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.

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.

Some very recent references

1. Francisco Malaca, Yves Lepage, Miguel Couceiro. Numerical analogies through generalized means:notion, properties and algorithmic approaches. Submitted.
2. Francisco Cunha, Yves Lepage, Zied Bouraoui, Miguel Couceiro. Generalizing Analogical Inference Across Boolean and Continuous Domains. Submitted.
3. Jakub Pillion, Miguel Couceiro, Yves Lepage. Analogical pooling for image reconstruction. Submitted.
4. Fadi Badra, Esteban Marquer, Marie-Jeanne Lesot, Miguel Couceiro, David Leake. EnergyCompress: A General Case Base Learning Strategy. To appear in IJCAI2025.
5. Yves Lepage, Miguel Couceiro. Any four real numbers are on all fours with analogy. CoRR abs/2407.18770 (2024)
6. Miguel Couceiro, Erkko Lehtonen. Galois theory for analogical classifiers. Ann. Math. Artif. Intell. 92(1): 29-47 (2024)
7. Pierre Monnin, Cherif-Hassan Nousradine, Lucas Jarnac, Laurel Zuckerman, Miguel Couceiro. KGPRUNE: A Web Application to Extract Subgraphs of Interest from Wikidata with Analogical Pruning. ECAI 2024: 4495-4498
8. Yves Lepage, Miguel Couceiro. Analogie et moyenne généralisée. JIAF-JFPDA 2024: 114-124
9. Lucas Jarnac, Miguel Couceiro, Pierre Monnin. Relevant Entity Selection: Knowledge Graph Bootstrapping via Zero-Shot Analogical Pruning. CIKM 2023: 934-944
10. N. Kumar, and S. Schockaert. Solving hard analogy questions with relation embedding chains. EMNLP 2023, 6224–6236. ACL

In this talk, I will discuss applications of deep neural networks as approximators. I will demonstrate an implementation of Gaussian processes for predicting baryon operator masses based on the meson spectrum of QCD, inspired by an idea from Witten. I will compare these results with those obtained from neural networks with finite width and depth. The second part of the talk will focus on using Physics-Informed Neural Networks (PINNs) to solve the Monge-Ampère equation on a Calabi-Yau manifold, including a comparison with approaches like Donaldson's algorithm.

In this talk, I will discuss applications of deep neural networks as approximators. I will demonstrate an implementation of Gaussian processes for predicting baryon operator masses based on the meson spectrum of QCD, inspired by an idea from Witten. I will compare these results with those obtained from neural networks with finite width and depth. The second part of the talk will focus on using Physics-Informed Neural Networks (PINNs) to solve the Monge-Ampère equation on a Calabi-Yau manifold, including a comparison with approaches like Donaldson's algorithm.

I will offer an introductory exploration into the field of Reinforcement Learning (RL) with a focus on Markov Decision Processes (MDPs). The first session provides a foundational understanding of RL, covering key concepts such as agents, environments, rewards, and actions. It explains the RL problem framework and introduces MDPs, exploring their role as the mathematical framework underpinning RL.

The second session delves into core algorithms, including Q-learning and policy gradients. The lecture highlights the connection between MDPs and dynamic programming techniques, emphasizing policy iteration and value iteration. Time allowing, I will finalize with a brief description of some recent research topics and results.