Yu Gui is currently a Postdoctoral Researcher in the Department of Statistics and Data Science at the Wharton School, University of Pennsylvania, working with Professors Dylan Small and Zhimei Ren. Yu obtained his PhD in Statistics at the University of Chicago, advised by Professors Rina Foygel Barber and Cong Ma. Prior to his PhD, he graduated from the School of the Gifted Young at the University of Science and Technology of China in 2020. His research centers around distribution-free inference, selective inference under distribution shifts, multi-modal learning, and observational studies, motivated by problems that both arise from and can further inform real-world applications. This lecture will take place in the Lawrence D. Brown PhD Student Award session at the IMS 2026 meeting, in Salzburg, July 6–9, 2026.

Distributionally robust risk evaluation with an isotonic constraint

Statistical learning under distribution shift is challenging when neither prior knowledge nor data from the target distribution is available.

As a mainstream technique in addressing distribution shifts, sample reweighting with the goal of learning the density ratio between distributions can be sensitive to the misspecification of weights, especially when available data from the target distribution is scarce. Alternatively, distributionally robust learning (DRL) aims to control out-of-distribution performance uniformly across a set of candidate target distributions, which may result in excessive conservativeness. In light of the limitations of existing methods, we aim to answer the following question: Can we leverage data-informed side information to balance the misspecification of reweighting methods and the excessive pessimism of DRL?

To enable distributional robustness without being overly conservative, we propose iso-DRL, a shape-constrained approach to DRL, which incorporates prior information about the way in which the unknown target distribution differs from its estimate—for instance, we may assume the unknown density ratio between the target distribution and its estimate is isotonic with respect to some partial order. One notable example of the partial order is the one determined by an estimated density ratio, in which case, if the estimated density ratio reveals useful side information of the underlying truth (e.g., correct relative magnitude that is evident for defining under- or over-represented regions in the covariate space), iso-DRL focuses on a fine-grained set of candidate distributions and the robust isotonic calibration of the estimated density ratio.

This paper aims to balance the misspecification of sample reweighting and the over-pessimism of DRL by incorporating the shape constraints as side information. More broadly, this work aims to find more practical, optimistic, and data-driven solutions to distribution shifts under certain structures. More concretely, our paper has the following contributions:

1. At the population level, explicitly solving iso-DRL is highly nontrivial as it involves a cone constraint in the function space. We provide an equivalent formulation of the shape-constrained optimization problem that can be solved without the challenge of an explicit isotonic constraint, in which case, iso-DRL is equivalent to a DRL problem with a risk function after isotonic projection, which simplifies computation and yields closed-form solutions for many choices of ambiguity sets of candidate distributions.

2. At the sample level, although the shape constraint can be written as a set of linear constraints, as the number of constraints is a function of sample size, it is unclear whether a consistent estimator of iso-DRL objective can be established. We provide consistency results for an empirical estimator of the target in a range of different settings.

3. We apply the proposed method to distribution-free uncertainty quantification and demonstrate its effectiveness through experiments on synthetic and real-world datasets subject to distribution shifts, achieving notable improvements over both sample reweighting and DRL methods.

This talk is based on my PhD work with advisors Rina Foygel Barber and Cong Ma.