When Learners Surpass their Sources: Mathematical Modeling of Learning from an Inconsistent Source

We present a new algorithm to model and investigate the learning process of a learner mastering a set of grammatical rules from an inconsistent source. The compelling interest of human language acquisition is that the learning succeeds in virtually every case, despite the fact that the input data are formally inadequate to explain the success of learning. Our model explains how a learner can successfully learn from or even surpass its imperfect source without possessing any additional biases or constraints about the types of patterns that exist in the language. We use the data collected by Singleton and Newport (2004) on the performance of a 7-year boy Simon, who mastered the American Sign Language (ASL) by learning it from his parents, both of whom were imperfect speakers of ASL. We show that the algorithm possesses a frequency-boosting property, whereby the frequency of the most common form of the source is increased by the learner. We also explain several key features of Simon's ASL.

Mathematics of learning

We study the convergence properties of a pair of learning algorithms (learning with and without memory). This leads us to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots uniformly at random in the interval [0, 1], although our results extend to other distributions). This, in turn, requires the study of the statistical behavior of the harmonic mean of random variables as above, which leads us to delicate question of the rate of convergence to stable laws and tail estimates for stable laws. The reader can find the proofs of most of the results announced here in the paper entitled "Harmonic mean, random polynomials, and random matrices", by the same authors.

Harmonic mean, random polynomials and stochastic matrices

Motivated by a problem in learning theory, we are led to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots uniformly at random in the interval [0, 1], although our results extend to other distributions). This, in turn, requires the study of the statistical behavior of the harmonic mean of random variables as above, and that, in turn, leads us to delicate question of the rate of convergence to stable laws and tail estimates for stable laws.