Job offer

Applications are invited for a full-time, 36-month PhD position in theoretical machine learning. The successful candidate will work on developing a new theoretical framework to understand feature learning and parameter dynamics in artificial neural networks.

This position is funded through a French National Research Agency (ANR) grant.

Project Goal & Research Objectives

The field of deep learning has grown at an unprecedented pace. Neural network models have become indispensable tools for applications in vision, language and across all scientific domains. As reliance on AI continues to increase, there is a growing urgency to understand how these models achieve such remarkable performance. Yet, the mathematical intractability of neural networks makes this one of the deepest challenges of modern machine learning.

Seminal work has identified two canonical learning regimes: networks may either learn low-dimensional features of their input data (the so-called “rich” learning regime), or random high-dimensional projections (the “lazy” regime). Importantly, rich learning has been shown to yield networks that better generalise to unseen data, leading to significant research efforts from the machine learning community towards finding the precise conditions under which networks exhibit such feature learning.

Adding nuance to this view, recent numerical experiments have suggested that the rich-lazy dichotomy may not characterise all possible learning regimes. For example, under certain conditions, networks can learn representations that are high-dimensional yet not fully random. Importantly, these new findings suggest that there may be other, yet unknown, learning regimes that could lead to fundamentally different network solutions.

This PhD will be targeted towards developing new mathematical tools embedded in Riemannian and information geometry to study feature learning.

Methodology & Key Tasks

The PhD project will combine theoretical analysis and computational modelling, focusing primarily on:

• Using Riemannian geometric tools to characterise feature learning
• Studying the information geometry of generative models
• Using this framework to promote feature learning in neural network models used for NeuroAI and broader biological data applications

Research Environment & Supervision

• Host Institution: The PhD candidate will be physically hosted at the Centre Sciences des Données (CSD) at École Normale Supérieure (ENS - PSL), Rue d’Ulm, Paris. The CSD provides a vibrant, world-class theoretical research environment bringing together experts across computer science, physics, and mathematics, with dedicated centre-wide access to high-performance computing resources (Jean-Zay supercomputer).
• Supervision:The PhD candidate will be supervised by Arthur Pellegrino (PI, AI Fellow at ENS/PR[AI]RIE) and co-supervised by Bruno Loureiro (HDR, CNRS / DI-ENS).
• Mentorship & Mobility: The candidate will benefit from a highly supportive supervision environment, including regular group seminars, opportunities to co-mentor Master’s students, dedicated funding to present research at top ML conferences (NeurIPS, ICML, COLT), and access to international collaborative networks.

• Degree: Master’s degree (or equivalent) in Theoretical Machine Learning, Mathematics, Physics, Computer Science, or a related quantitative field.
• Mathematical prerequisites: Strong foundations in linear algebra, multivariable calculus, and probability theory (familiarity with differential geometry or dynamical systems theory is a strong plus).
• Hard skills: Proficiency in Python and modern deep learning frameworks (PyTorch, JAX).
• Soft skills:High motivation to conduct research in both the mathematical foundations of AI and its applications., Research Field Mathematics » Applied mathematics

Education Level Master Degree or equivalent

Research Field Computer science » Informatics

Education Level Master Degree or equivalent, * Degree: Master’s degree (or equivalent) in Theoretical Machine Learning, Mathematics, Physics, Computer Science, or a related quantitative field.

• Mathematical prerequisites: Strong foundations in linear algebra, multivariable calculus, and probability theory (familiarity with differential geometry or dynamical systems theory is a strong plus).
• Technical skills: Proficiency in modern deep learning frameworks (PyTorch, JAX).
• Rsearch skills:High motivation to conduct foundational research in AI theory.

Years of Research Experience 1 - 4