Seminars

Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct when close to affine. These results enabled the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains.

In this series of seminars we will revisit analogical inference from a foundational perspective. After a brief motivation, we will first present a recently proposed formalism to model numerical analogies that relies on p-generalized means, and that enables a unifying framework that subsume the classical notions of arithmetic, geometric and harmonic analogies. We will derive several interesting properties such as transitivity of conformity, as well as present algorithmic approaches to detect and compute the parameter p.

In the second part of this series, we will leverage this unified formalism and lift analogical reasoning to real-valued domains and various ML&AI downstream tasks. In particular, we will see that it supports analogical inference over continuous functions, and thus both classification and regression tasks. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. If time allows, we will also discuss further applications, e.g., on image reconstruction and NLP downstream tasks.

These two seminars are based on several published and recently submitted by Miguel Couceiro and his collaborators, including Francisco Malaca and Francisco Vincente Cunha, respectively, graduate and undergraduate students at the DM@IST.