Table of Links

Abstract and 1. Introduction

2 Related Work

3 Preliminaries

4 Direct Preference Optimization

5 Theoretical Analysis of DPO

6 Experiments

7 Discussion, Acknowledgements, and References

Author Contributions

A Mathematical Derivations

A.1 Deriving the Optimum of the KL-Constrained Reward Maximization Objective

A.2 Deriving the DPO Objective Under the Bradley-Terry Model

A.3 Deriving the DPO Objective Under the Plackett-Luce Model

A.4 Deriving the Gradient of the DPO Objective and A.5 Proof of Lemma 1 and 2

A.6 Proof of Theorem 1

B DPO Implementation Details and Hyperparameters

C Further Details on the Experimental Set-Up and C.1 IMDb Sentiment Experiment and Baseline Details

C.2 GPT-4 prompts for computing summarization and dialogue win rates

C.3 Unlikelihood baseline

D Additional Empirical Results

D.1 Performance of Best of N baseline for Various N and D.2 Sample Responses and GPT-4 Judgments

D.3 Human study details

B DPO Implementation Details and Hyperparameters

DPO is relatively straightforward to implement; PyTorch code for the DPO loss is provided below:

Unless noted otherwise, we use a β = 0.1, batch size of 64 and the RMSprop optimizer with a learning rate of 1e-6 by default. We linearly warmup the learning rate from 0 to 1e-6 over 150 steps. For TL;DR summarization, we use β = 0.5, while rest of the parameters remain the same.