IMA & LMS Joint Conference on the Mathematical Foundations of AI

Artificial Intelligence is a rapidly evolving field that is already having a great societal impact thanks to the unprecedented performance of deep neural networks and AI generative models. The technological success of AI is boosting research on the mathematical foundations of AI which builds on an already rich history of important mathematical results and encompasses many areas, including physics inspired machine learning, statistical physics, Bayesian, deep, and reinforcement learning, among others.

The LMS/IMA Conference on The Mathematical Foundations of AI aims at bringing together top mathematicians and computer scientists working in both the academic and the private sectors to highlight key developments in the field. We foresee stimulating discussions and exchange of ideas on this exciting and fast-developing field that will contribute to changing the landscape of the field and to shape future research directions.

To register click here! 

Invited Speakers

David Saad – The Space of Functions Computed by Deep-learning Networks
Gitta Kutyniok – Reliable AI: Successes, Challenges, and Limitations
Lisa Kreusser – Differential equations for machine learning
Michael Bronstein – Physics-inspired graph neural networks
Petar Velickovic – Capturing Computation with Algorithmic Alignment

Abstracts

David Saad– The Space of Functions Computed by Deep-learning Networks
Recent engineering achievements of deep-learning machines have both impressed and intrigued the scientific community due to our limited theoretical understanding of the underlying reasons for their success. I will briefly review some of the challenges to be addressed and then focus on properties of the function space of different types of deeplearning machines, based on the generating functional analysis. This approach facilitates
studying the number of solution networks of a given error around a reference multi-layer network. Exploring the function landscape of densely-connected networks, we uncover a general layer-by-layer learning behaviour, while the study of sparsely-connected networks indicates the advantage in having more layers for increasing generalization ability in such models. This framework accommodates other network architectures and computing elements, including networks with correlated weights, convolutional networks and discretised variables. A similar approach also facilitates studying the distribution of Boolean functions computed by recurrent and layer-dependent architectures, which we find to be the
same. Depending on the initial conditions and computing elements used, we characterize the space of functions computed at the large depth limit and show that the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with the growing depth

Dr. Gitta Kutyniok – Reliable AI: Successes, Challenges, and Limitations
Artificial intelligence is currently leading to one breakthrough after the other, both in public life with, for instance, autonomous driving and speech recognition, and in the sciences in areas such as medical imaging or molecular dynamics. However, one current major drawback is the lack of reliability of such methodologies.

In this talk we will take a mathematical viewpoint towards this problem, showing the power of such approaches to reliability. We will first provide an introduction into this vibrant research area,
focussing specifically on deep neural networks. We will then survey recent advances, in particular, concerning generalization guarantees and explainability. Finally, we will discuss fundamental
limitations of deep neural networks and related approaches in terms of computability, which seriously affects their reliability.

Lisa Kreusser – Differential equations for machine learning.
Many computational methods for semi-supervised and unsupervised classification are based on variational models and PDEs. Since shortest path graph distances are widely used in data science and machine learning, it is natural to introduce the concept of information propagation to data classification and semi-supervised learning. The success of eikonal equations in the continuum setting motivates the development of similar tools on graphs. We propose and unify classes of different models for information propagation over graphs, and prove equivalences between them. Motivated by the connection between first arrival time model and the eikonal equation in the continuum setting, we derive mean field limits for graphs based on uniform grids in Euclidean space under grid refinement. For a specific parameter setting, we demonstrate that the solution on the grid approximates the Euclidean distance. Finally, we illustrate the use of front propagation on graphs to semi-supervised learning.

Petar Velickovic – Capturing Computation with Algorithmic Alignment 
Abstract: “What makes a neural network better, or worse, at fitting certain tasks? This question is arguably at the heart of neural network architecture design, and it is remarkably hard to answer rigorously. Over the past few years, there have been a plethora of attempts, using various facets of advanced mathematics, to answer this question under various assumptions. One of the most successful directions — algorithmic alignment — assumes that the target function, and a mechanism for computing it, are completely well-defined and known (i.e. the target is to learn to execute an algorithm). In this setting, fitting a task is equated to capturing the computations of an algorithm, inviting analyses from diverse branches of mathematics and computer science. I will present some of my personal favourite works in algorithmic alignment, along with their implications for building intelligent systems of the future.”

Michael Bronstein – Physics-inspired graph neural networks
The message-passing paradigm has been the “battle horse” of deep learning on graphs for several years, making graph neural networks a big success in a wide range of applications, from particle physics to protein design. From a theoretical viewpoint, it established the link to the Weisfeiler-Lehman hierarchy, allowing to analyse the expressive power of GNNs. We argue that the very “node-and-edge”-centric mindset of current graph deep learning schemes may hinder future progress in the field. As an alternative, we propose physics-inspired “continuous” learning models that open up a new trove of tools from the fields of differential geometry, algebraic topology, and differential equations so far largely unexplored in graph ML.

Committee

Andras Juhasz, Oxford
Ginestra Bianconi, QMUL
James Davenport , Bath

For further information or to register your interest, please contact the
Conferences Department: conferences@ima.org.uk

---

Related Posts

Interview with Andrew Oliver – Making video games

Interview with Professor Hannah Fry HonFREng FIMA

Artificial Intelligence in Aerospace Branch Event