Sim-GAIL: A generative adversarial imitation learning approach of student modelling for intelligent tutoring systems

The continuous application of artificial intelligence (AI) technologies in online education has led to significant progress, especially in the field of Intelligent Tutoring Systems (ITS), online courses and learning management systems (LMS). An important research direction of the field is to provide students with customised learning trajectories via student modelling. Previous studies have shown that customisation of learning trajectories could effectively improve students’ learning experiences and outcomes. However, training an ITS that can customise students’ learning trajectories suffers from cold-start, time-consumption, human labour-intensity, and cost problems. One feasible approach is to simulate real students’ behaviour trajectories through algorithms, to generate data that could be used to train the ITS. Nonetheless, implementing high-accuracy student modelling methods that effectively address these issues remains an ongoing challenge. Traditional simulation methods, in particular, encounter difficulties in ensuring the quality and diversity of the generated data, thereby limiting their capacity to provide intelligent tutoring systems (ITS) with high-fidelity and diverse training data. We thus propose Sim-GAIL, a novel student modelling method based on generative adversarial imitation learning (GAIL). To the best of our knowledge, it is the first method using GAIL to address the challenge of lacking training data, resulting from the issues mentioned above. We analyse and compare the performance of Sim-GAIL with two traditional Reinforcement Learning-based and Imitation Learning-based methods using action distribution evaluation, cumulative reward evaluation, and offline-policy evaluation. The experiments demonstrate that our method outperforms traditional ones on most metrics. Moreover, we apply our method to a domain plagued by the cold-start problem, knowledge tracing (KT), and the results show that our novel method could effectively improve the KT model’s prediction accuracy in a cold-start scenario.


Introduction
Intelligent tutoring systems (ITS) are increasingly incorporating artificial intelligence (AI) technologies, including machine learning and deep learning, which could effectively offer customised learning trajectories for each student based on their prior knowledge and learning activities [1].Research in cognitive science has shown that there is a strong relationship between, amongst others, the sequence of learning materials and learning outcomes [2].In a traditional online learning platform, there is only one single static linear learning trajectory provided to students.In this one-size-fits-all approach, students may lose their motivation and even drop out of the course, due to anxiety or boredom encountered in the learning process [3].Research on customised learning trajectories for students has been emerging in the ITS field.However, developing an ITS that can provide students with customised learning trajectories requires a large amount of data for training the system, which is time-consuming and costly [4], long known to be requiring a large amount of manual labour from education providers (instructors, authors, etc.) [5].Although many mature ITSs have sufficient data to train algorithms, a large number of emerging ITSs are still suffering from a lack of training data in the early stages of development, also known as the cold start problem [6].
To tackle these challenges, previous studies have proposed various methods for simulating student learning trajectories (i.e., generating massive student learning behavioural data) that can be used to train an ITS.Early simulated student behaviour proposals stemmed from the aim at automatic validation of educational interventions via a sandbox method [7].More recently, Jarboui et al. [8] attempted to model student trajectory sequences into a Markov Decision Process, but in real educational scenarios, only a few ITS can provide all the feature data consistent with a Markov Decision Process (e.g., the reward function of the ITS agent).Zimmer et al. [9] defined reward functions to build reinforcement learning agents to generate student trajectories, but this method requires building different reward functions for different datasets, which makes it difficult to generalise.Besides, humans' psychological responses to learning trajectories and reward mechanisms are difficult to simulate.This leads to circumstances where student simulation methods may not be able to simulate student learning trajectories sufficiently.Anderson et al. [10] proposed a student simulation method based on behavioural cloning (BC), the simplest form of Imitation Learning, which aims to solve the abovementioned problems where the reward is sparse and hard to define [11].Whilst promising, BC-based methods only learn from the few features collected in student data, and the actions that algorithms are able to model can be very limited.
Motivated by the gap in prior literature identified above, the research question of this paper is: How to build an efficient student simulation method that can generate massive student learning data, which can be used for ITS training?
To answer this research question, we propose Sim-GAIL, a generative adversarial imitation learning (GAIL) approach to student modelling.Our Sim-GAIL method can be used to generate simulated student data to solve the lack of data and cold-start problems in ITS training.
Furthermore, to showcase its efficiency, we compare our Sim-GAIL with the two main student modelling methods used in the ITS field, the RL-based and the BC-based student modelling approach, using data from the very recent and largest ITS dataset, EdNet [12].We extract action and state features to train the models.We analyse and compare performance using action distribution evaluation, cumulative reward evaluation (CRE), and two offline-policy evaluation (OPE) methods, which include importance sampling (IS) and Fitted Q Evaluation (FQE).Moreover, we apply our method's generated data in an ITS cold-start scenario.The experimental results show that our method outperforms the two traditional RL-based and BCbased baseline methods and could improve the training efficiency of the ITS in a cold-start scenario.
The main contributions of this work lie in the following three aspects: 1. We propose Sim-GAIL, a student modelling approach, to generate simulation data for ITS training.Thus, the advantages of Sim-GAIL include its ability to effectively generate data resembling real student behaviours, address the cold-start problem, demonstrate superior performance on various metrics, efficiently converge to an optimal policy, and offer scalability and generality across different datasets and applications.This paper is structured as follows.Section 2 introduces the background of reinforcement learning, imitation learning (including behavioural cloning), and student modelling.Section 3 demonstrates the dataset, data preprocessing, and model architecture.Section 4 outlines the experiments and baseline models.Section 5 discusses the evaluation methods and the experimental results based on action distribution, offline policy (OP) evaluation, expected cumulative rewards (ECR) evaluation, and knowledge tracing (KT).Section 6 discusses our findings and future works.Section 7 draws conclusions.

