Jeremias Knoblauch

Matias Altamirano, François-Xavier Briol, Jeremias Knoblauch

This paper proposes an online, provably robust, and scalable Bayesian approach for changepoint detection. The resulting algorithm has key advantages over previous work: it provides provable robustness by leveraging the generalised Bayesian perspective, and also addresses the scalability issues of previous attempts. Specifically, the proposed generalised Bayesian formalism leads to conjugate posteriors whose parameters are available in closed form by leveraging diffusion score matching. The resulting algorithm is exact, can be updated through simple algebra, and is more than 10 times faster than its closest competitor.

Matias Altamirano, François-Xavier Briol, Jeremias Knoblauch

To enable closed form conditioning, a common assumption in Gaussian process (GP) regression is independent and identically distributed Gaussian observation noise. This strong and simplistic assumption is often violated in practice, which leads to unreliable inferences and uncertainty quantification. Unfortunately, existing methods for robustifying GPs break closed-form conditioning, which makes them less attractive to practitioners and significantly more computationally expensive. In this paper, we demonstrate how to perform provably robust and conjugate Gaussian process (RCGP) regression at virtually no additional cost using generalised Bayesian inference. RCGP is particularly versatile as it enables exact conjugate closed form updates in all settings where standard GPs admit them. To demonstrate its strong empirical performance, we deploy RCGP for problems ranging from Bayesian optimisation to sparse variational Gaussian processes.

Joshua Rooijakkers, Leiv Rønneberg, François-Xavier Briol et al.

Multi-output Gaussian process (MOGP) regression allows modelling dependencies among multiple correlated response variables. Similarly to standard Gaussian processes, MOGPs are sensitive to model misspecification and outliers, which can distort predictions within individual outputs. This situation can be further exacerbated by multiple anomalous response variables whose errors propagate due to correlations between outputs. To handle this situation, we extend and generalise the robust and conjugate Gaussian process (RCGP) framework introduced by Altamirano et al. (2024). This results in the multi-output RCGP (MO-RCGP): a provably robust MOGP that is conjugate, and jointly captures correlations across outputs. We thoroughly evaluate our approach through applications in finance and cancer research.

Abdelhamid Ezzerg, Ilija Bogunovic, Jeremias Knoblauch

Bayesian Optimization is critically vulnerable to extreme outliers. Existing provably robust methods typically assume a bounded cumulative corruption budget, which makes them defenseless against even a single corruption of sufficient magnitude. To address this, we introduce a new adversary whose budget is only bounded in the frequency of corruptions, not in their magnitude. We then derive RCGP-UCB, an algorithm coupling the famous upper confidence bound (UCB) approach with a Robust Conjugate Gaussian Process (RCGP). We present stable and adaptive versions of RCGP-UCB, and prove that they achieve sublinear regret in the presence of up to $O(T^{1/2})$ and $O(T^{1/3})$ corruptions with possibly infinite magnitude. This robustness comes at near zero cost: without outliers, RCGP-UCB's regret bounds match those of the standard GP-UCB algorithm.

Veit Wild, James Wu, Dino Sejdinovic et al.

We propose a scalable inference algorithm for Bayes posteriors defined on a reproducing kernel Hilbert space (RKHS). Given a likelihood function and a Gaussian random element representing the prior, the corresponding Bayes posterior measure $Π_{\text{B}}$ can be obtained as the stationary distribution of an RKHS-valued Langevin diffusion. We approximate the infinite-dimensional Langevin diffusion via a projection onto the first $M$ components of the Kosambi-Karhunen-Loève expansion. Exploiting the thus obtained approximate posterior for these $M$ components, we perform inference for $Π_{\text{B}}$ by relying on the law of total probability and a sufficiency assumption. The resulting method scales as $O(M^3+JM^2)$, where $J$ is the number of samples produced from the posterior measure $Π_{\text{B}}$. Interestingly, the algorithm recovers the posterior arising from the sparse variational Gaussian process (SVGP) (see Titsias, 2009) as a special case, owed to the fact that the sufficiency assumption underlies both methods. However, whereas the SVGP is parametrically constrained to be a Gaussian process, our method is based on a non-parametric variational family $\mathcal{P}(\mathbb{R}^M)$ consisting of all probability measures on $\mathbb{R}^M$. As a result, our method is provably close to the optimal $M$-dimensional variational approximation of the Bayes posterior $Π_{\text{B}}$ in $\mathcal{P}(\mathbb{R}^M)$ for convex and Lipschitz continuous negative log likelihoods, and coincides with SVGP for the special case of a Gaussian error likelihood.

