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Generalization Analysis for Multi-Label Learning
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Derives Rademacher-complexity generalization bounds for multi-label learning with novel vector-contraction inequalities that remove dependence on the number of labels in the decoupled case, plus a bound for Macro-Averaged AUC linked to class imbalance.
Problem
Generalization theory for multi-label learning has bounds that depend strongly on the number of labels, and it lacks explicit treatment of label correlations and inductive biases such as norm regularizers.
Approach
Novel vector-contraction inequalities exploiting l2 and l∞ Lipschitz continuity of losses, using a projection function class, yield bounds independent of the number of labels in the decoupled (first-order correlation) case; coupled cases recover high-order correlation effects via norm regularizers; a label-based ranking Rademacher complexity handles Macro-Averaged AUC.
Key results
- ℓ2-Lipschitz loss: new vector-contraction yields bound R(f) ≤ R̂ + Õ(1/√n) with no dependence on the number of labels c, improving the O(c/√n) basic bound and the √c-dependent prior bounds.
- ℓ∞-Lipschitz loss in decoupled case: bound improves from Õ(√(c/n)) to Õ(1/√n) independent of c.
- Macro-Averaged AUC: excess risk bound Õ(√(c/r0)), where r0 = min over labels of min(positives, negatives), showing looser bounds under severe class imbalance.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the projection-function-class vector-contraction inequality instead of the typical vector-contraction inequality | Multi-label learning with ℓ2-Lipschitz loss · Multi-label learning, decoupling case | generalization-bound dependency on label count: O(c/√n) -> O(1/√n) | Helped |
| Use the projection-function-class vector-contraction inequality instead of the basic ℓ∞ bound | Multi-label learning with ℓ∞-Lipschitz loss · Multi-label learning, decoupling case | generalization-bound dependency on label count: eO(√c/n) -> eO(1/√n) | Helped |
| Preserve coupling among components using a norm regularizer | Multi-label learning with ℓ∞-Lipschitz loss · Multi-label learning, coupling case | generalization-bound dependency on label count: eO(1/√n) | Helped |
| Increase class imbalance, reducing the minimum positive/negative class count r0 | Macro-Averaged AUC · Macro-Averaged AUC | generalization bound: eO(√c/r0) | Hurt |
Surprising findings
- Authors argue the Ω(1/√n) lower bound is not tight and suggest a lower bound of Ω(1/√(nc)) may hold when label correlations exist.
Limitations
- Results are purely theoretical with no empirical validation.
- Lower bound remains open; the bounds are not shown to be tight.
- Does not specify particular models, so model-specific generalization differences (e.g., Lipschitz constants of deep vs shallow models) are not captured.
Experimental setup
Baselines: Wu & Zhu 2020 Wu et al. 2021a Lei et al. 2019 Lei et al. 2015 Wu et al. 2021b Yu et al. 2014 Xu et al. 2016 Wu et al. 2023
Builds on: Maurer vector-contraction inequality Foster & Rakhlin l∞ vector contraction Lei et al. 2019 vector-contraction Wu & Zhu 2020 multi-label generalization
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