On the Complexity of First-Order Methods in Stochastic Bilevel Optimization
<|reference_start|>On the Complexity of First-Order Methods in Stochastic Bilevel Optimization: We consider the problem of finding stationary points in Bilevel optimization when the lower-level problem is unconstrained and strongly convex. The problem has been extensively studied in recent years; the main technical challenge is to keep track of lower-level solutions $y^*(x)$ in response to the changes in the upper-level variables $x$. Subsequently, all existing approaches tie their analyses to a genie algorithm that knows lower-level solutions and, therefore, need not query any points far from them. We consider a dual question to such approaches: suppose we have an oracle, which we call $y^*$-aware, that returns an $O(\epsilon)$-estimate of the lower-level solution, in addition to first-order gradient estimators {\it locally unbiased} within the $\Theta(\epsilon)$-ball around $y^*(x)$. We study the complexity of finding stationary points with such an $y^*$-aware oracle: we propose a simple first-order method that converges to an $\epsilon$ stationary point using $O(\epsilon^{-6}), O(\epsilon^{-4})$ access to first-order $y^*$-aware oracles. Our upper bounds also apply to standard unbiased first-order oracles, improving the best-known complexity of first-order methods by $O(\epsilon)$ with minimal assumptions. We then provide the matching $\Omega(\epsilon^{-6})$, $\Omega(\epsilon^{-4})$ lower bounds without and with an additional smoothness assumption on $y^*$-aware oracles, respectively. Our results imply that any approach that simulates an algorithm with an $y^*$-aware oracle must suffer the same lower bounds.<|reference_end|>
@article{kwon2024on,
    title={On the Complexity of First-Order Methods in Stochastic Bilevel
  Optimization},
    author={Jeongyeol Kwon, Dohyun Kwon, Hanbaek Lyu},
    journal={arXiv preprint arXiv:2402.07101},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.07101},
    primaryClass={math.OC cs.LG}
}