Data-Driven Algorithm Design and Verification for Parametric Convex Optimization

Speaker: Asst. Prof. Bartolomeo Stellato (Princeton University)

Title: Data-Driven Algorithm Design and Verification for Parametric Convex Optimization

Abstract:
We present computational tools for analyzing and designing first-order methods in parametric convex optimization. These methods are popular for their low per-iteration cost and warm-starting capabilities. However, precisely quantifying the number of iterations required to compute high-quality solutions remains a key challenge, especially in real-time applications. First, we introduce a numerical framework for verifying the worst-case performance of first-order methods in parametric quadratic optimization. We formulate this as a mixed-integer linear program that maximizes the infinity norm of the fixed-point residual after a given number of iterations. Our approach captures a broad class of gradient, projection, and proximal iterations through affine or piecewise-affine constraints, with strong polyhedral formulations. To improve scalability, we incorporate bound-tightening techniques that exploit operator-theoretic bounds. Numerical results show that our method closely matches true worst-case performance, achieving significant reductions in worst-case fixed-point residuals compared to standard convergence analyses. Second, we present a data-driven approach for analyzing the performance of first-order methods using statistical learning theory. We establish generalization guarantees for classical optimizers using sample convergence bounds and for learned optimizers using the Probably Approximately Correct (PAC)-Bayes framework. We then apply this framework to learn accelerated first-order methods by directly minimizing the PAC-Bayes bound over key algorithmic parameters (e.g., gradient steps and warm-starts). Numerical experiments demonstrate that our approach provides strong generalization guarantees for both classical and learned optimizers, with statistical bounds that closely match true out-of-sample performance.

Bio:
Bartolomeo Stellato is an Assistant Professor in the Department of Operations Research and Financial Engineering at Princeton University. Previously, he was a Postdoctoral Associate at the MIT Sloan School of Management and Operations Research Center. He holds a DPhil (PhD) in Engineering Science from the University of Oxford, a MSc in Robotics, Systems and Control from ETH Zürich, and a BSc in Automation Engineering from Politecnico di Milano. He developed OSQP, a widely used solver in mathematical optimization. His awards include the Beale — Orchard-Hays Prize, the ONR Young Investigator Award, the NSF CAREER Award, the Princeton SEAS Howard B. Wentz Jr. Faculty Award, the Franco Strazzabosco Young Investigator Award from ISSNAF, the Princeton SEAS Innovation Award in Data Science, the Best Paper Award in Mathematical Programming Computation, and the First Place Prize Paper Award in IEEE Transactions on Power Electronics. His research focuses on data-driven computational tools for mathematical optimization, machine learning, and optimal control.