Criterion-Aware Factorial Optimization with Applications in Black-Box Text-to-Image Systems

Prompt optimization is commonly formulated as free-form search over instructions using a scalar reward, which obscures both criterion-level failures and the effects of individual prompt operations. We instead formulate prompt optimization as a finite factorial design problem. A transformation is composed of interpretable two-level factors, and each generated output is evaluated using multiple binary criteria. We introduce Factorial Upper Confidence Bound Algorithm (FACT-UCB), which fits criterion-specific factorial logistic models to share information across configurations, estimate main and interaction effects, and enforce explicit pass-probability constraints. The method combines uncertainty-aware adaptive sampling with forced exploration to maintain identifiability. We further establish fixed-confidence recovery of the best feasible transformation and an anytime-valid stopping rule. We conduct a text-to-image case study with seven factors, 128 configurations, nine criteria, and 999 adaptive queries, the fitted model suggests concise, Markdown-structured, positively phrased instructions on an industrial 3D pre-generation dataset. Together, these results provide an explicit framework for interpretable, criterion-aware optimization of black-box generative systems.