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Multi-Label Learning with Stronger Consistency Guarantees
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Studies surrogate losses for multi-label learning and proposes a multi-label logistic loss and multi-label comp-sum/constrained losses with H-consistency bounds, giving Bayes-consistency for any multi-label loss.
Problem
Existing Bayes-consistent surrogate for Hamming loss (binary relevance) has label-dependent H-consistency bounds with smooth losses and ignores label correlations; prior consistency results cover only specific multi-label losses.
Approach
Introduce a multi-label logistic loss that weights a softmax-style log-partition over label vectors by (1 - loss), giving label-independent H-consistency bounds; extend to multi-label comp-sum and constrained losses that are consistent for any multi-label loss, including a linear-fractional family and tree distance.
Key results
- Shows binary relevance with logistic loss has H-consistency bound of l^{1/2} epsilon^{1/2} for Hamming loss, dependent on number of labels l.
- Multi-label logistic loss yields label-independent bound: Hamming regret <= 2 * (surrogate regret)^{1/2}.
- Multi-label comp-sum losses with Psi(u)=log(u) or u-1 give label-independent bounds for any multi-label loss; gradient computable in O(l) time via weighted finite automaton.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace binary relevance with multi-label logistic loss | Multi-label learning with Hamming loss · Hamming loss | H-consistency bound dependence on label count: l^{\frac12} ...; 2 ... | Helped |
| Use generalized cross-entropy comp-sum instead of logistic or sum-exponential comp-sum | Multi-label comp-sum loss · General multi-label loss | H-consistency bound dependence on n: \Gamma(t) = 2\sqrt{t}; \Gamma(t) = 2\sqrt{n^{{q}}t} | Hurt |
| Use mean absolute error comp-sum instead of logistic or sum-exponential comp-sum | Multi-label comp-sum loss · General multi-label loss | H-consistency bound dependence on n: \Gamma(t) = 2\sqrt{t}; \Gamma(t) = n t | Hurt |
Limitations
- Other comp-sum choices (e.g., generalized cross-entropy, mean absolute error) yield bounds with factors exponential in l
- Theoretical paper; no empirical experiments reported in the provided text
- Analysis assumes complete hypothesis sets and threshold zero
Experimental setup
Baselines: binary relevance
Builds on: binary relevance comp-sum losses H-consistency bounds constrained losses
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