Background and literature review
Before analysing current competitors of the proposal for student modelling for generating training data presented in this paper, we show the current state of the underlying methodologies: Markov decision process, reinforcement learning, imitation learning, and finally, the method at the basis of our proposal, generative adversarial imitation learning.

Markov decision process and reinforcement learning
The Markov decision process (MDP) is the standard method for sequential decision-making (SDM) [13].The sequential decision-making models can generally be seen as an instance of the Markov decision process.Reinforcement learning is also typically regarded as an MDP [14].Therefore, in this section, we introduce MDP and then reinforcement learning.

Markov decision process (MDP)
MDP is a mathematical model of sequential decision used to generate stochastic policies and rewards, achievable by an agent in an environment where the system state exhibits Markov properties [15].MDPs are represented as a set of interacting objects, namely agents and environments, with components including states, actions, policies, and rewards.
In an MDP model, the agent observes the present state of the environment and takes actions on the environment in accordance with the policy, thereby changing the state of the environment and getting rewards.The ultimate goal of the agent is to reach the maximum cumulative reward, which is achieved using a reward function [16].Figure 1 shows the structure of the MDP.

Reinforcement learning (RL)
RL is a type of machine learning method that enables an agent to learn a policy by taking different actions in an interactive environment, in order to maximise cumulative rewards.It could be defined as the tuple of ðS; A; P; RÞ, where S is defined as the state of the environment, A represents actions of the agent, P : S Â A Â S !½0; 1 represents the transition probabilities of actions from the current state to the next state and R : S Â A Â S !R denotes the reward function.The goal of an RL agent is to achieve maximum cumulative rewards.However, the drawback of traditional RL methods lies in its computational overhead, brought by repeated interactions between the agent and the environment.

Imitation learning (IL)
Different from RL, where the agent learns by interacting with the environment to obtain the maximum rewards, IL is a method of learning policy that involves emulating the behaviour of experts' trajectories [17], instead of leveraging an explicit reward function as in RL.

Behavioural cloning (BC)
BC considers the learning of policy under supervised learning settings, leveraging state-action pairs [18,19].Albeit simple and effective, BC suffers from the heavy reliance on extremely large amounts of data [20,21], without which a distributional mismatch, often referred to as covariate shift [22,23], would occur, due to compounding errors and stochasticity in the environment during test time.

Apprenticeship learning (AL)
Different from BC, AL instead tries to identify features of the expert's trajectories that are more generalisable and to find a policy that matches the same feature expectations with respect to the expert [24].Its goal is to find a policy that performs no worse than the expert across a class of cost functions.The main limitation of AL is that it cannot imitate the expert trajectory well, due to the restricted class of cost functions.Specifically, when the true cost function does not lie within the cost function classes, the agent cannot be guaranteed to outperform the expert.GAIL addresses the drawbacks of RL and AL effectively [20], by borrowing the idea of Generative Adversarial Networks (GANs) [25].It is derived from a type of Imitation Learning, called Maximum Causal Entropy Inverse Reinforcement Learning (MaxEntIRL) [26].
Figure 2 shows the mechanism of GAIL.Integrating GANs into imitation learning allows for the Generator never to be exposed to real-world examples, enabling agents to learn only from experts' demonstrations.In GAIL, the Discriminator is trained with the objective of distinguishing the generated trajectories from real trajectories, while the Generator, on the other hand, attempts to imitate the real trajectories, to fool the Discriminator into thinking it is actually one of them.

Student modelling
As the traditional one-size-fits-all approach can no longer satisfy student learning needs, it leads to increased demands for customised learning [27,28].Various student modelling methods have been proposed, which are generally classified as integrating expert knowledge-based or data-driven methods [29,30].Knowledge-based methods refer to utilising human knowledge to address issues that would normally require human intelligence [7,31].Datadriven methods simulate students' learning trajectories through massive student learning records data [6,32,33].

Expert knowledge-based methods
The majority of the studies in this field involve building different forms of student models, to train a reinforcement learning (RL) agent [34].Glesias et al. proposed a Markov Decision Process based on expert knowledge, to train student models [35].Doroud et al. [34] suggested an RLbased agent method rooted in cognitive theory, to optimise the sequencing of the knowledge components (KCs).The reward function of this method is based on pre-and posttest scores, taken as a metric, and termed Normalised Learning Gain (NLG).However, this metric needs evaluation by human participants, which is excessively human Fig. 2 Mechanism of generative adversarial imitation learning resource-intensive.Yudelson et al. [36] proposed a 'Student Simulation' method based on Bayesian Knowledge Tracing (BKT), which could train a 'sim student' to imitate real students' mastery of different knowledge.Segal et al. suggested a student simulation method based on the Item Response Theory (IRT) [37], which could respond to different reactions to courses at different difficulty levels [38].Azhar et al. [39] introduced an application of reinforcement learning (RL) for optimising the learning sequence modelling of online learning materials, which is an end-toend pipeline to automatically derive and evaluate robust representations of students' interactions and policies for content sequencing in online education.

