Contrastive Normalizing Flows for Uncertainty-Aware Parameter Estimation

Estimating physical parameters from data is a crucial application of machine learning (ML) in the physical sciences. However, systematic uncertainties, such as detector miscalibration, induce data distribution distortions that can erode statistical precision. In both high-energy physics (HEP) and broader ML contexts, achieving uncertainty-aware parameter estimation under these domain shifts remains an open problem. In this work, we address this challenge of uncertainty-aware parameter estimation for a broad set of tasks critical for HEP. We introduce a novel approach based on Contrastive Normalizing Flows (CNFs), which achieves top performance on the HiggsML Uncertainty Challenge dataset. Building on the insight that a binary classifier can approximate the model parameter likelihood ratio, we address the practical limitations of expressivity and the high cost of simulating high-dimensional parameter grids by embedding data and parameters in a learned CNF mapping. This mapping yields a tunable contrastive distribution that enables robust classification under shifted data distributions. Through a combination of theoretical analysis and empirical evaluations, we demonstrate that CNFs, when coupled with a classifier and established frequentist techniques, provide principled parameter estimation and uncertainty quantification through classification that is robust to data distribution distortions.

Uncertainty Quantification From Scaling Laws in Deep Neural Networks

Quantifying the uncertainty from machine learning analyses is critical to their use in the physical sciences. In this work we focus on uncertainty inherited from the initialization distribution of neural networks. We compute the mean $\mu_{\mathcal{L}}$ and variance $\sigma_{\mathcal{L}}^2$ of the test loss $\mathcal{L}$ for an ensemble of multi-layer perceptrons (MLPs) with neural tangent kernel (NTK) initialization in the infinite-width limit, and compare empirically to the results from finite-width networks for three example tasks: MNIST classification, CIFAR classification and calorimeter energy regression. We observe scaling laws as a function of training set size $N_\mathcal{D}$ for both $\mu_{\mathcal{L}}$ and $\sigma_{\mathcal{L}}$, but find that the coefficient of variation $\epsilon_{\mathcal{L}} \equiv \sigma_{\mathcal{L}}/\mu_{\mathcal{L}}$ becomes independent of $N_\mathcal{D}$ at both infinite and finite width for sufficiently large $N_\mathcal{D}$. This implies that the coefficient of variation of a finite-width network may be approximated by its infinite-width value, and may in principle be calculable using finite-width perturbation theory.