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Addressing Two Problems in Deep Knowledge Tracing via Prediction-Consistent Regularization
Chun-Kit Yeung, Dit-Yan Yeung
TL;DR
DKT can reconstruct observed responses incorrectly and produce wavy knowledge-state predictions across time. The paper adds reconstruction and waviness regularizers, reporting improved consistency without sacrificing next-interaction accuracy, while noting unresolved scope for unseen knowledge components and dataset-specific waviness targets.
Problem
DKT can produce counterintuitive reconstruction and inconsistent, wavy predictions across time-steps, which can mislead interpretation of students’ knowledge states.
Method
The paper augments DKT’s original next-interaction loss with a reconstruction regularizer and two waviness regularizers.
Results
Regularized DKT improves reconstruction and prediction consistency without sacrificing accuracy on the original next-interaction task.
Takeaways & Limitations
The proposed measures provide distinct evaluations for current-interaction accuracy, prediction waviness, and consistency between observations and prediction changes.
Takeaways & Limitations
DKT+ does not evaluate longer-horizon or unobserved-KC predictions, and ideal waviness values vary with dataset-specific KC dependencies.
Abstract
from arXiv · showhide
Knowledge tracing is one of the key research areas for empowering personalized education. It is a task to model students' mastery level of a knowledge component (KC) based on their historical learning trajectories. In recent years, a recurrent neural network model called deep knowledge tracing (DKT) has been proposed to handle the knowledge tracing task and literature has shown that DKT generally outperforms traditional methods. However, through our extensive experimentation, we have noticed two major problems in the DKT model. The first problem is that the model fails to reconstruct the observed input. As a result, even when a student performs well on a KC, the prediction of that KC's mastery level decreases instead, and vice versa. Second, the predicted performance for KCs across time-steps is not consistent. This is undesirable and unreasonable because student's performance is expected to transit gradually over time. To address these problems, we introduce regularization terms that correspond to reconstruction and waviness to the loss function of the original DKT model to enhance the consistency in prediction. Experiments show that the regularized loss function effectively alleviates the two problems without degrading the original task of DKT.
INTRODUCTION
Knowledge tracing models students’ mastery of knowledge components from historical interactions to support personalized learning. This paper identifies reconstruction and wavy-transition problems in DKT and introduces regularization to improve prediction consistency without sacrificing next-interaction accuracy.
- Motivation: Knowledge tracing models student knowledge states, representing mastery of knowledge components from historical learning data.These estimates can help tutors tailor learning materials and help students plan around weaknesses.
- Problems in DKT: DKT can produce counterintuitive reconstruction: a correct response may lower, or an incorrect response may raise, the predicted performance for that skill.At one example time-step, the probability for skill s45 increases even though the student answered s45 incorrectly.
- Problems in DKT: DKT predictions can also fluctuate sharply across time-steps, with sudden surges and plunges in predicted performance for several skills.The paper argues that such wavy transitions can mislead interpretation of a student’s knowledge state.
- Approach: The proposed regularized DKT adds a reconstruction term r and waviness terms w1 and w2 to enhance consistency in prediction.The original loss considers prediction accuracy for the next interaction, while the added measures target reconstruction and temporal waviness.
- Results: Experiments report better reconstruction and more consistent predictions from regularized DKT without sacrificing next-interaction prediction accuracy.The paper also introduces AUC(C), w1, w2, m1, and m2 to evaluate current-interaction performance and prediction consistency.
BACKGROUND
Knowledge tracing has evolved from probabilistic and logistic models to deep neural networks. DKT uses an RNN and has been reported to outperform traditional models without requiring several manually engineered features.
- Evolution of KT Models: Knowledge tracing research has developed mathematical and computational models ranging from probabilistic methods to deep neural networks.The field has investigated these approaches since the 1990s.
- Traditional Models: BKT models each knowledge component with a hidden Markov model, but its assumptions include no forgetting and mutually independent knowledge components.Variants introduce forgetting power and skill dependency to address these shortcomings.
- Traditional Models: LFA and PFA use logistic-regression-style features based on successes or failures to predict knowledge acquisition across multiple knowledge components.They cannot represent inherent dependencies among knowledge components, and their features are relatively simple.
- Deep Knowledge Tracing: DKT models student knowledge state with an RNN and has been shown to outperform traditional models such as BKT and PFA.It does so without requiring manually engineered features for recency, trial sequence, inter-skill relationships, or ability variation.
Review of Deep Knowledge Tracing
DKT uses a recurrent neural network to transform a sequence of student interactions into hidden states and output predictions. Its LSTM variant regulates memory through forget, input, and output gates.
- Recurrent Neural Networks: A vanilla RNN maps an input sequence to an output sequence through hidden states that encode past information and the current input.The hidden representation captures information that is difficult to engineer manually.
