An Optimization Perspective on the Monotonicity of the Multiplicative Algorithm for Optimal Experimental Design

TL;DR

This paper offers a straightforward optimization-based proof for the monotonicity of the multiplicative algorithm in optimal experimental design, clarifying conditions for strict monotonicity and exploring its limitations through illustrative examples.

Contribution

Provides a simpler proof of the multiplicative algorithm's monotonicity, identifies conditions for strict monotonicity, and discusses limitations with illustrative examples.

Findings

Simpler proof of monotonicity for the multiplicative algorithm

Identification of conditions ensuring strict monotonicity

Examples revealing limitations of the algorithm under certain criteria

Abstract

We provide an optimization-based argument for the monotonicity of the multiplicative algorithm (MA) for a class of optimal experimental design problems considered in Yu (2010). Our proof avoids introducing auxiliary variables (or problems) and leveraging statistical arguments, and is much more straightforward and simpler compared to the proof in Yu (2010). The simplicity of our monotonicity proof also allows us to easily identify several sufficient conditions that ensure the strict monotonicity of MA. In addition, we provide two simple and similar-looking examples on which MA behaves very differently. These examples offer insight in the behaviors of MA, and also reveal some limitations of MA when applied to certain optimality criteria. We discuss these limitations, and pose open problems that may lead to deeper understanding of the behaviors of MA on these optimality criteria.