TCARD: Nearly Balanced Two-Level Designs with Treatment Cardinality Constraints with an Application to LLM Prompt Engineering

2026-05-20

AI Summary

This study addresses the statistical efficiency and practical feasibility of two-level experimental designs under cardinality constraints by proposing an approximately balanced TCARD construction method. The approach uniquely integrates factor-level replication balance and pairwise factor co-occurrence uniformity into a model-free objective function, denoted Φ_BCD, and establishes its intrinsic connections with several classical optimality criteria. Efficient optimization of Φ_BCD is achieved through a coordinate-exchange algorithm, incremental update strategies, and simulation-driven weight calibration. Numerical experiments demonstrate that the proposed designs consistently outperform existing methods across various problem scales and constraint intensities, yielding substantial improvements in projection properties, modeling efficiency, and real-world applicability.

Abstract

Modern experimental designs often face the so-called treatment cardinality constraint, which is the constraint on the number of included factors in each treatment. Experiments with such constraints are commonly encountered in engineering simulation, AI system tuning, and large-scale system verification. This calls for the development of adequate designs to enable statistical efficiency for modeling and analysis within feasible constraints. In this work, we study two-level designs under this $k$-treatment cardinality constraint (TCARD), where the design matrix $\mathbf{X} \in \{0,1\}^{n \times p}$ has constant row sums equal to $k$. Although TCARDs are closely related to balanced incomplete block designs (BIBDs), exact BIBD structure is unavailable for many practical $(n,p,k)$ combinations. This leads to the notion of nearly balanced TCARDs, which we prove minimize the first two components of the generalized word-length pattern. We also show that good projection behavior in this setting is governed by two count-based regularities: balanced factor replications and uniform pairwise concurrences. Motivated by this characterization, we then propose the Balanced Concurrence Deviation ($Φ_{\mathrm{BCD}}$), a model-free objective that jointly penalizes replication imbalance and concurrence dispersion. We further show that this criterion is closely connected to classical optimality principles, including $(M,S)$-optimality, centered $\mathrm{UE}(s^2)$ criterion, and Bayesian $D$-optimality. To construct designs minimizing $Φ_{\mathrm{BCD}}$, we develop a coordinate-exchange (CE) algorithm with efficient incremental updates, together with a simulation-based procedure for calibrating the criterion weights to the intended downstream task. Numerical experiments confirm that the proposed method compares favorably with existing alternatives across a range of problem sizes and constraint strengths.

Problem

• treatment cardinality constraint
• two-level designs
• experimental design
• balanced incomplete block designs
• statistical efficiency

Innovation

• TCARD
• balanced concurrence deviation
• treatment cardinality constraint
• nearly balanced design
• coordinate-exchange algorithm