Despite their successes, modern neural networks (NNs) still suffer from several shortcomings that limit their applicability in some settings.

Uncertainty quantification is an important prerequisite for the deployment of deep learning models in safety-critical areas. Yet, this hinges on the uncertainty estimates being useful to the extent the prediction intervals are well-calibrated and sharp. In the absence of inherent uncertainty estimates (e.g. pretrained models predicting only point estimates), popular approaches that operate post-hoc include Laplace's method and split conformal prediction (split-CP). However, Laplace's method can be miscalibrated when the model is misspecified and split-CP requires sample splitting, and thus comes at the expense of statistical efficiency. In this work, we construct prediction intervals for neural network regressors post-hoc without held-out data. This is achieved by approximating the full conformal prediction method (full-CP). Whilst full-CP nominally requires retraining the model for every test point and candidate label, we propose to train just once and locally perturb model parameters using Gauss-Newton influence to approximate the effect of retraining. Coupled with linearization of the network, we express the absolute residual nonconformity score as a piecewise linear function of the candidate label allowing for an efficient procedure that avoids the exhaustive search over the output space. On standard regression benchmarks and bounding box localization, we show the resulting prediction intervals are locally-adaptive and often tighter than those of split-CP.

Modern deep learning models generalize remarkably well in-distribution, despite being overparametrized and trained with little to no explicit regularization. Instead, current theory credits implicit regularization imposed by the choice of architecture, hyperparameters and optimization procedure. However, deploying deep learning models out-of-distribution, in sequential decision-making tasks, or in safety-critical domains, necessitates reliable uncertainty quantification, not just a point estimate. The machinery of modern approximate inference -- Bayesian deep learning -- should answer the need for uncertainty quantification, but its effectiveness has been challenged by our inability to define useful explicit inductive biases through priors, as well as the associated computational burden. Instead, in this work we demonstrate, both theoretically and empirically, how to regularize a variational deep network implicitly via the optimization procedure, just as for standard deep learning. We fully characterize the inductive bias of (stochastic) gradient descent in the case of an overparametrized linear model as generalized variational inference and demonstrate the importance of the choice of parametrization. Finally, we show empirically that our approach achieves strong in- and out-of-distribution performance without tuning of additional hyperparameters and with minimal time and memory overhead over standard deep learning.

Selecting hyperparameters in deep learning greatly impacts its effectiveness but requires manual effort and expertise. Recent works show that Bayesian model selection with Laplace approximations can allow to optimize such hyperparameters just like standard neural network parameters using gradients and on the training data. However, estimating a single hyperparameter gradient requires a pass through the entire dataset, limiting the scalability of such algorithms. In this work, we overcome this issue by introducing lower bounds to the linearized Laplace approximation of the marginal likelihood. In contrast to previous estimators, these bounds are amenable to stochastic-gradient-based optimization and allow to trade off estimation accuracy against computational complexity. We derive them using the function-space form of the linearized Laplace, which can be estimated using the neural tangent kernel. Experimentally, we show that the estimators can significantly accelerate gradient-based hyperparameter optimization.

Objectives: This paper develops two algorithms to achieve federated generalized linear mixed effect models (GLMM), and compares the developed model's outcomes with each other, as well as that from the standard R package (`lme4'). Methods: The log-likelihood function of GLMM is approximated by two numerical methods (Laplace approximation and Gaussian Hermite approximation), which supports federated decomposition of GLMM to bring computation to data. Results: Our developed method can handle GLMM to accommodate hierarchical data with multiple non-independent levels of observations in a federated setting. The experiment results demonstrate comparable (Laplace) and superior (Gaussian-Hermite) performances with simulated and real-world data. Conclusion: We developed and compared federated GLMMs with different approximations, which can support researchers in analyzing biomedical data to accommodate mixed effects and address non-independence due to hierarchical structures (i.e., institutes, region, country, etc.).