Consistency and Monotonicity Regularization for Neural Knowledge Tracing

1 INTRODUCTION . In recent years , Artificial Intelligence in Education ( AIEd ) has gained much attention as one of the currently emerging fields in educational technology . In particular , the recent COVID-19 pandemic has transformed the setting of education from classroom learning to online learning . As a result , AIEd has become more prominent because of its ability to diagnose students automatically and provide personalized learning paths . High-quality diagnosis and educational content recommendation require good understanding of students ’ current knowledge status , and it is essential to model their learning behavior precisely . Due to this , Knowledge Tracing ( KT ) , a task of modeling a student ’ s evolution of knowledge over time , has become one of the most central tasks in AIEd research . 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 & Schmidhuber , 1997 ) or Transformer ( Vaswani et al. , 2017 ) architectures , along with additional components that extract question textual information or students ’ forgetting behaviors ( Huang et al. , 2019 ; Pu et al. , 2020 ; Ghosh et al. , 2020 ) . However , as the number of parameters of these models increases , they may easily overfit on small datasets and hurt model ’ s generalizabiliy . Such an issue has been under-explored in the literature . 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 . Then , during training , we impose certain consistency ( for replacement ) and monotonicity ( for insertion/deletion ) bias on a model ’ s predictions by optimizing corresponding regularization losses that compares the original and the augmented interaction sequences . 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 correctly ( or incorrectly ) answered , i.e. , a student is more likely to answer correctly ( or incorrectly ) if the student did the same more in the past . By analyzing distribution of the previous correctness rates of interaction sequences , we can observe that the existing student interaction datasets indeed have monotonicity properties - see Figure 1 and Section A.2 for details . The overall augmentation and regularization strategies are sketched in Figure 2 . 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 . 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. , 2017b ) , 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 - 6.2 % 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 is dropped from 72.44 % to 66.48 % when we impose ‘ reversed ’ ( wrong ) monotonicity regularization . We believe that our work can be a strong guideline for other researchers attempting to improve the generalization ability of KT models . 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 & Khoshgoftaar , 2019 ) . More advanced techniques , e.g. , augmenting images by interpolation ( Zhang et al. , 2017a ; Yun et al. , 2019 ) or by using generative adversarial networks ( Huang et al. , 2018 ) , have been also investigated . For NLP models , it is popular to augment texts by replacing words with synonyms ( Zhang et al. , 2015 ) or words with similar ( contextualized ) embeddings ( Wang & Yang , 2015 ; Kobayashi , 2018 ) . As an alternative method , back translation ( Sennrich et al. , 2016 ; Yu et al. , 2018 ) generates an augmented sentence by translating a given sentence into a different language domain and translate it back to the original domain with machine translation models . Recently , Wei & Zou ( 2019 ) show that even simple methods like random insertion/swap/deletion could improve text classification performances . In the area of speech recognition , vocal tract length normalization ( Jaitly & Hinton , 2013 ) , synthesizing noisy audio ( Hannun et al. , 2014 ) , perturbing speed ( Ko et al. , 2015 ) , and augmenting spectrogram ( Park et al. , 2019 ) are popular as data augmentation methods . The aformentioned 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 ( Sajjadi et al. , 2016 ; Xie et al. , 2019 ; Berthelot et al. , 2019 ; Sohn et al. , 2020 ) 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 . L2-loss between outputs ) . Abu-Mostafa ( 1992 ; 1990 ) 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 ( I1 , . . . , IT ) , where each It = ( Qt , Rt ) is a pair of question id Qt and the student ’ s response correctness Rt ∈ { 0 , 1 } ( 1 means correct ) , KT aims to estimate the following probability P [ Rt = 1|I1 , I2 , . . . , It−1 , Qt ] , ( 1 ) i.e. , the probability that the student answers correctly to the question Qt at t-th step . Corbett & 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 & 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. , 2017b ; Abdelrahman & Wang , 2019 ) . Transformer based models ( Pandey & Karypis , 2019 ; Choi et al. , 2020a ; Ghosh et al. , 2020 ; Pu et al. , 2020 ) 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 ; Pandey & Srivastava , 2020 ) , prerequisite relations ( Chen et al. , 2018 ) and time information ( Nagatani et al. , 2019 ; Choi et al. , 2020a ; Pu et al. , 2020 ) is another way to improve performances . Recent works try to use graph neural networks ( Nakagawa et al. , 2019 ; Liu et al . ; Tong et al. , 2020 ; Yang et al. , 2020b ) and convolutional networks ( Yang et al. , 2020a ; Shen et al. , 2020 ) to model relations between questions and skills or extract individualized prior knowledge . 2 CONSISTENCY AND MONOTONICITY REGULARIZATION FOR KT . For a given set of data augmentations A , we train KT models with the following loss : Ltot = Lori + ∑ aug∈A ( λaugLaug + λreg-augLreg-aug ) , ( 2 ) where Lori is the commonly used binary cross-entropy ( BCE ) loss for original training sequences and Laug are the same BCE losses for augmented sequences generated by applying augmentation strategies aug ∈ A.1 Lreg-aug are the regularization losses that impose consistency and monotonicity bias on model ’ s predictions for the original and augmented sequence , which are going to be defined in the following sections . Finally , λaug , λreg-aug > 0 are hyperparameters to control the trade-off among Lori , Laug , and Lreg-aug. 1For replacement and insertion , we do not include outputs for augmented interactions in Laug . In the following sections , we introduce our three simple augmentation strategies , replacement , insertion and deletion with corresponding consistency and monotonicity regularization losses , Lreg-rep , Lreg-cor ins ( or Lreg-incor ins ) and Lreg-cor del ( or Lreg-incor del ) , respectively . 2.1 REPLACEMENT . Replacement , which is motivated by the synonym replacement 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 have overlapping skills attached to . Our hypothesis is that the predicted correctness probabilities for questions in an augmented interaction sequence will not change a lot from those in the original interaction sequence . Formally , for each interaction in the original interaction sequence ( I1 , . . . , IT ) , we randomly decide whether the interaction will be replaced or not , following the Bernoulli distribution with the probability αrep . If an interaction It = ( Qt , Rt ) with a set of skills St associated with the question Qt is set to be replaced , we determine Irept = ( Q rep t , Rt ) by selecting a question Q rep t with its associated set of skills Srept that satisfies St ∩ S rep t 6= ∅ . The resulting augmented sequence ( I rep 1 , . . . , I rep T ) is generated by replacing It with I rep t for t ∈ R ⊂ [ T ] = { 1 , 2 , . . . , T } , where R is a set of indices to replace . Then we consider the following consistency regularization loss : Lreg-rep = Et 6∈R [ ( pt − prept ) 2 ] ( 3 ) where pt and p rep t are model ’ s predicted correctness probabilities for t-th question of the original and augmented sequences , respectively . We do not include the output for the replaced interactions in the loss computation . For the replacement strategy itself we consider several variants . For instance , randomly selecting a question for Qrept from the question pool is an alternative strategy if a skill set for each question 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 .

Summary: This paper presents some enhancements for Knowledge Tracing (KT), in which predictions are made about the odds of a student answering a question correctly given a sequence of correct/incorrect responses to previous questions.  The authors observe that the predictive model should obey certain 3 common sense constraints. If a question is replaced in the student's data by a very similar question, the prediction should not change much. If an additional correct question is added to the data, the odds of the student being correct on the next question should go up, and the odds should go down for questions being removed and/or added with incorrect responses.  The learning algorithm's objective function is augmented with additional terms which encourage the model to obey these constraints.