Modeling the Relationship between Input Distributions and Learning Trajectories with the Tolerance Principle (2022)

Abstract: Child language learners develop with remarkable uniformity, both in their learning trajectories and ultimate outcomes, despite major differences in their learning environments. In this paper, we explore the role that the frequencies and distributions of irregular lexical items in the input plays in driving learning trajectories. We conclude that while the Tolerance Principle, a type-based model of productivity learning, accounts for inter-learner uniformity, it also interacts with input distributions to drive cross-linguistic variation in learning trajectories.

## introduction one of the most striking characteristics of child language acquisition is its uniformity @xcite . children in the same speech community acquire the same grammars despite the lexical variation in each child's individual input: a recent quantitative study of child-directed speech (cds) finds jaccard similarities of only 0.25-0.37 between individual portions of the providence corpus @xcite , not much higher than the lexical similarity between cds and adult genres @xcite . thus, to explain uniformity of outcomes, grammar learning must not depend primarily on lexical identity but on more general patterns in the learner's input. learners not only acquire the essentially same grammars but acquire them following similar trajectories. for example, english learners consistently acquire the verbal -s and -ing before the past -ed @xcite , the last of which shows a u-shaped developmental regression @xcite . individuals may show relative delays correlating to estimated working vocabulary size @xcite , but variability is otherwise limited. however, while individuals learning the same pattern show uniformity, expected learning paths vary across patterns. among english learners, for example, -ing does not show u-shaped learning, unlike -ed. children learning spanish verb stem alternations also show u-shaped learning, but they begin to over-regularize a year earlier than english past tense learners @xcite . one potential reason for this, differences in patterns' distributions in the input, is investigated here. this paper introduces a quantitative means of assessing the role that the distribution of linguistic patterns in learner input plays in shaping learning trajectories and variation even prior to the grammar and individual cognitive factors. adopting the tolerance principle (tp; yang, 2016) as a type-based model of productivity learning, we find that the type-frequency and (indirectly) token frequency of exceptions to linguistic patterns have a dramatic effect on the expected learning trajectories across patterns while also quantifying expected uniformity across individuals within a given pattern. ## the learning model the tolerance principle (tp; @xcite is a cognitively motivated type-based learning model which casts generalization in terms of productivity in the face of exceptions. the model has gained support in recent years through its successful application to problems in syntax and semantics (e.g., @xcite @xcite , morphology (e.g., @xcite @xcite , and phonology (e.g., @xcite @xcite . it has increasingly received backing from a range of psycholinguistic experiments @xcite @xcite . it is adopted here because it makes categorical and auditable predictions about productivity and thus provides a clear means for investigating and the relationship between distributions in the input and the dynamics of learning. the tp serves as a decision procedure for the learner. once the learner hypothesizes a generalization in the grammar, it establishes the threshold θ n at which it becomes more economical in terms of lexical access time to accept the hypothesis and exceptions rather than to just memorize items individually. (1) formalizes the tp. the tolerance threshold θ n is defined as the number of known types that a generalization should apply to divided by its natural logarithm. 1 (1) tolerance principle @xcite : if r is a productive rule applicable to n candidates, then the following relation holds between n and e, the number of exceptions that could but do not follow r: the derivation of the tp acknowledges that items in the input follow long-tailed zipfian frequency distributions @xcite in which few items are well-attested and others are rarely attested in the input. zipfian and other long-tailed distributions are quite common throughout language and are very prominent in lexical and inflectional frequencies (e.g., @xcite @xcite @xcite figure 1 provides a visualization of the tolerance principle over individual development. crucially, n depends on a learner's current working vocabulary and is not a comment on the language's vocabulary in general. an individual learner's n and e increase as they learn more vocabulary, and a pattern may fall in and out of productivity. ## input distributions driving trajectories this section uses the tolerance principle to calculate likely learning trajectories and variability in learning trajectories given distributions of regular and irregular forms in the input, and it discusses the impact that input distributions have on learning paths. it presents two illustrative examples and a case study from english past tense learning. for clarity, n tgt and e tgt are used here to represent the expected mature learner state, since n and e properly represent speaker-internal quantities and are not a description of the target language.