UC Berkeley

This dissertation investigates the design and analysis of methods for a few foundational problems in optimization, statistics, and automated discovery. We begin with a theoretical exploration of finding zeros of monotone operators—which generalize convex minimization and convex-concave saddle point problems—providing a geometric characterization of continuous-time oscillatory behavior and introducing a regularization framework for stabilization and acceleration of discrete methods. We then study robust mean estimation under affine equivariance constraints, proposing a high-dimensional median-based estimator and establishing that this natural stability property incurs a dimension-dependent statistical cost. Finally, we present a decision-making methodology that combines machine learning with adaptive experimentation to guide materials discovery, demonstrating improved efficiency in exploring chemical spaces.