Zeroth-Order Stackelberg Control in Combinatorial Congestion Games

2026-02-26

AI Summary

This work addresses the challenge of optimizing system-level objectives—such as total travel time—in combinatorial congestion games under Stackelberg leadership, where the leader’s objective becomes non-differentiable due to abrupt shifts in followers’ equilibrium strategies. To circumvent the need for explicit differentiation through equilibrium solutions, the paper proposes ZO-Stackelberg, the first method integrating zeroth-order optimization with a projection-free Frank–Wolfe equilibrium solver, enabling efficient Stackelberg control even with discrete follower strategies. A hierarchical sampling strategy is introduced to mitigate sampling inefficiencies caused by concentrated critical strategies, and the algorithm is theoretically shown to converge to a generalized Goldstein stationary point. Experiments on real-world traffic networks demonstrate orders-of-magnitude speedups over gradient-based baselines while accurately approximating follower equilibria.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Stackelberg control
• combinatorial congestion games
• zeroth-order optimization
• nonsmooth optimization
• equilibrium learning

Innovation

• Zeroth-order optimization
• Stackelberg control
• Combinatorial congestion games
• Frank-Wolfe method
• Stratified sampling