Georgia Institute of Technology College of Sciences

Part I (SAM as an Optimal Relaxation of Bayes) Dr. Thomas Moellenhoff

Sharpness-aware minimization (SAM) and related adversarial deep-learning methods can drastically improve generalization, but their underlying mechanisms are not yet fully understood. In this talk, I will show how SAM can be interpreted as optimizing a relaxation of the Bayes objective where the expected negative-loss is replaced by the optimal convex lower bound, obtained by using the so-called Fenchel biconjugate. The connection enables a new Adam-like extension of SAM to automatically obtain reasonable uncertainty estimates, while sometimes also improving its accuracy.

Part II (Lie Group updates for Learning Distributions on Machine Learning Parameters) Dr. Eren Mehmet Kıral

I will talk about our recent paper https://arxiv.org/abs/2303.04397 with Thomas Möllenhoff and Emtiyaz Khan, and other related results. Bayesian Learning learns a distribution over the model parameters, allowing for different descriptions of the same data. This is (contrary to classical learning which "bets-it-all" on a single set of parameters in describing a given dataset and making predictions. We focus on classes of distributions which have a transitive Lie group action on them given by pushforwards of an action on the parameter space. I will also specialize to a few concrete Lie groups and show distinct learning behavior.

A fundamental problem in analytic number theory is to calculate the maximal value of L-functions at a given point. For L-functions associated to quadratic Dirichlet characters at s = 1, the upper bounds and Ω-results of Littlewood differ by a factor of 2, and it is a long-standing (and still unsolved) problem to find out which one is closer to the truth. The important work of Granville and Soundararajan, who model the distribution of L(1, χ) by the distribution of random Euler products L(1, X) for random variables X(p) attached to each prime, shed more light to the question. We use similar techniques to study the distribution of L(1, χ) for cubic Dirichlet characters. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values. This is a joint work with P. Darbar, M. Lalin and A. Lumley.

A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$ Latin rectangle, where $k < (1/2 - \alpha)n$, contains a $\beta n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1 \pm \varepsilon}{n})^\ell$ for sufficiently large $n$ and sufficiently small $\beta$. Using this result, we prove that a uniformly random order-$n$ Latin square asymptotically almost surely has no Latin subsquare of order greater than $c\sqrt{n\log n}$ for an absolute constant $c$. This is joint work with Tom Kelly, Camille Kennedy, and Jasdeep Sidhu.