An Accelerated Gradient Method for Simple Bilevel Optimization with Convex Lower-level Problem

In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max{1/\sqrt{\epsilon{f}}, 1/\epsilong})$ iterations to find a solution that is $\epsilonf$-suboptimal and $\epsilong$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th H\"olderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max{\epsilon{f}^{{-\frac{2r-1}{2r}},\epsilon}{g}^{{-\frac{2r-1}{2r}}})$,} which matches the optimal complexity of single-level convex constrained optimization when $r=1$.