Data-driven methods
Compared with integrating expert knowledge-based methods, data-driven methods could better simulate real students' learning trajectories and more effectively reduce biases [13].There have been some studies [40][41][42] aiming to build student simulation methods based on data-driven MDP approaches.For example, Beck et al. proposed a population student model (PSM) based on a linear regression model that could simulate the probability of the student's correct response [43].However, this method requires a high-quality dataset from real ITS platforms.Limited by the quantity of high-quality datasets, the previous data-driven model struggled to keep up with the expanding requirements of ITS development.Li et al. proposed a student behaviour simulation method based on a Decision Transformer, to generate student behaviour data for ITS training [6,33].Emond et al. [44] proposed an adaptive instructional system (AIS) as a self-improvement system.It presented a methodological approach that incorporates three concurrent research activities: Bayesian networks modelling of learning processes, knowledge elicitation from expert instructors, and the use of simulated learners and tutors for exploring AIS design options.On the other hand, with the further development of ITS research, more and more high-quality datasets, such as EdNet [12], have been published in recent years, which can be used to achieve a high-quality data-driven student simulations.However, collecting data like the EdNet dataset is extremely time-consuming and labour-intensive.How to improve the effectiveness of ITS with small data volumes or in a cold-start scenario is still a problem that needs to be addressed.

Method
In this section, we introduce the methodology for the research described in this paper.First, we describe the EdNet dataset we use, in Sect.3.1.In Sect.3.2, we show how we preprocess the data in EdNet, to obtain the features we need.We then articulate the framework of our SIM-GAIL method in Sect.3.3.

Dataset
We adopt EdNet [12], the largest dataset in the ITS field, for our experiments.This dataset comprises students' interaction log data with an ITS, which can be used to extract the state and action representation.EdNet is a massive benchmark dataset of interactions between students and a MOOC learning platform called SANTA. 1ANTA is a TOEIC (Test of English for International Communication) learning platform in South Korea, and the EdNet dataset was collected by Riiid!AI Research. 2There are 131,417,236 interaction logs collected from 784,309 students in 13,169 exercises over two years, as shown in Table 1.The interaction logs for each student are recorded in an independent CSV (Comma-Separated Values) file.EdNet is a four-layer hierarchical dataset, structured from KT1 to KT4, according to the granularity of interactive actions.KT1 only contains simple information, such as question and answer pairs and elapsed time.Based on the information in KT1, to provide correlation information between student behaviour and question-and-answer sequences, EdNet adds detailed action records to KT2, such as watching video lectures and reading articles.In KT3, actions such as choosing response options and reviewing explanations are added to KT2, which can be used to infer the influence of different learning activities on students' knowledge states.KT4 includes the finest detailed action information, such as purchasing courses, and pausing and playing video lectures, which could be used to investigate the impact of sparse key actions on overall learning outcomes.

Data preprocessing
The problems involving decision-making processes are transformed into MDPs in general [8] (see Sect. 2.1.1).In this experiment, we view the students' sequential decisionmaking trajectories as a Markov Decision Process.Extracting the action space and state space of the real students' data is essential for building an effective student simulation method using MDP.Next, we show how we explore the data and extract the action space and state space.

Action space
There are 13,169 questions, 1021 lectures, and 293 kinds of skills in EdNet [12].However, there are no criteria for separating these courses into different parts.Bassen et al. [4] proposed a method to group knowledge concepts, based on the assumption that each part was grouped by domain experts' experience.Inspired by this method, we divide the lectures and questions space of the agent into 7 groups.However, as the division into 7 groups is of a too coarse granularity for the action space, we further use the method proposed in [38], and divide the difficulty of the questions from 1 to 4 by the answer correctness rate, obtained by comparing the students answer logs and the correct answers.Some lectures lack a difficulty ranking and are therefore assigned a default difficulty value of 0. Hence, all action spaces are divided into 5 difficulty levels, with 7 groups, and thus 35 action types in total.Figure 3 shows the distribution of the 35 types of actions in EdNet.In each group, the action types include 4 questions from difficulty levels 1 to 4, and 1 lecture.Taking Group 1, for example, actions 1 to 4 correspond to questions with different difficulty levels, and action 5 corresponds to lectures where the difficulty level cannot be defined, which is set as 0. As shown in Fig. 3, the rest of the groups follow this pattern.

State space
EdNet records the interaction data for each student with the system, in separate CSV files, via UNIX timestamps.Therefore, most of the state features obtained from EdNet are longitudinal and temporal.Previous works have shown that different state feature choices could make a large difference in the performance of the algorithms [40,45].We select state features that are widely chosen in similar simulated student works [4,35,40,42].Transitions between these selected states represent students' learning trajectories.Table 2 shows the features we select from EdNet: 'correct_so_far' is the proportion of the correct answer to the number of all activities attempted; 'av_time' is the cumulative average of the elapsed time spent on each action; 'av_fam' denotes the average familiarity of the 7 groups; 'topic_fam' denotes the familiarity with the current group; 'prev_correct' denotes the number of correct answers in the previous group; and 'steps_in_part' counts student learning steps in the current group.Compared to previous works [4,40], we select more state features, which could potentially simulate the students' trajectories in real situations more effectively.

