Variations on U-Shaped Learning

Abstract

The paper deals with the following problem: is returning to wrong conjectures necessary to achieve full power of algorithmic learning? Returning to wrong conjectures complements the paradigm of U-shaped learning [3,7,9,24,29] when a learner returns to old correct conjectures. We explore our problem for classical models of learning in the limit from positive data: explanatory learning (when a learner stabilizes in the limit on a correct grammar) and behaviourally correct learning (when a learner stabilizes in the limit on a sequence of correct grammars representing the target concept). In both cases we show that returning to wrong conjectures is necessary to achieve full learning power. In contrast, one can modify learners (without losing learning power) such that they never show inverted U-shaped learning behaviour, that is, never return to old wrong conjecture with a correct conjecture in-between. Furthermore, one can also modify a learner (without losing learning power) such that it does not return to old "overinclusive" conjectures containing non-elements of the target language. We also consider our problem in the context of vacillatory learning (when a learner stabilizes on a finite number of correct grammars) and show that each of the following four constraints is restrictive (that is, reduces learning power): the learner does not return to old wrong conjectures; the learner is not inverted U-shaped; the learner does not return to old overinclusive conjectures; the learner does not return to old overgeneralizing conjectures. We also show that learners that are consistent with the input seen so far can be made decisive [3,25]: on any text, they do not return to any old conjectures -wrong or right.

On the Necessity of U‐Shaped Learning

“…Furthermore, showing that U‐shapes are unnecessary for iterative learning of classes consisting of infinite languages turned out to be significantly easier than obtaining the result for arbitrary classes. In the context of Explanatory Learning, if a class does not contain an extension of every finite set , then that class can be learned by a decisive learner (Carlucci et al., )!…”

“…In Carlucci et al. (), a number of variants of nonmonotonic learning criteria have been investigated in the context of Gold's model. In particular, the following restrictions on the learner's behavior have been studied: (1) no return to previously abandoned wrong hypotheses, (2) no return to overinclusive hypotheses, (3) no return to overgeneralizing hypotheses, and (4) no inverted U‐shapes.…”

“…Herein we pursue, nonetheless for potential interesting insight into this problem, a mathematically precise version of the following question: Are there some formal learning tasks for which U‐shaped behavior is logically necessary? We discuss a large and growing body of results (Baliga, Case, Merkle, Stephan, & Wiehagen, ; Case & Moelius, ; Case & Kötzing, , ; Carlucci, Jain, Kinber, & Stephan, ; Carlucci, Case, Jain, & Stephan, ; Carlucci, Case, Jain, & Stephan, ) in the context of (extensions of) Gold's formal model of language learning from positive data (Gold, ) that suggest an answer to this latter question . In addition, the proofs of some of our results intriguingly may begin to shed light on the role played by exceptions on the one hand and rule‐governed behavior on the other in U‐shaped learning as discussed in Section .…”

Learning in Friedberg numberings

“…A learner is said to be U-shaped on L (see [3,7,8]), if on some text T for L, for some n, m, k with n < m < k, M (T [n]) and M (T [k]) are grammars for L (in the numbering being used as hypotheses space), but M (T [m]) is not a grammar for L. A learner is said to be non U-shaped on L if it is not Ushaped on L. A learner NUShI-identifies a class L if it I-identifies L and is non U-shaped on each L ∈ L.…”

“…." [3,7,8]. We denote the criteria of prudent, confident, consistent and non U-shaped learning with PrudentTxtEx, ConfTxtEx, ConsTxtEx and NUShTxtEx, respectively; accordingly for restricted variants.…”