Martin Jørgensen

Masaki Adachi, Satoshi Hayakawa, Martin Jørgensen et al. · oxford

Calculation of Bayesian posteriors and model evidences typically requires numerical integration. Bayesian quadrature (BQ), a surrogate-model-based approach to numerical integration, is capable of superb sample efficiency, but its lack of parallelisation has hindered its practical applications. In this work, we propose a parallelised (batch) BQ method, employing techniques from kernel quadrature, that possesses an empirically exponential convergence rate. Additionally, just as with Nested Sampling, our method permits simultaneous inference of both posteriors and model evidence. Samples from our BQ surrogate model are re-selected to give a sparse set of samples, via a kernel recombination algorithm, requiring negligible additional time to increase the batch size. Empirically, we find that our approach significantly outperforms the sampling efficiency of both state-of-the-art BQ techniques and Nested Sampling in various real-world datasets, including lithium-ion battery analytics.

Martin Jørgensen, Michael A. Osborne · oxford

Modern approximations to Gaussian processes are suitable for "tall data", with a cost that scales well in the number of observations, but under-performs on ``wide data'', scaling poorly in the number of input features. That is, as the number of input features grows, good predictive performance requires the number of summarising variables, and their associated cost, to grow rapidly. We introduce a kernel that allows the number of summarising variables to grow exponentially with the number of input features, but requires only linear cost in both number of observations and input features. This scaling is achieved through our introduction of the Bézier buttress, which allows approximate inference without computing matrix inverses or determinants. We show that our kernel has close similarities to some of the most used kernels in Gaussian process regression, and empirically demonstrate the kernel's ability to scale to both tall and wide datasets.

Masaki Adachi, Satoshi Hayakawa, Saad Hamid et al. · oxford

Batch Bayesian optimisation and Bayesian quadrature have been shown to be sample-efficient methods of performing optimisation and quadrature where expensive-to-evaluate objective functions can be queried in parallel. However, current methods do not scale to large batch sizes -- a frequent desideratum in practice (e.g. drug discovery or simulation-based inference). We present a novel algorithm, SOBER, which permits scalable and diversified batch global optimisation and quadrature with arbitrary acquisition functions and kernels over discrete and mixed spaces. The key to our approach is to reformulate batch selection for global optimisation as a quadrature problem, which relaxes acquisition function maximisation (non-convex) to kernel recombination (convex). Bridging global optimisation and quadrature can efficiently solve both tasks by balancing the merits of exploitative Bayesian optimisation and explorative Bayesian quadrature. We show that SOBER outperforms 11 competitive baselines on 12 synthetic and diverse real-world tasks.

Masaki Adachi, Satoshi Hayakawa, Martin Jørgensen et al. · oxford

Active learning parallelization is widely used, but typically relies on fixing the batch size throughout experimentation. This fixed approach is inefficient because of a dynamic trade-off between cost and speed -- larger batches are more costly, smaller batches lead to slower wall-clock run-times -- and the trade-off may change over the run (larger batches are often preferable earlier). To address this trade-off, we propose a novel Probabilistic Numerics framework that adaptively changes batch sizes. By framing batch selection as a quadrature task, our integration-error-aware algorithm facilitates the automatic tuning of batch sizes to meet predefined quadrature precision objectives, akin to how typical optimizers terminate based on convergence thresholds. This approach obviates the necessity for exhaustive searches across all potential batch sizes. We also extend this to scenarios with constrained active learning and constrained optimization, interpreting constraint violations as reductions in the precision requirement, to subsequently adapt batch construction. Through extensive experiments, we demonstrate that our approach significantly enhances learning efficiency and flexibility in diverse Bayesian batch active learning and Bayesian optimization applications.

Saad Hamid, Xingchen Wan, Martin Jørgensen et al.

Ensembling can improve the performance of Neural Networks, but existing approaches struggle when the architecture likelihood surface has dispersed, narrow peaks. Furthermore, existing methods construct equally weighted ensembles, and this is likely to be vulnerable to the failure modes of the weaker architectures. By viewing ensembling as approximately marginalising over architectures we construct ensembles using the tools of Bayesian Quadrature -- tools which are well suited to the exploration of likelihood surfaces with dispersed, narrow peaks. Additionally, the resulting ensembles consist of architectures weighted commensurate with their performance. We show empirically -- in terms of test likelihood, accuracy, and expected calibration error -- that our method outperforms state-of-the-art baselines, and verify via ablation studies that its components do so independently.

Chengkun Li, Grégoire Clarté, Martin Jørgensen et al.

In applied Bayesian inference scenarios, users may have access to a large number of pre-existing model evaluations, for example from maximum-a-posteriori (MAP) optimization runs. However, traditional approximate inference techniques make little to no use of this available information. We propose the framework of post-process Bayesian inference as a means to obtain a quick posterior approximation from existing target density evaluations, with no further model calls. Within this framework, we introduce Variational Sparse Bayesian Quadrature (VSBQ), a method for post-process approximate inference for models with black-box and potentially noisy likelihoods. VSBQ reuses existing target density evaluations to build a sparse Gaussian process (GP) surrogate model of the log posterior density function. Subsequently, we leverage sparse-GP Bayesian quadrature combined with variational inference to achieve fast approximate posterior inference over the surrogate. We validate our method on challenging synthetic scenarios and real-world applications from computational neuroscience. The experiments show that VSBQ builds high-quality posterior approximations by post-processing existing optimization traces, with no further model evaluations.

