Bounds on Lp errors in density ratio estimation via f-divergence loss functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917957496471552 |
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| author | Kitazawa, Yoshiaki |
| author_facet | Kitazawa, Yoshiaki |
| contents | Density ratio estimation (DRE) is a core technique in machine learning used to capture relationships between two probability distributions. $f$-divergence loss functions, which are derived from variational representations of $f$-divergence, have become a standard choice in DRE for achieving cutting-edge performance. This study provides novel theoretical insights into DRE by deriving upper and lower bounds on the $L_p$ errors through $f$-divergence loss functions. These bounds apply to any estimator belonging to a class of Lipschitz continuous estimators, irrespective of the specific $f$-divergence loss function employed. The derived bounds are expressed as a product involving the data dimensionality and the expected value of the density ratio raised to the $p$-th power. Notably, the lower bound includes an exponential term that depends on the Kullback--Leibler (KL) divergence, revealing that the $L_p$ error increases significantly as the KL divergence grows when $p > 1$. This increase becomes even more pronounced as the value of $p$ grows. The theoretical insights are validated through numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01516 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on Lp errors in density ratio estimation via f-divergence loss functions Kitazawa, Yoshiaki Machine Learning Density ratio estimation (DRE) is a core technique in machine learning used to capture relationships between two probability distributions. $f$-divergence loss functions, which are derived from variational representations of $f$-divergence, have become a standard choice in DRE for achieving cutting-edge performance. This study provides novel theoretical insights into DRE by deriving upper and lower bounds on the $L_p$ errors through $f$-divergence loss functions. These bounds apply to any estimator belonging to a class of Lipschitz continuous estimators, irrespective of the specific $f$-divergence loss function employed. The derived bounds are expressed as a product involving the data dimensionality and the expected value of the density ratio raised to the $p$-th power. Notably, the lower bound includes an exponential term that depends on the Kullback--Leibler (KL) divergence, revealing that the $L_p$ error increases significantly as the KL divergence grows when $p > 1$. This increase becomes even more pronounced as the value of $p$ grows. The theoretical insights are validated through numerical experiments. |
| title | Bounds on Lp errors in density ratio estimation via f-divergence loss functions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2410.01516 |