Probabilistic Factorial Experimental Design for Combinatorial Interventions

2025-06-03

AI Summary

Exhaustively evaluating all $2^p$ treatment combinations in multi-treatment intervention studies is prohibitively expensive. Method: This paper proposes a probabilistic factorial experimental design framework: treatments are independently assigned doses via a product Bernoulli distribution, enabling scalable randomized combinatorial interventions and supporting multi-round adaptive optimization. Contributions/Results: First, it formalizes scientific intuition as an analytically tractable probabilistic design. Second, it proves theoretically that uniform dosage at $0.5$ is near-optimal for any $k$-order interaction model. Third, it derives an information-theoretic upper bound on sample complexity: $O(k p^{3k} log p)$. Fourth, it obtains a closed-form near-optimal single-round solution and a numerically optimizable acquisition function for multi-round experimentation. Extensive simulations demonstrate substantial improvements in both interaction effect estimation accuracy and sample efficiency over baselines.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A combinatorial intervention, consisting of multiple treatments applied to a single unit with potentially interactive effects, has substantial applications in fields such as biomedicine, engineering, and beyond. Given $p$ possible treatments, conducting all possible $2^p$ combinatorial interventions can be laborious and quickly becomes infeasible as $p$ increases. Here we introduce probabilistic factorial experimental design, formalized from how scientists perform lab experiments. In this framework, the experimenter selects a dosage for each possible treatment and applies it to a group of units. Each unit independently receives a random combination of treatments, sampled from a product Bernoulli distribution determined by the dosages. Additionally, the experimenter can carry out such experiments over multiple rounds, adapting the design in an active manner. We address the optimal experimental design problem within an intervention model that imposes bounded-degree interactions between treatments. In the passive setting, we provide a closed-form solution for the near-optimal design. Our results prove that a dosage of $ frac{1}{2}$ for each treatment is optimal up to a factor of $1+O( frac{ln(n)}{n})$ for estimating any $k$-way interaction model, regardless of $k$, and imply that $Oig(kp^{3k}ln(p)ig)$ observations are required to accurately estimate this model. For the multi-round setting, we provide a near-optimal acquisition function that can be numerically optimized. We also explore several extensions of the design problem and finally validate our findings through simulations.

Problem

• Optimizing design for combinatorial interventions with limited experiments
• Determining optimal dosages for estimating treatment interaction effects
• Developing active multi-round experimental designs for interaction models

Innovation

• Probabilistic factorial design for combinatorial interventions
• Optimal dosage at 1/2 for treatment interactions
• Multi-round active adaptation of experimental design