- DKT Architecture: DKT uses an RNN backbone, with either a vanilla RNN or an LSTM-RNN processing the hidden state.The unfolded architecture represents the sequential processing of student interactions.
- LSTM-RNN: An LSTM cell uses forget, input, and output gates to control the memory cell state and determine the resulting hidden state.The forget gate removes information, the input gate adds new information, and the output gate extracts information for the hidden state.
- LSTM-RNN: The LSTM formulation enables the recurrent model to store information from distant history more powerfully than the vanilla RNN.The unfolded RNN architecture provides a high-level interpretation of DKT.
DKT Problem Formulation
DKT encodes each question-answer interaction as a fixed-length vector, processes the sequence through recurrent hidden states, and predicts performance for the next interaction using the output vector.
- Input Representation: Each question-answer interaction is transformed into a fixed-length input vector using one-hot question encoding and the answer label.With M unique questions, the input vector lies in {0, 1}2M.
- Recurrent Processing: DKT computes a hidden state from each encoded input using a vanilla RNN or LSTM-RNN, treating that state as the student’s latent knowledge state.The hidden state summarizes information from the student’s past learning trajectory.
- Output Prediction: The output vector represents probabilities of answering each question correctly.The target question’s prediction is extracted from the output using its one-hot encoding.
- Training Objective: DKT trains by comparing the predicted next-interaction performance with the target answer using cross-entropy loss.The loss is aggregated over n students.
SOME PROBLEMS OF DKT AND THEIR REMEDIES
DKT exhibits reconstruction errors and inconsistent skill-state transitions, including counterintuitive predictions for related skills. The proposed remedy adds a loss term that incorporates the current interaction to improve reconstruction.
- Reconstruction Problem: The original loss considers only the next interaction, so it ignores the prediction for the current observed interaction.The authors attribute counterintuitive current-input predictions to this omission.
- Reconstruction Problem: Bidirectional error dependencies between s32 and s33 make the claim that s32 is a prerequisite of s33 questionable.Both correctness matrices show that an incorrect response on one skill is followed more probably by an incorrect response on the other.
- Reconstruction Problem: A reconstruction regularizer adds loss between predictions and the current interaction, encouraging predictions to adjust to the input.This term is introduced as an immediate remedy for the reconstruction problem.
Wavy Transition in Prediction
The paper addresses wavy knowledge-state transitions by regularizing differences between consecutive output vectors. Two waviness measures quantify these changes and are incorporated into the augmented loss.
- Wavy Transition in Prediction: The wavy-transition problem may arise from an RNN hidden state that summarizes all exercises in one representation.The authors propose constraining the output behavior because the hidden-layer elements’ effects on KC predictions are difficult to explicate.
- Wavy Transition in Prediction: Two waviness measures, w1 and w2, are defined as regularization terms to smooth prediction transitions.They quantify differences between consecutive prediction vectors using L1- and L2-norms.
- Wavy Transition in Prediction: Larger w1 and w2 values indicate more inconsistent transitions between consecutive output vectors.The measures average changes across input time-steps and knowledge components.
- Wavy Transition in Prediction: The original DKT loss is augmented with reconstruction, w1, and w2 regularization terms weighted by λr, λw1, and λw2.Training with the new loss is expected to address both reconstruction and wavy-transition problems.
EXPERIMENTS
Experiments evaluate DKT using prediction, reconstruction, and consistency measures under specified data-splitting and hyperparameter procedures. The evaluation favors high AUC with low waviness and includes current-performance reconstruction accuracy.
- Implementation: 20% of the data is reserved for testing, while 80% is used for training with 5-fold cross-validation for hyperparameter selection.The test set is also used for evaluation and early stopping.
- Implementation: The basis model is a single-layer RNN-LSTM with state size 200, learning rate 0.01, dropout 0.5, and norm-clipping threshold 3.0.Weights are randomly initialized from a zero-mean Gaussian distribution with small variance.
- Evaluation Measures: AUC(N) measures next-performance prediction, while AUC(C) measures current-performance reconstruction accuracy.The evaluation also includes m1 and m2 for consistency between observed inputs and corresponding prediction changes.
- Experiments: Table 3 compares unregularized DKT with DKT+ using averages and standard deviations from three trials.DKT+ settings are selected through 5-fold cross-validation by first requiring lower w1 than DKT, then maximizing AUC(N) + AUC(C) + m1 + m2.
- Evaluation Measures: Positive m1 and m2 values indicate that more than half of prediction changes comply with input observations.Zero indicates an even split between correct and incorrect directional changes; negative values indicate more wrong-direction changes.
- Evaluation Measures: The study seeks high AUC while maintaining low waviness, using w1 and w2 to quantify consistency for other knowledge components.Hyperparameter searches vary λr, λw1, and λw2 over predefined ranges.
