Deriving the DPO Objective Under the Bradley-Terry Model
by Writings, Papers and Blogs on Text ModelsAugust 25th, 2024
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Learn how to derive the DPO objective under the bradley-terry model.
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.2 Deriving the DPO Objective Under the Bradley-Terry Model
It is straightforward to derive the DPO objective under the Bradley-Terry preference model as we have
In Section 4 we showed that we can express the (unavailable) ground-truth reward through its corresponding optimal policy:
Substituting Eq. 17 into Eq. 16 we obtain:
This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
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