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

Jacob Fein-Ashley

TL;DR

This work presents a unified framework for iterative reasoning in non-Euclidean spaces by leveraging Bregman divergences and higher-order averaging to achieve accelerated convergence. A generalized update with $s_{t+1}=(1-\alpha_t)s_t+\alpha_t\mathcal{T}(s_t,y_t)+\eta_t$ is shown to attain $D_{\phi}(s_t,s^*)=O\left(\frac{1}{t^2}\right)$ in the noise-free case under non-Euclidean contractivity and smoothness, with robustness to adaptive perturbations. A key depth-separation result establishes that iterative (feedback) architectures can approximate fixed-point functions in $t=O\left(\frac{1}{\sqrt{\epsilon}}\right)$ steps, while feedforward networks require exponential depth to achieve similar accuracy, highlighting the essential role of recurrence in complex reasoning. The findings bridge classical optimization acceleration with contemporary neural computation, offering geometry-aware design principles for robust, efficient iterative reasoning in both theoretical and practical settings.

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.

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

TL;DR

This work presents a unified framework for iterative reasoning in non-Euclidean spaces by leveraging Bregman divergences and higher-order averaging to achieve accelerated convergence. A generalized update with is shown to attain in the noise-free case under non-Euclidean contractivity and smoothness, with robustness to adaptive perturbations. A key depth-separation result establishes that iterative (feedback) architectures can approximate fixed-point functions in steps, while feedforward networks require exponential depth to achieve similar accuracy, highlighting the essential role of recurrence in complex reasoning. The findings bridge classical optimization acceleration with contemporary neural computation, offering geometry-aware design principles for robust, efficient iterative reasoning in both theoretical and practical settings.

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.

Paper Structure

This paper contains 35 sections, 2 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bregman Divergences and Non-Euclidean Geometry
  4. Iterative Operators and Convergence Metrics
  5. A Generalized Framework for Iterative/Sequential Reasoning
  6. Accelerated Convergence via Higher-Order Averaging
  7. Accelerated Iterative Update Scheme
  8. Main Theorem: Accelerated Convergence and Optimality
  9. Full Proof of Theorem \ref{['thm:accelerated']}
  10. Step 1: Reformulating the Update.
  11. Step 2: Expansion via the Three-Point Identity.
  12. Step 3: Descent Lemma for the Averaged Update.
  13. Step 4: Bounding the Cross-Term.
  14. Step 5: Combining the Estimates.
  15. Step 6: Bounding the Remainder Terms.
  16. ...and 20 more sections

Key Result

Theorem 4.1

Let $(\mathcal{S},d)$ be a complete metric space, and let $\phi:\mathcal{S}\to\mathbb{R}$ be a strictly convex, continuously differentiable function that is also $\mu$-strongly convex and $L$-smooth, inducing the Bregman divergence Assume that: Then, the sequence $\{s_t\}$ generated by eq:acc-update satisfies where $C>0$ is a constant depending on the initial error $D_\phi(s_0,s^*)$ and the con

Theorems & Definitions (4)

  • Theorem 4.1: Accelerated Convergence of Unified Iterative Reasoning
  • proof
  • Theorem 7.1: Expressiveness of Feedback Structures
  • proof