Gerardo Duran-Martin, Matias Altamirano, Alexander Y. Shestopaloff et al.

We derive a novel, provably robust, and closed-form Bayesian update rule for online filtering in state-space models in the presence of outliers and misspecified measurement models. Our method combines generalised Bayesian inference with filtering methods such as the extended and ensemble Kalman filter. We use the former to show robustness and the latter to ensure computational efficiency in the case of nonlinear models. Our method matches or outperforms other robust filtering methods (such as those based on variational Bayes) at a much lower computational cost. We show this empirically on a range of filtering problems with outlier measurements, such as object tracking, state estimation in high-dimensional chaotic systems, and online learning of neural networks.

Veit David Wild, Sahra Ghalebikesabi, Dino Sejdinovic et al.

We establish the first mathematically rigorous link between Bayesian, variational Bayesian, and ensemble methods. A key step towards this it to reformulate the non-convex optimisation problem typically encountered in deep learning as a convex optimisation in the space of probability measures. On a technical level, our contribution amounts to studying generalised variational inference through the lense of Wasserstein gradient flows. The result is a unified theory of various seemingly disconnected approaches that are commonly used for uncertainty quantification in deep learning -- including deep ensembles and (variational) Bayesian methods. This offers a fresh perspective on the reasons behind the success of deep ensembles over procedures based on parameterised variational inference, and allows the derivation of new ensembling schemes with convergence guarantees. We showcase this by proposing a family of interacting deep ensembles with direct parallels to the interactions of particle systems in thermodynamics, and use our theory to prove the convergence of these algorithms to a well-defined global minimiser on the space of probability measures.

Charita Dellaporta, Jeremias Knoblauch, Theodoros Damoulas et al.

Simulator-based models are models for which the likelihood is intractable but simulation of synthetic data is possible. They are often used to describe complex real-world phenomena, and as such can often be misspecified in practice. Unfortunately, existing Bayesian approaches for simulators are known to perform poorly in those cases. In this paper, we propose a novel algorithm based on the posterior bootstrap and maximum mean discrepancy estimators. This leads to a highly-parallelisable Bayesian inference algorithm with strong robustness properties. This is demonstrated through an in-depth theoretical study which includes generalisation bounds and proofs of frequentist consistency and robustness of our posterior. The approach is then assessed on a range of examples including a g-and-k distribution and a toggle-switch model.

Joel Jaskari, Jaakko Sahlsten, Theodoros Damoulas et al.

Automatic classification of diabetic retinopathy from retinal images has been widely studied using deep neural networks with impressive results. However, there is a clinical need for estimation of the uncertainty in the classifications, a shortcoming of modern neural networks. Recently, approximate Bayesian deep learning methods have been proposed for the task but the studies have only considered the binary referable/non-referable diabetic retinopathy classification applied to benchmark datasets. We present novel results by systematically investigating a clinical dataset and a clinically relevant 5-class classification scheme, in addition to benchmark datasets and the binary classification scheme. Moreover, we derive a connection between uncertainty measures and classifier risk, from which we develop a new uncertainty measure. We observe that the previously proposed entropy-based uncertainty measure generalizes to the clinical dataset on the binary classification scheme but not on the 5-class scheme, whereas our new uncertainty measure generalizes to the latter case.

Takuo Matsubara, Jeremias Knoblauch, François-Xavier Briol et al.

