Calibration without Ground Truth

TL;DR

This work tackles calibration and performance in data-scarce regimes where ground-truth labels are unavailable. It introduces a label-free post-processing framework that improves a strong primary model by leveraging a weaker yet better-calibrated reference, guaranteed to reduce a chosen proper loss via a Bregman-projection onto a reference-compatible set defined by calibration constraints. A central theoretical result shows that a strictly improving transformation exists if and only if the two predictors are not mutually calibrated, connecting calibration inconsistencies to arbitrage/no-trade opportunities. The approach is validated on open-source LLMs across multiple benchmarks, achieving substantial reductions in BS and ECE while preserving or marginally improving accuracy, and approaching supervised baselines despite no labeled data. Overall, the method provides a principled, general mechanism for calibration transfer and robust performance in label-scarce settings, with broad implications for trustworthy AI deployment.

Abstract

Villalobos et al. [2024] predict that publicly available human text will be exhausted within the next decade. Thus, improving models without access to ground-truth labels becomes increasingly important. We propose a label-free post-processing framework that improves a strong but miscalibrated model using a weaker yet better-calibrated reference. Our framework guarantees a strict performance improvement under any proper loss. Our approach is based on a characterization of when strict improvement is possible: when the strong and reference models are not mutually calibrated. We formalize this condition, connect it to arbitrage and no-trade results from economics, and develop an efficient Bregman projection algorithm that guarantees worst-case loss reduction without labels. Experiments on representative LLMs across varying scales demonstrate that our label-free method significantly reduces proper losses and calibration errors, achieving performance competitive with supervised baselines.

Figures (3)

• Figure 1: The projection to the reference-compatible set of predictors that are mutually calibrated with the reference model in \ref{['example: intro contradictory']}. The red point represents the "strong" but miscalibrated model. The blue line represents the constraints imposed by the "weak" but calibrated reference: the predictions to Groups A and B should imply a base rate of $0.5$, i.e., $\bm{q}_1 + \bm{q}_2 = 1$. By projecting the red point onto the blue line of reference-compatible predictors, we obtain a model (green point) that retains the informativeness of the strong model but aligns with the predictions of the weak model.
• Figure 2: Standard zero-shot prompt template utilized for Base models to directly elicit the first token probability following the question. The {subject} placeholder is instantiated with the specific domain for MMLU/MMLU-Redux or commonsense for CommonsenseQA, and {task_query},{choice} placeholders are instantiated with the question and choices of the specific sample.
• Figure 3: Two-step zero-shot CoT prompt templates designed for Instruct models. We first prompt the model to generate a reasoning chain (Step 1), and subsequently append an answer-triggering phrase to extract the next-token probability (Step 2). The {subject} placeholder is instantiated with the specific domain for MMLU/MMLU-Redux or commonsense for CommonsenseQA, and {task_query},{choice} placeholders are instantiated with the question and choices of the specific sample.

Theorems & Definitions (29)

• Example 1.1: Contradictory Predictions
• Example 1.2: Arbitrage in Contradictory Predictions
• Definition 2.2: Strictly Improving Transformation
• Example 2.3: Mode and Confidence
• Definition 3.1: Mutual Calibration
• Theorem 3.3
• Lemma A.1: Convex Representation of Proper Losses
• Remark A.2: Regret as Bregman Divergence
• proof : Proof of \ref{['obs:extendbreg']}
• Lemma A.4: Generalized Pythagorean Theorem