Is Multi-Distribution Learning as Easy as PAC Learning: Sharp Rates with Bounded Label Noise

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 $1\mathrm{/}\epsilon$ 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 $k\mathrm{/}{\epsilon }^2$, 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 $\mathit{\text{multiplicative}}$ penalty in $k$. This establishes a $\mathit{\text{statistical}}$ separation between random classification noise and Massart noise, highlighting a fundamental barrier unique to learning from multiple sources.