Deriving the DPO Objective Under the Plackett-Luce Model
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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.
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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
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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
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B DPO Implementation Details and Hyperparameters
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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
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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
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
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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:
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This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
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