting a question for $Q_{t}^{\mathrm{rep}}$ from
the question pool is an alternative strategy when the related skill set
information is not available. It is also possible to only consider outputs for
interactions that are replaced or consider outputs for all interactions in the
augmented sequence for the loss computation. We investigate the effectiveness
of each strategy in Section 3.

###  2.2 Insertion

Insertion strategy is based on the notion of a student’s knowledge to be
higher when more questions are answered correctly. Specifically, data is
augmented in a _monotonic_ manner by inserting new interactions into the
original interaction sequence. Formally, we generate an augmented interaction
sequence $(I_{1}^{\mathrm{ins}},\dots,I_{T^{\prime}}^{\mathrm{ins}})$ by
inserting a correctly (resp. incorrectly) answered interaction
$I_{t}^{\mathrm{ins}}=(Q_{t}^{\mathrm{ins}},1)$ (resp.
$I_{t}^{\mathrm{ins}}=(Q_{t}^{\mathrm{ins}},0)$) into the original interaction
sequence $(I_{1},\dots,I_{T})$ for $t\in\mathbf{I}\subset[T^{\prime}]$, where
the question $Q_{t}^{\mathrm{ins}}$ is randomly selected from the question
pool and $\mathbf{I}$ with the size $\alpha_{\mathrm{ins}}$ proportion of the
original sequence is a set of indices of inserted interactions. Then our
hypothesis is formulated as $p_{t}\leq p_{\sigma(t)}^{\mathrm{ins}}$ (resp.
$p_{t}\geq p_{\sigma(t)}^{\mathrm{ins}}$), where $p_{t}$ and
$p_{t}^{\mathrm{ins}}$ are model’s predicted correctness probabilities for
$t$-th question of the original and augmented sequences, respectively. Here,
$\sigma:[T]\to[T^{\prime}]-\mathbf{I}$ is the order-preserving bijection which
satisfies $I_{t}=I^{\mathrm{ins}}_{\sigma(t)}$ for $1\leq t\leq T$. (For
instance, in Figure 2, $\sigma$ sends $\\{1,2,3,4\\}$ to $\\{2,3,4,6\\}$) We
impose our hypothesis through the following losses:

$\displaystyle\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{cor}\\_\mathrm{ins}}$
$\displaystyle=\mathbb{E}_{t\in[T]}[\max(0,p_{t}-p_{\sigma(t)}^{\mathrm{ins}})],$
(4) $\displaystyle\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{incor}\\_\mathrm{ins}}$
$\displaystyle=\mathbb{E}_{t\in[T]}[\max(0,p_{\sigma(t)}^{\mathrm{ins}}-p_{t})],$
(5)

where $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{cor}\\_\mathrm{ins}}$ and $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{incor}\\_\mathrm{ins}}$ are losses for augmented interaction
sequences of inserting correctly and incorrectly answered interactions,
respectively.

###  2.3 Deletion

Similar to the insertion augmentation strategy, we enforce monotonicity by
removing some interactions in the original interaction sequence based on the
following hypothesis: if a student’s response record contains less correct
answers, the correctness probabilities for the remaining questions would
become decrease and vice versa. Formally, from the original interaction
sequence $(I_{1},\dots,I_{T})$, we randomly sample a set of indices
$\mathbf{D}\subset[T]$, where $R_{t}=1$ (resp. $R_{t}=0$) for
$t\in\mathbf{D}$, based on the Bernoulli distribution with the probability
$\alpha_{\mathrm{del}}$. We remove the index $t\in\mathbf{D}$ and impose the
hypothesis $p_{t}\geq p_{\sigma(t)}^{\mathrm{del}}$ (resp. $p_{t}\leq
p_{\sigma(t)}^{\mathrm{del}}$), where $p_{t}$ and $p_{t}^{\mathrm{del}}$ are
model’s predicted correctness probabilities for $t$-th question of the
original and augmented sequences, respectively. Here,
$\sigma:[T]-\mathbf{D}\to[T^{\prime}]$ is the order preserving bijection with
$I_{t}=I_{\sigma(t)}^{\mathrm{del}}$ for $t\in[T]-\mathbf{D}$. We impose the
hypothesis through the following losses:

