Stackelberg Control in Combinatorial Congestion Games without Differentiating Through Equilibria

Saeed Masiha, Sepehr Elahi, Negar Kiyavash, Patrick Thiran

Keywords: Stackelberg games, zeroth-order optimization, congestion games, zero-suppressed decision diagrams (ZDDs), compact combinatorial representations

TL;DR: Derivative-free bilevel learning for combinatorial congestion games: inner Frank–Wolfe equilibrium solver + exact/subsampled LMOs for scalable Stackelberg control, with convergence guarantees.

Abstract:

We study Stackelberg control in combinatorial congestion games, where a leader chooses parameters and a nonatomic population responds by a Wardrop equilibrium over an exponentially large strategy set.
Even with smooth primitives, the resulting hyper-objective is typically Lipschitz but nonsmooth, since small parameter changes can switch which combinatorial strategies are active at equilibrium.
We propose an oracle-based bilevel method that avoids differentiating through equilibria: an inner, projection-free Frank--Wolfe solver computes approximate equilibria using only gradient information and a linear minimization oracle, while an outer two-point zeroth-order scheme updates the leader using only objective evaluations at these approximate equilibria.
For the inner loop, we analyze a subsampled oracle and prove an $O(1/(\kappa_m T))$ rate under a mild optimizer-hit condition, together with stratified sampling schemes that keep $\kappa_m$ nontrivial in large, imbalanced strategy spaces.
For the full method, we establish convergence to generalized Goldstein stationary points of the hyper-objective, with explicit dependence on the equilibrium approximation error.
Overall, our results provide a convergence guarantee for Stackelberg control in CCGs while retaining scalability through combinatorial primitives such as shortest paths and decision-diagram-based oracles.
Experiments on real-world transportation networks corroborate the theory and highlight the practical gains of our oracle-based approach across both polynomial-time and NP-hard strategy families.

Submission Number: 37