Black-Box Optimization with Implicit Constraints for Public Policy

TL;DR

The paper tackles implicit constraints in black-box optimization for public policy and introduces CageBO, which learns a constraint-free latent space $\mathcal{Z}$ via a conditional variational autoencoder and performs Bayesian optimization there while decoding to feasible decisions in $\mathcal{X}$. It provides a post-decoding projection and proves a no-regret upper bound $\mathbb{E}[R_T] = \widetilde{O}(\sqrt{T \gamma_T} + \sqrt{d} (n+T)^{d/(d+1)})$ under standard GP-Bandit assumptions. The method is validated on synthetic benchmarks and a large-scale Atlanta police redistricting case, showing superior performance and efficiency over baselines. This approach enables efficient exploration of high-dimensional, implicitly constrained policy spaces with practical impact for policymaking, while acknowledging fairness considerations and the need for debiasing techniques.

Abstract

Black-box optimization (BBO) has become increasingly relevant for tackling complex decision-making problems, especially in public policy domains such as police redistricting. However, its broader application in public policymaking is hindered by the complexity of defining feasible regions and the high-dimensionality of decisions. This paper introduces a novel BBO framework, termed as the Conditional And Generative Black-box Optimization (CageBO). This approach leverages a conditional variational autoencoder to learn the distribution of feasible decisions, enabling a two-way mapping between the original decision space and a simplified, constraint-free latent space. The CageBO efficiently handles the implicit constraints often found in public policy applications, allowing for optimization in the latent space while evaluating objectives in the original space. We validate our method through a case study on large-scale police redistricting problems in Atlanta, Georgia. Our results reveal that our CageBO offers notable improvements in performance and efficiency compared to the baselines.

Figures (7)

• Figure 1: An illustrative example showing the difficult-to-define constraints using three districting plans for the Atlanta Police Department (APD). Gray lines represent the basic geographical units patrolled by the police, red lines outline the districting plans, and dashed blue lines highlight the changes made to the pre-2019 plan. (a) and (b) are feasible plans implemented by the APD pre and post 2019. (c) appears to be a feasible plan but was ultimately rejected by the APD because it overlooked traffic constraints and inadvertently cut off access to some highways with its zone boundaries.
• Figure 2: An illustration of the CageBO algorithm. Red dots represent observed feasible decisions, and grey dots denote observed infeasible decisions. The red dotted circle illustrates the complex feasible region that is not directly accessible. For each iteration, a new decision $z_{t}$ is then chosen in the feasible latent space $\hat{Z}$ by minimizing the lower confidence bound (LCB). This new decision $z_{t}$ is mapped back to the original space as $x_t$ using ${\mathrm{decoder}}_{\theta}$. If the CVAE model is well-trained, the newly generated $x_t$ is highly likely to reside within the feasible set $\mathcal{S}$. In case $x_t \notin \mathcal{S}$, the post-decoding process $\psi_{\mathcal{D}_f}$ will adjust it to the nearest observed feasible decision.
• Figure 3: Solution paths of the same optimization problem suggested by our CageBO algorithm in the (a) original and (b) latent spaces, respectively. The objective in this illustrative example is to minimize the Ackley function subject to an "implicit" constraint (a circle) in a 2-dimensional decision space.
• Figure 4: Comparisons of performance convergence with 95% confidence intervals for (a) 30-dimensional Keane's bump function, (b) 30-dimensional Michalewicz function, (c) police redistricting problem in a synthetic $6 \times 6$ grid, and (d) police redistricting problem in Atlanta, Georgia.
• Figure 5: Districting plans for the police redistricting problem in a synthetic $6 \times 6$ grid and a real-world scenario in Atlanta, Georgia. The workload in each region is indicated by the depth of the color. The CageBO plan achieves the lowest workload variance among all districts, indicating an optimal distribution of workload.

Theorems & Definitions (6)

• Theorem 1
• Remark 2
• Theorem : Restatement of Theorem \ref{['thm:main']}
• proof
• Lemma 5: Regret bound of GP-LCB (Theorem 1 of srinivas2010gaussian)
• Lemma 6: Expected minimum distance (Lemma 19.2 of shalev2014understanding)