$\displaystyle\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{cor}\\_\mathrm{del}}=\mathbb{E}_{t\not\in\mathbf{D}}[\max(0,p_{\sigma(t)}^{\mathrm{del}}-p_{t})],$
(6) $\displaystyle\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{incor}\\_\mathrm{del}}=\mathbb{E}_{t\not\in\mathbf{D}}[\max(0,p_{t}-p_{\sigma(t)}^{\mathrm{del}})],$
(7)

where $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{cor}\\_\mathrm{del}}$ and $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{incor}\\_\mathrm{del}}$ are losses for augmented interaction
sequences of deleting correctly and incorrectly answered interactions,
respectively.

dataset | model | no augmentation |  insertion \+ deletion |  insertion \+ deletion \+ replacement
---|---|---|---|---
ASSIST2015 | DKT |  72.01 $\pm$ 0.05 |  72.46 $\pm$ 0.06 |  72.39 $\pm$ 0.07
DKVMN |  71.21 $\pm$ 0.09 |  72.00 $\pm$ 0.18 |  72.23 $\pm$ 0.09
SAINT |  72.13 $\pm$ 0.09 |  72.78 $\pm$ 0.06 |  72.81 $\pm$ 0.04
ASSISTChall | DKT |  74.40 $\pm$ 0.16 |  75.98 $\pm$ 0.07 |  79.07 $\pm$ 0.08
DKVMN |  74.46 $\pm$ 0.11 |  75.06 $\pm$ 0.10 |  78.21 $\pm$ 0.05
SAINT |  77.01 $\pm$ 0.18 |  78.02 $\pm$ 0.09 |  80.18 $\pm$ 0.05
STATICS2011 | DKT |  86.43 $\pm$ 0.29 |  87.18 $\pm$ 0.12 |  87.27 $\pm$ 0.11
DKVMN |  84.89 $\pm$ 0.17 |  85.65 $\pm$ 0.94 |  87.17 $\pm$ 0.14
SAINT |  85.82 $\pm$ 0.50 |  86.53 $\pm$ 0.30 |  87.56 $\pm$ 0.06
EdNet-KT1 | DKT |  72.75 $\pm$ 0.09 |  74.04 $\pm$ 0.04 |  74.28 $\pm$ 0.06
DKVMN |  73.58 $\pm$ 0.08 |  73.94 $\pm$ 0.05 |  74.16 $\pm$ 0.11
SAINT |  74.78 $\pm$ 0.05 |  75.32 $\pm$ 0.05 |  75.26 $\pm$ 0.02

Table 1: Test AUCs of DKT, DKVMN, and SAINT models on 4 public benchmark
datasets. The results show the mean and standard deviation averaged over 5
runs and the best result for each dataset and model is indicated in bold.

##  3 Experiments

We demonstrated the effectiveness of the proposed method on 4 widely used
benchmark datasets: ASSISTments2015 Feng et al. (2009), ASSISTmentsChall,
STATICS2011, and EdNet-KT1 Choi et al. (2020b). We describe the details about
these benchmarks, their statistics, and pre-processing procedures in the
Appendix.

We test DKT Piech et al. (2015), DKVMN Zhang et al. (2017), and SAINT Choi et
al. (2020a) models. For DKT, we set the embedding dimension and the hidden
dimension as 256. For DKVMN, key, value, and summary dimension are all set to
be 256, and we set the number of latent concepts as 64. SAINT has 2 layers
with hidden dimension 256, 8 heads, and feed-forward dimension 1024. All the
models do not use any additional features of interactions except question ids
and responses as an input, and the model weights are initialized with Xavier
distribution Glorot and Bengio (2010). They are trained from scratch with
batch size 64, and we use the Adam optimizer with learning rate $0.001$ which
is scheduled by Noam scheme with warm-up step 4000. We set each model’s
maximum sequence size as 100 on ASSISTments2015 & EdNet-KT1 dataset and 200 on
ASSISTmentsChall & STATICS2011 dataset. Hyperparameters for augmentations,
$\alpha_{\mathrm{aug}},\lambda_{\mathrm{reg}\mathchar 45\relax\mathrm{aug}}$,
and $\lambda_{\mathrm{aug}}$ are searched over
$\alpha_{\mathrm{aug}}\in\\{0.1,0.3,0.5\\}$, $\lambda_{\mathrm{reg}\mathchar
45\relax\mathrm{aug}}\in\\{1,10,50,100\\}$, and
$\lambda_{\mathrm{aug}}\in\\{0,1\\}$. For all dataset, we evaluate our results
using 5-fold cross validation and use Area Under Curve (AUC) as an evaluation
metric.

