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Machine Learning

Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning

Benhao Huang, Zhengyang Geng, Zico Kolter

Featured June 8, 2026

Original Paper

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Simply

By teaching AI models to settle into stable "solution spots" (called attractors) through repeated internal thinking, they can solve much harder puzzles by thinking longer and trying different starting ideas.

In depth
The paper introduces Equilibrium Reasoners (EqR), a novel approach that frames iterative reasoning as learning task-conditioned attractors in a latent dynamical system. This allows models to achieve scalable and generalizable reasoning by driving latent states towards stable fixed points that correspond to valid solutions. The authors propose specific training interventions, randomized state initialization and path stochasticity via noise injection, to shape this attractor landscape, making correct solutions more reachable and stable, thereby enabling massive test-time compute scaling.

Key Takeaways

  • 1
    Iterative reasoning models can achieve scalable generalization by learning latent dynamical systems with stable fixed points (attractors) that align with task solutions.
  • 2
    The attractor landscape can be explicitly shaped during training using randomized state initialization and path stochasticity to make correct attractors more reachable and stable.
  • 3
    This approach, termed Equilibrium Reasoners (EqR), enables unprecedented test-time compute scaling (e.g., 40,000 effective layers) leading to significant accuracy boosts on complex reasoning tasks like Sudoku-Extreme.

Conceptual Flow

HIGH LEVEL
1
Methodology (The "Logic")

The model repeatedly updates its internal thinking state until it settles on a stable answer, like finding a comfortable resting spot.

Puzzle Input
Repeated Thinking
Stable Answer
2
Results (The "Impact")

This new way of thinking helps the AI solve very hard puzzles much better, even when it has to think for a very long time.

Hard Puzzles
Old Thinking
New Thinking Method
High Accuracy
Solves More