Authors:
(1) Corby Rosset, Microsoft Research and Correspondence to corbyrosset@microsoft.com;
(2) Ching-An Cheng, Microsoft Research;
(3) Arindam Mitra, Microsoft Research;
(4) Michael Santacroce, Microsoft Research;
(5) Ahmed Awadallah, Microsoft Research and Correspondence to hassanam@microsoft.com;
(6) Tengyang Xie, Microsoft Research and Correspondence to tengyangxie@microsoft.com. Table of Links Abstract and 1 Introduction 2 Preliminaries 2.1 RLHF Based on Reward Models 2.2 RLHF with General Preferences 3 Direct Nash Optimization and 3.1 Derivation of Algorithm 1 3.2 Theoretical Analysis 4 Practical Algorithm – Iterative Contrastive Self-Improvement 5 Experiments and 5.1 Experimental Setup 5.2 Results and Analysis 6 Related Work 7 Conclusion and References Appendix A Extension to Regularized Preferences B Detailed Proofs C Additional Experimental Details B Detailed Proofs In this section, we provide detailed proofs for our theoretical results. Note that, the definitions and assumptions presented heavily adopts the ideas related to version space and concentrability from reinforcement learning theory literature (esp., Xie et al., 2021, 2023). Nevertheless, the descriptions provided herein are intentionally simplified to elucidate the core insights into the algorithmic design. A full and exhaustive theoretical analysis falls outside the primary scope of this paper. We now make the following definitions and assumptions. Definition 2 can be viewed as a natural extension of concentrability from the (offline) reinforcement learning literature to our setup. Proof of Theorem 2. We will now present the proof using the following two-step procedure. Step 1: From regression with log loss to squared error bound. By standard results on the regression with the logarithmic loss, we know Note that similar results could also apply beyond finite Π. For simplicity, we omit the detailed discussion in our paper. For more in-depth discussions about regression with the logarithmic loss, the reader can refer to, e.g., Foster and Krishnamurthy (2021). On the other hand, we have This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Corby Rosset, Microsoft Research and Correspondence to corbyrosset@microsoft.com; (2) Ching-An Cheng, Microsoft Research; (3) Arindam Mitra, Microsoft Research; (4) Michael Santacroce, Microsoft Research; (5) Ahmed Awadallah, Microsoft Research and Correspondence to hassanam@microsoft.com; (6) Tengyang Xie, Microsoft Research and Correspondence to tengyangxie@microsoft.com. Authors: Authors: (1) Corby Rosset, Microsoft Research and Correspondence to corbyrosset@microsoft.com; (2) Ching-An Cheng, Microsoft Research; (3) Arindam Mitra, Microsoft Research; (4) Michael Santacroce, Microsoft Research; (5) Ahmed Awadallah, Microsoft Research and Correspondence to hassanam@microsoft.com; (6) Tengyang Xie, Microsoft Research and Correspondence to tengyangxie@microsoft.com. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminaries 2 Preliminaries 2.1 RLHF Based on Reward Models 2.1 RLHF Based on Reward Models 2.2 RLHF with General Preferences 2.2 RLHF with General Preferences 3 Direct Nash Optimization and 3.1 Derivation of Algorithm 1 3 Direct Nash Optimization and 3.1 Derivation of Algorithm 1 3.2 Theoretical Analysis 3.2 Theoretical Analysis 4 Practical Algorithm – Iterative Contrastive Self-Improvement 4 Practical Algorithm – Iterative Contrastive Self-Improvement 5 Experiments and 5.1 Experimental Setup 5 Experiments and 5.1 Experimental Setup 5.2 Results and Analysis 5.2 Results and Analysis 6 Related Work 6 Related Work 7 Conclusion and References 7 Conclusion and References Appendix Appendix A Extension to Regularized Preferences A Extension to Regularized Preferences B Detailed Proofs B Detailed Proofs C Additional Experimental Details C Additional Experimental Details B Detailed Proofs In this section, we provide detailed proofs for our theoretical results. Note that, the definitions and assumptions presented heavily adopts the ideas related to version space and concentrability from reinforcement learning theory literature (esp., Xie et al., 2021, 2023). Nevertheless, the descriptions provided herein are intentionally simplified to elucidate the core insights into the algorithmic design. A full and exhaustive theoretical analysis falls outside the primary scope of this paper. We now make the following definitions and assumptions. Definition 2 can be viewed as a natural extension of concentrability from the (offline) reinforcement learning literature to our setup. Proof of Theorem 2. We will now present the proof using the following two-step procedure. Proof of Theorem 2. Proof of Theorem 2. Step 1: From regression with log loss to squared error bound. By standard results on the regression with the logarithmic loss, we know Step 1: From regression with log loss to squared error bound. Note that similar results could also apply beyond finite Π. For simplicity, we omit the detailed discussion in our paper. For more in-depth discussions about regression with the logarithmic loss, the reader can refer to, e.g., Foster and Krishnamurthy (2021). On the other hand, we have This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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