Iterate to Accelerate: A Unified Framework for Iterative Reasoning and Feedback Convergence
Jacob Fein-Ashley

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 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.
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Taxonomy
TopicsLogic, programming, and type systems · AI-based Problem Solving and Planning · Logic, Reasoning, and Knowledge
