Loss Landscape Geometry and the Learning of Symmetries: Or, What Influence Functions Reveal About Robust Generalization

We study how neural emulators of partial differential equation solution operators internalize physical symmetries by introducing an influence-based diagnostic that measures the propagation of parameter updates between symmetry-related states, defined as the metric-weighted overlap of loss gradients evaluated along group orbits. This quantity probes the local geometry of the learned loss landscape and goes beyond forward-pass equivariance tests by directly assessing whether learning dynamics couple physically equivalent configurations. Applying our diagnostic to autoregressive fluid flow emulators, we show that orbit-wise gradient coherence provides the mechanism for learning to generalize over symmetry transformations and indicates when training selects a symmetry compatible basin. The result is a novel technique for evaluating if surrogate models have internalized symmetry properties of the known solution operator.

Statistical Uncertainty Quantification for Aggregate Performance Metrics in Machine Learning Benchmarks

Modern artificial intelligence is supported by machine learning models (e.g., foundation models) that are pretrained on a massive data corpus and then adapted to solve a variety of downstream tasks. To summarize performance across multiple tasks, evaluation metrics are often aggregated into a summary metric, e.g., average accuracy across 10 question-answering tasks. When aggregating evaluation metrics, it is useful to incorporate uncertainty in the aggregate metric in order to gain a more realistic understanding of model performance. Our objective in this work is to demonstrate how statistical methodology can be used for quantifying uncertainty in metrics that have been aggregated across multiple tasks. The methods we emphasize are bootstrapping, Bayesian hierarchical (i.e., multilevel) modeling, and the visualization of task weightings that consider standard errors. These techniques reveal insights such as the dominance of a specific model for certain types of tasks despite an overall poor performance. We use a popular ML benchmark, the Visual Task Adaptation Benchmark (VTAB), to demonstrate the usefulness of our approaches.

Posterior Mean Matching: Generative Modeling through Online Bayesian Inference

This paper introduces posterior mean matching (PMM), a new method for generative modeling that is grounded in Bayesian inference. PMM uses conjugate pairs of distributions to model complex data of various modalities like images and text, offering a flexible alternative to existing methods like diffusion models. PMM models iteratively refine noisy approximations of the target distribution using updates from online Bayesian inference. PMM is flexible because its mechanics are based on general Bayesian models. We demonstrate this flexibility by developing specialized examples: a generative PMM model of real-valued data using the Normal-Normal model, a generative PMM model of count data using a Gamma-Poisson model, and a generative PMM model of discrete data using a Dirichlet-Categorical model. For the Normal-Normal PMM model, we establish a direct connection to diffusion models by showing that its continuous-time formulation converges to a stochastic differential equation (SDE). Additionally, for the Gamma-Poisson PMM, we derive a novel SDE driven by a Cox process, which is a significant departure from traditional Brownian motion-based generative models. PMMs achieve performance that is competitive with generative models for language modeling and image generation.