Online Black-Box Prompt Optimization with Regret Guarantees under Noisy Feedback

Generative AI excels in various tasks through advanced language modeling techniques, with its performance heavily influenced by input prompts. This has driven significant research into prompt optimization, particularly in commercial generative AI platforms, where prompt optimization is treated as a black-box optimization problem. Most existing research on black-box prompt optimization primarily focuses on offline learning and overlooks the randomness in outputs. However, in real-world applications, black-box prompt optimization typically operates in an online learning setting, which remains largely unexplored, especially given the noisy outputs. To address these challenges, we propose an \textbf{A}daptive \textbf{O}nline \textbf{Z}eroth-order \textbf{P}rompt \textbf{T}uning (AOZPT) approach which integrates zeroth-order optimization with online learning in the non-convex setting. Specifically, we developed an uncertainty-scale-adjustment mechanism to mitigate the noise inherent in generative AI and the high variance associated with zeroth-order estimates. We conducted a comprehensive regret analysis of the AOZPT approach, and the results indicate that sublinear regret convergence is achievable. Extensive generative experiments demonstrate that AOZPT outperforms existing black-box prompt tuning methods, particularly in terms of stability in online scenarios.