# Effective Frontiers: A Unification of Neural Scaling Laws

arXiv: https://arxiv.org/abs/2602.02593 · HTML: https://mlsanity.com/paper/2602.02593/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 38 references that are in the map.
- First posted on arXiv 1 Feb 2026.

## Builds on (most cited first, 27 of 27)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [The Quantization Model of Neural Scaling](https://mlsanity.com/paper/2303.13506/) (2023)
- [A Solvable Model of Neural Scaling Laws](https://mlsanity.com/paper/2210.16859/) (2022)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Learning Curve Theory](https://mlsanity.com/paper/2102.04074/) (2021)
- [Small ReLU networks are powerful memorizers: a tight analysis of memorization capacity](https://mlsanity.com/paper/1810.07770/) (2018)
- [4+3 Phases of Compute-Optimal Neural Scaling Laws](https://mlsanity.com/paper/2405.15074/) (2024)
- [How Feature Learning Can Improve Neural Scaling Laws](https://mlsanity.com/paper/2409.17858/) (2024)
- [A Neural Scaling Law from the Dimension of the Data Manifold](https://mlsanity.com/paper/2004.10802/) (2020)
- [Emergence and scaling laws in SGD learning of shallow neural networks](https://mlsanity.com/paper/2504.19983/) (2025)
- [Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws](https://mlsanity.com/paper/2508.03688/) (2025)
- [A Multi-Power Law for Loss Curve Prediction Across Learning Rate Schedules](https://mlsanity.com/paper/2503.12811/) (2025)
- [Functional Scaling Laws in Kernel Regression: Loss Dynamics and Learning Rate Schedules](https://mlsanity.com/paper/2509.19189/) (2025)
- [Larger Datasets Can Be Repeated More: A Theoretical Analysis of Multi-Epoch Scaling in Linear Regression](https://mlsanity.com/paper/2511.13421/) (2025)
- [Precision Machine Learning](https://mlsanity.com/paper/2210.13447/) (2022)
- [Memory capacity of neural networks with threshold and ReLU activations](https://mlsanity.com/paper/2001.06938/) (2020)
- [Disentangling Feature Structure: A Mathematically Provable Two-Stage Training Dynamics in Transformers](https://mlsanity.com/paper/2502.20681/) (2025)

## Built on by (most cited first, 1 of 1)

- [Mirror Horizon: Viable Path Entropy as a Measure of Bounded Reflection](https://mlsanity.com/paper/2607.11937/) (2026)

## Abstract (arXiv)

Neural scaling laws govern the prediction power-law improvement of test loss with respect to model capacity (N), datasize (D), and compute (C). However, existing theoretical explanations often rely on specific architectures or complex kernel methods, lacking intuitive universality. In this paper, we propose a unified framework that abstracts general learning tasks as the progressive coverage of patterns from a long-tail (Zipfian) distribution. We introduce the Effective Frontier (k_star), a threshold in the pattern rank space that separates learned knowledge from the unlearned tail. We prove that reducible loss is asymptotically determined by the probability mass of the tail a resource-dependent frontier truncation. Based on our framework, we derive the precise scaling laws for N, D, and C, attributing them to capacity, coverage, and optimization bottlenecks, respectively. Furthermore, we unify these mechanisms via a Max-Bottleneck principle, demonstrating that the Kaplan and Chinchilla scaling laws are not contradictory, but equilibrium solutions to the same constrained optimization problem under different active bottlenecks.