Sim-GAIL model architecture
Our Sim-GAIL model is built upon generative adversarial imitation learning (GAIL) [20], which aims to solve the problem of Imitation Learning of having difficulty in dealing with constant regularisation and not being able to match occupancy measures in large environments.Equation (1) demonstrates the optimal negative log loss, distinguishing between the pair: state p and action p E .
where w GA Ã is the average of the real trajectories' data, and D is the discriminative classifier.Using causal entropy H as the policy regulariser, the following procedure can be derived: This equation combines imitation learning (IL) and generative adversarial networks (GAN) [25].Generator S generates trajectories that are passed to Discriminator D. The Generator's goal is to make it less likely for the Discriminator to differentiate the real trajectories and those generated by the Generator, whilst the Discriminator's goal is to distinguish between them.The Generator achieves the best learning effect when the Discriminator fails to Fig. 4 The Sim-GAIL pipeline recognise the generated trajectories.Lastly, q p E in Eq. ( 1) is the occupancy measure of the real trajectories.
There is a function approximation of p and D. TRPO [46] is used to find a saddle point ðp; DÞ, which decreases the value of Expression (3).To decrease the expected cost, we use the cost function cðs; aÞ ¼ log Dðs; aÞ.As classified by Discriminator, the cost function will move toward real trajectories-like regions of the state-action space, to achieve the training goal of Discriminator.
Figure 4 shows the pipeline of Sim-GAIL.Real student data from EdNet is processed by the methods introduced in Sect.3.2 and fed into the GAIL module (middle part) to create a simulation policy that could be used for training the 'sim student' (right part).The middle part is described in Algorithm 1.We start by initialising the policy h and Discriminator D. At each iteration, we sample real student trajectories from the dataset and update the Discriminator parameters using the Adam gradient [47].Then, we take a policy update step using the TRPO rule, to decrease the expected cost [46].At last, we take a KL-constrained natural gradient step, to train the Discriminator.

Experiments
In this section, we introduce the experimental setup in our Sim-GAIL method and the two baseline methods that serve as comparator.

Sim-GAIL
In order to simulate the real student learning behaviour on a real platform, we build a simulator, to play back the real student learning trajectories from EdNet, selected using a stochastic policy.Specifically, we first sample the real student trajectories from Ednet.The state includes 'correct_so_far', 'ave_time', 'av_fam', 'topic_fam', 'pre_correct', 'step_in_part', and 'lects_consumed'.Then, a subset of the trajectories is randomly picked and controlled with the policy.After that, for each student's trajectory, a set of action-state pairs, are extracted from the observation policy.The policy outputs a student action, responding to the state feature at each timestamp.In this way, we created the simulation that represents the groundtruth policy, used to train other methods on.
For the experimental setup, we use an auto-encoder to process the data.Sim-GAIL is implemented using the PyTorch framework.We train the model on the seven features mentioned before using the 1000 students' interaction logs.

Baseline models
Among the few studies that could be selected as baseline methods, the current state-of-the-art top performers so far are the Behavioural Cloning based method proposed by Torabi [48] and the Reinforcement Learning-based method proposed by Kumar [49].Therefore, we use these two methods as the baselines for the experiments.

Behavioural cloning (BC)
The first baseline is the behavioural cloning (BC)-based method, proposed by Torabi [48].This model has shown good performance in the task of simulating users' behaviour from observations.Similarly, we employ a mixture regression (MR) approach [50], which is a Gaussian mixture of the actions and states, to process the data features.For fairness of the comparison, we use the same actionstate pair data to train the Sim-GAIL and BC-methods, with data extracted from EdNet (see Sect. 4.1).The supervised learning method is applied to train the policy and Adam optimisation [47], with a batch size of 128.

Reinforcement learning (RL)
The second baseline is the reinforcement learning (RL)based method proposed by Kumar [49], which uses the conservative Q-learning (CQL) approach.EdNet does not contain any students' prior-or post-test scores.Hence, we use the method proposed by Azharet al. [51] to build a reward function, based on the historical logs of students' scores.More specifically, we use the correctness of the students' responses as the reward function.If a student's response is correct, a positive reward will be given; otherwise, a negative reward will be provided.Moreover, we integrate the difficulty levels of the questions.We set the rewards from 1 to 4, based on the difficulty level of the activity.Thus, if the student's responses match the correct answers, they get a positive reward of 1 to 4; and if no, they receive a negative reward of À 1 to À 4. The Dynamic Programming (DP) [52] method is used to train the model.More specifically, we utilise a Policy Iteration (PI) method to train the agent.This process could be separated into two repeated stages: the first is evaluating the value of every state in the finite MDP according to the current policy.The second is using the Bellman Optimality equation [53] to make the policy iteration based on the current policy.

Evaluation
Our evaluation includes two parts: The first part compares the Sim-GAIL with the two baseline models, and the second part uses Knowledge Tracing models to evaluate the effect of the Sim-GAIL.
In the first part of the evaluation, to better evaluate Sim-GAIL and its performance relative to traditional models, we develop our own comprehensive evaluation framework.Since the most critical elements for a Markov Decision Process are action, reward, and policy, as shown in Fig. 1, we build a novel framework, to evaluate the efficiency of Sim-GAIL and two baseline models from these three aspects, respectively.In particular, we identify action distribution, to evaluate the action, expected cumulative rewards, to evaluate the reward, and offline policy, to evaluate the policy.The first metric, the action distribution, is the similarity of distributions between the generated Fig. 5 Action distribution of the Sim-GAIL model actions and the real actions from the historical (groundtruth) data.We compare this metric amongst Sim-GAIL, the BC-based method, and the RL-based method with the original data, by using the Kullback-Leibler divergence method, which is generally used to measure the difference between two distributions [54].Second, we compare the expected cumulative rewards (ECR) for each of these three methods.Third, we use two off-line policy evaluation (OPE) methods, including Importance Sampling (IS) and Fitted Q Evaluation (FQE), to compare the policy of these three methods.Our comprehensive and nuanced evaluation framework is aimed at delivering a more detailed and informative assessment of Sim-GAIL and its performance relative to traditional models.
In the second part of the evaluation, we use three stateof-the-art Knowledge Tracing models to evaluate Sim-GAIL, to test whether our method could be efficaciously applied in a real-world cold-start scenario.We apply the generated data to a widely used ITS technique called knowledge tracing (KT) to verify the effectiveness of our model.KT could be used to predict the students' next actions, based on their historical behavioural trajectories [6].We apply the generated data in three state-of-the-art KT models, i.e., SSAKT, SAINT, and LTMTL, to test if the generated data mixed with the original data could improve their accuracy, when training on only a small set of student data.

