Leda Wang

Can we understand (with some mathematical rigorosity) the dynamics of learning algorithms used in modern machine learning, for different data structure/architectures/optimizers? What is the actual mathematical explanation for the success of deep learning, and what are the computational constraints on learning? Which aspects of the microscopic system remain relevant at the macroscopic scale (even for the simplest models in spin glasses and disordered systems), and what features are learnt by deep neural networks? (To be hotter, what is the mathematically interpretable mechanism of modern deep learning?)

Surprisingly, many complicated high-dimensional dynamics may admit much simpler descriptions, which is intrinsically very close to the goal of statistical physics. Tools like dynamical mean-field theory originating in statistical physics provide a unified route toward reliable descriptions of dynamics arising from both deep learning and high-dimensional probability tasks.

How should we sample from a high-dimensional distribution under realistic statistical and computational constraints? What mathematical guarantees show that Langevin dynamics and score-based/flow-based/transport-based generative models work and outperform in practice? What are the stochastic and deterministic dynamics for sampling and generative modeling approaches?

What feature of the landscape/optimization geometry/algorithmic dynamics is responsible for the computational obstruction in learning tasks or broader statistical models?