Learning Shrinks the Hard Tail: Training-Dependent Inference Scaling in a Solvable Linear Model

• URL: http://arxiv.org/abs/2601.03764v1
• Date: Wed, 07 Jan 2026 10:00:17 GMT
• Title: Learning Shrinks the Hard Tail: Training-Dependent Inference Scaling in a Solvable Linear Model
• Authors: Noam Levi,
• Abstract summary: We analyze neural scaling laws in a solvable model of last-layer fine-tuning where targets have intrinsic, instance-heterogeneous difficulty.<n>We show that learning shrinks the hard tail'' of the error distribution.
• Score: 2.7074235008521246
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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