Action distribution evaluation
As mentioned in Sect.3.2, we obtain the action distribution of EdNet by allocating the 35 actions into seven groups, resulting in five actions per group, as shown in Fig. 3.We can observe that actions 21, 22, 23, and 24 have higher frequencies than other actions.This pattern also appears in the action distribution generated by Sim-GAIL (shown in Fig. 5).The major difference in action distributions between the real data from EdNet and those generated by Sim-GAIL is that action 25 (i.e., one of the lecture actions) in the latter is not close to the average value of 0. In addition, action 26 in Sim-GAIL also exhibits a higher frequency.Figure 6 shows the action distribution of the simulated students generated by the RL-based method.The highest frequencies fall into groups 5 and 6, while group 6 contains most of the high-frequency actions.Unlike the action distribution of real data, the clustering of each group can not be clearly identified in the action distribution of the RL-based method.Figure 7 shows the action distribution of the simulated students generated by the behavioural cloning (BC)-based method.Within this distribution, actions in group 6 illustrate the highest frequencies, indicating that actions in group 6 are the most frequent ones.Figure 8 compares the action distribution amongst the data generated by these three different student simulation methods.We can see that the BC-based method outperforms the RL-based method in this metric, and the action Moreover, we use the Kullback-Leibler divergence (KL) method to measure whether the action distribution generated by these three methods conforms to the real action distribution from EdNet.Table 3 shows the KL values of the distribution of the actions generated by these three methods and that of the real actions, respectively.The KL value between the action distribution of the data generated by Sim-GAIL; and the action distribution of the real data (ground truth) is the lowest (0.297), which suggests that the action distribution generated by Sim-GAIL is the closest to the real action distribution.Thus, it performs the best in this metric.The result also shows that the BC-based method (0.391) performs worse than Sim-GAIL but better than the RL-based method (0.408) in this metric.
The state 'topic_fam' represents a student's familiarity with the current topic.It is an important indicator that can reflect a student's mastery of knowledge.We compare the action distribution of the state value 'topic_fam' from simulated students generated by three different methods, which is shown in Fig. 9. From left to right is the distribution of simulated student actions in the state of 'topic-fam' generated by Sim-GAIL, RL-based method, and BC-based method.It can be seen that data generated by the RL-based method is the most distributed in the most difficulty-level actions (the darkest bar in each figure).Within this policy generated by RL, the method could obtain the highest rewards in the short term.However, the distribution of actions in the lecture (the orange bar) is minimal.Such a distribution does not match the real learning trajectories of students, because students need to learn new knowledge through attending lectures.The BCbased method has a more average distribution of actions on all difficulty-level actions.However, the distributions of lecture actions are unstable, which is also inconsistent with the real students' learning trajectories.The action distribution of the simulated student method based on Sim-GAIL is the most in line with the real students' trajectories action distribution, and the counts of students' actions between lectures and questions are relatively stable.This indicates that the simulated students generated by the Sim-GAIL method can balance the data distribution and optimal policy to achieve a better simulation effect.Expected cumulative rewards (ECR) represents the average of the expected cumulative rewards under a given policy [55].ECR could effectively reflect the cumulative reward obtained by the method, which is a crucial indicator of the effect of the method.The equation for computing ECR is: where the Qðs 0 ; aÞ function is the 'action value' of the action a selected by policy p in the initial state s 0 .In this experimental setting, we set ECR to be simply equal to the unique initial state value ECR ¼ V p Ã s 0 ð Þ.We calculate the cumulative rewards for 100 rounds over 1000 steps starting from the initial state.The results of the expected cumulative rewards evaluation are shown in Fig. 10, and a higher ECR indicates better performance.
The ECR of Sim-GAIL grows the fastest among the three methods, suggesting its superior ability to accumulate rewards in the early stages of the simulation.This rapid growth could be attributed to the generative nature of the GAIL algorithm, which enables efficient exploration and exploitation of the simulation environment, leading to higher rewards.After 200 steps, Sim-GAIL's ECR reaches a plateau at around 400, indicating that the model has learned an optimal policy and further exploration does not significantly increase the total rewards.This illustrates the model's ability to converge to an optimal solution quickly, a key advantage in scenarios where computational resources or time are limited.
The RL method exhibits a slower ECR growth rate compared to Sim-GAIL.This could be due to the inherent challenge in reinforcement learning of balancing exploration and exploitation.Although RL eventually stabilises at a cumulative reward of approximately 290 after 500 steps, this indicates its lower efficiency compared to Sim-GAIL.BC displays the slowest ECR growth rate, stabilising at around 240 after 400 steps.This slower growth and lower final ECR compared to Sim-GAIL and RL reflect the limitations of the BC method, which may not fully capture the complex dynamics of the simulation environment.
These observations indicate that Sim-GAIL outperforms the traditional RL and BC methods in terms of ECR growth rate and final ECR value, highlighting the effectiveness of the GAIL approach in this context.This superior performance underscores the novelty and potential of our proposed Sim-GAIL as a powerful tool for generating simulated student data for ITS training.

