Projection-Free Stochastic Bi-Level Optimization

This work presents the first projection-free algorithm to solve stochastic bi-level optimization problems, where objective function depends on solution of another problem. The proposed $\textbf{S}$tochastic $\textbf{Bi}$-level $\textbf{F}$rank-$\textbf{W}$olfe ($\textbf{SBFW}$) can be applied streaming settings and does not make use large batches or checkpoints. sample complexity SBFW is shown $\mathcal{O}(\epsilon^{-3})$ for convex objectives $\mathcal{O}(\epsilon^{-4})$ non-convex objectives. Improved rates are derived compositional problem, which a special case entails minimizing composition two expected-value functions. $\textbf{C}$ompositional ($\textbf{SCFW}$) achieve $\mathcal{O}(\epsilon^{-2})$ objectives, at par with state-of-the-art complexities algorithms solving single-level problems. We demonstrate advantage methods by problem matrix completion denoising policy value evaluation in reinforcement learning.