Efficient Conditional Gradient Methods for Solving Stochastic Convex Bilevel Optimization Problems
Khanh-Hung (Bruce) Giang-Tran, Soroosh Shafiee, Nam Ho-Nguyen
We propose efficient methods for solving stochastic convex bilevel optimization problems, where the goal is to minimize an outer stochastic objective function subject to the solution set of an inner stochastic optimization problem. Existing methods often rely on costly projection or linear optimization oracles over complex sets, which limits scalability. To overcome this, we propose an iteratively regularized conditional gradient framework that leverages efficient linear optimization oracles exclusively over the base feasible set. Our proposed methods employ a vanishing regularization sequence that progressively emphasizes the inner problem while biasing towards desirable minimal outer objective solutions. Under standard convexity assumptions, we establish non-asymptotic convergence rates of $O(t^{-({1}/{2}-p)})$ for the outer objective and $O(t^{-p})$ for the inner objective, where $p \in (0,1/2)$ controls the regularization decay, in the one-sample stochastic setting, and $O(t^{-(1-p)})$ and $O(t^{-p})$ in the finite-sum setting using a mini-batch scheme, where $p \in (0,1)$. Experimental results on over-parametrized regression and $\ell_1$-constrained logistics regression tasks demonstrate the practical advantages of our approach over existing methods, confirming our theoretical findings.