Beyond Stable Convergence: Noise Covariance Design for Efficient Test-Time Scaling in Looped Models

Looped models scale test-time computation through recurrent refinement, but stable convergence alone does not ensure that additional iterations improve predictions. Even near an attracting fixed point, the fixed-point residual can decrease while the predictive loss increases. We formalize this discrepancy as misalignment between the attractor and task landscapes and develop a noise covariance learning method for efficient test-time depth scaling. Through a local analysis, we establish a noise covariance design framework that accounts for task-geometry adaptation under noisy training and optimizes a directional task-loss descent bound. Building on this framework, we propose Noise Designed Looped Models (NDLM), which employs an equivalent matrix-residual objective to train a lightweight network to generate structured perturbations efficiently. Experiments across multiple reasoning tasks demonstrate NDLM's competitive performance. Remarkably, NDLM enables test-time depth scaling with no observed overthinking and matches the isotropic-noise baseline's reference accuracy with an estimated $82.7\%$ reduction in recurrent depth on Maze-Unique.