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[[Probability | Back to Probability Group]]


= Fall 2020 =
* '''When''': Thursdays at 2:30 pm
* '''Where''': 901 Van Vleck Hall
* '''Organizers''': Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
* '''To join the probability seminar mailing list:''' email probsem+subscribe@g-groups.wisc.edu.
* '''To subscribe seminar lunch announcements:''' email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu


<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.
[[Past Seminars]]
<b>We  usually end for questions at 3:20 PM.</b>
 
<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]


If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==


'''Pre-Talk: (1:00pm)'''


'''Neural Networks for Probabilists''' 
= Spring 2025 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.
We usually end for questions at 3:20 PM.


'''Talk: (2:30pm)'''
== January 23, 2025: ==
No seminar 


'''Effective Theory of Deep Neural Networks'''  
== January 30, 2025: Promit Ghosal (UChicago) ==
'''Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond'''


Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.
Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near-optimal rates with polynomial dimensional dependence and extend these results to bilinear-enhanced kernels.  


== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==
== February 6, 2025: Subhabrata Sen (Harvard) ==
TBD 


'''Some new perspectives on moments of random matrices'''
== February 13, 2025: ==
TBD 


The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.
== February 20, 2025: Mustafa Alper Gunes (Princeton) ==
TBD 


== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==
== February 27, 2025: Souvik Dhara (Purdue) ==
 
'''Roots of random polynomials near the unit circle'''
TBD
 
It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle.  Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei.  We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle.  Based on joint work with Julian Sahasrabudhe.
 
== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==
 
'''Large deviations for dense random graphs: beyond mean-field'''
 
In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.
 
In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.
 
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.
 
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 
Title: '''Concentration in integrable polymer models'''
 
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}).
 
Joint work with Christian Noack.
 
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==
 
Title: '''The heat and the landscape'''
 
Abstract: The directed landscape is the conjectured universal scaling limit of the
most common random planar metrics. Examples are planar first passage
percolation, directed last passage percolation, distances in percolation
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.
 
We show that the KPZ fixed point is characterized by the Baik Ben-Arous
Peche statistics well-known from random matrix theory.
 
This provides a general and elementary method for showing convergence to
the KPZ fixed point. We apply this method to two models related to
random heat flow: the O'Connell-Yor polymer and the KPZ equation.
 
Note: there will be a follow-up talk with details about the proofs at 11am, Friday, October 23.
 
==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==
 
Title: '''Operator level hard-to-soft transition for β-ensembles'''
 
Abstract: It was shown that the soft and hard edge scaling limits of β-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. By tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. In this talk, I will present the corresponding limit on the level of the operators. This talk is based on joint work with Laure Dumaz and Benedek Valkó.
 
== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==
 
Title: '''Persistence and root detection algorithms in growing networks'''
 
Abstract: Motivated by questions in Network Archaeology, we investigate statistics of dynamic networks
that are ''persistent'', that is, they fixate almost surely after some random time as the network grows. We
consider ''generalized attachment models'' of network growth where at each time $n$, an incoming vertex
attaches itself to the network through $m_n$ edges attached one-by-one to existing vertices with probability
proportional to an ''arbitrary function'' $f$ of their degree. We identify the class of attachment functions $f$ for
which the ''maximal degree vertex'' persists and obtain asymptotics for its index when it does not. We also
show that for tree networks, the ''centroid'' of the tree persists and use it to device polynomial time root
finding algorithms and quantify their efficacy. Our methods rely on an interplay between dynamic
random networks and their continuous time embeddings.
 
This is joint work with Shankar Bhamidi.
 
== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==
 
Title: '''A forward-backward SDE from the 2D nonlinear stochastic heat equation'''
 
Abstract: I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover results of Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).


== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
== March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy) ==
'''What conditional probability could (probably) be'''


Title: '''TBA'''
According to orthodox probability theory, when B has probability zero, the conditional probability of A given B can depend on the partition or sub-sigma-field that B is relativized to. This relativization to sub-sigma-fields, a hallmark of Kolmogorov's theory of conditional expectation, is traditionally seen as appropriate in a treatment of conditioning with continuous variables, and it is what allows the theory to preserve Total Disintegrability, a generalization of the Law of Total Probability to uncountable partitions. In this talk, I will argue that although the relativization of conditional probability to sub-sigma-fields has advantages, it also has an underrecognized cost: it leads to puzzles for the treatment of ''iterated conditioning''. I will discuss these puzzles and some possible implications for the foundations of conditional probability.


Abstract: TBA
This talk is based on joint work with Snow Zhang (UC Berkeley).


== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==
== March 13, 2025: Klara Courteaut (Courant) ==
TBD 


Title: '''TBA'''
== March 20, 2025: Ewain Gwynne (UChicago) ==
TBD 


Abstract: TBA
== March 27, 2025: SPRING BREAK ==
No seminar 


== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==
== April 3, 2025: Jimme He (OSU) ==
TBD 


Title: '''TBA'''
== April 10, 2025: Evan Sorensen (Columbia) ==
TBD 


Abstract: TBA
== April 17, 2025: ==
TBD 


== April 24, 2025: William Leep (University of Minnesota, Twin Cities) ==
TBD 


[[Past Seminars]]
== May 1, 2025: ==
No seminar

Latest revision as of 23:09, 22 January 2025

Back to Probability Group

  • When: Thursdays at 2:30 pm
  • Where: 901 Van Vleck Hall
  • Organizers: Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
  • To join the probability seminar mailing list: email probsem+subscribe@g-groups.wisc.edu.
  • To subscribe seminar lunch announcements: email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu

Past Seminars


Spring 2025

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

January 23, 2025:

No seminar

January 30, 2025: Promit Ghosal (UChicago)

Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond

Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near-optimal rates with polynomial dimensional dependence and extend these results to bilinear-enhanced kernels.

February 6, 2025: Subhabrata Sen (Harvard)

TBD

February 13, 2025:

TBD

February 20, 2025: Mustafa Alper Gunes (Princeton)

TBD

February 27, 2025: Souvik Dhara (Purdue)

TBD

March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy)

What conditional probability could (probably) be

According to orthodox probability theory, when B has probability zero, the conditional probability of A given B can depend on the partition or sub-sigma-field that B is relativized to. This relativization to sub-sigma-fields, a hallmark of Kolmogorov's theory of conditional expectation, is traditionally seen as appropriate in a treatment of conditioning with continuous variables, and it is what allows the theory to preserve Total Disintegrability, a generalization of the Law of Total Probability to uncountable partitions. In this talk, I will argue that although the relativization of conditional probability to sub-sigma-fields has advantages, it also has an underrecognized cost: it leads to puzzles for the treatment of iterated conditioning. I will discuss these puzzles and some possible implications for the foundations of conditional probability.

This talk is based on joint work with Snow Zhang (UC Berkeley).

March 13, 2025: Klara Courteaut (Courant)

TBD

March 20, 2025: Ewain Gwynne (UChicago)

TBD

March 27, 2025: SPRING BREAK

No seminar

April 3, 2025: Jimme He (OSU)

TBD

April 10, 2025: Evan Sorensen (Columbia)

TBD

April 17, 2025:

TBD

April 24, 2025: William Leep (University of Minnesota, Twin Cities)

TBD

May 1, 2025:

No seminar

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