# Non-asymptotic estimates of the minimal risk in statistical learning

arXiv: https://arxiv.org/abs/2606.23295 · HTML: https://mlsanity.com/paper/2606.23295/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 54 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [High probability generalization bounds for uniformly stable algorithms with nearly optimal rate](https://mlsanity.com/paper/1902.10710/) (2019)
- [Sharper bounds for uniformly stable algorithms](https://mlsanity.com/paper/1910.07833/) (2019)
- [Generalization Bounds for Uniformly Stable Algorithms](https://mlsanity.com/paper/1812.09859/) (2018)
- [Stability and Deviation Optimal Risk Bounds with Convergence Rate O(1/n)](https://mlsanity.com/paper/2103.12024/) (2021)
- [Mathematical Perspective of Machine Learning](https://mlsanity.com/paper/2007.01503/) (2020)

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

- [Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions](https://mlsanity.com/paper/2609.32652/) (2026)

## Abstract (arXiv)

In this paper we prove some concentration inequalities for two types of error probabilities in the Empirical Risk Principle (ERP) in statistical learning, which provide a lower bound and an upper bound for the minimal risk (in terms of the minimal empirical risk) with non-asymptotic high confidence. The usual boundedness condition of the empirical risk function is relaxed to the Gaussian or exponential integrability condition. The confidence of the lower bound of the minimal risk is shown to be independent of the number of training parameters and the dimension of the input vectors, allowing one to detect the deficiency of a learning machine efficiently; and the confidence of the upper bound of the minimal risk is proved to be high provided that the sample size n is much greater than the box dimension of the parameter set Theta in the Orlicz metric dpsi1 associated with the risk functions. Our work is based on Talagrand's concentration inequalities (the sharp versions by Bousquet and Klein-Rio), transport-entropy inequalities and the recent progress in the theory of empirical processes and statistical learning.