Quantitative Methods Seminar

Abstract: We study the best way (according to appropriate criteria) to select a sample from a given population, when the selection is influenced by two antagonistic parties. Three examples are the selection of representative cases in multi-district litigations, the selection of jurors, and the selection of house committees.

Bio: Eilon Solan is the Dr. Irene Halmos Chair of Game Theory at the School of Mathematical Sciences in Tel-Aviv University, Israel, and the director of the Good To Know project.

Abstract: The long-standing four-thirds conjecture states that the standard linear programming (LP) relaxation of the traveling salesman problem (TSP) has integrality gap at most 4/3. It remains open even for half-integral instances, whose LP solutions take values in {0, 1/2, 1} and which are conjectured to capture the hardest cases. The best-known bound in this setting is 1.4983, due to Gupta et al. We identify a large class of half-integral instances for which the four-thirds conjecture holds. Previous approaches use induction on a hierarchy of critical tight sets in the LP solution’s support graph, classified as “cycle cuts” or “degree cuts.” We show that when this hierarchy consists entirely of cycle cuts, we can construct a distribution of tours with expected cost at most 4/3 times the LP value, yielding a randomized 4/3-approximation algorithm. This class includes known examples with integrality gap 4/3, so our bound is tight.

Bio: Billy Jin is an Assistant Professor in Quantitative Methods at Purdue. Before joining Purdue, he was a postdoctoral researcher with Baris Ata at the University of Chicago. He obtained his PhD in Operations Research at Cornell University, where he was advised by David Williamson. His research interests lie in the union of online decision-making, stochastic optimization, and approximation algorithms. His research has been recognized with several awards, including an NSERC graduate fellowship, the 2023 Student Paper Prize awarded by the INFORMS Decision Analysis Society, and runner-up in the 2023 Student Paper Prize awarded by the INFORMS Optimization Society. 

Abstract: Modern LLM evaluation increasingly relies on sparse and heterogeneous pairwise human preferences, making reliable estimation and uncertainty quantification challenging. In this talk, I will present a low-rank statistical framework that shares information across related tasks, models, and evaluation settings while preserving task-specific heterogeneity. I will first formulate LLM evaluation as a structured tensor-completion problem under pairwise comparisons and develop semiparametrically efficient inference for model abilities, score gaps, and other smooth functionals. A key technical ingredient is a score-whitening approach that handles the anisotropic Fisher information induced by non-uniform comparisons. I will then turn from score estimation to task-specific ranking, where ranks and top-K membership are nonsmooth functions of many dependent score gaps. We develop entrywise-accurate low-rank estimation, efficient debiased inference, and multiplier-bootstrap procedures that provide simultaneous rank intervals and uncertainty-aware top-K certificates. Together, these results provide statistical foundations for moving from sparse human feedback to reliable, task-specific LLM evaluation and ranking.

Bio: Will Wei Sun is a Professor of Quantitative Methods and University Faculty Scholar at Purdue University’s Daniels School of Business, with a courtesy appointment in Statistics. He is also a part-time Research Scientist at Meta. His research focuses on the statistical foundations of large language models and trustworthy reinforcement learning. His research has been supported by the National Science Foundation, the Office of Naval Research, and the Ross-Lynn Research Scholar Fund. Dr. Sun is an Associate Editor for the Annals of Applied Statistics and currently serves as Chair of the AI in Statistics initiative at the National Institute of Statistical Sciences (NISS).

Abstract: We study the unconstrained minimization of a smooth and strongly convex population loss function under a stochastic oracle that introduces both additive and multiplicative noise; this is a canonical and widely-studied setting that arises across operations research, signal processing, and machine learning. We begin by showing that standard approaches such as sample average approximation and robust (or averaged) stochastic approximation can lead to suboptimal --- and in some cases arbitrarily poor --- performance with realistic finite sample sizes. In contrast, we demonstrate that a carefully designed variance reduction strategy, which we term VISOR for short, can significantly outperform these approaches while using the same sample size. Our upper bounds are complemented by finite-sample, information-theoretic local minimax lower bounds, which highlight fundamental, instance-dependent factors that govern the performance of any estimator. Taken together, these results demonstrate that an accelerated variant of VISOR is instance-optimal, achieving the best possible sample complexity up to logarithmic factors while also attaining optimal oracle complexity. We apply our theory to generalized linear models and improve upon classical results. In particular, we obtain the best-known non-asymptotic, instance-dependent generalization error bounds for stochastic methods, even in linear regression.

Bio: Liwei Jiang is an Assistant Professor in the Edwardson School of Industrial Engineering at Purdue University. Before joining Purdue, he was a postdoctoral researcher at the Georgia Institute of Technology. He obtained his PhD in Operations Research and Information Engineering from Cornell University. His research interests lie at the intersection of optimization, statistics, and data science, with a particular focus on nonsmooth and stochastic optimization. His research has been recognized with several awards, including runner-up in the INFORMS Optimization Society Student Paper Prize and finalist in the 2026 INFORMS Junior Faculty Interest Group (JFIG) Paper Competition, with the winner to be announced.

