School of Mathematics

Geodesically Convex Optimization (or, can we prove P!=NP using gradient descent)

Avi Wigderson
Herbert H. Maass Professor, School of Mathematics
April 21, 2020
This talk aims to summarize a project I was involved in during the past 5 years, with the hope of explaining our most complete understanding so far, as well as challenges and open problems. The main messages of this project are summarized below; I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to them. No special background is assumed.

A variational approach to the regularity theory for the Monge-Ampère equation

Felix Otto
Max Planck Institute Leipzig
April 20, 2020
We present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced with M. Goldman. Following De Giorgi’s philosophy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, which leads to a One-Step Improvement Lemma, and feeds into a Campanato iteration on the C1,α-level for the displacement, capitalizing on affine invariance.

Equivariant quantum operations and relations between them

Nicholas Wilkins
University of Bristol
April 17, 2020
There is growing interest in looking at operations on quantum cohomology that take into account symmetries in the holomorphic spheres (such as the quantum Steenrod powers, using a Z/p-symmetry). In order to prove relations between them, one needs to generalise this to include equivariant operations with more marked points, varying domains and different symmetry groups. We will look at the general method of construction of these operations, as well as two distinct examples of relations between them.

Local-global compatibility in the crystalline case

Ana Caraiani
Imperial College
April 16, 2020
Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GL_n/F with p-torsion coefficients. These Galois representations are expected to satisfy local-global compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems. I will discuss joint work with J.

Do Simpler Models Exist and How Can We Find Them?

Cynthia Rudin
April 16, 2020
While the trend in machine learning has tended towards more complex hypothesis spaces, it is not clear that this extra complexity is always necessary or helpful for many domains. In particular, models and their predictions are often made easier to understand by adding interpretability constraints. These constraints shrink the hypothesis space; that is, they make the model simpler. Statistical learning theory suggests that generalization may be improved as a result as well. However, adding extra constraints can make optimization (exponentially) harder.

Deep equilibrium models via monotone operators

Zico Kolter
April 16, 2020
In this talk, I will first introduce our recent work on the Deep Equilibrium Model (DEQ). Instead of stacking nonlinear layers, as is common in deep learning, this approach finds the equilibrium point of the repeated iteration of a single non-linear layer, then backpropagates through the layer directly using the implicit function theorem. The resulting method achieves or matches state of the art performance in many domains (while consuming much less memory), and can theoretically express any "traditional" deep network with just a single layer.

The Peculiar Optimization and Regularization Challenges in Multi-Task Learning and Meta-Learning

Chelsea Finn
April 16, 2020
Despite the success of deep learning, much of its success has existed in settings where the goal is to learn one, single-purpose function from data. However, in many contexts, we hope to optimize neural networks for multiple, distinct tasks (i.e. multi-task learning), and optimize so that what is learned from these tasks is transferable to the acquisition of new tasks (e.g. as in meta-learning).