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David Belanger

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I&C Journal 2023 Journal Article

Learnability and positive equivalence relations

David Belanger
Ziyuan Gao
Sanjay Jain
Wei Li
Frank Stephan

Prior work of Gavryushkin, Khoussainov, Jain and Stephan investigated what algebraic structures can be realised in worlds given by a positive (= recursively enumerable) equivalence relation which partitions the natural numbers into infinitely many equivalence classes. The present work investigates the infinite one-one numbered recursively enumerable (r. e.) families realised by such relations and asks how the choice of the equivalence relation impacts the learnability properties of these classes when studying learnability in the limit from positive examples, also known as learning from text. For all choices of such positive equivalence relations, for each of the following entries, there are one-one numbered r. e. families which satisfy it: (a) they are behaviourally correctly learnable but not vacillatorily learnable; (b) they are explanatorily learnable but not confidently learnable; (c) they are not behaviourally correctly learnable. Furthermore, there is a positive equivalence relation which enforces that (d) every vacillatorily learnable one-one numbered family of languages closed under this equivalence relation is already explanatorily learnable and cannot be confidently learnable. In addition, if there is for a positive equivalence relation a confidently learnable class, then there is also a class which is confidently but not finitely learnable.

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NeurIPS Conference 2012 Conference Paper

MAP Inference in Chains using Column Generation

David Belanger
Alexandre Passos
Sebastian Riedel
Andrew McCallum

Linear chains and trees are basic building blocks in many applications of graphical models. Although exact inference in these models can be performed by dynamic programming, this computation can still be prohibitively expensive with non-trivial target variable domain sizes due to the quadratic dependence on this size. Standard message-passing algorithms for these problems are inefficient because they compute scores on hypotheses for which there is strong negative local evidence. For this reason there has been significant previous interest in beam search and its variants; however, these methods provide only approximate inference. This paper presents new efficient exact inference algorithms based on the combination of it column generation and pre-computed bounds on the model's cost structure. Improving worst-case performance is impossible. However, our method substantially speeds real-world, typical-case inference in chains and trees. Experiments show our method to be twice as fast as exact Viterbi for Wall Street Journal part-of-speech tagging and over thirteen times faster for a joint part-of-speed and named-entity-recognition task. Our algorithm is also extendable to new techniques for approximate inference, to faster two-best inference, and new opportunities for connections between inference and learning.

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