Energy-guided Recursive Model

2026-07-11

AI Summary

This work addresses the lack of a principled trajectory selection mechanism in existing recursive reasoning models during inference, which often compromises reasoning quality. To this end, the paper introduces, for the first time, an explicit Hopfield energy function into the recursive reasoning framework, formulating an energy-based trajectory selector. This selector leverages Hopfield-type memory structures to guide the filtering and refinement of candidate reasoning trajectories, while incorporating parallel tempering to enhance sampling efficiency. The proposed method achieves state-of-the-art performance across multiple benchmarks, significantly outperforming current recursive reasoning approaches with solution accuracies of 98.97% on Sudoku, 88.04% on PPBench, and 99.30% on Maze tasks.

Abstract

Recursive reasoning models address structured problems by repeatedly updating latent states of small neural networks. However, their test-time scaling lacks a principled inference mechanism: increasing depth or stochastic breadth generates more trajectories without a clear criterion for selection, and existing methods predominantly rely on additional q-heads or heuristic voting. Here, we develop the Energy-guided Recursive Model (ERM), which introduces an intrinsic selection principle based on explicit Hopfield energies. ERM leverages Hopfield-type memories of valid local or global structures to define the selector over candidate trajectories. The resulting energy seamlessly integrates with energy-based techniques such as parallel tempering to enhance sampling efficiency and ranking. With $D=64$ recurrent steps and $K=128$ candidates, ERM reaches optimal solutions on Sudoku ($98.97\%$), Pencil Puzzle Bench (PPBench, $88.04\%$) and Maze ($99.30\%$), improving upon recent Probabilistic Tiny Recursive Model and Equilibrium Reasoners. These results suggest that incorporating explicit energy functions into recursive reasoning offers a principled path toward more effective inference.

Problem

• recursive reasoning
• inference mechanism
• trajectory selection
• energy-based models
• structured problems

Innovation

• Energy-guided Recursive Model
• Hopfield energy
• recursive reasoning
• trajectory selection
• energy-based inference