The rising interest in Bayesian deep learning (BDL) has led to a plethora of methods for estimating the posterior distribution. However, efficient computation of inferences, such as predictions, has been largely overlooked with Monte Carlo integration remaining the standard. In this work we examine streamlining prediction in BDL through a single forward pass without sampling. For this we use local linearisation on activation functions and local Gaussian approximations at linear layers. Thus allowing us to analytically compute an approximation to the posterior predictive distribution. We showcase our approach for both MLP and transformers, such as ViT and GPT-2, and assess its performance on regression and classification tasks. Recent progress and adoption of deep learning models, has led to a sharp increase of interest in improving their reliability and robustness. In applications such as aided medical diagnosis (Begoli et al., 2019), autonomous driving (Michelmore et al., 2020), or supporting scientific discovery (Psaros et al., 2023); providing reliable and robust predictions as well as identifying failure modes is vital. A principled approach to address these challenges is the use of Bayesian deep learning (BDL, Wilson & Izmailov, 2020; Papamarkou et al., 2024) which promises a plug & play framework for uncertainty quantification. The key challenges associated with BDL, can roughly be divided into three parts: (i) defining a meaningful prior, (ii) estimating the posterior distribution, and (iii) performing inferences of interest, e.g., making predictions for unseen data, detecting out-of-distribution settings, or analysing model sensitivities. While constructing a meaningful prior is an important research direction (Nalisnick, 2018; Meronen et al., 2021; Fortuin et al., 2021; Tran et al., 2022), it has been argued that the differentiating aspect of Bayesian deep learning is marginalisation (Wilson & Izmailov, 2020; Wilson, 2020) rather than the prior itself. Figure 1: Our streamlined approach allows for practical outlier detection and sensitivity analysis. Locally linearizing the network function with local Gaussian approximations enables many relevant inference tasks to be solved analytically, helping render BDL a practical tool for downstream tasks.

The Distributionally Robust Markov Decision Process (DRMDP) is a popular framework for addressing dynamics shift in reinforcement learning by learning policies robust to the worst-case transition dynamics within a constrained set. However, solving its dual optimization oracle poses significant challenges, limiting theoretical analysis and computational efficiency. The recently proposed Robust Regularized Markov Decision Process (RRMDP) replaces the uncertainty set constraint with a regularization term on the value function, offering improved scalability and theoretical insights. Yet, existing RRMDP methods rely on unstructured regularization, often leading to overly conservative policies by considering transitions that are unrealistic. To address these issues, we propose a novel framework, the $d$-rectangular linear robust regularized Markov decision process ($d$-RRMDP), which introduces a linear latent structure into both transition kernels and regularization. For the offline RL setting, where an agent learns robust policies from a pre-collected dataset in the nominal environment, we develop a family of algorithms, Robust Regularized Pessimistic Value Iteration (R2PVI), employing linear function approximation and $f$-divergence based regularization terms on transition kernels. We provide instance-dependent upper bounds on the suboptimality gap of R2PVI policies, showing these bounds depend on how well the dataset covers state-action spaces visited by the optimal robust policy under robustly admissible transitions. This term is further shown to be fundamental to $d$-RRMDPs via information-theoretic lower bounds. Finally, numerical experiments validate that R2PVI learns robust policies and is computationally more efficient than methods for constrained DRMDPs.

In this paper, we investigate the concentration properties of cumulative rewards in Markov Decision Processes (MDPs), focusing on both asymptotic and non-asymptotic settings. We introduce a unified approach to characterize reward concentration in MDPs, covering both infinite-horizon settings (i.e., average and discounted reward frameworks) and finite-horizon setting. Our asymptotic results include the law of large numbers, the central limit theorem, and the law of iterated logarithms, while our non-asymptotic bounds include Azuma-Hoeffding-type inequalities and a non-asymptotic version of the law of iterated logarithms. Additionally, we explore two key implications of our results. First, we analyze the sample path behavior of the difference in rewards between any two stationary policies. Second, we show that two alternative definitions of regret for learning policies proposed in the literature are rate-equivalent. Our proof techniques rely on a novel martingale decomposition of cumulative rewards, properties of the solution to the policy evaluation fixed-point equation, and both asymptotic and non-asymptotic concentration results for martingale difference sequences.