Generalised Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and can therefore be used to confer robustness against possible mis-specification of the likelihood. Here we consider generalised Bayesian inference with a Stein discrepancy as a loss function, motivated by applications in which the likelihood contains an intractable normalisation constant. In this context, the Stein discrepancy circumvents evaluation of the normalisation constant and produces generalised posteriors that are either closed form or accessible using standard Markov chain Monte Carlo. On a theoretical level, we show consistency, asymptotic normality, and bias-robustness of the generalised posterior, highlighting how these properties are impacted by the choice of Stein discrepancy. Then, we provide numerical experiments on a range of intractable distributions, including applications to kernel-based exponential family models and non-Gaussian graphical models.

Sebastian M Schmon, Patrick W Cannon, Jeremias Knoblauch

Complex simulators have become a ubiquitous tool in many scientific disciplines, providing high-fidelity, implicit probabilistic models of natural and social phenomena. Unfortunately, they typically lack the tractability required for conventional statistical analysis. Approximate Bayesian computation (ABC) has emerged as a key method in simulation-based inference, wherein the true model likelihood and posterior are approximated using samples from the simulator. In this paper, we draw connections between ABC and generalized Bayesian inference (GBI). First, we re-interpret the accept/reject step in ABC as an implicitly defined error model. We then argue that these implicit error models will invariably be misspecified. While ABC posteriors are often treated as a necessary evil for approximating the standard Bayesian posterior, this allows us to re-interpret ABC as a potential robustification strategy. This leads us to suggest the use of GBI within ABC, a use case we explore empirically.

Juan Maroñas, Oliver Hamelijnck, Jeremias Knoblauch et al.

Gaussian Processes (GPs) can be used as flexible, non-parametric function priors. Inspired by the growing body of work on Normalizing Flows, we enlarge this class of priors through a parametric invertible transformation that can be made input-dependent. Doing so also allows us to encode interpretable prior knowledge (e.g., boundedness constraints). We derive a variational approximation to the resulting Bayesian inference problem, which is as fast as stochastic variational GP regression (Hensman et al., 2013; Dezfouli and Bonilla,2015). This makes the model a computationally efficient alternative to other hierarchical extensions of GP priors (Lazaro-Gredilla,2012; Damianou and Lawrence, 2013). The resulting algorithm's computational and inferential performance is excellent, and we demonstrate this on a range of data sets. For example, even with only 5 inducing points and an input-dependent flow, our method is consistently competitive with a standard sparse GP fitted using 100 inducing points.

Jeremias Knoblauch, Lara Vomfell

Models of discrete-valued outcomes are easily misspecified if the data exhibit zero-inflation, overdispersion or contamination. Without additional knowledge about the existence and nature of this misspecification, model inference and prediction are adversely affected. Here, we introduce a robust discrepancy-based Bayesian approach using the Total Variation Distance (TVD). In the process, we address and resolve two challenges: First, we study convergence and robustness properties of a computationally efficient estimator for the TVD between a parametric model and the data-generating mechanism. Second, we provide an efficient inference method adapted from Lyddon et al. (2019) which corresponds to formulating an uninformative nonparametric prior directly over the data-generating mechanism. Lastly, we empirically demonstrate that our approach is robust and significantly improves predictive performance on a range of simulated and real world data.

Jeremias Knoblauch, Hisham Husain, Tom Diethe

Continual Learning (CL) algorithms incrementally learn a predictor or representation across multiple sequentially observed tasks. Designing CL algorithms that perform reliably and avoid so-called catastrophic forgetting has proven a persistent challenge. The current paper develops a theoretical approach that explains why. In particular, we derive the computational properties which CL algorithms would have to possess in order to avoid catastrophic forgetting. Our main finding is that such optimal CL algorithms generally solve an NP-hard problem and will require perfect memory to do so. The findings are of theoretical interest, but also explain the excellent performance of CL algorithms using experience replay, episodic memory and core sets relative to regularization-based approaches.

