Probability Seminar: Difference between revisions
| (39 intermediate revisions by 6 users not shown) | |||
| Line 1: | Line 1: | ||
__NOTOC__ | __NOTOC__ | ||
[[Probability | Back to Probability Group]] | [[Probability | 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]] | [[Past Seminars]] | ||
= Spring | |||
= Spring 2025 = | |||
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> | <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> | ||
We usually end for questions at 3:20 PM. | We usually end for questions at 3:20 PM. | ||
== January | == 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) == | |||
== February | TBD | ||
== February 13, 2025: Hanbaek Lyu (UW-Madison) == | |||
'''Large random matrices with given margins''' | |||
We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization, Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such a margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the ''typical table''. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two potential functions, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$. We also propose a generalized Sinkhorn algorithm for computing typical tables and establish its linear convergence. We derive several new results for random contingency tables from our general framework. | |||
''' | |||
Based on a joint work with Sumit Mukherjee (Columbia) | |||
== February | == February 20, 2025: Mustafa Alper Gunes (Princeton) == | ||
TBD | |||
== February | == February 27, 2025: Souvik Dhara (Purdue) == | ||
''' | |||
'''Propagation of Shocks on Networks: Can Local Information Predict Survival?''' | |||
Abstract: Complex systems are often fragile, where minor disruptions can cascade into dramatic collapses. Epidemics serve as a prime example of this phenomenon, while the 2008 financial crisis highlights how a domino effect, originating from the small subprime mortgage sector, can trigger global repercussions. The mathematical theory underlying these phenomena is both elegant and foundational, profoundly shaping the field of Network Science since its inception. In this talk, I will present a unifying mathematical model for network fragility and cascading dynamics, and explore its deep connections to the theory of local-weak convergence, pioneered by Benjamini-Schramm and Aldous-Steele. | |||
== March | == 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 | == March 20, 2025: Ewain Gwynne (UChicago) == | ||
TBD | |||
== March | == March 27, 2025: SPRING BREAK == | ||
No seminar | |||
== April | == April 3, 2025: Jimme He (OSU) == | ||
TBD | |||
== April | == April 10, 2025: Evan Sorensen (Columbia) == | ||
TBD | |||
== April | == April 17, 2025: == | ||
TBD | |||
== April | == April 24, 2025: William Leep (University of Minnesota, Twin Cities) == | ||
TBD | |||
== May | == May 1, 2025: == | ||
No seminar | |||
Latest revision as of 07:27, 28 January 2025
- 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
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: Hanbaek Lyu (UW-Madison)
Large random matrices with given margins
We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization, Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such a margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the typical table. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two potential functions, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$. We also propose a generalized Sinkhorn algorithm for computing typical tables and establish its linear convergence. We derive several new results for random contingency tables from our general framework.
Based on a joint work with Sumit Mukherjee (Columbia)
February 20, 2025: Mustafa Alper Gunes (Princeton)
TBD
February 27, 2025: Souvik Dhara (Purdue)
Propagation of Shocks on Networks: Can Local Information Predict Survival?
Abstract: Complex systems are often fragile, where minor disruptions can cascade into dramatic collapses. Epidemics serve as a prime example of this phenomenon, while the 2008 financial crisis highlights how a domino effect, originating from the small subprime mortgage sector, can trigger global repercussions. The mathematical theory underlying these phenomena is both elegant and foundational, profoundly shaping the field of Network Science since its inception. In this talk, I will present a unifying mathematical model for network fragility and cascading dynamics, and explore its deep connections to the theory of local-weak convergence, pioneered by Benjamini-Schramm and Aldous-Steele.
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