**Authors:**

(1) Rafael Rafailo, Stanford University and Equal contribution; more junior authors listed earlier;

(2) Archit Sharma, Stanford University and Equal contribution; more junior authors listed earlier;

(3) Eric Mitchel, Stanford University and Equal contribution; more junior authors listed earlier;

(4) Stefano Ermon, CZ Biohub;

(5) Christopher D. Manning, Stanford University;

(6) Chelsea Finn, Stanford University.

## Table of Links

4 Direct Preference Optimization

7 Discussion, Acknowledgements, and References

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

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

D Additional Empirical Results

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

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

The Plackett-Luce model [30, 21] is a generalization of the Bradley-Terry model over rankings (rather than just pair-wise comparisons). Similar to to the Bradley-Terry model, it stipulates that when presented with a set of possible choices, people prefer a choice with probability proportional to the value of some latent reward function for that choice. In our context, when presented with a prompt x and a set of K answers y1, . . . , yK a user would output a permutation τ : [K] → [K], giving their ranking of the answers. The Plackett-Luce model stipulates that

Notice that when K = 2, Equation 18 reduces to the Bradley-Terry model. However, for the general Plackett-Luce model, we can still utilize the results of Eq. 5 and substitute the reward function parameterized by its optimal policy. Similarly to Appendix A.2, the normalization constant Z(x) cancels out and we’re left with:

This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.