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Deriving the Optimum of the KL-Constrained Reward Maximization Objective
by Writings, Papers and Blogs on Text ModelsAugust 25th, 2024
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The mathematical derivations in this appendix aim to derive Equation 4, which represents the objective function for optimizing the policy in the KL-constrained reward maximization problem. The objective is to find a policy that aligns well with human preferences while minimizing the KL-divergence from a reference policy. The partition function Z(x', π) plays a crucial role in normalizing the probability distribution.
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
Mathematical Derivations
A.1 Deriving the Optimum of the KL-Constrained Reward Maximization Objective
In this appendix, we will derive Eq. 4. Analogously to Eq. 3, we optimize the following objective:
where we have partition function:
This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license.
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