# Safe Convex Learning under Uncertain Constraints

## TL;DR

This paper introduces a robust Frank-Wolfe algorithm for convex optimization with uncertain linear constraints, ensuring feasibility and convergence in safety-critical applications like medicine and robotics.

## Contribution

A novel Frank-Wolfe variant that handles uncertain constraints with high-probability guarantees and convergence analysis.

## Key findings

- Guarantees feasibility of all iterates.
- Achieves convergence rate under uncertainty.
- Applicable to safety-critical domains.

## Abstract

We address the problem of minimizing a convex smooth function $f(x)$ over a compact polyhedral set $D$ given a stochastic zeroth-order constraint feedback model. This problem arises in safety-critical machine learning applications, such as personalized medicine and robotics. In such cases, one needs to ensure constraints are satisfied while exploring the decision space to find optimum of the loss function. We propose a new variant of the Frank-Wolfe algorithm, which applies to the case of uncertain linear constraints. Using robust optimization, we provide the convergence rate of the algorithm while guaranteeing feasibility of all iterates, with high probability.