Bayesian Optimization (BO) has been widely used to efficiently optimize expensive black-box functions with limited evaluations. In this paper, we investigate the use of BO for prompt engineering to enhance text classification with Large Language Models (LLMs). We employ an LLM-powered Gaussian Process (GP) as the surrogate model to estimate the performance of different prompt candidates. These candidates are generated by an LLM through the expansion of a set of seed prompts and are subsequently evaluated using an Upper Confidence Bound (UCB) acquisition function in conjunction with the GP posterior. The optimization process iteratively refines the prompts based on a subset of the data, aiming to improve classification accuracy while reducing the number of API calls by leveraging the prediction uncertainty of the LLM-based GP. The proposed BO-LLM algorithm is evaluated on two datasets, and its advantages are discussed in detail in this paper.

Prompt engineering is crucial for fully leveraging large language models (LLMs), yet most existing optimization methods follow a single trajectory, resulting in limited adaptability, gradient conflicts, and high computational overhead. We propose MAPGD (Multi-Agent Prompt Gradient Descent), a novel framework that reconceptualizes prompt optimization as a collaborative process among specialized agents. Each agent focuses on a distinct refinement dimension, such as instruction clarity, example selection, format structure, or stylistic adaptation, and their contributions are coordinated through semantic gradient embedding, conflict detection, and fusion. To further enhance robustness and stability, MAPGD introduces two new mechanisms: Hypersphere Constrained Gradient Clustering (HCGC), which enforces angular margin constraints for compact and well-separated clusters, and Channel Adaptive Agent Weighting (CAAW), which dynamically reweights agent contributions based on validation performance. Experiments on classification and reasoning benchmarks show that MAPGD consistently surpasses single-agent and random baselines in both accuracy and efficiency. Ablation studies confirm the effectiveness of gradient fusion, agent specialization, and conflict resolution. Together, these components establish MAPGD as a unified, gradient-based, and interpretable framework for robust prompt optimization with theoretical convergence guarantees.

Prompt design is a critical factor in the effectiveness of Large Language Models (LLMs), yet remains largely heuristic, manual, and difficult to scale. This paper presents the first comprehensive evaluation of Automatic Prompt Optimization (APO) methods for real-world, high-stakes multiclass classification in a commercial setting, addressing a critical gap in the existing literature where most of the APO frameworks have been validated only on benchmark classification tasks of limited complexity. We introduce APE-OPRO, a novel hybrid framework that combines the complementary strengths of APE and OPRO, achieving notably better cost-efficiency, around $18\%$ improvement over OPRO, without sacrificing performance. We benchmark APE-OPRO alongside both gradient-free (APE, OPRO) and gradient-based (ProTeGi) methods on a dataset of ~2,500 labeled products. Our results highlight key trade-offs: ProTeGi offers the strongest absolute performance at lower API cost but higher computational time as noted in~\cite{protegi}, while APE-OPRO strikes a compelling balance between performance, API efficiency, and scalability. We further conduct ablation studies on depth and breadth hyperparameters, and reveal notable sensitivity to label formatting, indicating implicit sensitivity in LLM behavior. These findings provide actionable insights for implementing APO in commercial applications and establish a foundation for future research in multi-label, vision, and multimodal prompt optimization scenarios.

Momentum-Aided Prompt Optimization (MAPO) enhances the efficiency and efficacy of prompt optimization for Large Language Models (LLMs). Building on ProTeGi, MAPO uses positive natural language "gradients" and a momentum-based extension to refine prompts effectively. By tracking gradient history, MAPO avoids local minima and oscillations. It also utilizes beam search and an Upper Confidence Bound (UCB) algorithm for balanced candidate expansion and selection. Benchmark testing shows that MAPO achieves faster convergence time with fewer API calls and higher F1 scores than ProTeGi, proving it as a robust and scalable solution for automated prompt engineering in LLMs.

Large Language Models (LLMs) have shown impressive capabilities in many scenarios, but their performance depends, in part, on the choice of prompt. Past research has focused on optimizing prompts specific to a task. However, much less attention has been given to optimizing the general instructions included in a prompt, known as a system prompt. To address this gap, we propose SPRIG, an edit-based genetic algorithm that iteratively constructs prompts from prespecified components to maximize the model's performance in general scenarios. We evaluate the performance of system prompts on a collection of 47 different types of tasks to ensure generalizability. Our study finds that a single optimized system prompt performs on par with task prompts optimized for each individual task. Moreover, combining system and task-level optimizations leads to further improvement, which showcases their complementary nature. Experiments also reveal that the optimized system prompts generalize effectively across model families, parameter sizes, and languages. This study provides insights into the role of system-level instructions in maximizing LLM potential.

Large Language Models (LLMs) have shown impressive performance as general purpose agents, but their abilities remain highly dependent on prompts which are hand written with onerous trial-and-error effort. We propose a simple and nonparametric solution to this problem, Automatic Prompt Optimization (APO), which is inspired by numerical gradient descent to automatically improve prompts, assuming access to training data and an LLM API. The algorithm uses minibatches of data to form natural language "gradients" that criticize the current prompt. The gradients are then "propagated" into the prompt by editing the prompt in the opposite semantic direction of the gradient. These gradient descent steps are guided by a beam search and bandit selection procedure which significantly improves algorithmic efficiency. Preliminary results across three benchmark NLP tasks and the novel problem of LLM jailbreak detection suggest that Automatic Prompt Optimization can outperform prior prompt editing techniques and improve an initial prompt's performance by up to 31%, by using data to rewrite vague task descriptions into more precise annotation instructions.