[Paper Review] Zeroth-Order Stackelberg Control in Combinatorial Congestion Games

Saeed Masiha, Sepehr Elahi+2 more|arXiv (Cornell University)|Feb 26, 2026

Game Theory and Applications0 citations

TL;DR

Proposes ZO-Stackelberg, a zero-order bilevel optimization approach that optimizes leader parameters in combinatorial congestion games by combining a Frank–Wolfe equilibrium solver with a zeroth-order outer update, avoiding differentiation through equilibria.

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 $κ_m$, then the Frank--Wolfe error decays as $\mathcal{O}(1/(κ_m T))$. We also propose stratified sampling as a practical way to avoid a vanishing $κ_m$ 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.

Motivation & Objective

• Steer follower discrete-route choices in combinatorial congestion games via leader parameter tuning (tolls, capacities, incentives).
• Optimize a system-level objective evaluated at Wardrop equilibrium, despite its potential nonsmoothness due to active-set changes.
• Provide convergence guarantees to generalized Goldstein stationary points for the true equilibrium objective.
• Develop practical, scalable algorithms that avoid differentiating through equilibria while maintaining accuracy.

Proposed method

• Couple a projection-free Frank–Wolfe equilibrium solver with a zeroth-order outer update on the leader parameter theta.
• Form the inner loop as FW to approximate y*(theta) with a linear minimization oracle (LMO) over the feasible load polytope C, optionally using a subsampled LMO_m.
• Update theta via a two-point finite-difference zeroth-order scheme using evaluations of Phi_hat_T(theta)=F(theta,y_T(theta)).
• Use stratified or structure-aware sampling to improve the probability kappa_m that the sampled LMO contains an exact minimizer, enabling faster convergence.
• Prove convergence to generalized Goldstein stationary points of the true, possibly nonsmooth, hyper-objective Phi, with explicit dependence on inner equilibrium error.
• Provide an efficient Python implementation with exact and subsampled LMOs for various strategy families (e.g., s–t paths, Hamiltonian paths, Steiner cycles).

Figure 1 : Leader objective vs outer iterations for Scenarios 1–3. For subsampled LMOs (US/UL/HL), lighter shades denote smaller sampling budgets $m$ (we use $m\in\{10,100,1000\}$ in Scenario 2 and 3); bands are 99% CIs over 10 runs, while Diff is deterministic.

Experimental results

Research questions

• RQ1 Can a zeroth-order bilevel approach converge to meaningful stationary points when the equilibrium map is nonsmooth due to active-set changes?
• RQ2 How does subsampling the LMO affect convergence rates, and can stratified sampling mitigate issues from huge combinatorial strategy spaces?
• RQ3 Does optimizing the true equilibrium objective without differentiating through equilibria yield practical speedups and comparable accuracy to differentiation-based baselines?
• RQ4 What are the convergence guarantees and rate dependencies (e.g., on inner approximation error) for ZO-Stackelberg in combinatorial congestion games?
• RQ5 How do different combinatorial strategy families affect LMO implementation and overall performance in real networks?

Key findings

• ZO-Stackelberg matches differentiation-based baselines in accuracy while delivering orders-of-magnitude speedups and lower memory usage.
• The inner Frank–Wolfe loop with LMO (exact or subsampled) provides convergence guarantees with rate O(1/(kappa_m T)) under a mild optimizer-hit assumption.
• The outer loop converges to a generalized Goldstein stationary point of the at-equilibrium objective Phi, with explicit dependence on the inner equilibrium error.
• Stratified sampling (e.g., length-debiased) keeps kappa_m nontrivial in large strategy spaces, improving practical performance.
• Exact LMOs are feasible for some families (e.g., s–t paths, Hamiltonian paths) via ZDD-based dynamic programming; subsampling can substitute when exact minimization is impractical.

Figure 2 : Final-iterate diagnostics: speedup vs Diff, peak RSS, FW gap, and social cost, for Scenarios 1–3. For subsampling-based variants, lighter shades denote smaller $m$ (same $m$ as in Figure 1 ); points are means and bars are 99% CIs over 10 runs.