Open-Ended Knowledge Tracing

Knowledge tracing refers to the problem of estimating each student's knowledge component/skill mastery level from their past responses to questions in educational applications. One direct benefit knowledge tracing methods provide is the ability to predict each student's performance on the future questions. However, one key limitation of most existing knowledge tracing methods is that they treat student responses to questions as binary-valued, i.e., whether the responses are correct or incorrect. Response correctness analysis/prediction is easy to navigate but loses important information, especially for open-ended questions: the exact student responses can potentially provide much more information about their knowledge states than only response correctness. In this paper, we present our first exploration into open-ended knowledge tracing, i.e., the analysis and prediction of students' open-ended responses to questions in the knowledge tracing setup. We first lay out a generic framework for open-ended knowledge tracing before detailing its application to the domain of computer science education with programming questions. We define a series of evaluation metrics in this domain and conduct a series of quantitative and qualitative experiments to test the boundaries of open-ended knowledge tracing methods on a real-world student code dataset.

Math Word Problem Generation with Mathematical Consistency and Problem Context Constraints

We study the problem of generating arithmetic math word problems (MWPs) given a math equation that specifies the mathematical computation and a context that specifies the problem scenario. Existing approaches are prone to generating MWPs that are either mathematically invalid or have unsatisfactory language quality. They also either ignore the context or require manual specification of a problem template, which compromises the diversity of the generated MWPs. In this paper, we develop a novel MWP generation approach that leverages i) pre-trained language models and a context keyword selection model to improve the language quality of the generated MWPs and ii) an equation consistency constraint for math equations to improve the mathematical validity of the generated MWPs. Extensive quantitative and qualitative experiments on three real-world MWP datasets demonstrate the superior performance of our approach compared to various baselines.

Math Operation Embeddings for Open-ended Solution Analysis and Feedback

Feedback on student answers and even during intermediate steps in their solutions to open-ended questions is an important element in math education. Such feedback can help students correct their errors and ultimately lead to improved learning outcomes. Most existing approaches for automated student solution analysis and feedback require manually constructing cognitive models and anticipating student errors for each question. This process requires significant human effort and does not scale to most questions used in homework and practices that do not come with this information. In this paper, we analyze students' step-by-step solution processes to equation solving questions in an attempt to scale up error diagnostics and feedback mechanisms developed for a small number of questions to a much larger number of questions. Leveraging a recent math expression encoding method, we represent each math operation applied in solution steps as a transition in the math embedding vector space. We use a dataset that contains student solution steps in the Cognitive Tutor system to learn implicit and explicit representations of math operations. We explore whether these representations can i) identify math operations a student intends to perform in each solution step, regardless of whether they did it correctly or not, and ii) select the appropriate feedback type for incorrect steps. Experimental results show that our learned math operation representations generalize well across different data distributions.

Scalable Neural Tangent Kernel of Recurrent Architectures

Kernels derived from deep neural networks (DNNs) in the infinite-width provide not only high performance in a range of machine learning tasks but also new theoretical insights into DNN training dynamics and generalization. In this paper, we extend the family of kernels associated with recurrent neural networks (RNNs), which were previously derived only for simple RNNs, to more complex architectures that are bidirectional RNNs and RNNs with average pooling. We also develop a fast GPU implementation to exploit its full practical potential. While RNNs are typically only applied to time-series data, we demonstrate that classifiers using RNN-based kernels outperform a range of baseline methods on 90 non-time-series datasets from the UCI data repository.

Wearing a MASK: Compressed Representations of Variable-Length Sequences Using Recurrent Neural Tangent Kernels

High dimensionality poses many challenges to the use of data, from visualization and interpretation, to prediction and storage for historical preservation. Techniques abound to reduce the dimensionality of fixed-length sequences, yet these methods rarely generalize to variable-length sequences. To address this gap, we extend existing methods that rely on the use of kernels to variable-length sequences via use of the Recurrent Neural Tangent Kernel (RNTK). Since a deep neural network with ReLu activation is a Max-Affine Spline Operator (MASO), we dub our approach Max-Affine Spline Kernel (MASK). We demonstrate how MASK can be used to extend principal components analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) and apply these new algorithms to separate synthetic time series data sampled from second-order differential equations.

