The Geometry of Multi-Task Grokking: Transverse Instability, Superposition, and Weight Decay Phase Structure

• URL: http://arxiv.org/abs/2602.18523v1
• Date: Thu, 19 Feb 2026 22:39:55 GMT
• Title: The Geometry of Multi-Task Grokking: Transverse Instability, Superposition, and Weight Decay Phase Structure
• Authors: Yongzhong Xu,
• Abstract summary: We study the abrupt transition from memorization to generalization long after near-zero training loss.<n>We extend geometric analysis to multi-task modular arithmetic.<n>Results support a dynamical picture in which multi-task grokking constructs a compact superposition subspace in parameter space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Grokking -- the abrupt transition from memorization to generalization long after near-zero training loss -- has been studied mainly in single-task settings. We extend geometric analysis to multi-task modular arithmetic, training shared-trunk Transformers on dual-task (mod-add + mod-mul) and tri-task (mod-add + mod-mul + mod-sq) objectives across a systematic weight decay sweep. Five consistent phenomena emerge. (1) Staggered grokking order: multiplication generalizes first, followed by squaring, then addition, with consistent delays across seeds. (2) Universal integrability: optimization trajectories remain confined to an empirically invariant low-dimensional execution manifold; commutator defects orthogonal to this manifold reliably precede generalization. (3) Weight decay phase structure: grokking timescale, curvature depth, reconstruction threshold, and defect lead covary systematically with weight decay, revealing distinct dynamical regimes and a sharp no-decay failure mode. (4) Holographic incompressibility: final solutions occupy only 4--8 principal trajectory directions yet are distributed across full-rank weights and destroyed by minimal perturbations; SVD truncation, magnitude pruning, and uniform scaling all fail to preserve performance. (5) Transverse fragility and redundancy: removing less than 10% of orthogonal gradient components eliminates grokking, yet dual-task models exhibit partial recovery under extreme deletion, suggesting redundant center manifolds enabled by overparameterization. Together, these results support a dynamical picture in which multi-task grokking constructs a compact superposition subspace in parameter space, with weight decay acting as compression pressure and excess parameters supplying geometric redundancy in optimization pathways.

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