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Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning

ICML 2026 · Large language models › Reasoning and chain of thought · approach: Efficient and adaptive thinking · 0 citations (OpenAlex)

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Proposes Equilibrium Reasoners (EqR), weight-tied iterative models trained with randomized state initialization and noise injection plus segmented online training, which turn latent convergence into a reliable test-time scaling signal and reach 99.8% exact accuracy on Sudoku-Extreme and 93.0% on Maze-Unique.

Problem

Understanding why iterative weight-tied latent reasoning models generalize beyond memorized patterns and how to make test-time compute scaling (depth and breadth) reliable on controlled reasoning benchmarks such as Sudoku and Maze.

Approach

Frames iterative reasoners as learned latent dynamical systems whose task-aligned attractors should be stable fixed points; shapes the landscape with randomized latent initialization and path noise, and trains with segmented online training so parameter updates reshape the attractor landscape along the trajectory, enabling depth and breadth scaling with convergence-based selection.

Key results

• EqR reaches 99.8% exact accuracy on Sudoku-Extreme and 93.0% on Maze-Unique with depth 64 and breadth 128, versus 84.8% (TRM) and 51.4% (URM) baselines at the base budget.
• Randomized state initialization plus noise injection raises Maze accuracy at D=16 from 44.9% (baseline) to 82.2%, and Sudoku from 84.8% to 86.4%.
• With ACT at D=1024 on Sudoku-Lite, Avg. NFE drops from 1024 to 58.7 (17.4x fewer) with accuracy changing from 96.1% to 95.3%.

Surprising findings

• Residual reduction can indicate convergence to spurious attractors for the baseline TRM, so top-1 convergence selection underperforms majority voting there; it only becomes reliable after landscape shaping.
• Full-trajectory intermediate supervision lowers accuracy (51.8% to 47.1%), while late-trajectory anchors improve it.

Limitations

• Evaluated only on controlled structured puzzles (Sudoku, Maze) and a small Mini-ARC/Transformer extension, so generality to natural reasoning tasks is untested.
• Convergence is a reliable selection signal only after landscape shaping; it is not a task-agnostic correctness certificate.
• Depth scaling with full backprop is memory-prohibitive, requiring segmented online training and truncation.

Experimental setup

Datasets: Sudoku-Extreme Maze-Unique Sudoku-Lite Mini-ARC

Benchmarks: Sudoku-Extreme Maze-Unique Sudoku-Lite Mini-ARC

Metrics: exact accuracy fixed-point residual average NFE top-1 convergence accuracy majority-vote accuracy

Baselines: HRM TRM URM feedforward MLP-mixer

Builds on: Hierarchical Reasoning Model Tiny Recursive Model Deep Equilibrium Models Universal Transformer Adaptive Computation Time

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