Linear-Core Surrogates: Smooth Loss Functions with Linear Rates for Classification and Structured Prediction
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| Format: | Preprint |
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2026
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| author | Mohri, Mehryar Zhong, Yutao |
| author_facet | Mohri, Mehryar Zhong, Yutao |
| contents | The choice of loss function in classification involves a fundamental trade-off: smooth losses (like Cross-Entropy) enable fast optimization rates but yield slow square-root consistency bounds, while piecewise-linear losses (like Hinge) offer fast linear consistency rates but suffer from non-differentiability. We propose Linear-Core (LC) Surrogates, a new family of convex loss functions that resolve this tension by stitching a linear core to a smooth tail. We prove that these surrogates are differentiable everywhere while retaining strict linear $H$-consistency bounds, effectively combining the optimization benefits of smoothness with the statistical efficiency of margin-based losses. In the structured prediction setting, we show that this smoothness unlocks a massive computational and energy advantage: it allows for an unbiased stochastic gradient estimator that bypasses the quadratic complexity $O(|\mathscr{Y}|^2)$ of exact inference (e.g., Viterbi). Empirically, our method achieves a 23$\times$ speedup over Structured SVMs on large-vocabulary sequence tagging tasks and demonstrates superior robustness to instance-dependent label noise, outperforming Cross-Entropy by 2.6% on corrupted CIFAR-10. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_27742 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear-Core Surrogates: Smooth Loss Functions with Linear Rates for Classification and Structured Prediction Mohri, Mehryar Zhong, Yutao Machine Learning The choice of loss function in classification involves a fundamental trade-off: smooth losses (like Cross-Entropy) enable fast optimization rates but yield slow square-root consistency bounds, while piecewise-linear losses (like Hinge) offer fast linear consistency rates but suffer from non-differentiability. We propose Linear-Core (LC) Surrogates, a new family of convex loss functions that resolve this tension by stitching a linear core to a smooth tail. We prove that these surrogates are differentiable everywhere while retaining strict linear $H$-consistency bounds, effectively combining the optimization benefits of smoothness with the statistical efficiency of margin-based losses. In the structured prediction setting, we show that this smoothness unlocks a massive computational and energy advantage: it allows for an unbiased stochastic gradient estimator that bypasses the quadratic complexity $O(|\mathscr{Y}|^2)$ of exact inference (e.g., Viterbi). Empirically, our method achieves a 23$\times$ speedup over Structured SVMs on large-vocabulary sequence tagging tasks and demonstrates superior robustness to instance-dependent label noise, outperforming Cross-Entropy by 2.6% on corrupted CIFAR-10. |
| title | Linear-Core Surrogates: Smooth Loss Functions with Linear Rates for Classification and Structured Prediction |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2604.27742 |