Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as 10^{108} x 10^{108}. So far, a complete mathematical explanation for their success has proven elusive. The family of random sparsification methods has not yet been extended to the important case of linear system solves. This talk proposes a new algorithm based on repeated random sparsification that is capable of solving linear systems in extremely high dimensions and provides a complete mathematical analysis of the new algorithm. The analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution vector is too large to store.

Robust Statistics is a classical topic that dates back to the seminal work of Huber in the 1980s. In essence, the main goal of the field is to account for the effect of outliers when performing estimation tasks, such as mean estimation. A recent line of research, inspired by the seminal work of Catoni, has revisited some classical problems in robust statistics from a non-asymptotic perspective. The goal of this short seminar series is to introduce the key ideas related to robust estimation and discuss various notions of robustness, including heavy-tailed distributions and adversarial contamination. The primary example will be the mean estimation problem, and if time permits, I will also cover covariance estimation

Robust Statistics is a classical topic that dates back to the seminal work of Huber in the 1980s. In essence, the main goal of the field is to account for the effect of outliers when performing estimation tasks, such as mean estimation. A recent line of research, inspired by the seminal work of Catoni, has revisited some classical problems in robust statistics from a non-asymptotic perspective. The goal of this short seminar series is to introduce the key ideas related to robust estimation and discuss various notions of robustness, including heavy-tailed distributions and adversarial contamination. The primary example will be the mean estimation problem, and if time permits, I will also cover covariance estimation

The Ramsey number of graphs G1 and G2, the smallest N so that any red/blue coloring of the N-clique contains either a red G1 or a blue G2, is one of the most studied notions in combinatorics. We study a related process called the online ordered Ramsey game, played between two players, Builder and Painter. Builder has two graphs, G1 and G2, each of which has a specific ordering on its vertices. Builder starts with an edgeless graph on an ordered vertex set (the integers) and attempts to build either an ordered red copy of G1 or an ordered blue copy of G2 by adding one edge at a time. When Builder adds an edge, Painter is required to decide, at the time of creation, whether an edge is red or blue. Ramsey’s Theorem tells us that Builder can eventually win; their objective is to do so using the minimum number of turns, and Painter’s objective is to delay them as long as possible. The *online ordered Ramsey number* of G1 and G2 is the number of turns taken when both players play optimally.

Online ordered Ramsey numbers were introduced by Perez-Gimenez, P. Pralat, and West in 2021. In this talk we will discuss their relation to other types of Ramsey numbers and present some results on the case when at least one of G1,G2 is sparse.

(Joint work with Emily Heath, Grace McCourt, Hannah Sheats, and Justin Wisby)

I will introduce some recent progress towards understanding the scalability of Markov chain Monte Carlo (MCMC) methods and their comparative advantage with respect to variational inference. I will fact-check the folklore that "variational inference is fast but biased, MCMC is unbiased but slow". I will then discuss a combination of the two via reverse diffusion, which holds promise of solving some of the multi-modal problems. This talk will be motivated by the need for Bayesian computation in reinforcement learning problems as well as the differential privacy requirements that we face.