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Tight and Fast Bounds for Multi-Label Learning
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Derives generalization bounds for multi-label learning with smooth base losses via new vector-contraction and local Rademacher complexity inequalities, giving bounds independent of the number of labels (up to logs) and O(1/n) rates, plus tight bounds for Macro-Averaged AUC.
Problem
Generalization analysis of multi-label learning has dependency on the number of labels c and slow O(1/sqrt(n)) rates, and the effect of base-loss smoothness on generalization bounds was unexplored.
Approach
Vector-contraction inequalities that exploit smoothness of base losses (eliminating the sqrt(c) factor from the l2 cover radius), combined with local Rademacher complexity to obtain fast rates, plus Lipschitz/smooth contraction for Macro-Averaged AUC.
Key results
- Tight O~(1/sqrt(n)) generalization bounds for Hamming, Subset and Ranking losses with smooth base losses, with no dependency on number of labels c up to logarithmic terms (improving O~(sqrt(c/n)) of Zhang & Zhang 2024a by sqrt(c)).
- Local vector-contraction yields O~(1/n) fast-rate generalization bounds for smooth base losses, independent of c up to logs.
- Macro-Averaged AUC bounds of O~(1/sqrt(r0)) with no dependency on c under both Lipschitz and smooth base losses.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use smooth base losses instead of relying on ℓ2 Lipschitz losses | Multi-label Hamming, Subset, and Ranking loss · Hamming, Subset, and Ranking loss | generalization bound: eO(√c/√n) -> eO(1/√n) | Helped |
| Use local Rademacher complexity with smooth base losses | Multi-label learning · Hamming, Subset, and Ranking loss | convergence rate: eO(1/√n) -> eO(1/n) | Helped |
| Use smooth base losses rather than the cited ℓ2 Lipschitz-loss approach | Multi-label learning · Hamming, Subset, and Ranking loss | label-count dependency: square-root to be independent | Helped |
| Use Lipschitz continuity of the base loss | Macro-Averaged AUC · Macro-Averaged AUC | generalization bound: eO(√c/r0) -> eO(1/√r0) | Helped |
| Use smoothness of the base loss | Macro-Averaged AUC · Macro-Averaged AUC | generalization bound: no dependency on c | Helped |
| Use smooth base losses with local Rademacher complexity | Multi-label learning · Hamming, Subset, and Ranking loss | label-count dependency: no dependency on c, up to logarithmic terms | Helped |
Surprising findings
- The square-root dependency on c in prior Lipschitz-loss bounds (Zhang & Zhang 2024a) is inevitable for l2-Lipschitz losses, due to a sqrt(c) factor in the covering radius, per the authors' analysis.
Limitations
- Does not cover cross-entropy loss since it is unbounded; only theory with no experiments.
- Fast rate O~(1/n) is shown only for smooth base losses; Lipschitz losses remain at O~(1/sqrt(n)).
- Explicit label correlations and large label spaces are not addressed in the analysis.
Experimental setup
Baselines: Wu & Zhu 2020 Wu et al. 2021a Zhang & Zhang 2024a Xu et al. 2016 Busa-Fekete et al. 2022 Wu et al. 2023
Builds on: Maurer vector-contraction inequality Zhang & Zhang 2024a vector-contraction for multi-label Bartlett et al. 2005 local Rademacher complexity Xu et al. 2016 local Rademacher bound
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