Zeroth-Order Stackelberg Control in Combinatorial Congestion Games

TL;DR

This paper introduces ZO-Stackelberg, a novel zeroth-order method for leader tuning in combinatorial congestion games that efficiently finds equilibrium solutions without requiring differentiation, significantly speeding up computations.

Contribution

The paper develops a zeroth-order Stackelberg control algorithm combining a projection-free equilibrium solver with a zeroth-order update, providing convergence guarantees and practical sampling strategies.

Findings

Achieves orders-of-magnitude speedups over differentiation-based methods.

Converges to equilibrium solutions with explicit error bounds.

Effective in real-world network experiments.

Abstract

We study Stackelberg (leader--follower) tuning of network parameters (tolls, capacities, incentives) in combinatorial congestion games, where selfish users choose discrete routes (or other combinatorial strategies) and settle at a congestion equilibrium. The leader minimizes a system-level objective (e.g., total travel time) evaluated at equilibrium, but this objective is typically nonsmooth because the set of used strategies can change abruptly. We propose ZO-Stackelberg, which couples a projection-free Frank--Wolfe equilibrium solver with a zeroth-order outer update, avoiding differentiation through equilibria. We prove convergence to generalized Goldstein stationary points of the true equilibrium objective, with explicit dependence on the equilibrium approximation error, and analyze subsampled oracles: if an exact minimizer is sampled with probability $\kappa_m$, then the Frank--Wolfe error decays as $\mathcal{O}(1/(\kappa_m T))$. We also propose stratified sampling as a practical way to avoid a vanishing $\kappa_m$ when the strategies that matter most for the Wardrop equilibrium concentrate in a few dominant combinatorial classes (e.g., short paths). Experiments on real-world networks demonstrate that our method achieves orders-of-magnitude speedups over a differentiation-based baseline while converging to follower equilibria.