Probabilistic Factorial Experimental Design for Combinatorial Interventions

TL;DR

This paper introduces a probabilistic factorial experimental design framework for efficiently studying combinatorial interventions, optimizing treatment dosages, and adaptively selecting experiments to estimate interaction effects with fewer observations.

Contribution

It formalizes a new probabilistic experimental design method inspired by lab experiments, providing closed-form solutions and near-optimal strategies for intervention modeling with interactions.

Findings

Optimal dosage of 1/2 per treatment for estimating interactions.

Requires O(kp^{3k} log p) observations for accurate modeling.

Validated through simulations demonstrating effectiveness.

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 $\tfrac{1}{2}$ for each treatment is optimal up to a factor of $1+O(\tfrac{\ln(n)}{n})$ for estimating any $k$-way interaction model, regardless of $k$, and imply that $O\big(kp^{3k}\ln(p)\big)$ 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.