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RL-STaR: Theoretical Analysis of Reinforcement Learning Frameworks for Self-Taught Reasoner

Fu-Chieh Chang, Yu-Ting Lee, Hui-Ying Shih, Yi Hsuan Tseng, Pei-Yuan Wu

TL;DR

This work provides a theoretical framework for understanding the effectiveness of reinforcement learning on CoT reasoning and STaR, and aims to bridge empirical findings with theoretical insights, advancing reinforcement learning approaches for reasoning in LLMs.

Abstract

The reasoning abilities of large language models (LLMs) have improved with chain-of-thought (CoT) prompting, allowing models to solve complex tasks stepwise. However, training CoT capabilities requires detailed reasoning data, which is often scarce. The self-taught reasoner (STaR) framework addresses this by using reinforcement learning to automatically generate reasoning steps, reducing reliance on human-labeled data. Although STaR and its variants have demonstrated empirical success, a theoretical foundation explaining these improvements is lacking. This work provides a theoretical framework for understanding the effectiveness of reinforcement learning on CoT reasoning and STaR. Our contributions are: (1) criteria for the quality of pre-trained models necessary to initiate effective reasoning improvement; (2) an analysis of policy improvement, showing why LLM reasoning improves iteratively with STaR; (3) conditions for convergence to an optimal reasoning policy; and (4) an examination of STaR's robustness, explaining how it can improve reasoning even when incorporating occasional incorrect steps; This framework aims to bridge empirical findings with theoretical insights, advancing reinforcement learning approaches for reasoning in LLMs.

RL-STaR: Theoretical Analysis of Reinforcement Learning Frameworks for Self-Taught Reasoner

TL;DR

This work provides a theoretical framework for understanding the effectiveness of reinforcement learning on CoT reasoning and STaR, and aims to bridge empirical findings with theoretical insights, advancing reinforcement learning approaches for reasoning in LLMs.

Abstract

The reasoning abilities of large language models (LLMs) have improved with chain-of-thought (CoT) prompting, allowing models to solve complex tasks stepwise. However, training CoT capabilities requires detailed reasoning data, which is often scarce. The self-taught reasoner (STaR) framework addresses this by using reinforcement learning to automatically generate reasoning steps, reducing reliance on human-labeled data. Although STaR and its variants have demonstrated empirical success, a theoretical foundation explaining these improvements is lacking. This work provides a theoretical framework for understanding the effectiveness of reinforcement learning on CoT reasoning and STaR. Our contributions are: (1) criteria for the quality of pre-trained models necessary to initiate effective reasoning improvement; (2) an analysis of policy improvement, showing why LLM reasoning improves iteratively with STaR; (3) conditions for convergence to an optimal reasoning policy; and (4) an examination of STaR's robustness, explaining how it can improve reasoning even when incorporating occasional incorrect steps; This framework aims to bridge empirical findings with theoretical insights, advancing reinforcement learning approaches for reasoning in LLMs.

Paper Structure

This paper contains 44 sections, 12 theorems, 61 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Theoretical Frameworks
  4. Problem Formulation
  5. Settings of Our Theoretical Analysis
  6. Markov Assumption of Language Models' Input:
  7. Simplification of STaR Algorithm:
  8. Assumption of the Ground-Truth Reasoner $\bar{\pi}$:
  9. Theoretical Results
  10. A Toy Example
  11. Main Theorem
  12. Experiments
  13. Theoretical Values of $J(P_t)$ Versus Experimental Values of $J(P_t)$
  14. Theoretical Values:
  15. Experimental Values:
  16. ...and 29 more sections

Key Result

Theorem 3.1

Given the toy example defined in the previous paragraph, in the RL-STaR algorithm, for every CoT step $n\in[2]$, we assume that $P_{0,n}$ represents the state transition estimated by a pre-trained LLM at this step, which is an interpolation between $\bar{P}_n$ and $P_{u,n}$ with a coefficient $0 \le In the RL-STaR algorithm, we assume that the training dataset is $\mathcal{D} = \{(s_{0,1}, s_{2,1}

Figures (2)

  • Figure 1: The first two figures on the left illustrate the comparison between theoretical values (blue dotted line) and experimental values (red dashed line) of $J(P_t)$, with the first figure corresponding to $\delta_0=0.2$ and the second figure corresponding to $\delta_0=0.1$. The remaining two figures on the right depict the comparison of transitions $P(S_1|S_0)$, directly extracted from dataset $\mathcal{D}_1$ (third figure), and the transitions $P_{1,1}(S_1|S_0)$ learned by LLMs during the RL-STaR algorithm (fourth figure).
  • Figure 2: Values of $J(P_t)$ when $(\delta_{0,1},\delta_{0,2},\delta_{0,3})=(0,0.2,0.2)$.

Theorems & Definitions (13)

  • Theorem 3.1: Sufficient Conditions for Pre-trained Models
  • Theorem 3.2: Conditions of Pre-trained Models
  • Theorem 3.3: Convergence Speed of $\delta_{t,n}$
  • Corollary 3.4: Policy Improvement
  • Corollary 3.5: Convergence to the Optimal Policy
  • Corollary 3.6: Diminishing of Incorrect Reasoning Trajectories
  • Remark 3.7
  • Theorem A.1: Convergence Speed of $\delta_t$
  • Corollary A.2: Policy Improvement in the Toy Example
  • Corollary A.3: Convergence to Optimal Policy in the Toy Example
  • ...and 3 more