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Why Self-Rewarding Works: Theoretical Guarantees for Iterative Alignment of Language Models

Shi Fu, Yingjie Wang, Shengchao Hu, Peng Wang, Dacheng Tao

TL;DR

This work provides the first rigorous theoretical foundation for Self-Rewarding Language Models (SRLMs) by analyzing iterative self-alignment without external feedback. It proves a fundamental single-step failure lower bound that highlights dependence on initialization, and develops finite-sample guarantees showing that iterative SRLMs achieve a convergence rate of $\tilde{O}(1/\sqrt{n})$ while the influence of the initial model decays exponentially with the number of iterations $T$. The authors identify a contraction mechanism on the policy condition number $\kappa_t$ that underpins a two-stage dynamic: Stage I self-correction toward internal consistency, followed by Stage II efficient statistical learning. They instantiate the framework for linear softmax models, deriving bounds that scale with the model’s effective dimension $d_{eff}(\lambda)$ under various spectral decays, and show how spectral properties can substantially tighten rates (e.g., logarithmic dependence under exponential decay). Overall, the results formalize why SRLMs can robustly improve alignment without external feedback and guide resource allocation for iterative self-improvement, while outlining clear directions for extending the theory to Transformer architectures.

Abstract

Self-Rewarding Language Models (SRLMs) achieve notable success in iteratively improving alignment without external feedback. Yet, despite their striking empirical progress, the core mechanisms driving their capabilities remain unelucidated, leaving a critical gap in theoretical understanding. This paper provides the first rigorous theoretical guarantees for SRLMs. We first establish a lower bound that characterizes the fundamental limits of a single update step, revealing a critical dependence on the quality of the initial model. We then derive finite-sample error bounds for the full iterative paradigm, showing that performance improves at a rate of $\widetilde{\mathcal{O}}\left(1/\sqrt{n}\right)$ with sample size $n$. Crucially, our analysis reveals that the dependence on the initial model decays exponentially with the number of iterations $T$. This provides a formal explanation for why self-rewarding succeeds: it robustly overcomes poor initialization by steering the dynamics toward internal stability and consistency. Finally, we instantiate our theoretical framework for the linear softmax model class, yielding tailored guarantees that connect our high-level insights to practical model architectures.

Why Self-Rewarding Works: Theoretical Guarantees for Iterative Alignment of Language Models

TL;DR

This work provides the first rigorous theoretical foundation for Self-Rewarding Language Models (SRLMs) by analyzing iterative self-alignment without external feedback. It proves a fundamental single-step failure lower bound that highlights dependence on initialization, and develops finite-sample guarantees showing that iterative SRLMs achieve a convergence rate of while the influence of the initial model decays exponentially with the number of iterations . The authors identify a contraction mechanism on the policy condition number that underpins a two-stage dynamic: Stage I self-correction toward internal consistency, followed by Stage II efficient statistical learning. They instantiate the framework for linear softmax models, deriving bounds that scale with the model’s effective dimension under various spectral decays, and show how spectral properties can substantially tighten rates (e.g., logarithmic dependence under exponential decay). Overall, the results formalize why SRLMs can robustly improve alignment without external feedback and guide resource allocation for iterative self-improvement, while outlining clear directions for extending the theory to Transformer architectures.

Abstract

Self-Rewarding Language Models (SRLMs) achieve notable success in iteratively improving alignment without external feedback. Yet, despite their striking empirical progress, the core mechanisms driving their capabilities remain unelucidated, leaving a critical gap in theoretical understanding. This paper provides the first rigorous theoretical guarantees for SRLMs. We first establish a lower bound that characterizes the fundamental limits of a single update step, revealing a critical dependence on the quality of the initial model. We then derive finite-sample error bounds for the full iterative paradigm, showing that performance improves at a rate of with sample size . Crucially, our analysis reveals that the dependence on the initial model decays exponentially with the number of iterations . This provides a formal explanation for why self-rewarding succeeds: it robustly overcomes poor initialization by steering the dynamics toward internal stability and consistency. Finally, we instantiate our theoretical framework for the linear softmax model class, yielding tailored guarantees that connect our high-level insights to practical model architectures.
Paper Structure (100 sections, 15 theorems, 307 equations)

This paper contains 100 sections, 15 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Notations and Setup.
  5. Self-Reward Mechanism and Data Generation.
  6. Model Update via DPO.
  7. Fundamental Limitations of Single-Step Self-Rewarding
  8. Finite-Sample Analysis for Iterative Self-Rewarding Alignment
  9. Application to Parametric Models: The Linear Softmax Case
  10. Conclusion
  11. Additional Related Work
  12. Limitation
  13. Proof of Theorem \ref{['thm:main results']}: Finite-Sample Guarantee for Iterative Self-Rewarding Alignment
  14. Notation and Assumptions.
  15. From Performance Gap to Suboptimality Probability.
  16. ...and 85 more sections

Key Result

Theorem 4.1

Let any self-rewarding algorithm produce a model $\pi_1$ using $n$ samples generated from a base model $\pi_0$. There exists a hard problem instance, characterized by a model class $\Pi$ and an initial policy condition number $\kappa_0$, such that the failure rate of any algorithm with a fixed budge where $\gtrsim$ hides logarithmic factors with respect to $n$ and $\kappa_0$.

Theorems & Definitions (38)

  • Theorem 4.1: Single-Step Failure Rate Lower Bound
  • Remark 4.2
  • Proposition 4.3: Greedy Decoding Fails for Unaligned Models
  • Remark 4.4
  • Definition 5.2: Confidence and Margin Constants
  • Theorem 5.3: Finite-Sample Guarantee for Iterative Self-Rewarding Alignment
  • Remark 5.4
  • Remark 5.5
  • Remark 5.6: Proof Sketch for Theorem \ref{['thm:main results']}
  • Corollary 5.7: Iterations to Suppress Initialization Effects
  • ...and 28 more