$p1$: Better Prompt Optimization with Fewer Prompts

TL;DR

This paper introduces $p1$, a prompt filtering method that enhances prompt optimization by selecting high-variance user prompts, leading to better system prompts especially on reasoning benchmarks.

Contribution

The paper reveals how prompt optimization effectiveness depends on prompt and response variance and proposes $p1$, a simple filtering approach that improves optimization outcomes.

Findings

Scaling to more prompts can reduce variance among prompts, hindering optimization.

Filtering prompts to select high-variance prompts improves prompt optimization.

Training on just two prompts can produce a system prompt that generalizes well.

Abstract

Prompt optimization improves language models without updating their weights by searching for a better system prompt, but its effectiveness varies widely across tasks. We study what makes a task amenable to prompt optimization. We show that the reward variance across different system prompts can be decomposed into two components: variance among responses, which captures generation stochasticity, and variance among system prompts, which captures differences in system prompt quality. Prompt optimization succeeds when variance among system prompts is sufficiently large, but fails when variance among responses dominates the variance of the system prompts. Surprisingly, we further show that scaling to more user prompts can hurt optimization by reducing variance among system prompts, especially on heterogeneous datasets where different user prompts favor different system prompts. Motivated by this insight, we propose $p1$, a simple user prompt filtering method that selects a small subset of user prompts with high variance across candidate system prompts. This subset of user prompts allows one to distinguish a good system prompt from a bad one, making system optimization easier. Experiments on reasoning benchmarks show that $p1$ substantially improves prompt optimization over training on the full dataset and outperforms strong baselines such as GEPA. Notably, training on only two prompts from AIME 24 yields a system prompt that generalizes well to other reasoning benchmarks.