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 rare-visit assumption governed by a parameter $\delta \in (0,\frac{1}{2}]$, SDBPG finds an $(\epsilon ,\epsilon )$-stationary point in $\mathcal{O}({\epsilon }^{-1/\delta })$ iterations, with sample gradient complexities $\mathcal{O}({\epsilon }^{-2/\delta })$ and $\mathcal{O}({\epsilon }^{-3/\delta })$ for the upper- and lower-level objectives, where larger $\delta$ corresponds to rarer visits to the bad region describing the negative alignment between the two objectives when the lower-level gradient is small. 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 $({\epsilon }_f,{\epsilon }_g)$-stationarity guarantees for stochastic nonconvex-nonconvex simple bilevel optimization.

The new version resolves the issue in the proof of Theorem 3.3 and modifies Assumption 3.1 to impose the rare visit assumption on the true bad region only