Vincent Fortuin

Mrinank Sharma, Tom Rainforth, Yee Whye Teh et al.

Conventional Bayesian Neural Networks (BNNs) are unable to leverage unlabelled data to improve their predictions. To overcome this limitation, we introduce Self-Supervised Bayesian Neural Networks, which use unlabelled data to learn models with suitable prior predictive distributions. This is achieved by leveraging contrastive pretraining techniques and optimising a variational lower bound. We then show that the prior predictive distributions of self-supervised BNNs capture problem semantics better than conventional BNN priors. In turn, our approach offers improved predictive performance over conventional BNNs, especially in low-budget regimes.

Jonas Rothfuss, Martin Josifoski, Vincent Fortuin et al.

Meta-Learning aims to speed up the learning process on new tasks by acquiring useful inductive biases from datasets of related learning tasks. While, in practice, the number of related tasks available is often small, most of the existing approaches assume an abundance of tasks; making them unrealistic and prone to overfitting. A central question in the meta-learning literature is how to regularize to ensure generalization to unseen tasks. In this work, we provide a theoretical analysis using the PAC-Bayesian theory and present a generalization bound for meta-learning, which was first derived by Rothfuss et al. (2021a). Crucially, the bound allows us to derive the closed form of the optimal hyper-posterior, referred to as PACOH, which leads to the best performance guarantees. We provide a theoretical analysis and empirical case study under which conditions and to what extent these guarantees for meta-learning improve upon PAC-Bayesian per-task learning bounds. The closed-form PACOH inspires a practical meta-learning approach that avoids the reliance on bi-level optimization, giving rise to a stochastic optimization problem that is amenable to standard variational methods that scale well. Our experiments show that, when instantiating the PACOH with Gaussian processes and Bayesian Neural Networks models, the resulting methods are more scalable, and yield state-of-the-art performance, both in terms of predictive accuracy and the quality of uncertainty estimates.

Julyan Arbel, Konstantinos Pitas, Mariia Vladimirova et al.

Neural networks have achieved remarkable performance across various problem domains, but their widespread applicability is hindered by inherent limitations such as overconfidence in predictions, lack of interpretability, and vulnerability to adversarial attacks. To address these challenges, Bayesian neural networks (BNNs) have emerged as a compelling extension of conventional neural networks, integrating uncertainty estimation into their predictive capabilities. This comprehensive primer presents a systematic introduction to the fundamental concepts of neural networks and Bayesian inference, elucidating their synergistic integration for the development of BNNs. The target audience comprises statisticians with a potential background in Bayesian methods but lacking deep learning expertise, as well as machine learners proficient in deep neural networks but with limited exposure to Bayesian statistics. We provide an overview of commonly employed priors, examining their impact on model behavior and performance. Additionally, we delve into the practical considerations associated with training and inference in BNNs. Furthermore, we explore advanced topics within the realm of BNN research, acknowledging the existence of ongoing debates and controversies. By offering insights into cutting-edge developments, this primer not only equips researchers and practitioners with a solid foundation in BNNs, but also illuminates the potential applications of this dynamic field. As a valuable resource, it fosters an understanding of BNNs and their promising prospects, facilitating further advancements in the pursuit of knowledge and innovation.

Fedor Sergeev, Paola Malsot, Gunnar Rätsch et al. · eth-zurich

Knowing which features of a multivariate time series to measure and when is a key task in medicine, wearables, and robotics. Better acquisition policies can reduce costs while maintaining or even improving the performance of downstream predictors. Inspired by the maximization of conditional mutual information, we propose an approach to train acquirers end-to-end using only the downstream loss. We show that our method outperforms random acquisition policy, matches a model with an unrestrained budget, but does not yet overtake a static acquisition strategy. We highlight the assumptions and outline avenues for future work.

Agustinus Kristiadi, Alexander Immer, Runa Eschenhagen et al.

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.

Tristan Cinquin, Marvin Pförtner, Vincent Fortuin et al.

Laplace approximations are popular techniques for endowing deep networks with epistemic uncertainty estimates as they can be applied without altering the predictions of the trained 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 the 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 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 (as is the case in many scientific inference tasks). At the same time, it stays competitive for black-box supervised learning problems, where neural networks typically excel.

Eliot Wong-Toi, Alex Boyd, Vincent Fortuin et al.

Deep, overparameterized regression models are notorious for their tendency to overfit. This problem is exacerbated in heteroskedastic models, which predict both mean and residual noise for each data point. At one extreme, these models fit all training data perfectly, eliminating residual noise entirely; at the other, they overfit the residual noise while predicting a constant, uninformative mean. We observe a lack of middle ground, suggesting a phase transition dependent on model regularization strength. Empirical verification supports this conjecture by fitting numerous models with varying mean and variance regularization. To explain the transition, we develop a theoretical framework based on a statistical field theory, yielding qualitative agreement with experiments. As a practical consequence, our analysis simplifies hyperparameter tuning from a two-dimensional to a one-dimensional search, substantially reducing the computational burden. Experiments on diverse datasets, including UCI datasets and the large-scale ClimSim climate dataset, demonstrate significantly improved performance in various calibration tasks.

