Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inference

Model selection in Gaussian processes scales prohibitively with the size of the training dataset, both in time and memory. While many approximations exist, all incur inevitable approximation error. Recent work accounts for this error in the form of computational uncertainty, which enables -- at the cost of quadratic complexity -- an explicit tradeoff between computation and precision. Here we extend this development to model selection, which requires significant enhancements to the existing approach, including linear-time scaling in the size of the dataset. We propose a novel training loss for hyperparameter optimization and demonstrate empirically that the resulting method can outperform SGPR, CGGP and SVGP, state-of-the-art methods for GP model selection, on medium to large-scale datasets. Our experiments show that model selection for computation-aware GPs trained on 1.8 million data points can be done within a few hours on a single GPU. As a result of this work, Gaussian processes can be trained on large-scale datasets without significantly compromising their ability to quantify uncertainty -- a fundamental prerequisite for optimal decision-making.

Debiasing Mini-Batch Quadratics for Applications in Deep Learning

Quadratic approximations form a fundamental building block of machine learning methods. E.g., second-order optimizers try to find the Newton step into the minimum of a local quadratic proxy to the objective function; and the second-order approximation of a network's loss function can be used to quantify the uncertainty of its outputs via the Laplace approximation. When computations on the entire training set are intractable - typical for deep learning - the relevant quantities are computed on mini-batches. This, however, distorts and biases the shape of the associated stochastic quadratic approximations in an intricate way with detrimental effects on applications. In this paper, we (i) show that this bias introduces a systematic error, (ii) provide a theoretical explanation for it, (iii) explain its relevance for second-order optimization and uncertainty quantification via the Laplace approximation in deep learning, and (iv) develop and evaluate debiasing strategies.

Efficient Weight-Space Laplace-Gaussian Filtering and Smoothing for Sequential Deep Learning

Efficiently learning a sequence of related tasks, such as in continual learning, poses a significant challenge for neural nets due to the delicate trade-off between catastrophic forgetting and loss of plasticity. We address this challenge with a grounded framework for sequentially learning related tasks based on Bayesian inference. Specifically, we treat the model's parameters as a nonlinear Gaussian state-space model and perform efficient inference using Gaussian filtering and smoothing. This general formalism subsumes existing continual learning approaches, while also offering a clearer conceptual understanding of its components. Leveraging Laplace approximations during filtering, we construct Gaussian posterior measures on the weight space of a neural network for each task. We use it as an efficient regularizer by exploiting the structure of the generalized Gauss-Newton matrix (GGN) to construct diagonal plus low-rank approximations. The dynamics model allows targeted control of the learning process and the incorporation of domain-specific knowledge, such as modeling the type of shift between tasks. Additionally, using Bayesian approximate smoothing can enhance the performance of task-specific models without needing to re-access any data.

FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning

Laplace approximations are popular techniques for endowing deep networks with epistemic uncertainty estimates as they can be applied without altering the predictions of the neural network, and they scale to large models and datasets. While the choice of prior strongly affects the resulting posterior distribution, computational tractability and lack of interpretability of weight space typically limit the Laplace approximation to isotropic Gaussian priors, which are known to cause pathological behavior as depth increases. As a remedy, we directly place a prior on function space. More precisely, since Lebesgue densities do not exist on infinite-dimensional function spaces, we have to recast training as finding the so-called weak mode of the posterior measure under a Gaussian process (GP) prior restricted to the space of functions representable by the neural network. Through the GP prior, one can express structured and interpretable inductive biases, such as regularity or periodicity, directly in function space, while still exploiting the implicit inductive biases that allow deep networks to generalize. After model linearization, the training objective induces a negative log-posterior density to which we apply a Laplace approximation, leveraging highly scalable methods from matrix-free linear algebra. Our method provides improved results where prior knowledge is abundant, e.g., in many scientific inference tasks. At the same time, it stays competitive for black-box regression and classification tasks where neural networks typically excel.

Uncertainty-Guided Optimization on Large Language Model Search Trees

Beam search is a standard tree search algorithm when it comes to finding sequences of maximum likelihood, for example, in the decoding processes of large language models. However, it is myopic since it does not take the whole path from the root to a leaf into account. Moreover, it is agnostic to prior knowledge available about the process: For example, it does not consider that the objective being maximized is a likelihood and thereby has specific properties, like being bound in the unit interval. Taking a probabilistic approach, we define a prior belief over the LLMs' transition probabilities and obtain a posterior belief over the most promising paths in each iteration. These beliefs are helpful to define a non-myopic Bayesian-optimization-like acquisition function that allows for a more data-efficient exploration scheme than standard beam search. We discuss how to select the prior and demonstrate in on- and off-model experiments with recent large language models, including Llama-2-7b, that our method achieves higher efficiency than beam search: Our method achieves the same or a higher likelihood while expanding fewer nodes than beam search.

