Online Reasoning Calibration: Test-Time Training Enables Generalizable Conformal LLM Reasoning

While test-time scaling has enabled large language models to solve highly difficult tasks, state-of-the-art results come at exorbitant compute costs. These inefficiencies can be attributed to the miscalibration of post-trained language models, and the lack of calibration in popular sampling techniques. Here, we present Online Reasoning Calibration (ORCA), a framework for calibrating the sampling process that draws upon conformal prediction and test-time training. Specifically, we introduce a meta-learning procedure that updates the calibration module for each input. This allows us to provide valid confidence estimates under distributional shift, e.g. in thought patterns that occur across different stages of reasoning, or in prompt distributions between model development and deployment. ORCA not only provides theoretical guarantees on conformal risks, but also empirically shows higher efficiency and generalization across different reasoning tasks. At risk level $δ=0.1$, ORCA improves Qwen2.5-32B efficiency on in-distribution tasks with savings up to 47.5% with supervised labels and 40.7% with self-consistency labels. Under zero-shot out-of-domain settings, it improves MATH-500 savings from 24.8% of the static calibration baseline to 67.0% while maintaining a low empirical error rate, and the same trend holds across model families and downstream benchmarks. Our code is publicly available at https://github.com/wzekai99/ORCA.

Conformal Prediction for Generative Models via Adaptive Cluster-Based Density Estimation

Conditional generative models map input variables to complex, high-dimensional distributions, enabling realistic sample generation in a diverse set of domains. A critical challenge with these models is the absence of calibrated uncertainty, which undermines trust in individual outputs for high-stakes applications. To address this issue, we propose a systematic conformal prediction approach tailored to conditional generative models, leveraging density estimation on model-generated samples. We introduce a novel method called CP4Gen, which utilizes clustering-based density estimation to construct prediction sets that are less sensitive to outliers, more interpretable, and of lower structural complexity than existing methods. Extensive experiments on synthetic datasets and real-world applications, including climate emulation tasks, demonstrate that CP4Gen consistently achieves superior performance in terms of prediction set volume and structural simplicity. Our approach offers practitioners a powerful tool for uncertainty estimation associated with conditional generative models, particularly in scenarios demanding rigorous and interpretable prediction sets.

On the design of scalable, high-precision spherical-radial Fourier features

Approximation using Fourier features is a popular technique for scaling kernel methods to large-scale problems, with myriad applications in machine learning and statistics. This method replaces the integral representation of a shift-invariant kernel with a sum using a quadrature rule. The design of the latter is meant to reduce the number of features required for high-precision approximation. Specifically, for the squared exponential kernel, one must design a quadrature rule that approximates the Gaussian measure on $\mathbb{R}^d$. Previous efforts in this line of research have faced difficulties in higher dimensions. We introduce a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions by exploiting its isotropy. These rules are constructed as a tensor product of a radial quadrature rule and a spherical quadrature rule. Compared to previous work, our approach leverages a thorough analysis of the approximation error, which suggests natural choices for both the radial and spherical components. We demonstrate that this family of Fourier features yields improved approximation bounds.