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Bilevel optimization is a fundamental tool in hierarchical decision-makingand has been widely applied to machine learning tasks such as hyperparametertuning, meta-learning, and continual learning. While significant progress hasbeen made in bilevel optimization, existing methods predominantly focus on the{nonconvex-strongly convex, or the} nonconvex-PL settings, leaving the moregeneral nonconvex-nonconvex framework underexplored. In this paper, we addressthis gap by developing an efficient gradient-based method inspired by therecently proposed Relaxed Gradient Flow (RXGF) framework with a continuous-timedynamic. In particular, we introduce a discretized variant of RXGF andformulate convex quadratic program subproblems with closed-form solutions. Weprovide a rigorous convergence analysis, demonstrating that under the existenceof a KKT point and a regularity assumption {(lower-level gradient PLassumption)}, our method achieves an iteration complexity of$\mathcal{O}(1/\epsilon^{1.5})$ in terms of the squared norm of the KKTresidual for the reformulated problem. Moreover, even in the absence of theregularity assumption, we establish an iteration complexity of$\mathcal{O}(1/\epsilon^{3})$ for the same metric. Through extensive numericalexperiments on convex and nonconvex synthetic benchmarks and a hyper-datacleaning task, we illustrate the efficiency and scalability of our approach.

Perturbed Gradient Descent via Convex Quadratic Approximation for Nonconvex Bilevel Optimization