Jeremias Knoblauch

This paper investigates Frequentist consistency properties of the posterior distributions constructed via Generalized Variational Inference (GVI). A number of generic and novel strategies are given for proving consistency, relying on the theory of $Γ$-convergence. Specifically, this paper shows that under minimal regularity conditions, the sequence of GVI posteriors is consistent and collapses to a point mass at the population-optimal parameter value as the number of observations goes to infinity. The results extend to the latent variable case without additional assumptions and hold under misspecification. Lastly, the paper explains how to apply the results to a selection of GVI posteriors with especially popular variational families. For example, consistency is established for GVI methods using the mean field normal variational family, normal mixtures, Gaussian process variational families as well as neural networks indexing a normal (mixture) distribution.

Jeremias Knoblauch

This report provides an in-depth overview over the implications and novelty Generalized Variational Inference (GVI) (Knoblauch et al., 2019) brings to Deep Gaussian Processes (DGPs) (Damianou & Lawrence, 2013). Specifically, robustness to model misspecification as well as principled alternatives for uncertainty quantification are motivated with an information-geometric view. These modifications have clear interpretations and can be implemented in less than 100 lines of Python code. Most importantly, the corresponding empirical results show that DGPs can greatly benefit from the presented enhancements.

Jeremias Knoblauch, Jack Jewson, Theodoros Damoulas

We advocate an optimization-centric view on and introduce a novel generalization of Bayesian inference. Our inspiration is the representation of Bayes' rule as infinite-dimensional optimization problem (Csiszar, 1975; Donsker and Varadhan; 1975, Zellner; 1988). First, we use it to prove an optimality result of standard Variational Inference (VI): Under the proposed view, the standard Evidence Lower Bound (ELBO) maximizing VI posterior is preferable to alternative approximations of the Bayesian posterior. Next, we argue for generalizing standard Bayesian inference. The need for this arises in situations of severe misalignment between reality and three assumptions underlying standard Bayesian inference: (1) Well-specified priors, (2) well-specified likelihoods, (3) the availability of infinite computing power. Our generalization addresses these shortcomings with three arguments and is called the Rule of Three (RoT). We derive it axiomatically and recover existing posteriors as special cases, including the Bayesian posterior and its approximation by standard VI. In contrast, approximations based on alternative ELBO-like objectives violate the axioms. Finally, we study a special case of the RoT that we call Generalized Variational Inference (GVI). GVI posteriors are a large and tractable family of belief distributions specified by three arguments: A loss, a divergence and a variational family. GVI posteriors have appealing properties, including consistency and an interpretation as approximate ELBO. The last part of the paper explores some attractive applications of GVI in popular machine learning models, including robustness and more appropriate marginals. After deriving black box inference schemes for GVI posteriors, their predictive performance is investigated on Bayesian Neural Networks and Deep Gaussian Processes, where GVI can comprehensively improve upon existing methods.

Jeremias Knoblauch, Jack Jewson, Theodoros Damoulas

We present the very first robust Bayesian Online Changepoint Detection algorithm through General Bayesian Inference (GBI) with $β$-divergences. The resulting inference procedure is doubly robust for both the parameter and the changepoint (CP) posterior, with linear time and constant space complexity. We provide a construction for exponential models and demonstrate it on the Bayesian Linear Regression model. In so doing, we make two additional contributions: Firstly, we make GBI scalable using Structural Variational approximations that are exact as $β\to 0$. Secondly, we give a principled way of choosing the divergence parameter $β$ by minimizing expected predictive loss on-line. Reducing False Discovery Rates of CPs from more than 90% to 0% on real world data, this offers the state of the art.

Jeremias Knoblauch, Theodoros Damoulas

Bayesian On-line Changepoint Detection is extended to on-line model selection and non-stationary spatio-temporal processes. We propose spatially structured Vector Autoregressions (VARs) for modelling the process between changepoints (CPs) and give an upper bound on the approximation error of such models. The resulting algorithm performs prediction, model selection and CP detection on-line. Its time complexity is linear and its space complexity constant, and thus it is two orders of magnitudes faster than its closest competitor. In addition, it outperforms the state of the art for multivariate data.