Optimality of Sub-network Laplace Approximations: New Results and Methods

Approximate Bayesian inference typically revolves around computing the posterior parameter distribution. In practice, however, the main object of interest is often a model's predictions rather than its parameters. In this work, we propose to bypass the parameter posterior and focus directly on approximating the posterior predictive distribution. We achieve this by drawing inspiration from self-training within self-supervised and semi-supervised learning. Essentially, we quantify a Bayesian model's predictive uncertainty by refitting on self-predicted data. The idea is strikingly simple: If a model assigns high likelihood to self-predicted data, these predictions are of low uncertainty, and vice versa. This yields a deterministic, sampling-free approximation of the posterior predictive. The modular structure of our Self-Supervised Laplace Approximation (SSLA) further allows us to plug in different prior specifications, enabling classical Bayesian sensitivity (w.r.t. prior choice) analysis. In order to bypass expensive refitting, we further introduce an approximate version of SSLA, called ASSLA. We study (A)SSLA both theoretically and empirically in regression models ranging from Bayesian linear models to Bayesian neural networks. Across a wide array of regression tasks with simulated and real-world datasets, our methods outperform classical Laplace approximations in predictive calibration while remaining computationally efficient.

Existing methods for quantifying predictive uncertainty in neural networks are either computationally intractable for large language models or require access to training data that is typically unavailable. We derive a lightweight alternative through two approximations: a first-order Taylor expansion that expresses uncertainty in terms of the gradient of the prediction and the parameter covariance, and an isotropy assumption on the parameter covariance. Together, these yield epistemic uncertainty as the squared gradient norm and aleatoric uncertainty as the Bernoulli variance of the point prediction, from a single forward-backward pass through an unmodified pretrained model. We justify the isotropy assumption by showing that covariance estimates built from non-training data introduce structured distortions that isotropic covariance avoids, and that theoretical results on the spectral properties of large networks support the approximation at scale. Validation against reference Markov Chain Monte Carlo estimates on synthetic problems shows strong correspondence that improves with model size. We then use the estimates to investigate when each uncertainty type carries useful signal for predicting answer correctness in question answering with large language models, revealing a benchmark-dependent divergence: the combined estimate achieves the highest mean AUROC on TruthfulQA, where questions involve genuine conflict between plausible answers, but falls to near chance on TriviaQA's factual recall, suggesting that parameter-level uncertainty captures a fundamentally different signal than self-assessment methods.

Learning neural set functions is pivotal to a wide range of important applications, including compound selection in AI-driven drug discovery and product recommendation. Recent work has introduced optimal subset oracles to implicitly learn set functions under practical weakly supervised settings, where model parameters are optimized through mean-field variational inference. However, these frameworks rely on Monte Carlo sampling to estimate gradients of the evidence lower bound when updating the variational distribution. Repeated sampling across iterations incurs substantial computational overhead, while the resulting stochasticity can destabilize the optimization trajectory. In this work, we reinterpret the evidence lower bound as a continuous relaxation of the set function and learn a surrogate objective that replaces sampling-based ELBO gradient estimation during variational optimization. The learned surrogate provides stable and efficient gradients throughout the continuous domain, thereby reducing computational overhead and accelerating inference. Furthermore, we establish an approximation guarantee for the proposed framework under submodular maximization and characterize its connection to variational free energy. Experiments on a variety of real-world tasks demonstrate consistent improvements over existing baselines.