Convergence Bounds for Language Evolution by Iterated Learning

Convergence Bounds for Language Evolution by Iterated Learning Anna N. Rafferty (rafferty@cs.berkeley.edu) Computer Science Division, University of California, Berkeley, CA 94720 USA Thomas L. Griffiths (tom griffiths@berkeley.edu) Department of Psychology, University of California, Berkeley, CA 94720 USA Dan Klein (klein@cs.berkeley.edu) Computer Science Division, University of California, Berkeley, CA 94720 USA Abstract Similarities between human languages are often taken as ev- idence of constraints on language learning. However, such similarities could also be the result of descent from a com- mon ancestor. In the framework of iterated learning, language evolution converges to an equilibrium that is independent of its starting point, with the effect of shared ancestry decaying over time. Therefore, the central question is the rate of this conver- gence, which we formally analyze here. We show that conver- gence occurs in a number of generations that is O(n log n) for Bayesian learning of the ranking of n constraints or the values of n binary parameters. We also present simulations confirm- ing this result and indicating how convergence is affected by the entropy of the prior distribution over languages. Introduction Human languages share a surprising number of properties, ranging from high level characteristics like compositional mapping between sound and meaning to relatively low- level syntactic regularities (Comrie, 1981; Greenberg, 1963; Hawkins, 1988). One explanation for these universal proper- ties is that they reflect constraints on human language learn- ing, with the mechanisms by which we acquire language be- ing restricted to languages with these properties (e.g., Chom- sky, 1965). However, if all modern languages are descended from a common ancestor, these similarities could just reflect the properties of that ancestor. Evaluating these different pos- sibilities requires establishing how constraints on learning in- fluence the properties of languages, and how long it takes for this process to remove the influence of a common ancestor. In this paper, we explore these questions using a simple model of language evolution. We model language evolution as a process of iterated learning (Kirby, 2001). This model assumes that each gen- eration of people learns language from utterances generated by the previous generation. While this model makes certain simplifying assumptions, such as a lack of interaction be- tween learners in the same generation, it has the advantage that it can be analyzed mathematically. Previous research has shown that after some number of generations, the distribution over languages produced by learners converges to an equilib- rium that reflects the constraints that guide learning (Griffiths & Kalish, 2007). After convergence, the behavior of learners is independent of the language spoken by the first generation. These results provide a way to relate constraints on learn- ing to linguistic universals. However, convergence to the equilibrium has to occur in order for these constraints to be the sole factor influencing the languages learners acquire. Our key contribution is providing bounds on the number of generations required for convergence, known as the conver- gence time, which we obtain by analyzing Markov chains as- sociated with iterated learning. Bounding the convergence time is a step towards understanding the source of linguistic universals: If convergence occurs in relatively few genera- tions, it suggests constraints on learning are more likely than common descent to be responsible for linguistic universals. To bound the number of generations required for iterated learning to converge, we need to make some assumptions about the algorithms and representations used by learners. Following previous analyses (Griffiths & Kalish, 2007), we assume that learners update their beliefs about the plausibil- ity of a set of linguistic hypotheses using Bayesian inference. We outline how this approach can be applied using two kinds of hypothesis spaces that appear in prominent formal linguis- tic theories: constraint rankings, as used in Optimality Theory (Prince & Smolensky, 2004), and vectors of binary parame- ter values, consistent with a simple Principles and Parameters model (Chomsky & Lasnik, 1993). In each case, we show that iterated learning with a uniform prior reaches equilib- rium after O(n log n) generations, where n is the number of constraints or parameters. Analyzing Iterated Learning Iterated learning has been used to model a variety of aspects of language evolution, providing a simple way to explore the effects of cultural transmission on the structure of languages (Kirby, 2001; Smith, Kirby, & Brighton, 2003). The basic assumption behind the model – that each learner learns from somebody who was themselves a learner – captures a phe- nomenon we see in nature: Parents pass on language to their children, and these children in turn pass on language to their own children. The sounds that the children hear are the in- put, and the child produces language (creates output) based on this input, as well as prior constraints on the form of the language. Formally, we conceptualize iterated learning as follows (see Figure 1). A first learner receives data, forms a hypothe- sis about the process that generated these data, and then pro- duces output based on this hypothesis. A second learner re- ceives the output of the first learner as data and produces a new output that is in turn provided as data to a third learner.