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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.15615 |
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| _version_ | 1866908331695669248 |
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| author | Tang, Jingwu Wu, Jiayun Wu, Zhiwei Steven Zhang, Jiahao |
| author_facet | Tang, Jingwu Wu, Jiayun Wu, Zhiwei Steven Zhang, Jiahao |
| contents | When model predictions inform downstream decision making, a natural question is under what conditions can the decision-makers simply respond to the predictions as if they were the true outcomes. Calibration suffices to guarantee that simple best-response to predictions is optimal. However, calibration for high-dimensional prediction outcome spaces requires exponential computational and statistical complexity. The recent relaxation known as decision calibration ensures the optimality of the simple best-response rule while requiring only polynomial sample complexity in the dimension of outcomes. However, known results on calibration and decision calibration crucially rely on linear loss functions for establishing best-response optimality. A natural approach to handle nonlinear losses is to map outcomes $y$ into a feature space $ϕ(y)$ of dimension $m$, then approximate losses with linear functions of $ϕ(y)$. Unfortunately, even simple classes of nonlinear functions can demand exponentially large or infinite feature dimensions $m$. A key open problem is whether it is possible to achieve decision calibration with sample complexity independent of~$m$. We begin with a negative result: even verifying decision calibration under standard deterministic best response inherently requires sample complexity polynomial in~$m$. Motivated by this lower bound, we investigate a smooth version of decision calibration in which decision-makers follow a smooth best-response. This smooth relaxation enables dimension-free decision calibration algorithms. We introduce algorithms that, given $\mathrm{poly}(|A|,1/ε)$ samples and any initial predictor~$p$, can efficiently post-process it to satisfy decision calibration without worsening accuracy. Our algorithms apply broadly to function classes that can be well-approximated by bounded-norm functions in (possibly infinite-dimensional) separable RKHS. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension-Free Decision Calibration for Nonlinear Loss Functions Tang, Jingwu Wu, Jiayun Wu, Zhiwei Steven Zhang, Jiahao Machine Learning When model predictions inform downstream decision making, a natural question is under what conditions can the decision-makers simply respond to the predictions as if they were the true outcomes. Calibration suffices to guarantee that simple best-response to predictions is optimal. However, calibration for high-dimensional prediction outcome spaces requires exponential computational and statistical complexity. The recent relaxation known as decision calibration ensures the optimality of the simple best-response rule while requiring only polynomial sample complexity in the dimension of outcomes. However, known results on calibration and decision calibration crucially rely on linear loss functions for establishing best-response optimality. A natural approach to handle nonlinear losses is to map outcomes $y$ into a feature space $ϕ(y)$ of dimension $m$, then approximate losses with linear functions of $ϕ(y)$. Unfortunately, even simple classes of nonlinear functions can demand exponentially large or infinite feature dimensions $m$. A key open problem is whether it is possible to achieve decision calibration with sample complexity independent of~$m$. We begin with a negative result: even verifying decision calibration under standard deterministic best response inherently requires sample complexity polynomial in~$m$. Motivated by this lower bound, we investigate a smooth version of decision calibration in which decision-makers follow a smooth best-response. This smooth relaxation enables dimension-free decision calibration algorithms. We introduce algorithms that, given $\mathrm{poly}(|A|,1/ε)$ samples and any initial predictor~$p$, can efficiently post-process it to satisfy decision calibration without worsening accuracy. Our algorithms apply broadly to function classes that can be well-approximated by bounded-norm functions in (possibly infinite-dimensional) separable RKHS. |
| title | Dimension-Free Decision Calibration for Nonlinear Loss Functions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2504.15615 |