December CDT in Mathematics of Random Systems Seminars

Adaptive agents through active inference: The main fields of research that are used to model and realise adaptive agents are optimal control, reinforcement learning and active inference. Active inference is a probabilistic description of adaptive agents that is relatively less known to mathematicians, as it originated from neuroscience in the last decade. This talk presents the mathematical underpinnings of active inference, starting from fundamental considerations about agents that maintain their structural integrity in the face of environmental perturbations. Through this, we derive a probability distribution over actions, that describes decision-making under uncertainty in adaptive agents . Interestingly, this distribution has an interesting information geometric structure, combining, for instance, drives for exploration and exploitation, which may yield a principled answer to the exploration-exploitation trade-off. Preserving this geometric structure enables to realise adaptive agents in practice. We illustrate their behaviour with simulation examples and empirical comparisons with reinforcement learning.

Scaling Limits of Random Graphs: The scaling limit of directed random graphs remains relatively unexplored compared to their undirected counterparts. In contrast, many real-world networks, such as links on the world wide web, financial transactions and “follows” on Twitter, are inherently directed. Previous work by Goldschmidt and Stephenson established the scaling limit for the strongly connected components (SCCs) of the Erdős -- Rényi model in the critical window when appropriately rescaled. In this talk, we present a result showing the SCCs of another class of critical random directed graphs will converge when rescaled to the same limit. Central to the proof is an exploration of the directed graph and subsequent encodings of the exploration as real valued random processes. We aim to present this exploration algorithm and other key components of the proof.

From Mathematics to Data Science and Back: We give an overview of the interaction between rough path theory and data science at the current time.