Richard D. Paul, Alessio Quercia, Vincent Fortuin et al.

State-of-the-art computer vision tasks, like monocular depth estimation (MDE), rely heavily on large, modern Transformer-based architectures. However, their application in safety-critical domains demands reliable predictive performance and uncertainty quantification. While Bayesian neural networks provide a conceptually simple approach to serve those requirements, they suffer from the high dimensionality of the parameter space. Parameter-efficient fine-tuning (PEFT) methods, in particular low-rank adaptations (LoRA), have emerged as a popular strategy for adapting large-scale models to down-stream tasks by performing parameter inference on lower-dimensional subspaces. In this work, we investigate the suitability of PEFT methods for subspace Bayesian inference in large-scale Transformer-based vision models. We show that, indeed, combining BitFit, DiffFit, LoRA, and CoLoRA, a novel LoRA-inspired PEFT method, with Bayesian inference enables more robust and reliable predictive performance in MDE.

Alexander Möllers, Alexander Immer, Elvin Isufi et al.

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.

Alexander Möllers, Alexander Immer, Vincent Fortuin et al.

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.

Fedor Sergeev, Manuel Burger, Polina Leshetkina et al.

Clinical time series data are critical for patient monitoring and predictive modeling. These time series are typically multivariate and often comprise hundreds of heterogeneous features from different data sources. The grouping of features based on similarity and relevance to the prediction task has been shown to enhance the performance of deep learning architectures. However, defining these groups a priori using only semantic knowledge is challenging, even for domain experts. To address this, we propose a novel method that learns feature groups by clustering weights of feature-wise embedding layers. This approach seamlessly integrates into standard supervised training and discovers the groups that directly improve downstream performance on clinically relevant tasks. We demonstrate that our method outperforms static clustering approaches on synthetic data and achieves performance comparable to expert-defined groups on real-world medical data. Moreover, the learned feature groups are clinically interpretable, enabling data-driven discovery of task-relevant relationships between variables.

Szilvia Ujváry, Gergely Flamich, Vincent Fortuin et al.

An important yet underexplored question in the PAC-Bayes literature is how much tightness we lose by restricting the posterior family to factorized Gaussian distributions when optimizing a PAC-Bayes bound. We investigate this issue by estimating data-independent PAC-Bayes bounds using the optimal posteriors, comparing them to bounds obtained using MFVI. Concretely, we (1) sample from the optimal Gibbs posterior using Hamiltonian Monte Carlo, (2) estimate its KL divergence from the prior with thermodynamic integration, and (3) propose three methods to obtain high-probability bounds under different assumptions. Our experiments on the MNIST dataset reveal significant tightness gaps, as much as 5-6\% in some cases.

Arik Reuter, Tim G. J. Rudner, Vincent Fortuin et al.

Transformers have emerged as the dominant architecture in the field of deep learning, with a broad range of applications and remarkable in-context learning (ICL) capabilities. While not yet fully understood, ICL has already proved to be an intriguing phenomenon, allowing transformers to learn in context -- without requiring further training. In this paper, we further advance the understanding of ICL by demonstrating that transformers can perform full Bayesian inference for commonly used statistical models in context. More specifically, we introduce a general framework that builds on ideas from prior fitted networks and continuous normalizing flows and enables us to infer complex posterior distributions for models such as generalized linear models and latent factor models. Extensive experiments on real-world datasets demonstrate that our ICL approach yields posterior samples that are similar in quality to state-of-the-art MCMC or variational inference methods that do not operate in context. The source code for this paper is available at https://github.com/ArikReuter/ICL_for_Full_Bayesian_Inference.

Ben Riegler, James Odgers, Vincent Fortuin

Asynchronous Bayesian optimization is widely used for gradient-free optimization in domains with independent parallel experiments and varying evaluation times. Existing methods posit that standard acquisitions lead to redundant and repeated queries, proposing complex solutions to enforce diversity in queries. Challenging this fundamental premise, we show that methods, like the Upper Confidence Bound, can in fact achieve theoretical guarantees essentially equivalent to those of sequential Thompson sampling. A conceptual analysis of asynchronous Bayesian optimization reveals that existing works neglect intermediate posterior updates, which we find to be generally sufficient to avoid redundant queries. Further investigation shows that by penalizing busy locations, diversity-enforcing methods can over-explore in asynchronous settings, reducing their performance. Our extensive experiments demonstrate that simple standard acquisition functions match or outperform purpose-built asynchronous methods across synthetic and real-world tasks.

Elif Akata, Konstantinos Voudouris, Vincent Fortuin et al.

