Mathematics

The Graphon Mean Field Game equations consist of a collection of parameterized Hamilton-Jacobi-Bellman equations, and a collection of parameterized Fokker-Planck-Kolmogorov equations coupled through a given graphon. In this talk, we will discuss the sensitivity of the gradient of HJB solutions with respect to the coefficients, which can be used for the solvability of Graphon Mean Field Game equation. It's based on the joint work with Peter Caines, Daniel Ho, Minyi Huang, and Jiamin Jian, see https://arxiv.org/pdf/2009.12144.pdf.

Large-scale systems comprising of multiple subsystems connected over a network arise in a number of applications including power systems, traffic networks, communication networks, and some economic systems. A common feature of such systems is that the state evolutions and costs are coupled, i.e., the state evolution and local cost of one subsystems depend not only on its own state and control action, but also on the state and control actions of other subsystems in the network.

We consider the problem of designing control strategies for such systems when some of the parameters of the system model are not known. Due to the unknown parameters, the control problem is also a learning problem. Directly using existing reinforcement learning algorithms on such network coupled subsystems would incur $O(n^{1.5} \sqrt{T})$ regret over a horizon $T$, where the $n$ is the number of subsystems. This superlinear dependence on $n$ is prohibitive in large scale networked systems because the regret per subsystem (which is $O(\sqrt{nT})$ grows with the size of the network.

We consider networks where the dynamics coupling may be represented by a symmetric matrix (e.g., the adjacency or Laplacian matrix corresponding to a undirected weighted graph) and the cost coupling matrix have the same eigenvalues as the dynamics coupling. We use spectral decomposition of the coupling matrices to decompose the system into (L + 1) systems which are only coupled through the noise, where L is the rank of the coupling matrix. We show that, when the system model is known, the optimal control input at each subsystem can be computing by solving (L+1) decoupled Riccati equations.

Using this structure of the planning solution, we propose a new Thompson sampling algorithm and show that its regret is bounded by $O(n\sqrt{T})$, which increases linearly in the size of the network. We present numerical simulations to illustrate our results on mean-field control and general low-rank networks.

Joint work with Shuang Gao, Sagar Sudhakara Ashutosh Nayyar, and Ouyang Yi

Real data collected from different applications that have additional topological structures and connection information are amenable to be represented as a weighted graph.
Considering the node labeling problem, Graph Neural Networks (GNNs) is a powerful tool, which can mimic experts' decisions on node labeling.
GNNs combine node features, connection patterns, and graph structure by using a neural network to embed node information and pass it through edges in the graph.
We want to identify the patterns in the input data used by the GNN model to make a decision and examine if the model works as we desire.
However, due to the complex data representation and non-linear transformations, explaining decisions made by GNNs is challenging.
In this work, we propose new graph features' explanation methods to identify the informative components and important node features. Besides, we propose a pipeline to identify the key factors used for node classification. We use four datasets (two synthetic and two real) to validate our methods. Our results demonstrate that our explanation approach can mimic data patterns used for node classification by human interpretation and disentangle different features in the graphs. Furthermore, our explanation methods can be used for understanding data, debugging GNN models, and examine model decisions.

This work introduces a new N-player dynamic routing game that extend current Markovian traffic static assignment model.
It extends the N-player dynamic routing game to the corresponding mean field routing game, which models congestion in its dynamics.
Therefore, this new mean field routing game does not need to model congestion in the player cost function as done in the existing literature.
Both games are implemented in the open source library OpenSpiel.
The mean field game is used to solve the N-player dynamic game which leads to efficient computation of a approximate dynamic user equilibrium of the dynamic routing game.

The PIMS-UBC Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in Mathematics and related areas with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion.

Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?

Panelists:

Dan Coombs, Head of Mathematics, UBC

Pamela Harris, Associate Professor, Williams College, and Faculty Fellow of the Office of Institutional Diversity, Equity and Inclusion

Eugene Li, Chair of Mathematics and Statistics, Langara

Luis Serrano, Quantum AI Research Scientist at Zapata Computing, and ex-Google, ex-Apple