Case Studies in Mathematical Optimization Using LLMs

:::info Authors:

(1) Chengrun Yang, Google DeepMind and Equal contribution;

(2) Xuezhi Wang, Google DeepMind;

(3) Yifeng Lu, Google DeepMind;

(4) Hanxiao Liu, Google DeepMind;

(5) Quoc V. Le, Google DeepMind;

(6) Denny Zhou, Google DeepMind;

(7) Xinyun Chen, Google DeepMind and Equal contribution.

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Table of Links

Abstract and 1. Introduction

2 Opro: Llm as the Optimizer and 2.1 Desirables of Optimization by Llms

2.2 Meta-Prompt Design

3 Motivating Example: Mathematical Optimization and 3.1 Linear Regression

3.2 Traveling Salesman Problem (TSP)

4 Application: Prompt Optimization and 4.1 Problem Setup

4.2 Meta-Prompt Design

5 Prompt Optimization Experiments and 5.1 Evaluation Setup

5.2 Main Results

5.3 Ablation Studies

5.4 Overfitting Analysis in Prompt Optimization and 5.5 Comparison with Evoprompt

6 Related Work

7 Conclusion, Acknowledgments and References

A Some Failure Cases

B Prompting Formats for Scorer Llm

C Meta-Prompts and C.1 Meta-Prompt for Math Optimization

C.2 Meta-Prompt for Prompt Optimization

D Prompt Optimization Curves on the Remaining Bbh Tasks

E Prompt Optimization on Bbh Tasks – Tabulated Accuracies and Found Instructions

3 MOTIVATING EXAMPLE: MATHEMATICAL OPTIMIZATION

We first demonstrate the potential of LLMs in serving as optimizers for mathematical optimization. In particular, we present a case study on linear regression as an example of continuous optimization, and on the Traveling Salesman Problem (TSP) as an example of discrete optimization. On both tasks, we see LLMs properly capture the optimization directions on small-scale problems merely based on the past optimization trajectory provided in the meta-prompt.

3.1 LINEAR REGRESSION

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:::info This paper is available on arxiv under CC0 1.0 DEED license.

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