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Consistency Conditions for Differentiable Surrogate Losses
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Shows indirect elicitation equals calibration for 1-d convex differentiable surrogates, gives a 2-d counterexample, introduces strong indirect elicitation which implies calibration for differentiable surrogates and is necessary and sufficient for strongly convex ones, and uses it to design a 1-d Huber-like surrogate for ordinal regression.
Problem
Verifying calibration (equivalently consistency) of surrogate losses for discrete targets is cumbersome; it is unknown whether the easier indirect elicitation (IE) condition implies calibration for non-polyhedral, differentiable convex surrogates.
Approach
Shows IE equals calibration for 1-d differentiable convex surrogates, that this fails in higher dimensions, and proposes strong IE (every surrogate level set lies within a single target level-set-intersection cell), which implies calibration for differentiable surrogates and is necessary and sufficient for strongly convex ones.
Key results
- For 1-d convex differentiable surrogates, indirect elicitation is equivalent to calibration under the compactness assumption.
- A 2-d differentiable, strongly convex surrogate satisfies IE for a target but violates calibration for every link, showing the equivalence fails in higher dimensions.
- Strong IE implies calibration for differentiable surrogates and is necessary and sufficient for calibration when components are strongly convex.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use indirect elicitation (IE) with a convex differentiable surrogate in one dimension | One-dimensional convex differentiable surrogates · Arbitrary discrete target | calibration | Helped |
| Use IE without strong IE in dimensions greater than one | Differentiable surrogates · Higher-dimensional discrete prediction | calibration | Hurt |
| Require strong indirect elicitation instead of IE | Convex differentiable surrogates · Arbitrary discrete target | calibration | Helped |
| Require strong IE for a strongly convex differentiable surrogate | Strongly convex differentiable surrogates · Arbitrary discrete target | calibration | Helped |
| Smooth the cusp in the one-dimensional abstain surrogate | Cusp surrogate for binary classification with abstain level 1/4 · Abstain loss | calibration | Helped |
| Use the cusp surrogate without smoothing | Cusp surrogate for binary classification with abstain level 1/4 · Abstain loss | calibration | Hurt |
| Use the two-dimensional differentiable surrogate that satisfies IE but not strong IE | Two-dimensional surrogate L_ce · Target loss ℓ₂ | calibration | Hurt |
| Shift the target boundary so the surrogate level set lies entirely within a target cell, satisfying strong IE | Two-dimensional surrogate L_ce · Target loss ℓ₃ | calibration | Helped |
| Construct a one-dimensional convex differentiable surrogate for an orderable target | Orderable discrete targets · Arbitrary orderable target | calibration | Helped |
| Apply the one-dimensional construction to ordinal regression | Huber-like surrogate L_H · Three-class ordinal regression | calibration | Helped |
Surprising findings
- Indirect elicitation does not imply calibration in dimensions above one, even for strongly convex differentiable surrogates.
- No polyhedral surrogate satisfies strong IE if it indirectly elicits a target.
Limitations
- Results assume compactness of surrogate minimizer sets, which excludes common unbounded-minimizer losses such as squared hinge.
- Strong IE for non-minimizable losses is not addressed; calibration there requires different tools.
Experimental setup
Builds on: polyhedral indirect elicitation calibration framework for discrete targets orderable properties
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