Diagnostic Questions:The NeurIPS 2020 Education Challenge

Digital technologies are becoming increasingly prevalent in education, enabling personalized, high quality education resources to be accessible by students across the world. Importantly, among these resources are diagnostic questions: the answers that the students give to these questions reveal key information about the specific nature of misconceptions that the students may hold. Analyzing the massive quantities of data stemming from students' interactions with these diagnostic questions can help us more accurately understand the students' learning status and thus allow us to automate learning curriculum recommendations. In this competition, participants will focus on the students' answer records to these multiple-choice diagnostic questions, with the aim of 1) accurately predicting which answers the students provide; 2) accurately predicting which questions have high quality; and 3) determining a personalized sequence of questions for each student that best predicts the student's answers. These tasks closely mimic the goals of a real-world educational platform and are highly representative of the educational challenges faced today. We provide over 20 million examples of students' answers to mathematics questions from Eedi, a leading educational platform which thousands of students interact with daily around the globe. Participants to this competition have a chance to make a lasting, real-world impact on the quality of personalized education for millions of students across the world.

An Improved Semi-Supervised VAE for Learning Disentangled Representations

Learning interpretable and disentangled representations is a crucial yet challenging task in representation learning. In this work, we focus on semi-supervised disentanglement learning and extend work by Locatello et al. (2019) by introducing another source of supervision that we denote as label replacement. Specifically, during training, we replace the inferred representation associated with a data point with its ground-truth representation whenever it is available. Our extension is theoretically inspired by our proposed general framework of semi-supervised disentanglement learning in the context of VAEs which naturally motivates the supervised terms commonly used in existing semi-supervised VAEs (but not for disentanglement learning). Extensive experiments on synthetic and real datasets demonstrate both quantitatively and qualitatively the ability of our extension to significantly and consistently improve disentanglement with very limited supervision.

The Recurrent Neural Tangent Kernel

The study of deep networks (DNs) in the infinite-width limit, via the so-called Neural Tangent Kernel (NTK) approach, has provided new insights into the dynamics of learning, generalization, and the impact of initialization. One key DN architecture remains to be kernelized, namely, the Recurrent Neural Network (RNN). In this paper we introduce and study the Recurrent Neural Tangent Kernel (RNTK), which sheds new insights into the behavior of overparametrized RNNs, including how different time steps are weighted by the RNTK to form the output under different initialization parameters and nonlinearity choices, and how inputs of different lengths are treated. We demonstrate via a number of experiments that the RNTK offers significant performance gains over other kernels, including standard NTKs across a range of different data sets. A unique benefit of the RNTK is that it is agnostic to the length of the input, in stark contrast to other kernels.

LaRVAE: Label Replacement VAE for Semi-Supervised Disentanglement Learning

Learning interpretable and disentangled representations is a crucial yet challenging task in representation learning. In this work, we develop LaRVAE, a novel semi-supervised variational auto-encoder (VAE) for learning disentangled representations by more effectively exploiting the limited labeled data. Specifically, during training, we replace the inferred representation associated with a data point with its ground-truth representation whenever it is available. LaRVAE is theoretically inspired by our proposed general framework of semi-supervised disentanglement learning in the context of VAEs. It naturally motivates both the novel representation replacement strategy in LaRVAE and a supervised regularization term commonly (albeit in an ad-hoc way) used in existing semi-supervised VAEs. Extensive experiments on synthetic and real datasets demonstrate both quantitatively and qualitatively the ability of LaRVAE to significantly and consistently improve disentanglement with very limited supervision.