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Trading off Consistency and Dimensionality of Convex Surrogates for Multiclass Classification

NeurIPS 2024 · Learning theory and algorithms › Prediction and decision losses · approach: Consistency of surrogate losses · 0 citations (OpenAlex)

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Studies trade-offs between consistency and surrogate dimension for multiclass classification via polytope embeddings, giving partial-consistency results under low-noise assumptions and a multi-instance method that elicits the mode in dimension d with O(d^2) problem instances.

Problem

Consistent convex surrogate losses for multiclass classification over n outcomes need prediction dimension d ≈ n−1, which is computationally infeasible for large n, while low-dimensional embeddings used in practice lack theoretical guarantees.

Approach

Analyzes polytope embeddings with Bregman-divergence induced losses and MAP links in dimension d ≪ n, defining hallucination regions and partial calibration under low-noise assumptions; also shows the mode can be fully elicited by aggregating comparisons from many low-dimensional cross-polytope surrogate instances run in parallel.

Key results

• Every polytope embedding is calibrated for the 0-1 loss under a sufficiently strong low-noise assumption (Theorem 5).
• Hallucination regions always exist when embedding n outcomes into d < n-1 dimensions (Theorem 3).
• The mode over n outcomes can be elicited via O(n^2) low-dimensional problem instances with dimension about n/2 (Theorem 10).

Surprising findings

• Calibration regions for the permutahedron embedding shrink exponentially in d under low noise, so factorial embeddings are not advisable without a power-law prior.
• Hallucinations can be forced by the target loss itself, e.g. Hamming loss, not just by the embedding.

Limitations

• Results are asymptotic and assume infinite data and model expressivity.
• Strict calibration regions are hard to characterize computationally.
• Hallucinations can occur in the surrogate-to-label link for d < n-1, a risk in high-stakes settings.

Experimental setup

Builds on: Agarwal and Agarwal 2015 calibration Ramaswamy and Agarwal 2016 Blondel 2019 projection-based losses Wainwright et al. 2008

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