Near-Optimal Convex Simple Bilevel Optimization with a Bisection Method

This paper studies a class of simple bilevel optimization problems where we minimize a composite convex function at the upper-level subject to a composite convex lower-level problem. Existing methods either provide asymptotic guarantees for the upper-level objective or attain slow sublinear convergence rates. We propose a bisection algorithm to find a solution that is εf-optimal for the upper-level objective and εg-optimal for the lower-level objective. In each iteration, the binary search narrows the interval by assessing inequality system feasibility. Under mild conditions, the total operation complexity of our method is tilde Oleft(max√Lf1/εf,√Lg1/εg right). Here, a unit operation can be a function evaluation, gradient evaluation, or the invocation of the proximal mapping, Lf1 and Lg1 are the Lipschitz constants of the upper- and lower-level objectives' smooth components, and tilde O hides logarithmic terms. Our approach achieves a near-optimal rate, matching the optimal rate in unconstrained smooth or composite convex optimization when disregarding logarithmic terms. Numerical experiments demonstrate the effectiveness of our method.