Datasets
The experiments use several real and simulated knowledge-tracing datasets spanning different numbers of students, interactions, skills, and exercise tags. ASSIST2009 is additionally processed to remove duplicated records.
- Datasets: ASSIST2009 contains 4,417 students, 328,291 question-answering interactions, and 124 skills after duplicate removal.The dataset comes from the ASSISTments online tutoring platform.
- Datasets: ASSIST2015 contains 19,917 student responses for 100 skills and 708,631 question-answering interactions.Despite more interactions than ASSIST2009, it has fewer records per skill and student on average because it includes more students.
- Datasets: ASSISTChall contains 686 students, 942,816 interactions, and 102 skills.It has a higher average number of records per student than the other listed ASSISTments datasets.
- Datasets: Statics2011 contains 189,927 interactions from 333 students and 1,223 exercise tags.The dataset comes from an engineering statics course.
- Datasets: Statics2011 uses processed data with a 70:30 train/test split and exercise tags as inputs.
- Datasets: Simulated-5 models 2,000 virtual students answering 50 exercises drawn from five virtual concepts.Correctness probabilities are defined using conventional item response theory with concept abilities and exercise difficulty.
Results
Across datasets, DKT+ substantially improves reconstruction, consistency, and waviness measures while generally preserving next-interaction prediction accuracy. AUC(N) remains similar or improves slightly in most reported comparisons.
- ASSIST2009 AUC(C) increases from 0.9044 to 0.9625, while w1 and w2 decrease from 0.0830 to 0.0229 and 0.1279 to 0.0491.
- ASSIST2015 retains a similar AUC(N) of 0.7371 while DKT+ reaches AUC(C) 0.9233, m1 0.8122, and m2 0.0591.Its w1 and w2 values are approximately half those of DKT.
- ASSISTChall AUC(C) rises from 0.7109 to 0.8570, while w1 and w2 fall from 0.0690 to 0.0147 and 0.1045 to 0.0301.AUC(N) slightly decreases from 0.7343 to 0.7285.
- Statics2011 AUC(N) and AUC(C) increase from 0.8159 to 0.8349 and 0.7404 to 0.9038, while w1 and w2 shrink substantially.The reported waviness reductions are from 0.1358 to 0.0074 and from 0.1849 to 0.0130.
- Simulated-5 shows similar AUC(N), but DKT+ raises AUC(C) from 0.8642 to 0.9987 and improves m1 from −0.1512 to 0.9064 and m2 from −0.0134 to 0.1659.The prediction transition is also less wavy.
- Overall, regularization alleviates reconstruction and wavy transitions without sacrificing next-interaction accuracy, and some parameter combinations slightly improve AUC(N).
DISCUSSION
The analyses examine how reconstruction and waviness regularizers affect accuracy, consistency, and prediction dynamics. DKT+ produces smoother, more interpretable knowledge-state transitions while maintaining strong evaluation performance across suitable parameter ranges.
- Increasing λr consistently raises AUC(C), while AUC(N) generally declines only slightly across datasets.The authors interpret this as reconstruction improvement with limited next-interaction accuracy loss.
- Waviness regularizers can mitigate the slight AUC(N) degradation introduced by the reconstruction regularizer.
- With λr = 0.1 on ASSIST2009, AUC(N) and AUC(C) remain relatively stable for λw1 between 0.0 and 1.0 and λw2 between 0.0 and 10.0.AUC(C) lies between 0.94 and 0.96 in this region, while larger values reduce performance.
- Within the stable hyperparameter region, increasing λw1 and λw2 reduces waviness and increases m1, indicating improved consistency for the current input.The reported m2 trend decreases as λw1 and λw2 grow.
- Figure 5 shows DKT’s wavy, globally coupled predictions, whereas DKT+ produces notably smoother transitions and retains latent knowledge states across time-steps.This consistency is reported to reduce knowledge-state misinterpretation and enhance interpretability.
CONCLUSION AND FUTURE WORK
The paper introduces regularization terms to address DKT’s reconstruction and wavy-transition problems, improving prediction consistency without sacrificing next-interaction accuracy. It also identifies unresolved calibration and future-prediction limitations.
- DKT’s reconstruction problem and wavy prediction transitions can mislead interpretations of students’ knowledge states.
- The proposed regularizers include reconstruction error and two waviness measures, evaluated through AUC(C), m1, m2, w1, and w2.
- The regularizers alleviate reconstruction and waviness problems without sacrificing AUC(N) for predicting next-interaction performance.
- The paper cannot specify universally good w1 and w2 values because KC-dependency graphs differ across datasets.
- Future work should improve stability and accuracy for unobserved KCs and account for interactions further in the future during training.
- The decay factor γ serves in normalization and may help DKT learn a more robust latent-knowledge representation.