Dimension-Free Decision Calibration for Nonlinear Loss Functions

2025-04-22

AI Summary

This paper addresses decision calibration under nonlinear loss functions in high-dimensional prediction spaces. To overcome the curse of dimensionality inherent in conventional approaches, we propose a novel paradigm—*smooth decision calibration*. Our method constructs a smooth best-response mapping to avoid exponential dependence on feature dimensionality and employs functional approximation within a reproducing kernel Hilbert space (RKHS), coupled with a polynomial-time post-processing algorithm. Theoretically, we prove that only $ ext{poly}(|mathcal{A}|, 1/varepsilon)$ samples are required to achieve $varepsilon$-calibration for any initial predictor—achieving *dimension-free* sample complexity without compromising predictive accuracy. This is the first result establishing dimension-independent decision calibration, lifting the restrictive assumption of linear losses and extending to bounded-norm function classes in infinite-dimensional separable RKHSs. Our framework provides downstream decision-makers with near-optimal response guarantees under general nonlinear losses.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Sequential Decision Making
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

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 $phi(y)$ of dimension $m$, then approximate losses with linear functions of $phi(y)$. 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 $mathrm{poly}(|A|,1/epsilon)$ 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.

Problem

• Ensures optimal decision-making with nonlinear loss functions
• Achieves dimension-free decision calibration efficiently
• Handles high-dimensional outcome spaces with polynomial complexity

Innovation

• Dimension-free decision calibration for nonlinear losses
• Smooth best-response enables efficient calibration
• Poly sample complexity post-processing for calibration