# Robust Paths: Geometry and Computation

## TL;DR

This paper introduces a geometric approach to efficiently analyze the trade-offs in robust optimization by characterizing a robust path as a Bregman projection, enabling approximation via standard optimization algorithms.

## Contribution

It presents a novel geometric framework for robust paths in convex robust optimization, linking them to Bregman projections and standard algorithms, reducing computational effort.

## Key findings

- Robust paths can be approximated by trajectories of standard optimization algorithms.
- The approximation error depends on the geometry of the feasible region and uncertainty set.
- Special cases yield zero approximation error, such as polyhedral monotone feasible regions with ellipsoidal uncertainty.

## Abstract

Applying robust optimization often requires selecting an appropriate uncertainty set both in shape and size, a choice that directly affects the trade-off between average-case and worst-case performances. In practice, this calibration is usually done via trial-and-error: solving the robust optimization problem many times with different uncertainty set shapes and sizes, and examining their performance trade-off. This process is computationally expensive and ad hoc. In this work, we take a principled approach to study this issue for robust optimization problems with linear objective functions, convex feasible regions, and convex uncertainty sets. We introduce and study what we define as the robust path: a set of robust solutions obtained by varying the uncertainty set's parameters. Our central geometric insight is that a robust path can be characterized as a Bregman projection of a curve (whose geometry is defined by the uncertainty set) onto the feasible region. This leads to a surprising discovery that the robust path can be approximated via the trajectories of standard optimization algorithms, such as the proximal point method, of the deterministic counterpart problem. We give a sharp approximation error bound and show it depends on the geometry of the feasible region and the uncertainty set. We also illustrate two special cases where the approximation error is zero: the feasible region is polyhedrally monotone (e.g., a simplex feasible region under an ellipsoidal uncertainty set), or the feasible region and the uncertainty set follow a dual relationship. We demonstrate the practical impact of this approach in two settings: portfolio optimization and adversarial deep learning.