5102550100$68$$70$$72$ASSISTments20155102550100$64$$68$$72$$76$$80$ASSISTmentsChall5102550100$74$$78$$82$$86$STATICS20115102550100$64$$68$$72$EdNet-
KT1 Figure 3: Test AUCs with various sizes of training data under the DKT
model. $x$ axis stands for the portion of the training set we use for training
(relative to the full train set) and $y$ axis is the AUC. Blue line represents
the AUCs of the vanilla DKT model, and red line represents the AUCs of the DKT
model trained with augmentations and regularizations.

###  3.1 Main results

The results (AUCs) are shown in Table 1 that compares models without and with
augmentations, and we report the best results for each strategy. (The detailed
hyperparameters for these results are given in Supplementary materials.) The
4th column represents results using both insertion and deletion, and the last
column shows the results with all 3 augmentations. Since there’s no big
difference on performance gain between insertion and deletion, we only report
the performance that uses one or both of them together. We use skill-based
replacement if skill information for each question in the dataset is
available, and use question-random replacement that that selects new questions
among all questios if not (e.g. ASSISTments2015). As one can see, the models
trained with consistency and monotonicity regularizations outperforms the
models without augmentations in a large margin, regardless of model’s
architectures or datasets. Using all three augmentations gives the best
performances for most of the cases. For instance, there exists 6.3% gain in
AUC on ASSISTmentsChall dataset under the DKT model. Furthermore, not only
enhancing the prediction performances, our training scheme also resolves the
vanilla model’s issue where the monotonicity condition on the predictions of
original and augmented sequences is violated. As in Figure 4, the predictions
of the model trained with monotonicity regularization (correct insertion) are
increased after insertion, which contrasts to the vanilla DKT model’s outputs.

Since overfitting is expected to be more severe when using a smaller dataset,
we conduct experiments using various fractions of the existing training
datasets (5%, 10%, 25%, 50%) and show that our augmentations yield more
significant improvements for smaller training datasets. Figure 3 shows
performances of DKT model on various datasets, with and without augmentations.
For example, on ASSISTmentsChall dataset, using 100% of the training data
gives AUC 74.4%, while the same model trained with augmentations achieved AUC
75.44% with only 25% of the training dataset.

dataset | loss | replacement | insertion, O | insertion, X | deletion, O | deletion, X
---|---|---|---|---|---|---
ASSIST2015 |  (8) |  72.03 $\pm$ 0.06 |  71.98 $\pm$ 0.06 |  71.98 $\pm$ 0.05 |  72.05 $\pm$ 0.04 |  72.04 $\pm$ 0.02
(72.01 $\pm$ 0.05) |  (2) |  72.06 $\pm$ 0.03 |  72.09 $\pm$ 0.06 |  72.35 $\pm$ 0.11 |  72.53 $\pm$ 0.08 |  72.26 $\pm$ 0.04
ASSISTChall |  (8) |  75.13 $\pm$ 0.04 |  74.61 $\pm$ 0.17 |  74.57 $\pm$ 0.14 |  74.92 $\pm$ 0.12 |  74.42 $\pm$ 0.20
(74.40 $\pm$ 0.16) |  (2) |  78.85 $\pm$ 0.08 |  75.98 $\pm$ 0.07 |  75.64 $\pm$ 0.12 |  75.60 $\pm$ 0.06 |  74.77 $\pm$ 0.11
STATICS2011 |  (8) |  86.89 $\pm$ 0.23 |  86.45 $\pm$ 0.26 |  86.40 $\pm$ 0.22 |  86.53 $\pm$ 0.29 |  86.55 $\pm$ 0.25
(86.43 $\pm$ 0.29) |  (2) |  87.27 $\pm$ 0.11 |  86.72 $\pm$ 0.23 |  87.18 $\pm$ 0.12 |  87.07 $\pm$ 0.33 |  86.97 $\pm$ 0.26
EdNet-KT1 |  (8) |  73.04 $\pm$ 0.10 |  72.81 $\pm$ 0.08 |  72.88 $\pm$ 0.09 |  72.99 $\pm$ 0.07 |  73.28 $\pm$ 0.04
(72.75 $\pm$ 0.09) |  (2) |  73.89 $\pm$ 0.06 |  73.73 $\pm$ 0.06 |  73.52 $\pm$ 0.06 |  74.04 $\pm$ 0.04 |  73.76 $\pm$ 0.04
Table 2: Comparison of the test AUCs of the DKT model, trained with only data
augmentation (i.e., using the loss (8)) and with consistency and monotonicity
regularizations (i.e., using the loss (2)). AUCs of the vanilla DKT model are
given in parentheses below the dataset names. O (resp. X) represents correct
(resp. incorrect) response. Figure 4:  Response correctness prediction for a
student in the ASSISTmentsChall dataset. We randomly insert interactions with
correct responses (interactions with yellow boundaries). In case of the
vanilla DKT model, the predictions for the original interactions (especially
the interactions with green boundaries) are decreased, even if the student
answered more questions correctly. However, such problem is resolved when we
train the model with monotonicity regularization. Unlike the vanilla DKT
model, predicted correctness probabilities for the original interactions are
increased after insertion.

