Closing the Accuracy Gap in Stochastic First-Order Bilevel Optimization
We characterize the optimal accuracy dependence for smooth nonconvex--strongly-convex bilevel optimization with stochastic first-order oracles. For globally unbiased fresh-gradient observations with bounded variance, we prove an $Ω(ε^{-6})$ lower bound matching the accuracy exponent of existing upper bounds. We consider globally Lipschitz gradients, a bounded upper gradient in the lower variable, and Lipschitz lower Hessian blocks, with target $\mathbb{E}|\nabla F(\widehat{x})|\leε$. Let $L$ be the common first-order scale, $μ$ the lower strong-convexity modulus, $ρ$ the lower Hessian-variation budget, $κ=L/μ$, and $χ=1+ρ/μ$. For $ρ\gtrsimμ$, $χ\lesssimκ$, sufficiently high accuracy, and sufficiently large dimension, we establish $Ω\left(Lκ^2Δε^{-2}+L^3χ^2κ^8σ_g^2Δε^{-6}\right)$, where $Δ$ is the initial gap budget and $σ_g^2$ is the lower-gradient variance budget. The bound holds over full Euclidean spaces against arbitrary randomized adaptive algorithms, even when every sample is the gradient of a scalar function. In the common-scale regime $ρ=Θ(L)$, the leading stochastic term has condition-number dependence $κ^{10}$, compared with the achievable $κ^{11}$ dependence. The proof embeds a sequential hard objective into an exact lower response while preserving global regularity and finite gap. A multiscale decomposition limits the gradient signal carrying each new direction, yielding the required sample complexity. We complement this lower bound with a frozen-linear-tilt estimator whose bias is proportional to lower Hessian variation. Its analysis makes this structural dependence explicit and yields matching leading rates in the specified curvature-controlled and small-gap regimes.