A Spectral Learning Approach to Knowledge Tracing

Knowledge Tracing (BKT) is a common way of determining student knowledge of skills in adaptive educational systems and cognitive tutors. The basic BKT is a Hidden Markov Model (HMM) that models student knowledge based on five parameters: prior, learn rate, forget, guess, and slip. Expectation Maximization (EM) is often used to learn these parameters from training data. However, EM is a time-consuming process, and is prone to converging to erroneous, implausible local optima depending on the initial values of the BKT parameters. In this paper we address these two problems by using spectral learning to learn a Predictive State Representation (PSR) that represents the BKT HMM. We then use a heuristic to extract the BKT parameters from the learned PSR using basic matrix operations. The spectral learning method is based on an approximate factorization of the estimated covariance of windows from students' sequences of correct and incorrect responses; it is fast, local-optimum-free, and statistically consistent. In the past few years, spectral techniques have been used on real-world problems involving latent variables in dynamical systems, computer vision, and natural language processing. Our results suggest that the parameters learned by the spectral algorithm can replace the parameters learned by EM; the results of our study show that the spectral algorithm can improve knowledge tracing parameter- fitting time significantly while maintaining the same prediction accuracy, or help to improve accuracy while still keeping parameter-fitting time equivalent to EM.