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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.17607 |
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| _version_ | 1866915354608926720 |
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| author | Zhang, Chicheng Zhou, Yihan |
| author_facet | Zhang, Chicheng Zhou, Yihan |
| contents | Multi-distribution learning extends agnostic Probably Approximately Correct (PAC) learning to the setting in which a family of $k$ distributions, $\{D_i\}_{i\in[k]}$, is considered and a classifier's performance is measured by its error under the worst distribution. This problem has attracted a lot of recent interests due to its applications in collaborative learning, fairness, and robustness. Despite a rather complete picture of sample complexity of passive multi-distribution learning, research on active multi-distribution learning remains scarce, with algorithms whose optimality remaining unknown.
In this paper, we develop new algorithms for active multi-distribution learning and establish improved label complexity upper and lower bounds, in distribution-dependent and distribution-free settings. Specifically, in the near-realizable setting we prove an upper bound of $\widetilde{O}\Bigl(θ_{\max}(d+k)\ln\frac{1}{\varepsilon}\Bigr)$ and $\widetilde{O}\Bigl(θ_{\max}(d+k)\Bigl(\ln\frac{1}{\varepsilon}+\frac{ν^2}{\varepsilon^2}\Bigr)+\frac{kν}{\varepsilon^2}\Bigr)$ in the realizable and agnostic settings respectively, where $θ_{\max}$ is the maximum disagreement coefficient among the $k$ distributions, $d$ is the VC dimension of the hypothesis class, $ν$ is the multi-distribution error of the best hypothesis, and $\varepsilon$ is the target excess error. Moreover, we show that the bound in the realizable setting is information-theoretically optimal and that the $kν/\varepsilon^2$ term in the agnostic setting is fundamental for proper learners. We also establish instance-dependent sample complexity bound for passive multidistribution learning that smoothly interpolates between realizable and agnostic regimes~\citep{blum2017collaborative,zhang2024optimal}, which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards Fundamental Limits for Active Multi-distribution Learning Zhang, Chicheng Zhou, Yihan Machine Learning Multi-distribution learning extends agnostic Probably Approximately Correct (PAC) learning to the setting in which a family of $k$ distributions, $\{D_i\}_{i\in[k]}$, is considered and a classifier's performance is measured by its error under the worst distribution. This problem has attracted a lot of recent interests due to its applications in collaborative learning, fairness, and robustness. Despite a rather complete picture of sample complexity of passive multi-distribution learning, research on active multi-distribution learning remains scarce, with algorithms whose optimality remaining unknown. In this paper, we develop new algorithms for active multi-distribution learning and establish improved label complexity upper and lower bounds, in distribution-dependent and distribution-free settings. Specifically, in the near-realizable setting we prove an upper bound of $\widetilde{O}\Bigl(θ_{\max}(d+k)\ln\frac{1}{\varepsilon}\Bigr)$ and $\widetilde{O}\Bigl(θ_{\max}(d+k)\Bigl(\ln\frac{1}{\varepsilon}+\frac{ν^2}{\varepsilon^2}\Bigr)+\frac{kν}{\varepsilon^2}\Bigr)$ in the realizable and agnostic settings respectively, where $θ_{\max}$ is the maximum disagreement coefficient among the $k$ distributions, $d$ is the VC dimension of the hypothesis class, $ν$ is the multi-distribution error of the best hypothesis, and $\varepsilon$ is the target excess error. Moreover, we show that the bound in the realizable setting is information-theoretically optimal and that the $kν/\varepsilon^2$ term in the agnostic setting is fundamental for proper learners. We also establish instance-dependent sample complexity bound for passive multidistribution learning that smoothly interpolates between realizable and agnostic regimes~\citep{blum2017collaborative,zhang2024optimal}, which may be of independent interest. |
| title | Towards Fundamental Limits for Active Multi-distribution Learning |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2506.17607 |