Shaving Weights with Occam's Razor: Bayesian Sparsification for Neural Networks Using the Marginal Likelihood

Neural network sparsification is a promising avenue to save computational time and memory costs, especially in an age where many successful AI models are becoming too large to na\"ively deploy on consumer hardware. While much work has focused on different weight pruning criteria, the overall sparsifiability of the network, i.e., its capacity to be pruned without quality loss, has often been overlooked. We present Sparsifiability via the Marginal likelihood (SpaM), a pruning framework that highlights the effectiveness of using the Bayesian marginal likelihood in conjunction with sparsity-inducing priors for making neural networks more sparsifiable. Our approach implements an automatic Occam's razor that selects the most sparsifiable model that still explains the data well, both for structured and unstructured sparsification. In addition, we demonstrate that the pre-computed posterior Hessian approximation used in the Laplace approximation can be re-used to define a cheap pruning criterion, which outperforms many existing (more expensive) approaches. We demonstrate the effectiveness of our framework, especially at high sparsity levels, across a range of different neural network architectures and datasets.

Position Paper: Bayesian Deep Learning in the Age of Large-Scale AI

In the current landscape of deep learning research, there is a predominant emphasis on achieving high predictive accuracy in supervised tasks involving large image and language datasets. However, a broader perspective reveals a multitude of overlooked metrics, tasks, and data types, such as uncertainty, active and continual learning, and scientific data, that demand attention. Bayesian deep learning (BDL) constitutes a promising avenue, offering advantages across these diverse settings. This paper posits that BDL can elevate the capabilities of deep learning. It revisits the strengths of BDL, acknowledges existing challenges, and highlights some exciting research avenues aimed at addressing these obstacles. Looking ahead, the discussion focuses on possible ways to combine large-scale foundation models with BDL to unlock their full potential.

Uncertainty in Graph Contrastive Learning with Bayesian Neural Networks

Graph contrastive learning has shown great promise when labeled data is scarce, but large unlabeled datasets are available. However, it often does not take uncertainty estimation into account. We show that a variational Bayesian neural network approach can be used to improve not only the uncertainty estimates but also the downstream performance on semi-supervised node-classification tasks. Moreover, we propose a new measure of uncertainty for contrastive learning, that is based on the disagreement in likelihood due to different positive samples.

Kronecker-Factored Approximate Curvature for Modern Neural Network Architectures

The core components of many modern neural network architectures, such as transformers, convolutional, or graph neural networks, can be expressed as linear layers with $\textit{weight-sharing}$. Kronecker-Factored Approximate Curvature (K-FAC), a second-order optimisation method, has shown promise to speed up neural network training and thereby reduce computational costs. However, there is currently no framework to apply it to generic architectures, specifically ones with linear weight-sharing layers. In this work, we identify two different settings of linear weight-sharing layers which motivate two flavours of K-FAC -- $\textit{expand}$ and $\textit{reduce}$. We show that they are exact for deep linear networks with weight-sharing in their respective setting. Notably, K-FAC-reduce is generally faster than K-FAC-expand, which we leverage to speed up automatic hyperparameter selection via optimising the marginal likelihood for a Wide ResNet. Finally, we observe little difference between these two K-FAC variations when using them to train both a graph neural network and a vision transformer. However, both variations are able to reach a fixed validation metric target in $50$-$75\%$ of the number of steps of a first-order reference run, which translates into a comparable improvement in wall-clock time. This highlights the potential of applying K-FAC to modern neural network architectures.

Learning Layer-wise Equivariances Automatically using Gradients

Convolutions encode equivariance symmetries into neural networks leading to better generalisation performance. However, symmetries provide fixed hard constraints on the functions a network can represent, need to be specified in advance, and can not be adapted. Our goal is to allow flexible symmetry constraints that can automatically be learned from data using gradients. Learning symmetry and associated weight connectivity structures from scratch is difficult for two reasons. First, it requires efficient and flexible parameterisations of layer-wise equivariances. Secondly, symmetries act as constraints and are therefore not encouraged by training losses measuring data fit. To overcome these challenges, we improve parameterisations of soft equivariance and learn the amount of equivariance in layers by optimising the marginal likelihood, estimated using differentiable Laplace approximations. The objective balances data fit and model complexity enabling layer-wise symmetry discovery in deep networks. We demonstrate the ability to automatically learn layer-wise equivariances on image classification tasks, achieving equivalent or improved performance over baselines with hard-coded symmetry.

