IWR Colloquium Summer Semester 2026 Sharp Tradeoffs Between Accuracy and Robustness of Estimators Through the Lens of Differential Geometry

The fragility of many modern learning models in the face of data perturbations, adversarial examples, and distributional shifts is by now nearly as familiar as their success stories. In light of this, many works have proposed replacing strategies such as standard risk minimization with more robust alternatives. However, there is no free lunch, and robust classifiers frequently exhibit degraded performance on clean data. Because of this, identifying frameworks which balance performance and robustness is of pressing interest to the machine learning and data sciences communities. 

To shed light on how to study this tradeoff, however, we will take a step back and revisit the simpler but perhaps more fundamental setting of statistical inference. This will motivate us to revisit some of the deep interactions between differential geometry and statistics that, for example, can be appreciated in the Cramér-Rao theory of estimation, which studies the variance of estimators by viewing statistical models as submanifolds in the Hellinger geometry. By considering an analogue geometric perspective, now viewing statistical models as submanifolds in the Wasserstein geometry, we will access a powerful mathematical toolkit to study the notion of “sensitivity” of an estimator (a quantity that measures how much an estimators changes under small data perturbations), obtaining fundamental lower bounds for it, and deriving estimation procedures that are provably sensitivity efficient. This same geometric perspective will, finally, allow us to concretely formulate and answer the talk’s main question: how can we construct estimators that, at least asymptotically, balance between accuracy (variance) and robustness (sensitivity) optimally?

The talk is based on joint works with Adam Quinn Jaffe (Columbia), Bodhisattva Sen (Columbia), and my PhD student Congwei Yang (UWisconsin).