Offline policy evaluation
As a robust policy evaluation method that does not require human participation, the offline policy evaluation (OPE) is often used to evaluate reinforcement learning (RL), which has shown great potential in decision-making tasks, such as robotics [56] and games [57].In these tasks, RL optimal  12 The FQE-loss strategies could be evaluated in either the environment or the simulator.There are various ways of evaluation, such as maximum cumulative reward, optimal policy, and evaluating the score in games, and the score could be high or low, and a high score indicates a better performance [58].However, in human-participating tasks, evaluation becomes very difficult.First, human subjectivity may lead to bias in the results.Second, the simulator cannot consider every feature in a complex environment.Finally, experiments, where humans are involved, may make the evaluation process expensive, time-consuming, and resourceintensive.The OPE methods [59] were proposed to address these problems, where the evaluation of the policy is only based on the collected historical offline log data.They are mainly applied in scenarios where online interactions involve high-risk and expensive settings, such as stock trading, medical recommendation, and educational systems [60].In this paper, we employed a combination of two OPE methods: the Importance Sampling (including three variants, OIS, WIS, and PIS) [61] and the Fitted Q Evaluation method [62], which allows for testing the policy performance of the three models.

Importance sampling
As one of the OPE methods, importance sampling (IS) is used in situations where it is difficult to sample directly from the original data distribution.It is a method that uses a simple and collectable distribution to calculate the expected value of the desired distribution [61].There are many works using IS to evaluate the target policy (the policy derived from the RL algorithms) and the behaviour policy (the policy used to gather the data) when dealing with MDPs [63,64].However, the basic IS method may suffer from high variance, due to the huge difference between those two policies.In our experiment, we used three IS methods: the general IS (i.e., ordinary importance sampling (OIS)) and two variants of the general IS, including weighted importance sampling (WIS) and per-decision importance sampling (PDIS).WIS employs a weighted average to mitigate the variance [65].The Per-Decision Importance Sampling modifies the sampling ratio and makes the reward dependent only upon the previous action in each timestamp [62].The combination of the three methods can better observe the policy distribution of the generated data.Table 4 shows the results of the Importance Sampling evaluation.On the OIS criteria, the BC-based method outperforms the RL-based method but is worsen than Sim-GAIL.On the PDIS criteria, the Sim-GAIL method outperforms both RL-based and BC-based methods and the BC-based method performs better than the RL-based method.Sim-GAIL outperforms the other two baseline models, and the RL-based method performs better than the BC-based method on the WIS criteria.In summary, Sim-GAIL outperforms the other baseline models on every criterion.

Fitted Q evaluation
The FQE algorithm regards the MDP as a supervised learning problem.This method uses a function approximator to fit the Q function under a specified policy, based on the observation of the dataset [62].
Figure 11 shows the Fitted Q Evaluation results on the initial state.Sim-GAIL outshines the other two methods, affirming its superior performance.This is likely due to the strengths of the GAIL approach, which efficiently captures the complex dynamics of the environment and generates more robust policies.Sim-GAIL's ISV peaks in the third epoch, indicating rapid learning and optimisation.Despite subsequent oscillations, Sim-GAIL's performance consistently surpasses that of RL and BC methods, showcasing its robustness and stability.The RL method exhibits better ISV performance than the BC method.Both methods show a steady increase, with their maximum ISV reached in the 9th epoch.However, their peak performance still falls short of Sim-GAIL's average level, underscoring the superior efficiency and effectiveness of Sim-GAIL.In summary, Fig. 11 highlights the efficacy of Sim-GAIL in terms of policy quality and learning speed, as evidenced by its superior Fitted Q Evaluation results.This underscores the potential of Sim-GAIL as an efficient and robust approach for generating simulated student data for ITS training.
Figure 12 shows the FQE loss of the three methods.Sim-GAIL's FQE loss increases rapidly, peaking in the third and fourth epochs.It then swiftly declines but starts to ascend again after the fifth epoch.This rapid fluctuation reflects the model's active learning and adaptation process.In contrast, RL and BC methods exhibit relatively stable, slower FQE loss growth.In particular, RL shows moderate growth, while BC displays the slowest growth.This slower and more stable growth could be indicative of a more conservative learning process compared to Sim-GAIL.
Despite generating the highest Qðs0; pðs0ÞÞ values, Sim-GAIL also incurs higher and less stable validation loss compared to the RL and BC methods.This suggests that while Sim-GAIL is efficient in learning and optimising the policy, it may overfit the training data, leading to higher validation loss.While Sim-GAIL outperforms RL and BC methods overall, the results also indicate a need for parameter tuning to reduce the loss, highlighting an area for further improvement in Sim-GAIL's implementation.
In summary, Fig. 12 underscores the dynamic and efficient learning capability of Sim-GAIL, as well as the need for further tuning to optimise its performance.Despite the higher and less stable validation loss, Sim-GAIL's overall superiority in generating higher Q-values reaffirms its potential as a robust tool for generating simulated student data for Intelligent Tutoring System training.

