用几何方法统一加速推理，理论证明收敛更快且必要。
Iterate to Accelerate: A Unified Framework for Iterative Reasoning and Feedback Convergence
- 基于非欧几何与自适应反馈，统一多种推理算法
- 无持续扰动下收敛率达 $O(1/t^2)$，优于传统方法
- 解释大模型链式思考为何需要迭代，适合优化与推理研究者
我们提出一种基于Bregman散度的统一迭代推理框架，结合高阶算子平均与自适应反馈机制。在温和光滑性和压缩性假设下，该通用更新方案不仅统一了镜面下降与动态规划等经典方法，还捕捉了大型语言模型中的现代思维链推理过程。理论上证明，在无持续扰动时，加速迭代更新可达到 $O(1/t^2)$ 的收敛速度；进一步表明，反馈（迭代）架构是高效逼近特定不动点函数所必需的。这些理论洞见将经典加速技术与神经计算和优化的当代应用相连接。
原文摘要 · Abstract (English)
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.
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