# Feature Normalization under Finite Training Budgets

arXiv: https://arxiv.org/abs/2610.16845 · HTML: https://mlsanity.com/paper/2610.16845/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 13 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 9 of 9)

- [A Continuous-Time View of Early Stopping for Least Squares](https://mlsanity.com/paper/1810.10082/) (2018)
- [The Implicit Regularization of Stochastic Gradient Flow for Least Squares](https://mlsanity.com/paper/2003.07802/) (2020)
- [Near-Optimal Linear Regression under Distribution Shift](https://mlsanity.com/paper/2106.12108/) (2021)
- [When Does Preconditioning Help or Hurt Generalization?](https://mlsanity.com/paper/2006.10732/) (2020)
- [The Power and Limitation of Pretraining-Finetuning for Linear Regression under Covariate Shift](https://mlsanity.com/paper/2208.01857/) (2022)
- [Whitening and second order optimization both make information in the dataset unusable during training, and can reduce or prevent generalization](https://mlsanity.com/paper/2008.07545/) (2020)
- [Optimal Algorithms in Linear Regression under Covariate Shift: On the Importance of Precondition](https://mlsanity.com/paper/2502.09047/) (2025)
- [Improving Implicit Regularization of SGD with Preconditioning for Least Square Problems](https://mlsanity.com/paper/2403.08585/) (2024)
- [The Choice of Normalization Influences Shrinkage in Regularized Regression](https://mlsanity.com/paper/2501.03821/) (2025)

## Abstract (arXiv)

Standardizing a rare feature can help stochastic gradient descent learn its effect within a finite budget, but it also changes the noise cost of learning. We characterize the lowest rare-subgroup risks attainable by standardized and unscaled training when each chooses a nonnegative constant learning rate and a deterministic stopping time under the same observation cap. For two independent binary features, we derive both optimized risks as prevalence vanishes while the expected number of rare observations remains finite. Uniform control over rates and stopping times reduces each limiting oracle to a scalar minimization. With positive noise, the preferred scaling changes exactly once as we move signal to the rare feature while the limiting risk at initialization stays fixed and positive. The formulas also explain how noise and budget affect the comparison. Separate finite-prevalence bounds favor standardization on a rare-dominant signal family and unscaled training on a common-dominant family. The results concern final iterates with population preprocessing and fresh observations. Exact-moment searches assess the approximation at finite prevalence. In a controlled study, both orderings persist under observation reuse, estimated preprocessing, and intercept fitting when tuning choices are fixed before evaluation. Selecting a predictor after observing training can reverse an ordering, even when selection uses exact population risk.