Tree-Based Stochastic Optimization for Solving Large-Scale Urban Network Security Games

2025-11-13

AI Summary

Large-scale urban network security games (UNSGs) suffer from intractably large action spaces, rendering Nash equilibrium (NE) computation prohibitively expensive. To address this, we propose a tree-structured stochastic optimization framework. Our key contributions are: (1) a tree-based action representation that structurally encodes combinatorial action spaces, overcoming neural networks’ limitations in modeling high-dimensional discrete actions; (2) a novel differentiable loss mechanism provably equivalent to an unbiased loss function, enabling gradient-based optimization; and (3) a sampling-based pruning strategy that enhances optimization stability and ensures global convergence. Experiments on real-world urban road networks demonstrate that our method significantly outperforms PSRO and state-of-the-art neural stochastic optimization baselines—achieving superior scalability, faster convergence, and higher-quality NE approximations.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Urban Network Security Games (UNSGs), which model the strategic allocation of limited security resources on city road networks, are critical for urban safety. However, finding a Nash Equilibrium (NE) in large-scale UNSGs is challenging due to their massive and combinatorial action spaces. One common approach to addressing these games is the Policy-Space Response Oracle (PSRO) framework, which requires computing best responses (BR) at each iteration. However, precisely computing exact BRs is impractical in large-scale games, and employing reinforcement learning to approximate BRs inevitably introduces errors, which limits the overall effectiveness of the PSRO methods. Recent advancements in leveraging non-convex stochastic optimization to approximate an NE offer a promising alternative to the burdensome BR computation. However, utilizing existing stochastic optimization techniques with an unbiased loss function for UNSGs remains challenging because the action spaces are too vast to be effectively represented by neural networks. To address these issues, we introduce Tree-based Stochastic Optimization (TSO), a framework that bridges the gap between the stochastic optimization paradigm for NE-finding and the demands of UNSGs. Specifically, we employ the tree-based action representation that maps the whole action space onto a tree structure, addressing the challenge faced by neural networks in representing actions when the action space cannot be enumerated. We then incorporate this representation into the loss function and theoretically demonstrate its equivalence to the unbiased loss function. To further enhance the quality of the converged solution, we introduce a sample-and-prune mechanism that reduces the risk of being trapped in suboptimal local optima. Extensive experimental results indicate the superiority of TSO over other baseline algorithms in addressing the UNSGs.

Problem

• Solving large-scale urban security games with massive combinatorial action spaces
• Overcoming neural network limitations in representing vast action spaces
• Improving Nash Equilibrium approximation quality while avoiding local optima

Innovation

• Tree-based action representation for large-scale games
• Sample-and-prune mechanism to avoid local optima
• Stochastic optimization with equivalent loss function