Evaluation using knowledge tracing (KT) models
Knowledge tracing (KT) is an emerging research direction and has been widely applied in intelligent educational applications, where students' historical trajectories are used to model and predict their knowledge states [31].However, the lack of student interaction data in the early stage of using a system, known as the cold-start problem, limits the performance of KT models.It has been one massive obstacle to the development and application of KT.In this experiment, we applied the original data and the data generated from the Sim-GAIL method to the state-of-theart KT models to test whether our model could improve the performance of KT models in a cold-start scenario.This in turn proves the efficiency of our proposed Sim-GAIL method's ability to simulate and generate students' historical trajectory data.
In the KT research area, there is a Riiid Answer Correctness Prediction Competition on Kaggle, 3 which compares the state-of-the-art KT models using the EdNet dataset.The current top three models in this competition are SAINT, SSAKT, and LTMTI. 4The prediction competition provides a dataset of 2500 students to train the KT model.Therefore, we assume that the volume of 2500 students is sufficient for KT models to get good prediction performances.Thus, in our experiments, we considered the case of a data size of no more than 2500.Therefore, we selected datasets of sizes 500, 1000, 1500, 2000, and 2500 student records.Each student record contains the student's sequence of discrete learning actions.In our experiment, we first used Sim-GAIL to generate simulated data whose size is equal to the original data size, and then we mixed it with the original real data to build a new dataset.After that, we fed this mixed dataset into the 3 KT models, respectively.For example, in the case of the original data size being equal to 500, we input the 500 student records to Sim-GAIL, which generated equally-sized (i.e., 500) simulated student records.Then, we mixed these 500 generated student records with the original 500 student records, to build a new dataset of size 1000.This new mixed dataset was finally used to train the KT models.We compared the performance of the KT models between using this mixed dataset and using only the original data.The metric we used here is AUC.
Figure 13 shows the pairwise AUC comparisons of the three KT models trained on only the original students' data (SAINT, SSAKT, and LTMTL; in grey) and trained on the mixed dataset (SAINT*, SSAKT*, and LTMTL*; in red).The curves of SSAKT* and LTMTL* are constantly higher than the curves of SSAKT and LTMTL in all the cases, i.e., 1000, 2000, 3000, 4000, and 5000 sizes of the mixed dataset.The curve of SAINT* is higher than the curve of SAINT in the cases of 1000, 2000, and 3000 sizes of data.Although the curve of SAINT* is very close to SAINT in the cases of 5000 sizes of data, the former still outperforms the latter.In all those three pairwise comparisons, especially in the cases of smaller data sizes (1000, 2000, and 3000), obviously, training on mixed data (a combination of the original and generated data) could improve the KT models.The graphical representation of these results would likely show an upward trend for all KT models, demonstrating that the accuracy of the KT models can be improved with more data and iterations.The lines representing the training on mixed data would be above those of the original KT models, indicating our method's superior performance.This suggests that the data generated by our Sim-GAIL method can help improve the KT models, especially in cold-start scenarios, where the size of the available data is small.
6 Discussions and future work

Result analysis
From the results of the experiment, we observe that Sim-GAIL outperforms the baseline methods on the metrics of Action Distribution Evaluation, Expected Cumulative Rewards Evaluation, and Offline Policy Evaluation.The satisfying fit simulation results may come from the fact that there is no need to define a reward function for Sim-GAIL, compared with other baseline models.Defining reward functions manually may be too complex to fit the real student trajectories', thus a reward function built by algorithms instead of humans might result in a better policy [20].The results of the evaluation using the KT models show that Sim-GAIL could be applied to real-world educational scenarios and improve the efficiency of current educational technologies.More specifically, our method could effectively alleviate the cold-start problem of KT models.
Our Sim-GAIL method outperforms the baseline models on every metric.The RL-based method outperforms the BC-based method in terms of offline policy evaluation.This indicates that a suitable setting of the reward function could generate better policies.This result is also reflected in the distribution of 'topic_fam' actions.The policy generated by the RL-based method places more emphasis on high-difficulty and high-reward actions.Such a policy works well for obtaining higher cumulative rewards, but it does not match the action distribution of real students' trajectories.Besides, the distribution of 'lecture' actions whose default reward value is 0, is very small and unstable.Thus, the action distribution generated by the RL-based method is inconsistent with the action distribution of real students' trajectories.The BC-based method outperforms the RL-based method in action distribution, but is worse in offline policy evaluation.This suggests that, although the BC-based method can render the action distribution more aligned with the real action distribution, it is difficult to obtain a better learning policy.Therefore, Sim-GAIL is a more advanced student simulation method than those two traditional ones.Besides, as Sim-GAIL does not require a dedicated reward function to fit different datasets, compared with traditional student simulation methods, our method could be easily transferred and applied to another ITS.
In the evaluation using KT models, we apply our method to three different state-of-the-art KT models.The results indicate that our method could improve training efficiency in cold-start scenarios.In Fig. 13, every KT model trained on the mixed data (a combination of the original data and the data generated by our Sim-GAIL method) performs better in each group.The results suggest that it could improve training efficiency in small-sized data scenarios, proving that it could alleviate the cold-start problem in the early stages of ITS development.For instance, in the above experiments, every KT* model performs better when the original data size is smaller than 2000.After the data size is larger than 2000, the performance of using the original dataset (KT) is close to that of using a mixed dataset (KT*), but the KT* still outperforms the KT.

Advantages
Our proposed model, Sim-GAIL, brings several significant advantages to the field of student modelling for intelligent tutoring systems (ITS).A fundamental strength of Sim-GAIL lies in its underlying mechanism, that of generative adversarial imitation learning (GAIL), which endows the model with the capacity to generate new data instances that closely resemble actual student behaviour data.This generative modelling capability of Sim-GAIL is crucial for creating a rich, diverse dataset needed for effective ITS training.Additionally, Sim-GAIL offers a solution to a common issue encountered in the early stages of ITS development -the cold-start problem.The ability to generate simulated student data allows Sim-GAIL to effectively tackle this problem, accelerating the training process of ITS.
In terms of performance, Sim-GAIL has demonstrated superiority over traditional reinforcement learning (RL) and behavioural cloning (BC) based methods across various metrics, including action distribution evaluation, cumulative reward evaluation, and offline-policy evaluation.This implies that Sim-GAIL can simulate student behaviours with higher accuracy and effectiveness.Furthermore, the efficiency of Sim-GAIL is evident from the rapid convergence to an optimal policy whilst simulating real student learning trajectories, providing a significant advantage in scenarios where computational resources or time are limited.
Beyond these, the scalability and generality of Sim-GAIL further enhance its appeal.As a data-driven model, Sim-GAIL does not rely on expert knowledge for defining the reward function, which contrasts with some RL-based methods.This attribute allows Sim-GAIL to scale and generalise across different datasets and applications, seamlessly.
In essence, Sim-GAIL represents a novel, effective, and efficient approach to student modelling.By offering a promising tool for generating simulated student data, Sim-GAIL contributes to enhancing the efficacy of ITS training.

