Iterate to Accelerate: A Unified Framework for Iterative Reasoning and Feedback Convergence
This bridges classical acceleration techniques with modern applications in neural computation and optimization, though it appears incremental as it unifies existing methods.
The paper tackles the problem of iterative reasoning by introducing a unified framework that combines non-Euclidean geometry, operator averaging, and adaptive feedback, proving an O(1/t^2) convergence rate and showing feedback architectures are necessary for efficient fixed-point approximation.
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.