Towards Training Without Depth Limits: Batch Normalization Without Gradient Explosion

Normalization layers are one of the key building blocks for deep neural networks. Several theoretical studies have shown that batch normalization improves the signal propagation, by avoiding the representations from becoming collinear across the layers. However, results on mean-field theory of batch normalization also conclude that this benefit comes at the expense of exploding gradients in depth. Motivated by these two aspects of batch normalization, in this study we pose the following question: "Can a batch-normalized network keep the optimal signal propagation properties, but avoid exploding gradients?" We answer this question in the affirmative by giving a particular construction of an Multi-Layer Perceptron (MLP) with linear activations and batch-normalization that provably has bounded gradients at any depth. Based on Weingarten calculus, we develop a rigorous and non-asymptotic theory for this constructed MLP that gives a precise characterization of forward signal propagation, while proving that gradients remain bounded for linearly independent input samples, which holds in most practical settings. Inspired by our theory, we also design an activation shaping scheme that empirically achieves the same properties for certain non-linear activations.

Hodge-Aware Contrastive Learning

Simplicial complexes prove effective in modeling data with multiway dependencies, such as data defined along the edges of networks or within other higher-order structures. Their spectrum can be decomposed into three interpretable subspaces via the Hodge decomposition, resulting foundational in numerous applications. We leverage this decomposition to develop a contrastive self-supervised learning approach for processing simplicial data and generating embeddings that encapsulate specific spectral information.Specifically, we encode the pertinent data invariances through simplicial neural networks and devise augmentations that yield positive contrastive examples with suitable spectral properties for downstream tasks. Additionally, we reweight the significance of negative examples in the contrastive loss, considering the similarity of their Hodge components to the anchor. By encouraging a stronger separation among less similar instances, we obtain an embedding space that reflects the spectral properties of the data. The numerical results on two standard edge flow classification tasks show a superior performance even when compared to supervised learning techniques. Our findings underscore the importance of adopting a spectral perspective for contrastive learning with higher-order data.

Stochastic Marginal Likelihood Gradients using Neural Tangent Kernels

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.

Laplace-Approximated Neural Additive Models: Improving Interpretability with Bayesian Inference

Deep neural networks (DNNs) have found successful applications in many fields, but their black-box nature hinders interpretability. This is addressed by the neural additive model (NAM), in which the network is divided into additive sub-networks, thus making apparent the interaction between input features and predictions. In this paper, we approach the additive structure from a Bayesian perspective and develop a practical Laplace approximation. This enhances interpretability in three primary ways: a) It provides credible intervals for the recovered feature interactions by estimating function-space uncertainty of the sub-networks; b) it yields a tractable estimate of the marginal likelihood, which can be used to perform an implicit selection of features through an empirical Bayes procedure; and c) it can be used to rank feature pairs as candidates for second-order interactions in fine-tuned interaction models. We show empirically that our proposed Laplace-approximated NAM (LA-NAM) improves performance and interpretability on tabular regression and classification datasets and challenging real-world medical tasks.

Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization

The linearized-Laplace approximation (LLA) has been shown to be effective and efficient in constructing Bayesian neural networks. It is theoretically compelling since it can be seen as a Gaussian process posterior with the mean function given by the neural network's maximum-a-posteriori predictive function and the covariance function induced by the empirical neural tangent kernel. However, while its efficacy has been studied in large-scale tasks like image classification, it has not been studied in sequential decision-making problems like Bayesian optimization where Gaussian processes -- with simple mean functions and kernels such as the radial basis function -- are the de-facto surrogate models. In this work, we study the usefulness of the LLA in Bayesian optimization and highlight its strong performance and flexibility. However, we also present some pitfalls that might arise and a potential problem with the LLA when the search space is unbounded.