Large language models (LLMs) can learn from a few demonstrations provided at inference time. We study this in-context learning phenomenon through the lens of Gaussian Processes (GPs). We build controlled experiments where models observe sequences of multivariate scalar-valued function samples drawn from known GP priors. We evaluate prediction error in relation to the number of demonstrations and compare against two principled references: (i) an empirical GP-regression learner that gives a lower bound on achievable error, and (ii) the expected error of a 1-nearest-neighbor (1-NN) rule, which gives a data-driven upper bound. Across model sizes, we find that LLM learning curves are strongly influenced by the function-generating kernels and approach the GP lower bound as the number of demonstrations increases. We then study the inductive biases of these models using a likelihood-based analysis. We find that LLM predictions are most likely under less smooth GP kernels. Finally, we explore whether post-training can shift these inductive biases and improve sample-efficiency on functions sampled from GPs with smoother kernels. We find that both reinforcement learning and supervised fine-tuning can effectively shift inductive biases in the direction of the training data. Together, our framework quantifies the extent to which LLMs behave like GP learners and provides tools for steering their inductive biases for continuous function learning tasks.

Theodore Papamarkou, Maria Skoularidou, Konstantina Palla et al.

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.

Laura Manduchi, Clara Meister, Kushagra Pandey et al.

The field of deep generative modeling has grown rapidly in the last few years. With the availability of massive amounts of training data coupled with advances in scalable unsupervised learning paradigms, recent large-scale generative models show tremendous promise in synthesizing high-resolution images and text, as well as structured data such as videos and molecules. However, we argue that current large-scale generative AI models exhibit several fundamental shortcomings that hinder their widespread adoption across domains. In this work, our objective is to identify these issues and highlight key unresolved challenges in modern generative AI paradigms that should be addressed to further enhance their capabilities, versatility, and reliability. By identifying these challenges, we aim to provide researchers with insights for exploring fruitful research directions, thus fostering the development of more robust and accessible generative AI solutions.

Theodore Papamarkou, Pierre Alquier, Matthias Bauer et al.

LLMs excel at predictive tasks and complex reasoning tasks, but many high-value deployments rely on decisions under uncertainty, for example, which tool to call, which expert to consult, or how many resources to invest. While the usefulness and feasibility of Bayesian approaches remain unclear for LLM inference, this position paper argues that the control layer of an agentic AI system (that orchestrates LLMs and tools) is a clear case where Bayesian principles should shine. Bayesian decision theory provides a framework for agentic systems that can help to maintain beliefs over task-relevant latent quantities, to update these beliefs from observed agentic and human-AI interactions, and to choose actions. Making LLMs themselves explicitly Bayesian belief-updating engines remains computationally intensive and conceptually nontrivial as a general modeling target. In contrast, this paper argues that coherent decision-making requires Bayesian principles at the orchestration level of the agentic system, not necessarily the LLM agent parameters. This paper articulates practical properties for Bayesian control that fit modern agentic AI systems and human-AI collaboration, and provides concrete examples and design patterns to illustrate how calibrated beliefs and utility-aware policies can improve agentic AI orchestration.

Rayen Dhahri, Alexander Immer, Betrand Charpentier et al.

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 naïvely 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.

Tommy Rochussen, Vincent Fortuin

One of the core facets of Bayesianism is in the updating of prior beliefs in light of new evidence$\text{ -- }$so how can we maintain a Bayesian approach if we have no prior beliefs in the first place? This is one of the central challenges in the field of Bayesian deep learning, where it is not clear how to represent beliefs about a prediction task by prior distributions over model parameters. Bridging the fields of Bayesian deep learning and probabilistic meta-learning, we introduce a way to $\textit{learn}$ a weights prior from a collection of datasets by introducing a way to perform per-dataset amortised variational inference. The model we develop can be viewed as a neural process whose latent variable is the set of weights of a BNN and whose decoder is the neural network parameterised by a sample of the latent variable itself. This unique model allows us to study the behaviour of Bayesian neural networks under well-specified priors, use Bayesian neural networks as flexible generative models, and perform desirable but previously elusive feats in neural processes such as within-task minibatching or meta-learning under extreme data-starvation.

Michal Kmicikiewicz, Vincent Fortuin, Ewa Szczurek

Designing protein sequences of both high fitness and novelty is a challenging task in data-efficient protein engineering. Exploration beyond wild-type neighborhoods often leads to biologically implausible sequences or relies on surrogate models that lose fidelity in novel regions. Here, we propose ProSpero, an active learning framework in which a frozen pre-trained generative model is guided by a surrogate updated from oracle feedback. By integrating fitness-relevant residue selection with biologically-constrained Sequential Monte Carlo sampling, our approach enables exploration beyond wild-type neighborhoods while preserving biological plausibility. We show that our framework remains effective even when the surrogate is misspecified. ProSpero consistently outperforms or matches existing methods across diverse protein engineering tasks, retrieving sequences of both high fitness and novelty.

Klemens Flöge, Srisruthi Udayakumar, Johanna Sommer et al.

Recent advances in Artificial Intelligence have enabled multi-modal systems to model and translate diverse information spaces. Extending beyond text and vision, we introduce OneProt, a multi-modal AI for proteins that integrates structural, sequence, text, and binding site data. Using the ImageBind framework, OneProt aligns the latent spaces of protein modality encoders in a lightweight fine-tuning scheme that focuses on pairwise alignment with sequence data rather than requiring full matches. This novel approach comprises a mix of Graph Neural Networks and transformer architectures. It demonstrates strong performance in retrieval tasks and showcases the efficacy of multi-modal systems in Protein Machine Learning through a broad spectrum of downstream baselines, including enzyme function prediction and binding site analysis. Furthermore, OneProt enables the transfer of representational information from specialized encoders to the sequence encoder, enhancing capabilities for distinguishing evolutionarily related and unrelated sequences and exhibiting representational properties where evolutionarily related proteins align in similar directions within the latent space. In addition, we extensively investigate modality ablations to identify the encoders that contribute most to predictive performance, highlighting the significance of the binding site encoder, which has not been used in similar models previously. This work expands the horizons of multi-modal protein models, paving the way for transformative applications in drug discovery, biocatalytic reaction planning, and protein engineering.

Eliot Wong-Toi, Alex Boyd, Vincent Fortuin et al.

Uncertainty quantification is vital for decision-making and risk assessment in machine learning. Mean-variance regression models, which predict both a mean and residual noise for each data point, provide a simple approach to uncertainty quantification. However, overparameterized mean-variance models struggle with signal-to-noise ambiguity, deciding whether prediction targets should be attributed to signal (mean) or noise (variance). At one extreme, models fit all training targets perfectly with zero residual noise, while at the other, they provide constant, uninformative predictions and explain the targets as noise. We observe a sharp phase transition between these extremes, driven by model regularization. Empirical studies with varying regularization levels illustrate this transition, revealing substantial variability across repeated runs. To explain this behavior, we develop a statistical field theory framework, which captures the observed phase transition in alignment with experimental results. This analysis reduces the regularization hyperparameter search space from two dimensions to one, significantly lowering computational costs. Experiments on UCI datasets and the large-scale ClimSim dataset demonstrate robust calibration performance, effectively quantifying predictive uncertainty.

Tommy Rochussen, Vincent Fortuin

Despite significant recent advances in probabilistic meta-learning, it is common for practitioners to avoid using deep learning models due to a comparative lack of interpretability. Instead, many practitioners simply use non-meta-models such as Gaussian processes with interpretable priors, and conduct the tedious procedure of training their model from scratch for each task they encounter. While this is justifiable for tasks with a limited number of data points, the cubic computational cost of exact Gaussian process inference renders this prohibitive when each task has many observations. To remedy this, we introduce a family of models that meta-learn sparse Gaussian process inference. Not only does this enable rapid prediction on new tasks with sparse Gaussian processes, but since our models have clear interpretations as members of the neural process family, it also allows manual elicitation of priors in a neural process for the first time. In meta-learning regimes for which the number of observed tasks is small or for which expert domain knowledge is available, this offers a crucial advantage.

Klemens Flöge, Mohammed Abdul Moeed, Vincent Fortuin

Deep neural network ensembles are powerful tools for uncertainty quantification, which have recently been re-interpreted from a Bayesian perspective. However, current methods inadequately leverage second-order information of the loss landscape, despite the recent availability of efficient Hessian approximations. We propose a novel approximate Bayesian inference method that modifies deep ensembles to incorporate Stein Variational Newton updates. Our approach uniquely integrates scalable modern Hessian approximations, achieving faster convergence and more accurate posterior distribution approximations. We validate the effectiveness of our method on diverse regression and classification tasks, demonstrating superior performance with a significantly reduced number of training epochs compared to existing ensemble-based methods, while enhancing uncertainty quantification and robustness against overfitting.

Agustinus Kristiadi, Felix Strieth-Kalthoff, Sriram Ganapathi Subramanian et al.

Bayesian optimization (BO) is an integral part of automated scientific discovery -- the so-called self-driving lab -- where human inputs are ideally minimal or at least non-blocking. However, scientists often have strong intuition, and thus human feedback is still useful. Nevertheless, prior works in enhancing BO with expert feedback, such as by incorporating it in an offline or online but blocking (arrives at each BO iteration) manner, are incompatible with the spirit of self-driving labs. In this work, we study whether a small amount of randomly arriving expert feedback that is being incorporated in a non-blocking manner can improve a BO campaign. To this end, we run an additional, independent computing thread on top of the BO loop to handle the feedback-gathering process. The gathered feedback is used to learn a Bayesian preference model that can readily be incorporated into the BO thread, to steer its exploration-exploitation process. Experiments on toy and chemistry datasets suggest that even just a few intermittent, asynchronous expert feedback can be useful for improving or constraining BO. This can especially be useful for its implication in improving self-driving labs, e.g. making them more data-efficient and less costly.

Emre Onal, Klemens Flöge, Emma Caldwell et al.

Fine-tuned Large Language Models (LLMs) often suffer from overconfidence and poor calibration, particularly when fine-tuned on small datasets. To address these challenges, we propose a simple combination of Low-Rank Adaptation (LoRA) with Gaussian Stochastic Weight Averaging (SWAG), facilitating approximate Bayesian inference in LLMs. Through extensive testing across several Natural Language Processing (NLP) benchmarks, we demonstrate that our straightforward and computationally efficient approach improves model generalization and calibration competitively with comparable, more sophisticated methods for Bayesian inference in LLMs. We further show that our method exhibits greater robustness against distribution shift, as reflected in its improved performance on out-of-distribution tasks.

Kouroche Bouchiat, Alexander Immer, Hugo Yèche et al.

Neural additive models (NAMs) enhance the transparency of deep neural networks by handling input features in separate additive sub-networks. However, they lack inherent mechanisms that provide calibrated uncertainties and enable selection of relevant features and interactions. Approaching NAMs from a Bayesian perspective, we augment them in three primary ways, namely by a) providing credible intervals for the individual additive sub-networks; b) estimating the marginal likelihood to perform an implicit selection of features via an empirical Bayes procedure; and c) facilitating the ranking of feature pairs as candidates for second-order interaction in fine-tuned models. In particular, we develop Laplace-approximated NAMs (LA-NAMs), which show improved empirical performance on tabular datasets and challenging real-world medical tasks.

Alexander Immer, Tycho F. A. van der Ouderaa, Gunnar Rätsch et al.

Data augmentation is commonly applied to improve performance of deep learning by enforcing the knowledge that certain transformations on the input preserve the output. Currently, the data augmentation parameters are chosen by human effort and costly cross-validation, which makes it cumbersome to apply to new datasets. We develop a convenient gradient-based method for selecting the data augmentation without validation data during training of a deep neural network. Our approach relies on phrasing data augmentation as an invariance in the prior distribution on the functions of a neural network, which allows us to learn it using Bayesian model selection. This has been shown to work in Gaussian processes, but not yet for deep neural networks. We propose a differentiable Kronecker-factored Laplace approximation to the marginal likelihood as our objective, which can be optimised without human supervision or validation data. We show that our method can successfully recover invariances present in the data, and that this improves generalisation and data efficiency on image datasets.

Alexander Immer, Lucas Torroba Hennigen, Vincent Fortuin et al.

Pre-trained contextual representations have led to dramatic performance improvements on a range of downstream tasks. Such performance improvements have motivated researchers to quantify and understand the linguistic information encoded in these representations. In general, researchers quantify the amount of linguistic information through probing, an endeavor which consists of training a supervised model to predict a linguistic property directly from the contextual representations. Unfortunately, this definition of probing has been subject to extensive criticism in the literature, and has been observed to lead to paradoxical and counter-intuitive results. In the theoretical portion of this paper, we take the position that the goal of probing ought to be measuring the amount of inductive bias that the representations encode on a specific task. We further describe a Bayesian framework that operationalizes this goal and allows us to quantify the representations' inductive bias. In the empirical portion of the paper, we apply our framework to a variety of NLP tasks. Our results suggest that our proposed framework alleviates many previous problems found in probing. Moreover, we are able to offer concrete evidence that -- for some tasks -- fastText can offer a better inductive bias than BERT.

Tristan Cinquin, Alexander Immer, Max Horn et al.

In recent years, the transformer has established itself as a workhorse in many applications ranging from natural language processing to reinforcement learning. Similarly, Bayesian deep learning has become the gold-standard for uncertainty estimation in safety-critical applications, where robustness and calibration are crucial. Surprisingly, no successful attempts to improve transformer models in terms of predictive uncertainty using Bayesian inference exist. In this work, we study this curiously underpopulated area of Bayesian transformers. We find that weight-space inference in transformers does not work well, regardless of the approximate posterior. We also find that the prior is at least partially at fault, but that it is very hard to find well-specified weight priors for these models. We hypothesize that these problems stem from the complexity of obtaining a meaningful mapping from weight-space to function-space distributions in the transformer. Therefore, moving closer to function-space, we propose a novel method based on the implicit reparameterization of the Dirichlet distribution to apply variational inference directly to the attention weights. We find that this proposed method performs competitively with our baselines.

James Urquhart Allingham, Florian Wenzel, Zelda E Mariet et al.

Machine learning models based on the aggregated outputs of submodels, either at the activation or prediction levels, often exhibit strong performance compared to individual models. We study the interplay of two popular classes of such models: ensembles of neural networks and sparse mixture of experts (sparse MoEs). First, we show that the two approaches have complementary features whose combination is beneficial. This includes a comprehensive evaluation of sparse MoEs in uncertainty related benchmarks. Then, we present Efficient Ensemble of Experts (E$^3$), a scalable and simple ensemble of sparse MoEs that takes the best of both classes of models, while using up to 45% fewer FLOPs than a deep ensemble. Extensive experiments demonstrate the accuracy, log-likelihood, few-shot learning, robustness, and uncertainty improvements of E$^3$ over several challenging vision Transformer-based baselines. E$^3$ not only preserves its efficiency while scaling to models with up to 2.7B parameters, but also provides better predictive performance and uncertainty estimates for larger models.

Vincent Fortuin, Mark Collier, Florian Wenzel et al.

Uncertainty estimation in deep learning has recently emerged as a crucial area of interest to advance reliability and robustness in safety-critical applications. While there have been many proposed methods that either focus on distance-aware model uncertainties for out-of-distribution detection or on input-dependent label uncertainties for in-distribution calibration, both of these types of uncertainty are often necessary. In this work, we propose the HetSNGP method for jointly modeling the model and data uncertainty. We show that our proposed model affords a favorable combination between these two types of uncertainty and thus outperforms the baseline methods on some challenging out-of-distribution datasets, including CIFAR-100C, ImageNet-C, and ImageNet-A. Moreover, we propose HetSNGP Ensemble, an ensembled version of our method which additionally models uncertainty over the network parameters and outperforms other ensemble baselines.

Lauro Langosco di Langosco, Vincent Fortuin, Heiko Strathmann

Particle-based approximate Bayesian inference approaches such as Stein Variational Gradient Descent (SVGD) combine the flexibility and convergence guarantees of sampling methods with the computational benefits of variational inference. In practice, SVGD relies on the choice of an appropriate kernel function, which impacts its ability to model the target distribution -- a challenging problem with only heuristic solutions. We propose Neural Variational Gradient Descent (NVGD), which is based on parameterizing the witness function of the Stein discrepancy by a deep neural network whose parameters are learned in parallel to the inference, mitigating the necessity to make any kernel choices whatsoever. We empirically evaluate our method on popular synthetic inference problems, real-world Bayesian linear regression, and Bayesian neural network inference.

Nikolaos Mourdoukoutas, Marco Federici, Georges Pantalos et al.

We propose a novel Bayesian neural network architecture that can learn invariances from data alone by inferring a posterior distribution over different weight-sharing schemes. We show that our model outperforms other non-invariant architectures, when trained on datasets that contain specific invariances. The same holds true when no data augmentation is performed.

Francesco D'Angelo, Vincent Fortuin

Deep ensembles have recently gained popularity in the deep learning community for their conceptual simplicity and efficiency. However, maintaining functional diversity between ensemble members that are independently trained with gradient descent is challenging. This can lead to pathologies when adding more ensemble members, such as a saturation of the ensemble performance, which converges to the performance of a single model. Moreover, this does not only affect the quality of its predictions, but even more so the uncertainty estimates of the ensemble, and thus its performance on out-of-distribution data. We hypothesize that this limitation can be overcome by discouraging different ensemble members from collapsing to the same function. To this end, we introduce a kernelized repulsive term in the update rule of the deep ensembles. We show that this simple modification not only enforces and maintains diversity among the members but, even more importantly, transforms the maximum a posteriori inference into proper Bayesian inference. Namely, we show that the training dynamics of our proposed repulsive ensembles follow a Wasserstein gradient flow of the KL divergence with the true posterior. We study repulsive terms in weight and function space and empirically compare their performance to standard ensembles and Bayesian baselines on synthetic and real-world prediction tasks.

Francesco D'Angelo, Vincent Fortuin, Florian Wenzel

Ensembles of deep neural networks have achieved great success recently, but they do not offer a proper Bayesian justification. Moreover, while they allow for averaging of predictions over several hypotheses, they do not provide any guarantees for their diversity, leading to redundant solutions in function space. In contrast, particle-based inference methods, such as Stein variational gradient descent (SVGD), offer a Bayesian framework, but rely on the choice of a kernel to measure the similarity between ensemble members. In this work, we study different SVGD methods operating in the weight space, function space, and in a hybrid setting. We compare the SVGD approaches to other ensembling-based methods in terms of their theoretical properties and assess their empirical performance on synthetic and real-world tasks. We find that SVGD using functional and hybrid kernels can overcome the limitations of deep ensembles. It improves on functional diversity and uncertainty estimation and approaches the true Bayesian posterior more closely. Moreover, we show that using stochastic SVGD updates, as opposed to the standard deterministic ones, can further improve the performance.

Seth Nabarro, Stoil Ganev, Adrià Garriga-Alonso et al.

Bayesian neural networks that incorporate data augmentation implicitly use a ``randomly perturbed log-likelihood [which] does not have a clean interpretation as a valid likelihood function'' (Izmailov et al. 2021). Here, we provide several approaches to developing principled Bayesian neural networks incorporating data augmentation. We introduce a ``finite orbit'' setting which allows likelihoods to be computed exactly, and give tight multi-sample bounds in the more usual ``full orbit'' setting. These models cast light on the origin of the cold posterior effect. In particular, we find that the cold posterior effect persists even in these principled models incorporating data augmentation. This suggests that the cold posterior effect cannot be dismissed as an artifact of data augmentation using incorrect likelihoods.

Vincent Fortuin, Adrià Garriga-Alonso, Mark van der Wilk et al.

Bayesian neural networks have shown great promise in many applications where calibrated uncertainty estimates are crucial and can often also lead to a higher predictive performance. However, it remains challenging to choose a good prior distribution over their weights. While isotropic Gaussian priors are often chosen in practice due to their simplicity, they do not reflect our true prior beliefs well and can lead to suboptimal performance. Our new library, BNNpriors, enables state-of-the-art Markov Chain Monte Carlo inference on Bayesian neural networks with a wide range of predefined priors, including heavy-tailed ones, hierarchical ones, and mixture priors. Moreover, it follows a modular approach that eases the design and implementation of new custom priors. It has facilitated foundational discoveries on the nature of the cold posterior effect in Bayesian neural networks and will hopefully catalyze future research as well as practical applications in this area.

Vincent Fortuin

While the choice of prior is one of the most critical parts of the Bayesian inference workflow, recent Bayesian deep learning models have often fallen back on vague priors, such as standard Gaussians. In this review, we highlight the importance of prior choices for Bayesian deep learning and present an overview of different priors that have been proposed for (deep) Gaussian processes, variational autoencoders, and Bayesian neural networks. We also outline different methods of learning priors for these models from data. We hope to motivate practitioners in Bayesian deep learning to think more carefully about the prior specification for their models and to provide them with some inspiration in this regard.

Alexander Immer, Matthias Bauer, Vincent Fortuin et al.

Marginal-likelihood based model-selection, even though promising, is rarely used in deep learning due to estimation difficulties. Instead, most approaches rely on validation data, which may not be readily available. In this work, we present a scalable marginal-likelihood estimation method to select both hyperparameters and network architectures, based on the training data alone. Some hyperparameters can be estimated online during training, simplifying the procedure. Our marginal-likelihood estimate is based on Laplace's method and Gauss-Newton approximations to the Hessian, and it outperforms cross-validation and manual-tuning on standard regression and image classification datasets, especially in terms of calibration and out-of-distribution detection. Our work shows that marginal likelihoods can improve generalization and be useful when validation data is unavailable (e.g., in nonstationary settings).

Vincent Fortuin, Adrià Garriga-Alonso, Sebastian W. Ober et al.

Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. However, it is unclear whether these priors accurately reflect our true beliefs about the weight distributions or give optimal performance. To find better priors, we study summary statistics of neural network weights in networks trained using stochastic gradient descent (SGD). We find that convolutional neural network (CNN) and ResNet weights display strong spatial correlations, while fully connected networks (FCNNs) display heavy-tailed weight distributions. We show that building these observations into priors can lead to improved performance on a variety of image classification datasets. Surprisingly, these priors mitigate the cold posterior effect in FCNNs, but slightly increase the cold posterior effect in ResNets.

Simon Bing, Vincent Fortuin, Gunnar Rätsch

Complex multivariate time series arise in many fields, ranging from computer vision to robotics or medicine. Often we are interested in the independent underlying factors that give rise to the high-dimensional data we are observing. While many models have been introduced to learn such disentangled representations, only few attempt to explicitly exploit the structure of sequential data. We investigate the disentanglement properties of Gaussian process variational autoencoders, a class of models recently introduced that have been successful in different tasks on time series data. Our model exploits the temporal structure of the data by modeling each latent channel with a GP prior and employing a structured variational distribution that can capture dependencies in time. We demonstrate the competitiveness of our approach against state-of-the-art unsupervised and weakly-supervised disentanglement methods on a benchmark task. Moreover, we provide evidence that we can learn meaningful disentangled representations on real-world medical time series data.

Adrià Garriga-Alonso, Vincent Fortuin

Stochastic gradient Markov Chain Monte Carlo algorithms are popular samplers for approximate inference, but they are generally biased. We show that many recent versions of these methods (e.g. Chen et al. (2014)) cannot be corrected using Metropolis-Hastings rejection sampling, because their acceptance probability is always zero. We can fix this by employing a sampler with realizable backwards trajectories, such as Gradient-Guided Monte Carlo (Horowitz, 1991), which generalizes stochastic gradient Langevin dynamics (Welling and Teh, 2011) and Hamiltonian Monte Carlo. We show that this sampler can be used with stochastic gradients, yielding nonzero acceptance probabilities, which can be computed even across multiple steps.

Francesco D'Angelo, Vincent Fortuin

Particle based optimization algorithms have recently been developed as sampling methods that iteratively update a set of particles to approximate a target distribution. In particular Stein variational gradient descent has gained attention in the approximate inference literature for its flexibility and accuracy. We empirically explore the ability of this method to sample from multi-modal distributions and focus on two important issues: (i) the inability of the particles to escape from local modes and (ii) the inefficacy in reproducing the density of the different regions. We propose an annealing schedule to solve these issues and show, through various experiments, how this simple solution leads to significant improvements in mode coverage, without invalidating any theoretical properties of the original algorithm.

Metod Jazbec, Michael Pearce, Vincent Fortuin

Variational autoencoders often assume isotropic Gaussian priors and mean-field posteriors, hence do not exploit structure in scenarios where we may expect similarity or consistency across latent variables. Gaussian process variational autoencoders alleviate this problem through the use of a latent Gaussian process, but lead to a cubic inference time complexity. We propose a more scalable extension of these models by leveraging the independence of the auxiliary features, which is present in many datasets. Our model factorizes the latent kernel across these features in different dimensions, leading to a significant speed-up (in theory and practice), while empirically performing comparably to existing non-scalable approaches. Moreover, our approach allows for additional modeling of global latent information and for more general extrapolation to unseen input combinations.

Metod Jazbec, Matthew Ashman, Vincent Fortuin et al.

Conventional variational autoencoders fail in modeling correlations between data points due to their use of factorized priors. Amortized Gaussian process inference through GP-VAEs has led to significant improvements in this regard, but is still inhibited by the intrinsic complexity of exact GP inference. We improve the scalability of these methods through principled sparse inference approaches. We propose a new scalable GP-VAE model that outperforms existing approaches in terms of runtime and memory footprint, is easy to implement, and allows for joint end-to-end optimization of all components.

Matthew Ashman, Jonathan So, Will Tebbutt et al.

Large, multi-dimensional spatio-temporal datasets are omnipresent in modern science and engineering. An effective framework for handling such data are Gaussian process deep generative models (GP-DGMs), which employ GP priors over the latent variables of DGMs. Existing approaches for performing inference in GP-DGMs do not support sparse GP approximations based on inducing points, which are essential for the computational efficiency of GPs, nor do they handle missing data -- a natural occurrence in many spatio-temporal datasets -- in a principled manner. We address these shortcomings with the development of the sparse Gaussian process variational autoencoder (SGP-VAE), characterised by the use of partial inference networks for parameterising sparse GP approximations. Leveraging the benefits of amortised variational inference, the SGP-VAE enables inference in multi-output sparse GPs on previously unobserved data with no additional training. The SGP-VAE is evaluated in a variety of experiments where it outperforms alternative approaches including multi-output GPs and structured VAEs.

Jonas Rothfuss, Vincent Fortuin, Martin Josifoski et al.

Meta-learning can successfully acquire useful inductive biases from data. Yet, its generalization properties to unseen learning tasks are poorly understood. Particularly if the number of meta-training tasks is small, this raises concerns about overfitting. We provide a theoretical analysis using the PAC-Bayesian framework and derive novel generalization bounds for meta-learning. Using these bounds, we develop a class of PAC-optimal meta-learning algorithms with performance guarantees and a principled meta-level regularization. Unlike previous PAC-Bayesian meta-learners, our method results in a standard stochastic optimization problem which can be solved efficiently and scales well. When instantiating our PAC-optimal hyper-posterior (PACOH) with Gaussian processes and Bayesian Neural Networks as base learners, the resulting methods yield state-of-the-art performance, both in terms of predictive accuracy and the quality of uncertainty estimates. Thanks to their principled treatment of uncertainty, our meta-learners can also be successfully employed for sequential decision problems.

Andreas Kopf, Vincent Fortuin, Vignesh Ram Somnath et al.

Clustering high-dimensional data, such as images or biological measurements, is a long-standingproblem and has been studied extensively. Recently, Deep Clustering has gained popularity due toits flexibility in fitting the specific peculiarities of complex data. Here we introduce the Mixture-of-Experts Similarity Variational Autoencoder (MoE-Sim-VAE), a novel generative clustering model.The model can learn multi-modal distributions of high-dimensional data and use these to generaterealistic data with high efficacy and efficiency. MoE-Sim-VAE is based on a Variational Autoencoder(VAE), where the decoder consists of a Mixture-of-Experts (MoE) architecture. This specific architecture allows for various modes of the data to be automatically learned by means of the experts.Additionally, we encourage the lower dimensional latent representation of our model to follow aGaussian mixture distribution and to accurately represent the similarities between the data points. Weassess the performance of our model on the MNIST benchmark data set and challenging real-worldtasks of clustering mouse organs from single-cell RNA-sequencing measurements and defining cellsubpopulations from mass cytometry (CyTOF) measurements on hundreds of different datasets.MoE-Sim-VAE exhibits superior clustering performance on all these tasks in comparison to thebaselines as well as competitor methods.