Abstract: How can stochastic optimization algorithms achieve the best statistical accuracy while keeping computation inexpensive? In this talk, I will introduce a local minimax framework in the sense of Hájek and Le Cam that characterizes problem-specific limits on statistical accuracy and relates them to optimization geometry. A classical example of such optimality is Polyak–Juditsky averaging: for smooth unconstrained problems, averaged stochastic-gradient iterates satisfy a central limit theorem with the optimal covariance. I will then discuss how, in the constrained problems, active constraints alter this covariance and the algorithms needed to attain it. Existing methods, including those developed in my earlier work, achieve statistical optimality but require repeated, potentially costly projections onto the feasible set. Finally, I will present a stochastic primal–dual method based on the classical augmented Lagrangian framework that avoids these projections while preserving statistical optimality: under suitable regularity conditions, its averaged primal iterates satisfy a central limit theorem with the optimal asymptotic covariance.

Bio: Feng Ruan is currently an assistant professor in Department of Statistics and Data Science at Northwestern University. His research focuses on the foundations of feature learning and stochastic and nonsmooth optimization, and is supported by an NSF CAREER Award. 

Abstract: Causal discovery is the fundamental task of inferring underlying cause-and-effect relationships from observational data, a crucial step to enable robust prediction and effective intervention in complex systems. Applications range from deciphering protein signaling networks in biology to identifying root causes in industrial systems. This talk presents optimization-based approaches for learning Directed Acyclic Graphs (Bayesian networks) from continuous data generated by linear Gaussian structural equation models.

First, we introduce a mixed-integer optimization framework that overcomes the limitations of state-of-the-art methods in terms of optimality guarantees and restrictive noise assumptions. We demonstrate that this framework handles heteroscedastic noise and achieves optimality via a branch-and-bound procedure equipped with a novel early stopping criterion. Second, we address the computational challenges of the L0-penalized maximum likelihood estimator for learning Bayesian networks. We propose a new coordinate descent algorithm that is scalable and, despite the non-convexity of the loss function and the combinatorial constraints, converges asymptotically to the optimal objective value. Finally, we review recent results on related mixed-integer convex optimization problems, focusing on the convexification of resulting sets and efficient algorithms for special cases.

Bio: Simge Küçükyavuz is Chair and David A. and Karen Richards Sachs Professor in the Industrial Engineering and Management Sciences Department at Northwestern University. An expert in mixed-integer, large-scale, and stochastic optimization, she is an INFORMS Fellow and the recipient of the NSF CAREER Award and the INFORMS Computing Society (ICS) Prize. In addition to serving as past chair of ICS, president of the Association of Chairs of OR Departments (ACORD), and chair-elect of the INFORMS Optimization Society, she has served on the editorial boards of Mathematics of Operations Research, Mathematical Programming, Operations Research, SIAM Journal on Optimization, and the INFORMS Journal on Computing.

Abstract: Flexible probabilistic models are often inconsistent with known distributional information about low-dimensional, noisy transformations of their outputs. Examples include (1) generative modeling, where samples may be biased toward particular classes; (2) federated learning, where a reference distribution must be updated using distributional information from multiple local sources; and (3) deconvolution, where a latent distribution must be consistent with one or more noisy marginals. We consider these settings through a generalized static Schrödinger bridge formulation: find the distribution closest to a reference distribution in Kullback–Leibler divergence, while satisfying constraints on noisy functionals of its marginals. We formulate and optimize the dual problem by parameterizing the dual variables with a neural network. For score-based models, the dual variables yield a lightweight additive correction to the reference score.

Simulating from the corrected model involves a stochastic-gradient Langevin dynamics (SGLD) scheme, in which the drift of a diffusion is approximated by an unbiased, noisy estimate. In the second part of the talk, we analyze how this gradient noise affects the error between the discretized noisy chain and the exact diffusion. For strongly log-concave targets, existing Wasserstein-2 analyses yield a noise-induced error of order O(√α), where α is the step size, matching the order of the Euler discretization error. We show that for sub-Gaussian gradient noise, and with a more careful coupling, we can establish an O(α) noise contribution. Replacing the Euler discretization with a randomized midpoint method results in a sampling algorithm that achieves an overall O(α) rate.

(Joint work with Shishang Wu, Changhe Ji, Bingjing Tang and Yifei Xiong)

Bio: Vinayak Rao is an Associate Professor in the Department of Statistics at Purdue University, with a courtesy appointment in Computer Science. He earned his Ph.D. from University College London in the UK and subsequently completed postdoctoral research at Duke University. He received the International Society for Bayesian Analysis’s Savage Award in Theory and Methods for his doctoral thesis. He regularly serves as a Senior Area Chair and Area Chair for leading machine learning conferences, including NeurIPS, ICML, ICLR, and AISTATS. His research interests lie in the methodological, computational, and theoretical aspects of Bayesian, stochastic process and generative models.

Abstract:

Bio:

Abstract: 

Bio: 

Abstract: 

Bio: 

Abstract: 

Bio: 

Abstract: 

Bio: