# Multiclass learning with margin: exponential rates with no bias-variance trade-off

arXiv: https://arxiv.org/abs/2202.01773 · HTML: https://mlsanity.com/paper/2202.01773/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 25 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 4 of 4)

- [Exponential convergence of testing error for stochastic gradient methods](https://mlsanity.com/paper/1712.04755/) (2017)
- [Sharp Analysis of Learning with Discrete Losses](https://mlsanity.com/paper/1810.06839/) (2018)
- [Fast rates in structured prediction](https://mlsanity.com/paper/2102.00760/) (2021)
- [Stochastic Gradient Descent with Exponential Convergence Rates of Expected Classification Errors](https://mlsanity.com/paper/1806.05438/) (2018)

## Built on by (most cited first, 5 of 5)

- [A Case of Exponential Convergence Rates for SVM](https://mlsanity.com/paper/2205.10055/) (2022)
- [From Weakly Supervised Learning to Active Learning](https://mlsanity.com/paper/2209.11629/) (2022)
- [Accounting for Missing Covariates in Heterogeneous Treatment Estimation](https://mlsanity.com/paper/2410.15655/) (2024)
- [Super-fast Rates of Convergence for Neural Network Classifiers under the Hard Margin Condition](https://mlsanity.com/paper/2505.08262/) (2025)
- [Near-Exponential Convergence Rates for kNN Classification based on Boltzmann Margin](https://mlsanity.com/paper/2606.10361/) (2026)

## Abstract (arXiv)

We study the behavior of error bounds for multiclass classification under suitable margin conditions. For a wide variety of methods we prove that the classification error under a hard-margin condition decreases exponentially fast without any bias-variance trade-off. Different convergence rates can be obtained in correspondence of different margin assumptions. With a self-contained and instructive analysis we are able to generalize known results from the binary to the multiclass setting.