Masaki Adachi, Satoshi Hayakawa, Martin Jørgensen et al. · oxford

Parallelisation in Bayesian optimisation is a common strategy but faces several challenges: the need for flexibility in acquisition functions and kernel choices, flexibility dealing with discrete and continuous variables simultaneously, model misspecification, and lastly fast massive parallelisation. To address these challenges, we introduce a versatile and modular framework for batch Bayesian optimisation via probabilistic lifting with kernel quadrature, called SOBER, which we present as a Python library based on GPyTorch/BoTorch. Our framework offers the following unique benefits: (1) Versatility in downstream tasks under a unified approach. (2) A gradient-free sampler, which does not require the gradient of acquisition functions, offering domain-agnostic sampling (e.g., discrete and mixed variables, non-Euclidean space). (3) Flexibility in domain prior distribution. (4) Adaptive batch size (autonomous determination of the optimal batch size). (5) Robustness against a misspecified reproducing kernel Hilbert space. (6) Natural stopping criterion.

Pola Schwöbel, Martin Jørgensen, Sebastian W. Ober et al.

Data augmentation is often used to incorporate inductive biases into models. Traditionally, these are hand-crafted and tuned with cross validation. The Bayesian paradigm for model selection provides a path towards end-to-end learning of invariances using only the training data, by optimising the marginal likelihood. Computing the marginal likelihood is hard for neural networks, but success with tractable approaches that compute the marginal likelihood for the last layer only raises the question of whether this convenient approach might be employed for learning invariances. We show partial success on standard benchmarks, in the low-data regime and on a medical imaging dataset by designing a custom optimisation routine. Introducing a new lower bound to the marginal likelihood allows us to perform inference for a larger class of likelihood functions than before. On the other hand, we demonstrate failure modes on the CIFAR10 dataset, where the last layer approximation is not sufficient due to the increased complexity of our neural network. Our results indicate that once more sophisticated approximations become available the marginal likelihood is a promising approach for invariance learning in neural networks.

Frederik Warburg, Martin Jørgensen, Javier Civera et al.

Uncertainty quantification in image retrieval is crucial for downstream decisions, yet it remains a challenging and largely unexplored problem. Current methods for estimating uncertainties are poorly calibrated, computationally expensive, or based on heuristics. We present a new method that views image embeddings as stochastic features rather than deterministic features. Our two main contributions are (1) a likelihood that matches the triplet constraint and that evaluates the probability of an anchor being closer to a positive than a negative; and (2) a prior over the feature space that justifies the conventional l2 normalization. To ensure computational efficiency, we derive a variational approximation of the posterior, called the Bayesian triplet loss, that produces state-of-the-art uncertainty estimates and matches the predictive performance of current state-of-the-art methods.

Martin Jørgensen, Søren Hauberg

Causal discovery estimates the underlying physical process that generates the observed data: does X cause Y or does Y cause X? Current methodologies use structural conditions to turn the causal query into a statistical query, when only observational data is available. But what if these statistical queries are sensitive to causal invariants? This study investigates one such invariant: the causal relationship between X and Y is invariant to the marginal distributions of X and Y. We propose an algorithm that uses a non-parametric estimator that is robust to changes in the marginal distributions. This way we may marginalize the marginals, and inspect what relationship is intrinsically there. The resulting causal estimator is competitive with current methodologies and has high emphasis on the uncertainty in the causal query; an aspect just as important as the query itself.

Martin Jørgensen, Marc Peter Deisenroth, Hugh Salimbeni

We present a Bayesian non-parametric way of inferring stochastic differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the stochastic part of the differential equation, also known as the diffusion, and modelling it by means of Wishart processes. Further, we present a semi-parametric approach that allows the framework to scale to high dimensions. This successfully lead us onto how to model both latent and auto-regressive temporal systems with conditional heteroskedastic noise. We provide experimental evidence that modelling diffusion often improves performance and that this randomness in the differential equation can be essential to avoid overfitting.

Martin Jørgensen, Søren Hauberg

We present a probabilistic model where the latent variable respects both the distances and the topology of the modeled data. The model leverages the Riemannian geometry of the generated manifold to endow the latent space with a well-defined stochastic distance measure, which is modeled locally as Nakagami distributions. These stochastic distances are sought to be as similar as possible to observed distances along a neighborhood graph through a censoring process. The model is inferred by variational inference based on observations of pairwise distances. We demonstrate how the new model can encode invariances in the learned manifolds.

Pola Schwöbel, Frederik Warburg, Martin Jørgensen et al.

Spatial Transformer Networks (STNs) estimate image transformations that can improve downstream tasks by `zooming in' on relevant regions in an image. However, STNs are hard to train and sensitive to mis-predictions of transformations. To circumvent these limitations, we propose a probabilistic extension that estimates a stochastic transformation rather than a deterministic one. Marginalizing transformations allows us to consider each image at multiple poses, which makes the localization task easier and the training more robust. As an additional benefit, the stochastic transformations act as a localized, learned data augmentation that improves the downstream tasks. We show across standard imaging benchmarks and on a challenging real-world dataset that these two properties lead to improved classification performance, robustness and model calibration. We further demonstrate that the approach generalizes to non-visual domains by improving model performance on time-series data.

Nicki S. Detlefsen, Martin Jørgensen, Søren Hauberg

We propose and investigate new complementary methodologies for estimating predictive variance networks in regression neural networks. We derive a locally aware mini-batching scheme that result in sparse robust gradients, and show how to make unbiased weight updates to a variance network. Further, we formulate a heuristic for robustly fitting both the mean and variance networks post hoc. Finally, we take inspiration from posterior Gaussian processes and propose a network architecture with similar extrapolation properties to Gaussian processes. The proposed methodologies are complementary, and improve upon baseline methods individually. Experimentally, we investigate the impact on predictive uncertainty on multiple datasets and tasks ranging from regression, active learning and generative modeling. Experiments consistently show significant improvements in predictive uncertainty estimation over state-of-the-art methods across tasks and datasets.