# A Rate Separation for Agnostic Direct Sums

arXiv: https://arxiv.org/abs/2608.06951 · HTML: https://mlsanity.com/paper/2608.06951/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 8 references that are in the map.
- First posted on arXiv 7 Aug 2026.

## Builds on (most cited first, 3 of 3)

- [The Optimal Sample Complexity of Multiclass and List Learning](https://mlsanity.com/paper/2604.24749/) (2026)
- [Sample Complexity of Agnostic Multiclass Classification: Natarajan Dimension Strikes Back](https://mlsanity.com/paper/2511.12659/) (2025)
- [Uniform Laws of Large Numbers in Product Spaces](https://mlsanity.com/paper/2603.24493/) (2026)

## Abstract (arXiv)

Hanneke, Moran, and Waknine citeHannekeMoranWaknine2024 asked how the agnostic PAC learning curve of the direct sum Cr depends on the single-instance learning curve epsagn(nmid C) and on r. We show that the single-instance learning rate does not determine the direct-sum rate. Let F be the class of the two constant binary functions and let G consist of the zero function and the identity function. Both classes have agnostic learning curve of order n-1/2.