Stochastic Dynamic Barrier Perturbed Gradient Methods for Nonconvex Simple Bilevel Optimization

Abstract

We study stochastic simple bilevel optimization with smooth, possibly nonconvex upper- and lower-level objectives accessed only through stochastic gradient oracles. A key challenge is that the dual multiplier induced by the lower-level constraint may become unbounded near lower-level stationary points, invalidating bounded-dual analyses and destabilizing stochastic gradient estimates. To address this, we propose \emph{Stochastic Dynamic Barrier Perturbed Gradient} (SDBPG), a single-loop method that adaptively perturbs the dual formulation to regularize this degeneracy. The perturbation stabilizes the multiplier and yields controlled bias and variance even near the lower-level stationarity region. Under a mild rare-visit assumption, SDBPG finds an $(ε_f,ε_g)$-stationary point in $\mathcal{O}(\max\{ε_f^{-2},ε_g^{-2}\})$ iterations, with sample gradient complexities $\mathcal{O}(ε^{-4})$ and $\mathcal{O}(ε^{-6})$ for the upper- and lower-level objectives where $ε=\max\{ε_f,ε_g\}$. We further develop PR-SDBPG, a penalty-regularized variant that eliminates the rare-visit assumption, and VR-PR-SDBPG, which improves the resulting sample complexities entirely through variance reduction. To our knowledge, these are the first explicit $(ε_f,ε_g)$-stationarity guarantees for stochastic nonconvex-nonconvex simple bilevel optimization.