Causal Bayesian Optimization via Exogenous Distribution Learning (2402.02277v8)

Abstract: Maximizing a target variable as an operational objective in a structural causal model is an important problem. Existing Causal Bayesian Optimization~(CBO) methods either rely on hard interventions that alter the causal structure to maximize the reward; or introduce action nodes to endogenous variables so that the data generation mechanisms are adjusted to achieve the objective. In this paper, a novel method is introduced to learn the distribution of exogenous variables, which is typically ignored or marginalized through expectation by existing methods. Exogenous distribution learning improves the approximation accuracy of structural causal models in a surrogate model that is usually trained with limited observational data. Moreover, the learned exogenous distribution extends existing CBO to general causal schemes beyond Additive Noise Models~(ANM). The recovery of exogenous variables allows us to use a more flexible prior for noise or unobserved hidden variables. We develop a new CBO method by leveraging the learned exogenous distribution. Experiments on different datasets and applications show the benefits of our proposed method.

• The paper introduces EXCBO, a unified framework that combines Structural Causal Models with Bayesian optimization by learning exogenous variable distributions from observational data.
• It establishes a rigorous theoretical foundation that yields sublinear cumulative regret bounds and improved sample efficiency over traditional methods.
• Empirical evaluations demonstrate that EXCBO outperforms conventional approaches by offering better convergence rates and robustness in complex, noisy environments.

Introduction to Causal Bayesian Optimization

Bayesian Optimization (BO) is a fundamental technique for optimizing black-box functions that are expensive to evaluate. It has been widely used in domains needing efficient optimization without the complete understanding of the underlying function, from tuning hyperparameters in machine learning models to designing new materials. Conventionally, BO assumes that all input variables to the objective function are independent. However, this independence assumption is often unrealistic in real-world problems where the variables may be causally related.

Incorporating Structural Causal Models

The paper under review introduces an innovative approach that leverages the structures known as Structural Causal Models (SCMs) to enhance Bayesian Optimization. SCMs provide a principled way to represent and reason about cause-and-effect relationships between variables. By combining the SCM's structural knowledge and the conventional BO, Causal Bayesian Optimization (CBO) has been developed. This unified framework takes advantage of causal insights, allowing for a more nuanced exploration of the input space.

The key contribution of this work is a method that recovers and learns the distribution of exogenous variables for each endogenous node in an SCM using observational data. This presents a significant advancement over existing CBO methods that primarily focus on simpler Additive Noise Models (ANMs) and assume Gaussian exogenous variables. The broader applicability of the proposed method to non-Gaussian and multi-modal exogenous variables suggests a more general and robust approach to causal inference and optimization.

Theoretical Framework and Algorithm Development

The paper goes beyond empirical methods to contribute a rigorous theoretical foundation for the recovery of exogenous variables, showing that under certain conditions, the distribution of these variables can be approximated with high fidelity. This theoretical underpinning is a key asset, allowing for sublinear cumulative regret bounds for the proposed EXCBO algorithm, which is significant for establishing the convergence and reliability of the method in practice.

The authors have developed an EXCBO algorithm, which is a specific instantiation of the proposed CBO framework leveraging the learned exogenous distribution. Moreover, the paper reports a detailed empirical validation, showcasing the superiority of EXCBO over several baseline approaches across various datasets. It demonstrates the proposed method improves the surrogate model's accuracy, facilitates better causal inference regarding CBO updating, and enhances sample efficiency.

Empirical Validation and Results

The experimental results illustrate the performance benefits of the EXCBO approach. Applications to the calibration of epidemic models, for instance, highlight the proposed framework's capability to optimize soft interventions more accurately and robustly, particularly in the face of noise. Across the board, the EXCBO algorithm shows favorable properties in terms of convergence rate and resistance to different noise levels compared to traditional BO and other state-of-the-art CBO methods.

The extensive experimental evaluation provides substantial evidence supporting the key claims of the paper, especially the flexibility and robustness offered by learning the exogenous distributions in different problem contexts, such as dynamic systems represented within SCMs.

Conclusion and Impact on Future Research

In summary, the work presented in this paper marks a substantial forward leap in the domain of Causal Bayesian Optimization. By methodically learning the distribution of exogenous variables, it opens a new avenue for optimization in complex causal environments. The implications are far-reaching, potentially influencing how future research may approach optimization problems in fields where causal relationships naturally exist among the decision variables. With rigorous theoretical and empirical backing, the proposed EXCBO algorithm stands as a promising alternative for practitioners and researchers alike.