Iterate to Accelerate: A Unified Framework for Iterative Reasoning and Feedback Convergence

TL;DR

This paper presents a unified iterative reasoning framework using non-Euclidean geometry, achieving accelerated convergence and highlighting the importance of feedback mechanisms in neural and optimization contexts.

Contribution

It introduces a generalized update scheme that unifies classical methods and modern reasoning processes, with theoretical convergence guarantees and insights into feedback architectures.

Findings

Achieves $O(1/t^2)$ convergence rate without perturbations.

Unifies mirror descent, dynamic programming, and chain-of-thought reasoning.

Demonstrates feedback architectures are essential for fixed-point approximation.

Abstract

We introduce a unified framework for iterative reasoning that leverages non-Euclidean geometry via Bregman divergences, higher-order operator averaging, and adaptive feedback mechanisms. Our analysis establishes that, under mild smoothness and contractivity assumptions, a generalized update scheme not only unifies classical methods such as mirror descent and dynamic programming but also captures modern chain-of-thought reasoning processes in large language models. In particular, we prove that our accelerated iterative update achieves an $O(1/t^2)$ convergence rate in the absence of persistent perturbations, and we further demonstrate that feedback (iterative) architectures are necessary to approximate certain fixed-point functions efficiently. These theoretical insights bridge classical acceleration techniques with contemporary applications in neural computation and optimization.