###  3.2 Ablation study

Are constraint losses necessary? One might think that data augmentations are
enough for boosting up the performance, and imposing consistency and
monotonicity are not necessary. However, we found that including such
regularization losses for training is essential for further performance gain.
To see this, we compare the performances of the model trained only with KT
losses for both original and augmented sequences

$\displaystyle\mathcal{L}_{\mathrm{tot}}=\mathcal{L}_{\mathrm{ori}}+\sum_{\mathrm{aug}\in\mathcal{A}}\lambda_{\mathrm{aug}}\mathcal{L}_{\mathrm{aug}}$
(8)

(where $\lambda_{\mathrm{aug}}=1$) and with consistency and monotonicity
regularization losses (2) where $\mathcal{A}$ is a set that contains a single
augmentation. Training a model with the loss (8) can be thought as using
augmentations without imposing any consistency or monotonicity biases.

Table 2 shows results under the DKT model. Using only data augmentation
(training the model with the loss (8)) gives a marginal gain in performances
or even worse performances. However, training with both data augmentation and
consistency or monotonicity regularization losses (2) give significantly
higher performance gain. Under ASSISTmentsChall dataset, using replacement
along with consistency regularization improves AUC by 6%, which is much higher
than the 1% improvement only using data augmentation.

Ablation on monotonicity constraints We perform an ablation study to compare
the effects of monotonicity regularization and _reversed_ monotonocity
regularization. Monotonocity regularization introduces constraint loss to
align the inserted or deleted sequence in order to modify the probability of
correctness of the original sequence to follow insertion or deletion. For
example, when a correct response is inserted to the sequence, the probability
of correctness for the original sequence increases. Reversed monotonocity
regularization modifies the probability of correctness in the opposite manner,
where inserting a correct response would decrease the probability of
correctness in the original sequence.

For each
$\mathrm{aug}\in\\{\mathrm{cor}\\_\mathrm{ins},\mathrm{incor}\\_\mathrm{ins},\mathrm{cor}\\_\mathrm{del},\mathrm{incor}\\_\mathrm{del}\\}$,
we can define reversed version of the monotonicity regularization loss
$\mathcal{L}^{\mathrm{rev}}_{\mathrm{reg}\mathchar 45\relax\mathrm{aug}}$
which impose the opposite constraint on the model’s output, e.g. we define
$\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{cor}\\_\mathrm{ins}}^{\mathrm{rev}}$ as

(9)

which forces model’s output of correctness probability to _decrease_ when
_correct_ responses are inserted. In this experiments, we do not include KT
loss for augmented sequences (set $\lambda_{\mathrm{aug}}=0$) to observe the
effects of consistency loss only. Also, the same hyperparameters
($\alpha_{\mathrm{aug}}$ and $\lambda_{\mathrm{reg}\mathchar
45\relax\mathrm{aug}}$) are used for both the original and reversed
constraints.

augmentation | direction | ASSIST2015 | ASSISTChall | STATICS2011 | EdNet-KT1
---|---|---|---|---|---
- | - |  72.01 $\pm$ 0.05 |  74.40 $\pm$ 0.16 |  86.43 $\pm$ 0.29 |  72.75 $\pm$ 0.09
insertion, O | $\uparrow$ |  72.08 $\pm$ 0.02 |  75.98 $\pm$ 0.06 |  86.72 $\pm$ 0.23 |  73.70 $\pm$ 0.08
insertion, X | $\downarrow$ |  72.31 $\pm$ 0.04 |  75.34 $\pm$ 0.16 |  87.18 $\pm$ 0.12 |  73.40 $\pm$ 0.06
[0.5pt/2pt] deletion, O | $\downarrow$ |  72.44 $\pm$ 0.05 |  75.60 $\pm$ 0.06 |  87.07 $\pm$ 0.33 |  74.01 $\pm$ 0.05
deletion, X | $\uparrow$ |  72.26 $\pm$ 0.04 |  74.77 $\pm$ 0.11 |  86.68 $\pm$ 0.27 |  73.71 $\pm$ 0.04
insertion, O | $\downarrow$ |  71.79 $\pm$ 0.06 |  75.42 $\pm$ 0.17 |  86.58 $\pm$ 0.50 |  69.67 $\pm$ 0.06
insertion, X | $\uparrow$ |  70.73 $\pm$ 0.10 |  74.92 $\pm$ 0.11 |  86.22 $\pm$ 0.18 |  71.95 $\pm$ 0.15
[0.5pt/2pt] deletion, O | $\uparrow$ |  66.48 $\pm$ 0.10 |  74.68 $\pm$ 0.13 |  86.76 $\pm$ 0.27 |  71.23 $\pm$ 0.81
deletion, X | $\downarrow$ |  67.34 $\pm$ 0.17 |  73.91 $\pm$ 0.14 |  86.58 $\pm$ 0.28 |  69.99 $\pm$ 0.11

