Is Multi-Distribution Learning as Easy as PAC Learning: Sharp Rates with Bounded Label Noise
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arXiv
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| Format: | Preprint |
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2026
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| author | Hanashiro, Rafael Shetty, Abhishek Jaillet, Patrick |
| author_facet | Hanashiro, Rafael Shetty, Abhishek Jaillet, Patrick |
| contents | Towards understanding the statistical complexity of learning from heterogeneous sources, we study the problem of multi-distribution learning. Given $k$ data sources, the goal is to output a classifier for each source by exploiting shared structure to reduce sample complexity. We focus on the bounded label noise setting to determine whether the fast $1/ε$ rates achievable in single-task learning extend to this regime with minimal dependence on $k$. Surprisingly, we show that this is not the case. We demonstrate that learning across $k$ distributions inherently incurs slow rates scaling with $k/ε^2$, even under constant noise levels, unless each distribution is learned separately. A key technical contribution is a structured hypothesis-testing framework that captures the statistical cost of certifying near-optimality under bounded noise-a cost we show is unavoidable in the multi-distribution setting.
Finally, we prove that when competing with the stronger benchmark of each distribution's optimal Bayes error, the sample complexity incurs a \textit{multiplicative} penalty in $k$. This establishes a \textit{statistical} separation between random classification noise and Massart noise, highlighting a fundamental barrier unique to learning from multiple sources. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21039 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Is Multi-Distribution Learning as Easy as PAC Learning: Sharp Rates with Bounded Label Noise Hanashiro, Rafael Shetty, Abhishek Jaillet, Patrick Machine Learning Towards understanding the statistical complexity of learning from heterogeneous sources, we study the problem of multi-distribution learning. Given $k$ data sources, the goal is to output a classifier for each source by exploiting shared structure to reduce sample complexity. We focus on the bounded label noise setting to determine whether the fast $1/ε$ rates achievable in single-task learning extend to this regime with minimal dependence on $k$. Surprisingly, we show that this is not the case. We demonstrate that learning across $k$ distributions inherently incurs slow rates scaling with $k/ε^2$, even under constant noise levels, unless each distribution is learned separately. A key technical contribution is a structured hypothesis-testing framework that captures the statistical cost of certifying near-optimality under bounded noise-a cost we show is unavoidable in the multi-distribution setting. Finally, we prove that when competing with the stronger benchmark of each distribution's optimal Bayes error, the sample complexity incurs a \textit{multiplicative} penalty in $k$. This establishes a \textit{statistical} separation between random classification noise and Massart noise, highlighting a fundamental barrier unique to learning from multiple sources. |
| title | Is Multi-Distribution Learning as Easy as PAC Learning: Sharp Rates with Bounded Label Noise |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.21039 |