Robust Optimization with Diffusion Models for Green Security

TL;DR

This work introduces a conditional diffusion model to express complex adversary behavior in green security and embeds it into a robust patrol optimization framework called DiffOracle. By reformulating uncertainty over distributions as a mixed strategy over mixed strategies and leveraging a double oracle flow with Twisted SMC sampling, the approach achieves convergence to an $oldsymbol{ extup{epsilon}}$-equilibrium in a distributionally robust setting. The authors prove finite-sample convergence guarantees and demonstrate substantial reductions in decision regret on both synthetic and real poaching datasets, highlighting improved resilience to model misspecification. Overall, the paper provides a principled, scalable method for robust wildlife protection planning under rich, multimodal adversary behavior.

Abstract

In green security, defenders must forecast adversarial behavior, such as poaching, illegal logging, and illegal fishing, to plan effective patrols. These behavior are often highly uncertain and complex. Prior work has leveraged game theory to design robust patrol strategies to handle uncertainty, but existing adversarial behavior models primarily rely on Gaussian processes or linear models, which lack the expressiveness needed to capture intricate behavioral patterns. To address this limitation, we propose a conditional diffusion model for adversary behavior modeling, leveraging its strong distribution-fitting capabilities. To the best of our knowledge, this is the first application of diffusion models in the green security domain. Integrating diffusion models into game-theoretic optimization, however, presents new challenges, including a constrained mixed strategy space and the need to sample from an unnormalized distribution to estimate utilities. To tackle these challenges, we introduce a mixed strategy of mixed strategies and employ a twisted Sequential Monte Carlo (SMC) sampler for accurate sampling. Theoretically, our algorithm is guaranteed to converge to an epsilon equilibrium with high probability using a finite number of iterations and samples. Empirically, we evaluate our approach on both synthetic and real-world poaching datasets, demonstrating its effectiveness.

Figures (3)

• Figure 1: Overview of DiffOracle. We begin by initializing the strategy set for each player. At the $i$-th iteration, we use SMC sampler to obtain a set of empirical distributions $\hat{\mathcal{T}}_{i-1}$. Next, a mixed Nash solver computes the equilibrium $\pi^*_{i-1}$ and $\hat{\sigma}^*_{i-1}$. We then compute each player’s best response against the opponent’s mixed strategy and update the players’ strategy sets. This procedure is repeated until convergence.
• Figure 2: Experimental Results on both synthetic and real-world datasets. Following ho2020denoising, we average the results over 5 random seeds.
• Figure 3: Parameter Study on DiffOracle-SMC on poaching data under Budget 5.

Theorems & Definitions (17)

• Definition 5.1: Mixed Strategy over Mixed Strategies
• Proposition 5.1
• Proposition 5.2
• Proposition 5.2
• Proposition 5.3
• Theorem 5.1
• Theorem 5.2
• proof
• proof
• Proposition 5.3: Twisted SMC