Table 3: Ablation test on the directions of monotonicity regularizations with
the DKT model. 2nd to 5th rows show the results with the original
regularization losses, and the last 4 rows show the results with the reversed
regularization losses. $\uparrow$ (resp. $\downarrow$) means that the loss
impose increasing (resp. decreasing) bias on a model.

dataset | vanilla | replaced inters | remaining inters | full inters
---|---|---|---|---
ASSIST2015 |  72.01 $\pm$ 0.05 |  70.53 $\pm$ 0.07 |  72.07 $\pm$ 0.02 |  71.39 $\pm$ 0.09
ASSISTChall |  74.40 $\pm$ 0.16 |  74.68 $\pm$ 0.09 |  78.45 $\pm$ 0.08 |  75.91 $\pm$ 0.07
STATICS2011 |  86.43 $\pm$ 0.29 |  82.97 $\pm$ 0.27 |  87.17 $\pm$ 0.15 |  83.49 $\pm$ 0.10
EdNet-KT1 |  72.75 $\pm$ 0.09 |  65.52 $\pm$ 0.07 |  73.87 $\pm$ 0.10 |  68.77 $\pm$ 0.11

Table 4: Test AUCs of the DKT model with variations of replacements and qDKT with Lapacian regularization. Best result for each dataset is indicated in bold. dataset | no augmentation | question-random | interaction-random | skill-set | skill
---|---|---|---|---|---
ASSIST2015 |  72.01 $\pm$ 0.05 |  72.07 $\pm$ 0.02 |  71.77 $\pm$ 0.05 | - | -
ASSISTChall |  74.40 $\pm$ 0.16 |  78.39 $\pm$ 0.09 |  74.27 $\pm$ 0.38 |  77.57 $\pm$ 0.08 |  78.45 $\pm$ 0.08
STATICS2011 |  86.43 $\pm$ 0.29 |  86.35 $\pm$ 0.06 |  84.50 $\pm$ 0.28 | - |  87.17 $\pm$ 0.15
EdNet-KT1 |  72.75 $\pm$ 0.09 |  73.84 $\pm$ 0.05 |  72.62 $\pm$ 0.17 |  73.80 $\pm$ 0.07 |  73.87 $\pm$ 0.10
Table 5: Test AUCs of the DKT model with different type of replacements - question-random replacement, interaction-random replacement, skill-set-based replacement, and skill-based replacement. Best result for each dataset is indicated in bold. dataset | DKT | DKT+ | qDKT | ours
---|---|---|---|---
ASSIST2015 |  72.01 $\pm$ 0.05 |  72.28 $\pm$ 0.08 | - |  72.46 $\pm$ 0.06
ASSISTChall |  74.40 $\pm$ 0.16 |  74.70 $\pm$ 0.08 |  75.03 $\pm$ 0.08 |  79.07 $\pm$ 0.08
STATICS2011 |  86.43 $\pm$ 0.29 |  85.66 $\pm$ 0.08 |  86.65 $\pm$ 0.12 |  87.27 $\pm$ 0.11
EdNet-KT1 |  72.75 $\pm$ 0.09 |  73.43 $\pm$ 0.05 |  69.77 $\pm$ 0.22 |  74.28 $\pm$ 0.06
Table 6: Test AUCs of the vanilla DKT, DKT+, qDKT, and DKT with our
regularizations. Best result for each dataset is indicated in bold. Detailed
hyperparameters are given in the Appendix.

