# Risk-Monotonicity in Statistical Learning

arXiv: https://arxiv.org/abs/2011.14126 · HTML: https://mlsanity.com/paper/2011.14126/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Nov 2020.

## Built on by (most cited first, 9 of 9)

- [The Shape of Learning Curves: a Review](https://mlsanity.com/paper/2103.10948/) (2021)
- [Learning Curves for Decision Making in Supervised Machine Learning: A Survey](https://mlsanity.com/paper/2201.12150/) (2022)
- [Fully Unconstrained Online Learning](https://mlsanity.com/paper/2405.20540/) (2024)
- [Monotone Learning](https://mlsanity.com/paper/2202.05246/) (2022)
- [A Survey of Learning Curves with Bad Behavior: or How More Data Need Not Lead to Better Performance](https://mlsanity.com/paper/2211.14061/) (2022)
- [Sharp Matrix Empirical Bernstein Inequalities](https://mlsanity.com/paper/2411.09516/) (2024)
- [RiskProp: Collision-Anchored Self-Supervised Risk Propagation for Early Accident Anticipation](https://mlsanity.com/paper/2603.27165/) (2026)
- [Monotonic Learning in the PAC Framework: A New Perspective](https://mlsanity.com/paper/2501.05493/) (2025)
- [Risk reversal for least squares estimators under nested convex constraints](https://mlsanity.com/paper/2601.16041/) (2026)

## Abstract (arXiv)

Acquisition of data is a difficult task in many applications of machine learning, and it is only natural that one hopes and expects the population risk to decrease (better performance) monotonically with increasing data points. It turns out, somewhat surprisingly, that this is not the case even for the most standard algorithms that minimize the empirical risk. Non-monotonic behavior of the risk and instability in training have manifested and appeared in the popular deep learning paradigm under the description of double descent. These problems highlight the current lack of understanding of learning algorithms and generalization. It is, therefore, crucial to pursue this concern and provide a characterization of such behavior. In this paper, we derive the first consistent and risk-monotonic (in high probability) algorithms for a general statistical learning setting under weak assumptions, consequently answering some questions posed by Viering et al. 2019 on how to avoid non-monotonic behavior of risk curves. We further show that risk monotonicity need not necessarily come at the price of worse excess risk rates. To achieve this, we derive new empirical Bernstein-like concentration inequalities of independent interest that hold for certain non-i.i.d.~processes such as Martingale Difference Sequences.