BayesPO: Bayesian Prompt Optimization via Parallel-Tempered Gradient-Guided Discrete MCMC

Prompt optimizers automate the search for prompts that improve language-model performance, but existing methods rely on a predefined optimization procedure: the algorithm determines which candidates to explore and how the search progresses, while the language model generates or refines prompt proposals. We introduce RLMOpt, a prompt optimizer that makes the search policy itself language-model-driven through a recursive language model (RLM). The RLM agent operates over a tool-based environment, inspecting task information, analyzing failures, generating candidates, allocating evaluation budget, and deciding when to stop. A deterministic harness complements the agent by enforcing objective scoring, Pareto-based selection, and regression constraints. We evaluate RLMOpt across four benchmarks spanning structured clinical information extraction (Chia), multi-hop question answering (HotpotQA), verifiable instruction following (IFBench-2025), and multi-turn tool-calling agents (BFCL). In a matched comparison at a single seed, RLMOpt obtains the best held-out score on all four benchmarks and leads the four-task mean (0.610 against 0.589 for GEPA). Repeating each benchmark across seeds yields 11 matched benchmark-seed comparisons, in which RLMOpt outperforms GEPA in 9 cases. Across all 11 runs, it never produced a prompt that underperformed its seed, whereas GEPA fell below its starting point twice. It is also more efficient, achieving these results with fewer search rollouts while producing prompts that are 27-79% the size of those produced by GEPA. Our results further show that optimization gains are determined primarily by the headroom available in the seed prompt, rather than by the search budget. Efficient optimization therefore depends on reaching the available headroom reliably and with minimal search

Automatic prompt optimization (APO) has driven significant gains in LLM-based agentic workflows. However, existing methods treat each task's prompt as a monolithic, instance-blind string optimized through global edits, producing brittle updates and preventing the reuse of learned sub-behaviors. We propose Prompt Codebooks (PCO), a novel compositional prompt optimization framework that recasts APO as discrete learning over a finite vocabulary of natural-language instincts - atomic, reusable instruction units. PCO organizes prompt-construction knowledge in a discrete codebook and routes each input to a small subset of entries via an LLM-based encoder; a generator composes them into a prompt for the frozen target model; a critic emits a structured verdict that decomposes by attribution into per-variable textual gradients, jointly training the encoder, generator, and codebook under a language-valued min-max objective. The resulting routing is per-instance: different inputs in the same task receive different instinct compositions, a regime structurally inexpressible under instance-blind methods. Across six benchmarks on Qwen3-8B and LLaMA-3.1-8B, PCO improves over zero-shot by up to +30.36 points, surpasses the strongest prior baseline (GEPA) by +3.34 on HotpotQA and +1.11 in aggregate, and reduces deployed prompt length by up to 14.1x versus MIPROv2 and 3.0x versus GEPA using only K=16 instincts.

Gradient-free post-training has emerged as a compelling alternative to gradient-based optimization for large language models (LLMs), but existing approaches remain costly. We ask whether structured search can identify a strong single expert under a modest evaluation budget. Motivated by evidence that useful weight updates lie in low-dimensional subspaces, we apply Bayesian optimization within a random linear embedding of weight space. Our method requires no backpropagation and uses a Gaussian process surrogate to guide candidate evaluations efficiently. Across several reasoning benchmarks with Qwen2.5-Instruct models from 0.5B to 3B parameters, Bayesian optimization using five times less candidate evaluations matches or exceeds RandOpt. These results show that surrogate-guided search can substantially reduce the evaluation cost of gradient-free post-training while producing stronger deployable single experts.