May 11. Soumik Pal (University of Washington)

Title: Scaling limit of stochastic optimization over large networks

Abstract: Wasserstein gradient flows often arise from mean-field interactions of exchangeable particles. In many interesting applications however, the "particles" are edge weights in a graph whose vertex labels are exchangeable but not the edges themselves. We investigate the optimization of functions over this class of symmetries. Popular applications include training of large computational graphs like Neural Networks and commonly used Markov chain algorithms such Metropolis Hastings. We show that discrete noisy stochastic optimization algorithms over finite graphs have a well-defined analytical scaling limit as the size of the network grows to infinity. The limiting space is that of graphons, a notion introduced by Lovász and Szegedy to describe limits of dense graph sequences. The limiting curves are given by a novel notion of McKean-Vlasov equation on graphons and a propagation of chaos phenomenon, predicted by this theory, can be observed to hold. In the asymptotically zero-noise case, the limit is a L-2 gradient flow on the metric space of graphons.

May 12.  Sumit Mukherjee (Columbia University)

Title: Persistence of weighted sums of GSPs

Abstract: With $\{\xi_i\}_{i\ge 0}$ a centered discrete time stationary Gaussian stochastic process (GSP) with non-negative correlation function $\rho(i)$ and $\{\sigma(i)\}_{i\ge 1}$ a sequence of positive reals, we study the persistence probability of the weighted sum $\sum_{i=1}^n\sigma(i)\xi_i$. For summable correlations $\rho$, we show that the persistence exponent is universal. On the contrary, for non-summable $\rho$, even for polynomial weight functions $\sigma(i)\sim i^p$ the persistence exponent depends on $p$, and on the rate of decay of correlations. In this case, we show the existence of a persistence exponent, and study some of its properties.

Joint work with Frank Aurzada, University of Darmstadt.