Scaling up Probabilistic PDE Simulators with Structured Volumetric Information

Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning analogues. Any such numerical solution is subject to multiple sources of uncertainty, both from limited computational resources and limited data (including unknown parameters). Gaussian process analogues to classic PDE simulation methods have recently emerged as a framework to construct fully probabilistic estimates of all these types of uncertainty. So far, much of this work focused on theoretical foundations, and as such is not particularly data efficient or scalable. Here we propose a framework combining a discretization scheme based on the popular Finite Volume Method with complementary numerical linear algebra techniques. Practical experiments, including a spatiotemporal tsunami simulation, demonstrate substantially improved scaling behavior of this approach over previous collocation-based techniques.

Linearization Turns Neural Operators into Function-Valued Gaussian Processes

Modeling dynamical systems, e.g. in climate and engineering sciences, often necessitates solving partial differential equations. Neural operators are deep neural networks designed to learn nontrivial solution operators of such differential equations from data. As for all statistical models, the predictions of these models are imperfect and exhibit errors. Such errors are particularly difficult to spot in the complex nonlinear behaviour of dynamical systems. We introduce a new framework for approximate Bayesian uncertainty quantification in neural operators using function-valued Gaussian processes. Our approach can be interpreted as a probabilistic analogue of the concept of currying from functional programming and provides a practical yet theoretically sound way to apply the linearized Laplace approximation to neural operators. In a case study on Fourier neural operators, we show that, even for a discretized input, our method yields a Gaussian closure--a structured Gaussian process posterior capturing the uncertainty in the output function of the neural operator, which can be evaluated at an arbitrary set of points. The method adds minimal prediction overhead, can be applied post-hoc without retraining the neural operator, and scales to large models and datasets. We showcase the efficacy of our approach through applications to different types of partial differential equations.

Reparameterization invariance in approximate Bayesian inference

Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i.e. BNNs assign different posterior densities to different parametrizations of identical functions. This creates a fundamental flaw in the application of Bayesian principles as it breaks the correspondence between uncertainty over the parameters with uncertainty over the parametrized function. In this paper, we investigate this issue in the context of the increasingly popular linearized Laplace approximation. Specifically, it has been observed that linearized predictives alleviate the common underfitting problems of the Laplace approximation. We develop a new geometric view of reparametrizations from which we explain the success of linearization. Moreover, we demonstrate that these reparameterization invariance properties can be extended to the original neural network predictive using a Riemannian diffusion process giving a straightforward algorithm for approximate posterior sampling, which empirically improves posterior fit.

Flexible inference in heterogeneous and attributed multilayer networks

Networked datasets are often enriched by different types of information about individual nodes or edges. However, most existing methods for analyzing such datasets struggle to handle the complexity of heterogeneous data, often requiring substantial model-specific analysis. In this paper, we develop a probabilistic generative model to perform inference in multilayer networks with arbitrary types of information. Our approach employs a Bayesian framework combined with the Laplace matching technique to ease interpretation of inferred parameters. Furthermore, the algorithmic implementation relies on automatic differentiation, avoiding the need for explicit derivations. This makes our model scalable and flexible to adapt to any combination of input data. We demonstrate the effectiveness of our method in detecting overlapping community structures and performing various prediction tasks on heterogeneous multilayer data, where nodes and edges have different types of attributes. Additionally, we showcase its ability to unveil a variety of patterns in a social support network among villagers in rural India by effectively utilizing all input information in a meaningful way.

Computation-Aware Kalman Filtering and Smoothing

Kalman filtering and smoothing are the foundational mechanisms for efficient inference in Gauss-Markov models. However, their time and memory complexities scale prohibitively with the size of the state space. This is particularly problematic in spatiotemporal regression problems, where the state dimension scales with the number of spatial observations. Existing approximate frameworks leverage low-rank approximations of the covariance matrix. Since they do not model the error introduced by the computational approximation, their predictive uncertainty estimates can be overly optimistic. In this work, we propose a probabilistic numerical method for inference in high-dimensional Gauss-Markov models which mitigates these scaling issues. Our matrix-free iterative algorithm leverages GPU acceleration and crucially enables a tunable trade-off between computational cost and predictive uncertainty. Finally, we demonstrate the scalability of our method on a large-scale climate dataset.