A Probabilistic Inference Scaling Theory for LLM Self-Correction

TL;DR

This paper introduces a probabilistic theory modeling how LLMs improve their accuracy through self-correction over multiple rounds, providing a mathematical framework that predicts accuracy evolution.

Contribution

It presents a novel probabilistic model explaining the dynamics of accuracy improvement in LLM self-correction, validated by experiments across various models and datasets.

Findings

Accuracy follows an exponential convergence pattern.

The model accurately predicts accuracy after a single self-correction round.

Theoretical predictions closely match empirical results.

Abstract

Large Language Models (LLMs) have demonstrated the capability to refine their generated answers through self-correction, enabling continuous performance improvement over multiple rounds. However, the mechanisms underlying how and why accuracy evolves during this iterative process remain unexplored. To fill this gap, we propose a probabilistic theory to model the dynamics of accuracy change and explain the performance improvements observed in multi-round self-correction. Through mathematical derivation, we establish that the accuracy after the $t^{th}$ round of self-correction is given by: $Acc_t = Upp - \alpha^t(Upp - Acc_0),$ where $Acc_0$ denotes the initial accuracy, $Upp$ represents the upper bound of accuracy convergence, and $\alpha$ determines the rate of convergence. Based on our theory, these parameters can be calculated and the predicted accuracy curve then can be obtained through only a single round of self-correction. Extensive experiments across diverse models and datasets demonstrate that our theoretical predictions align closely with empirical accuracy curves, validating the effectiveness of the theory. Our work provides a theoretical foundation for understanding LLM self-correction, thus paving the way for further explorations.