Bilevel optimization involves a hierarchical structure where one problem is nested within another, leading to complex interdependencies between levels. We propose a single-loop, tuning-free algorithm that guarantees anytime feasibility, i.e., approximate satisfaction of the lower-level optimality condition, while ensuring descent of the upper-level objective. At each iteration, a convex quadratically-constrained quadratic program (QCQP) with a closed-form solution yields the search direction, followed by a backtracking line search inspired by control barrier functions to ensure safe, uniformly positive step sizes. The resulting method is scalable, requires no hyperparameter tuning, and converges under mild local regularity assumptions. We establish an O(1/k) ergodic convergence rate in terms of a first-order stationary metric and demonstrate the algorithm's effectiveness on representative bilevel tasks.

We present an approach that eliminates this problem.

This greedy behavior may hinder an algorithm's

Several recent works have aimed to design algorithms that are robust to adversarial perturbations. In practice, first order methods are a popular choice for adversarial training. These methods are appealing since they are generally applicable to many network architecture and only rely on black box access to gradient information. Hence, analyzing this algorithm is interesting in its own right. We first discuss the setting of the hyperparameters in our experiments.

Changepoint (CP) detection is a fundamental problem and has been studied in many areas.