Iterate to Accelerate: A Unified Framework for Iterative Reasoning and Feedback Convergence (2502.03787v1)

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 LLMs. 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.

• The paper presents a unified framework using Bregman divergences and adaptive feedback to achieve an optimal O(1/t^2) convergence rate.
• It integrates operator averaging and feedback architectures to efficiently approximate fixed-point functions in non-Euclidean spaces.
• The framework bridges classical methods with modern iterative reasoning, suggesting promising extensions to stochastic and reinforcement learning settings.

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

This paper presents a unified framework advancing iterative reasoning processes by integrating Bregman divergences, operator averaging, and adaptive feedback mechanisms. It establishes important theoretical convergence rates and examines the necessity of feedback architectures for efficiently approximating fixed-point functions.

Introduction and Framework Development

Iterative techniques are crucial in optimization algorithms such as mirror descent and dynamic programming. Modern applications, particularly neural networks, often involve complex iterative processes like chain-of-thought reasoning. Standard acceleration methods such as Nesterov's momentum improve convergence rates significantly in convex scenarios, but don't always extend smoothly into non-Euclidean spaces where perturbations are prevalent.

This research develops a generalized framework for iterative reasoning using Bregman divergences. The framework recasts update dynamics in non-Euclidean terms and presents an accelerated convergence scheme achieving an optimal $O(1/t^2)$ rate in noise-free conditions. The analysis underscores the indispensable role of feedback, demonstrating that recurrent architectures approximate fixed-point functions efficiently while feedforward models with similar accuracy require exponential layer depth.

Mathematical Preliminaries

Bregman Divergences and Non-Euclidean Geometry

Bregman divergence, using a strictly convex and differentiable function $\phi$, generalizes squared Euclidean distance in non-Euclidean geometries. It is particularly useful in settings where convexity and smoothness assumptions hold, providing quadratic bounds necessary for establishing strong theoretical guarantees.

Iterative Operators

The iterative process, utilizing state $s \in \mathcal{S}$ and auxiliary information $y \in \mathcal{Y}$, aims to find a unique fixed point $s^*$ satisfying $\mathcal{T}(s^*, y)=s^*$. Convergence is tracked using Bregman divergence $D_\phi(s_t, s^*)$, and the operator $\mathcal{T}$ brings together classical methods and modern reasoning approaches.

Accelerated Convergence Through Higher-Order Averaging

Iterative Update Scheme

The research adopts an update rule integrating operator dynamics, state averaging, and perturbation handling to achieve accelerated convergence. Setting the averaging parameter $\alpha_t = \frac{2}{t+2}$, the iterative update efficiently accounts for operator geometry and adaptive perturbations.

Theoretical Convergence Proof

Under standard assumptions like non-Euclidean contractivity and smoothness, the research proves the convergence of the iterates $\{s_t\}$ toward $s^*$ with $O(1/t^2)$ rate in noise-free conditions.

Feedback Structures Necessity

The paper demonstrates that feedback architectures fundamentally enhance expressiveness, enabling efficient approximation of complex functions that feedforward models cannot emulate without exponential depth. This expressiveness theorem solidifies the role of feedback in handling intricate reasoning tasks inherent in modern applications.

Discussion

The unified perspective offered by this framework bridges different iterative methodologies, ensuring robust performance in non-Euclidean spaces while enabling accelerated convergence amidst perturbations. Extending the framework to stochastic settings and more application-specific challenges remains promising.

Future Directions

Potential enhancements include adopting stochastic noise adaptations, exploring multi-agent scenarios, and refining averaging strategies. Theoretical groundwork laid by this paper suggests expanding into areas like reinforcement learning and LLMs where iterative refinement and feedback mechanisms are crucial.

Conclusion

The paper establishes a comprehensive framework for iterative reasoning processes, aligning traditional techniques with modern applications through Bregman divergences and adaptive feedback mechanisms. These theoretical insights promise continued advancement in domains requiring intricate iterative reasoning, notably neural computation and optimization.

In closing, researchers are encouraged to explore extending this framework to real-world systems, reinforcing the convergence benefits amid practical complexities within iterative reasoning tasks.