Table 3 shows the performances of DKT model with the original and reversed
monotonicity regularizations. Second row represents the performance with no
augmentations, the 3rd to the 6th rows represent the results from using
original (aligned) insertion/deletion monotonicity regularization losses, and
the last four rows represent the results when the reversed monotonicity
regularization losses are used. The results demonstrate that using aligned
monotonicity regularization loss outperforms the model with reversed
monotonicity regularization loss. Also, the performances of reversed
monotonicity shows large decrease in performance on several datasets even
compared to the model with no augmentation. In case of the EdNet-KT1 dataset,
the model’s performance with correct insertion along with original
regularization improves the AUC from 72.75% to 73.70%, while using the
reversed regularization drops the performance to 69.67%.

Ablation on replacement We compare our consistency regularization with the
other two variations of replacements, consistency regularization on replaced
interactions and full interactions, correspond to the following losses:

$\displaystyle\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{rep}\\_\mathrm{ro}}$
$\displaystyle=\mathbb{E}_{t\in\mathbf{R}}[(p_{t}-p_{t}^{\mathrm{rep}})^{2}],$
(10) $\displaystyle\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{rep}\\_\mathrm{full}}$
$\displaystyle=\mathbb{E}_{t\in[T]}[(p_{t}-p_{t}^{\mathrm{rep}})^{2}],$ (11)

where $\mathrm{ro}$ stands for _replaced only_. We compared such variations
with the original consistency loss $\mathcal{L}_{\mathrm{reg}\mathchar
45\relax\mathrm{rep}}$ that does not include predictions for the replaced
interactions. For all variations, we used the same replacement probability
$\alpha_{\mathrm{rep}}$ and loss weight $\lambda_{\mathrm{reg}\mathchar
45\relax\mathrm{rep}}$, and we do not include KT loss for replaced sequences
as before. Table 4 shows that including the replaced interactions’ outputs
hurt performances.

To see the effect of using the skill information of questions for replacement,
we compared skill-based replacement with three different random versions of
replacement: _question random replacement_ , _interaction random replacement_
, and _skill-set-based replacement_. For _question random replacement_ , we
replace questions with different ones randomly (without considering skill
information), while _interaction random replacement_ changes both question and
responses (sample each response with 0.5 probability). _Skill-set-based
replacement_ is almost the same as the original skill-based replacement, but
the candidates of the questions to be replaced are chosen as ones with exactly
same set of skills are associated, not only have common skills
($S=S^{\mathrm{rep}}$). The results in Table 5 show that the performances of
the question random replacements depends on the nature of dataset. It shows
similar performance with skill-based replacement on ASSISTmentsChall and
EdNet-KT1 datasets, but only give a minor gain or even dropped the performance
on other datasets. However, applying interaction-random replacement
significantly hurt performances over all datasets, e.g. the AUC is decreased
from 86.43% to 84.50% on STATICS2011 dataset. This demonstrates the importance
of fixing responses of the interactions for consistency regularization. At
last, skill-set-based replacement works similar or even worse than the
original skill-based replacement. Note that each question of the STATICS2011
dataset has single skill attached to, so the performance of skill-based and
skill-set-based replacement coincide on the dataset.

Comparison with other regularization methods in KT We also compare our
regularization scheme with previous works: DKT+ Yeung and Yeung (2018) and
qDKT Sonkar et al. (2020). DKT+ uses two types of regularization losses:
_reconstruction loss_ and _waviness loss_. Reconstruction loss enable a model
to recover the current interaction’s label, and waviness loss make model’s
prediction to be consistent over all timesteps. qDKT uses the Laplacian loss
that regularizes the variance of predicted correctness probabilities for
questions that fall under the same skill, which is similar to the variation
$\mathcal{L}_{\mathrm{reg}\mathchar 45\relax\mathrm{rep}\\_\mathrm{ro}}$ of
the consistency loss. We explain these losses in detail in the Appendix.

Results in Table 6 shows that our regularization approach yields the largest
performance gain over all benchmarks compared to other methods. In some cases,
using DKT+ or qDKT even harm the performances, while consistency and
monotonocity regularization yields substantial performance gain over all
datasets.

##  4 Conclusion

We propose simple augmentation strategies with corresponding constraint
regularization losses for KT and show their efficacy. We only considered the
most basic features of interactions, question and response correctness, and
other features like elapsed time or question texts enables us to exploit
diverse augmentation strategies if available. Furthermore, exploring
applicability of our idea on other AIEd tasks (dropout prediction or at-risk
student prediction) is another interesting future direction.

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