Searching for the Limit Point of Self-Improvement in Large Language Models Through Changes in the Probability of Their Own Answers

Abstract

This paper proposes a research hypothesis concerning the possibility of identifying the point at which further iterative reconsideration of a task by a large language model ceases to produce substantial improvement. The central idea is based on an iterative tool that does not simply ask a model to answer the same prompt repeatedly. After each generated answer, the tool analyzes the result and constructs a new prompt containing the original task and the previous answer. The model then receives this new prompt, reconsiders the task, checks its previous solution, and generates a new version of the answer. The resulting process can be represented as: Q₀ → A₀ → Q₁ → A₁ → Q₂ → A₂ → … where Q₀ is the original prompt, Aₙ is the model's answer at iteration n, and Qₙ₊₁ is a new prompt constructed from the previous solution. The original prompt Q₀ remains unchanged and serves as the common reference point for evaluating every subsequent answer. For each answer, a measurable quantity Wₙ = W(Q₀,Aₙ) is proposed. The main hypothesis is that, for many tasks, Wₙ will initially increase as the model repeatedly revisits its solution, after which the rate of increase will decrease and eventually reach a stable plateau. This plateau is proposed as a possible indicator of a limit point of iterative self-improvement, beyond which additional computation provides diminishing returns. The paper does not claim that a higher internal score necessarily corresponds to a correct answer. One of the primary experimental objectives is to determine how strongly changes in Wₙ correlate with actual improvements in answer correctness. The proposed approach is presented as a research hypothesis for adaptive determination of reasoning depth and requires empirical validation across different models, task types, and methods for measuring internal confidence.