# A Fundamental Accuracy--Robustness Trade-off in Regression and Classification

arXiv: https://arxiv.org/abs/2411.05853 · HTML: https://mlsanity.com/paper/2411.05853/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 11 references that are in the map.
- First posted on arXiv 6 Nov 2024.

## Builds on (most cited first, 7 of 7)

- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [On Evaluating Adversarial Robustness](https://mlsanity.com/paper/1902.06705/) (2019)
- [Adversarially Robust Generalization Requires More Data](https://mlsanity.com/paper/1804.11285/) (2018)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [Precise Tradeoffs in Adversarial Training for Linear Regression](https://mlsanity.com/paper/2002.10477/) (2020)
- [Analysis of classifiers' robustness to adversarial perturbations](https://mlsanity.com/paper/1502.02590/) (2015)
- [Precise Statistical Analysis of Classification Accuracies for Adversarial Training](https://mlsanity.com/paper/2010.11213/) (2020)

## Abstract (arXiv)

We derive a fundamental trade-off between standard and adversarial risk in a rather general situation that formalizes the following simple intuition: "If no (nearly) optimal predictor is smooth, adversarial robustness comes at the cost of accuracy." As a concrete example, we evaluate the derived trade-off in regression with polynomial ridge functions under mild regularity conditions. Generalizing our analysis of this example, we formulate a necessary condition under which adversarial robustness can be achieved without significant degradation of the accuracy. This necessary condition is expressed in terms of a quantity that resembles the Poincar'e constant of the data distribution.