Limitations
The limitations of this work mainly lie in the following aspects.First, our work adopts a general state representation method from other studies [4,51], where Sim-GAIL outperforms other baseline methods on most metrics.As discussed in Sect.3.2, the selection of state representation may impact the models' performance.However, the experimental design of our work does not consider the potential impact of different state combinations on various methods.Second, in the experiments of evaluation using KT models, when a KT model moves beyond the cold-start stage and has sufficient data, the increase in the amount of simulated data may lead to a decrease in the prediction accuracy of the KT model, which may be the bias caused by Sim-GAIL not considering all the features of student actions.

Future work
While our proposed Sim-GAIL method shows promising results in student simulation for Intelligent Tutoring Systems (ITS), there are several avenues for future exploration and improvement.
Fine-grained simulations In our current implementation, Sim-GAIL focuses on generating simulated student behaviour data at a coarse level.Future work can explore methods to capture more fine-grained details, such as students' cognitive processes, metacognitive strategies, and affective states.Incorporating these aspects could lead to more accurate and comprehensive student modelling.
Adaptive simulation Currently, Sim-GAIL generates simulated student data based on predefined models.Future research can investigate methods to make the simulation adaptive, allowing sim students to learn and evolve based on feedback from the ITS.This adaptive simulation approach can provide more dynamic and personalised student trajectories.
Transfer learning and generalisation Sim-GAIL has been evaluated on the EdNet dataset, but its generalisability to other domains and datasets remains an open question.Future work can explore transfer learning techniques to enhance the model's ability to generalise across different educational contexts and datasets, enabling wider applicability of Sim-GAIL in various ITS settings.
Human-in-the-loop simulations Although Sim-GAIL offers an efficient alternative to collecting real student data, it is crucial to acknowledge the limitations of fully replacing human students with sim-students.Future research can investigate human-in-the-loop simulation methods, where sim students are combined with real student interactive data, allowing for iterative refinement and validation of the simulated trajectories.
By pursuing these future research directions, we can further enhance Sim-GAIL's capabilities and contribute to the advancement of student modelling techniques in the field of Intelligent Tutoring Systems.

Conclusion
In this study, we have introduced Sim-GAIL, a pioneering student simulation method founded on the generative adversarial imitation learning (GAIL) algorithm.It stands as the first of its kind that trains ITS using simulated student behaviour data, effectively addressing the challenges of high-cost, resource-intensive real student data collection, and the cold-start problem encountered during earlystage ITS training.Sim-GAIL demonstrates superior performance in comparison with traditional Reinforcement Learning-based and Imitation Learning-based methods, marking a significant advancement in state-of-the-art student modelling for Intelligent Tutoring Systems.
Our student simulation method, Sim-GAIL, leverages the EdNet dataset and outperforms the baseline methods: a Reinforcement Learning method based on Conservative Q-learning and an Imitation Learning method based on Behavioural Cloning.We have thoroughly evaluated our method from four aspects: action distribution discrepancy based on the Kullback-Leibler divergence, reward function using expected cumulative rewards (ECR), and two offline policy evaluation (OPE) methods-Importance Sampling and Fitted Q Evaluation.Our results convincingly demonstrate that Sim-GAIL outperforms the baseline models in all these aspects.
Further, we have applied Sim-GAIL to state-of-the-art knowledge tracing models and observed a noticeable improvement in their performance, especially in cold-start scenarios.This underlines Sim-GAIL's efficiency in simulating and generating students' historical trajectory data, further emphasising its novelty and potential to contribute to the field of student modelling for Intelligent Tutoring Systems.
Moving forward, research can explore fine-grained simulations, adaptive simulation techniques, transfer learning and generalisation, and human-in-the-loop simulations, to enhance Sim-GAIL's capabilities in student modelling even further, as discussed in Sect.6.This study paves the way for these future endeavours by providing a robust, novel method for generating simulated student data for ITS training.

Fig. 1
Fig. 1 Framework of the Markov decision process

Fig. 6 Fig. 7 Fig. 8
Fig. 6 Action distribution of the reinforcement learning-based model

Fig. 13
Fig. 13 Pairwise AUC comparisons of the three KT models trained on only original students' data (SAINT, SSAKT, LTMTL, in grey) and trained on the mixed dataset (SAINT*, SSAKT*, LTMTL*, in red).On the horizontal axis, 500, 1000,...,2500 indicate that the grey curve model uses the original dataset, and (1000),(2000),...,(5000) indicate that the red curve model uses the mixed dataset 2. It is the first method, to the best of our knowledge, that uses Generative Adversarial Imitation Learning (GAIL) to implement student modelling to address the challenge of lacking training data and the cold-start problem.3. The experiments demonstrate that a trained Sim-GAIL could simulate real student learning trajectories very well.Our method outperforms traditional RL-based and BC-based methods on most metrics and can improve the training efficiency in cold-start scenarios.

Table 1
Statistics of the EdNet Fig. 3 Analysis of the action distribution of the EdNet dataset

Table 3
Kullback-Leibler divergence of action distribution distribution of Sim-GAIL generated data is closest to the real data's distribution.

Table 4
Importance sampling evaluation results Fig. 11 Initial state value estimate of the FQE Fig.