Learning Shrinks the Hard Tail: Training-Dependent Inference Scaling in a Solvable Linear Model
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909983374835712 |
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| author | Levi, Noam |
| author_facet | Levi, Noam |
| contents | We analyze neural scaling laws in a solvable model of last-layer fine-tuning where targets have intrinsic, instance-heterogeneous difficulty. In our Latent Instance Difficulty (LID) model, each input's target variance is governed by a latent ``precision'' drawn from a heavy-tailed distribution. While generalization loss recovers standard scaling laws, our main contribution connects this to inference. The pass@$k$ failure rate exhibits a power-law decay, $k^{-β_\text{eff}}$, but the observed exponent $β_\text{eff}$ is training-dependent. It grows with sample size $N$ before saturating at an intrinsic limit $β$ set by the difficulty distribution's tail. This coupling reveals that learning shrinks the ``hard tail'' of the error distribution: improvements in the model's generalization error steepen the pass@$k$ curve until irreducible target variance dominates. The LID model yields testable, closed-form predictions for this behavior, including a compute-allocation rule that favors training before saturation and inference attempts after. We validate these predictions in simulations and in two real-data proxies: CIFAR-10H (human-label variance) and a maths teacher-student distillation task. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_03764 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning Shrinks the Hard Tail: Training-Dependent Inference Scaling in a Solvable Linear Model Levi, Noam Machine Learning Disordered Systems and Neural Networks Artificial Intelligence We analyze neural scaling laws in a solvable model of last-layer fine-tuning where targets have intrinsic, instance-heterogeneous difficulty. In our Latent Instance Difficulty (LID) model, each input's target variance is governed by a latent ``precision'' drawn from a heavy-tailed distribution. While generalization loss recovers standard scaling laws, our main contribution connects this to inference. The pass@$k$ failure rate exhibits a power-law decay, $k^{-β_\text{eff}}$, but the observed exponent $β_\text{eff}$ is training-dependent. It grows with sample size $N$ before saturating at an intrinsic limit $β$ set by the difficulty distribution's tail. This coupling reveals that learning shrinks the ``hard tail'' of the error distribution: improvements in the model's generalization error steepen the pass@$k$ curve until irreducible target variance dominates. The LID model yields testable, closed-form predictions for this behavior, including a compute-allocation rule that favors training before saturation and inference attempts after. We validate these predictions in simulations and in two real-data proxies: CIFAR-10H (human-label variance) and a maths teacher-student distillation task. |
| title | Learning Shrinks the Hard Tail: Training-Dependent Inference Scaling in a Solvable Linear Model |
| topic | Machine Learning Disordered Systems and Neural Networks Artificial Intelligence |
| url | https://arxiv.org/abs/2601.03764 |