The Passive Student Is Really Weaker (a Parametrization Scheme for Classifying Models'of Learnability)

Abslract We present a systematic framework: for classifying, comparing and defining models of computadonalleamabili1y. Apart from the obvious 'uniformity' parameters we present a novel 'passiveness' notion that captures the difference between 'Guess and Test' learning algorithms and leamability notions for which consistency with the samples guaranlU3 success. We analyze known models in terms of our new parameterization scheme and investigate the relative strength of notions of leamability that correspond to different parameter values. In the last secdon we consider 'proximity' between concept classes. We def. . notions of 'covering' one class by another and show that with respect to leamability they play a role similar to the role of reductions in computational complexity ; If a class is covcrable by a learnable one, then, the farst class too is leamable. We apply the covering technique to resolve some open questions raised in [BI2] and [LMR]. The discussion is carried out in the context of information-theoretic complexity rather than algorithmic complexity. For the sake of concreteness we have also limited the discussion to the setting of learning through randomIy.drawn samples. Nevertheless, our parameters, and consequently our analysis, are relevant to a much wider variety of leammg silUa-dons.