The reliable recovery and uncertainty quantification of a fixed effect function $\mu$ in a functional mixed model, for modelling population- and object-level variability in noisily observed functional data, is a notoriously challenging task: variations along the $x$ and $y$ axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of $\mu$ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable $\mu$ under a Bayesian functional mixed model. The size-and-shape of $\mu$ is a geometric property invariant to a family of space-time unitary transformations, viewed as rotations of the Hilbert space, that jointly transform the $x$ and $y$ axes. A random object-level unitary transformation then captures size-and-shape \emph{preserving} deviations of $\mu$ from an individual function, while a random linear term and measurement error capture size-and-shape \emph{altering} deviations. The model is regularized by appropriate priors on the unitary transformations, posterior summaries of which may then be suitably interpreted as optimal data-driven rotations of a fixed orthonormal basis for the Hilbert space. Our numerical experiments demonstrate utility of the proposed model, and superiority over the current state-of-the-art.

Federated Learning (FL) has emerged as a promising method to collaboratively learn from decentralized and heterogeneous data available at different clients without the requirement of data ever leaving the clients. Recent works on FL have advocated taking a Bayesian approach to FL as it offers a principled way to account for the model and predictive uncertainty by learning a posterior distribution for the client and/or server models. Moreover, Bayesian FL also naturally enables personalization in FL to handle data heterogeneity across the different clients by having each client learn its own distinct personalized model. In particular, the hierarchical Bayesian approach enables all the clients to learn their personalized models while also taking into account the commonalities via a prior distribution provided by the server. However, despite their promise, Bayesian approaches for FL can be computationally expensive and can have high communication costs as well because of the requirement of computing and sending the posterior distributions. We present a novel Bayesian FL method using an efficient second-order optimization approach, with a computational cost that is similar to first-order optimization methods like Adam, but also provides the various benefits of the Bayesian approach for FL (e.g., uncertainty, personalization), while also being significantly more efficient and accurate than SOTA Bayesian FL methods (both for standard as well as personalized FL settings). Our method achieves improved predictive accuracies as well as better uncertainty estimates as compared to the baselines which include both optimization based as well as Bayesian FL methods.

Existing approaches to differentiable structure learning of directed acyclic graphs (DAGs) rely on strong identifiability assumptions in order to guarantee that global minimizers of the acyclicity-constrained optimization problem identifies the true DAG. Moreover, it has been observed empirically that the optimizer may exploit undesirable artifacts in the loss function. We explain and remedy these issues by studying the behaviour of differentiable acyclicity-constrained programs under general likelihoods with multiple global minimizers. By carefully regularizing the likelihood, it is possible to identify the sparsest model in the Markov equivalence class, even in the absence of an identifiable parametrization or even faithfulness. We first study the Gaussian case in detail, showing how proper regularization of the likelihood defines a score that identifies the sparsest model. These results are then generalized to general models and likelihoods, where the same claims hold. Furthermore, under standard faithfulness assumptions, our approach also recovers the Markov equivalence class. These theoretical results are validated empirically, showing how this can be done using standard gradient-based optimizers, thus paving the way for differentiable structure learning under general models and losses.

Markov chains are simple yet powerful mathematical structures to model temporally dependent processes. They generally assume stationary data, i.e., fixed transition probabilities between observations/states. However, live, real-world processes, like in the context of activity tracking, biological time series, or industrial monitoring, often switch behavior over time. Such behavior switches can be modeled as transitions between higher-level \emph{modes} (e.g., running, walking, etc.). Yet all modes are usually not previously known, often exhibit vastly differing transition probabilities, and can switch unpredictably. Thus, to track behavior changes of live, real-world processes, this study proposes an online and efficient method to construct Evolving Markov chains (EMCs). EMCs adaptively track transition probabilities, automatically discover modes, and detect mode switches in an online manner. In contrast to previous work, EMCs are of arbitrary order, the proposed update scheme does not rely on tracking windows, only updates the relevant region of the probability tensor, and enjoys geometric convergence of the expected estimates. Our evaluation of synthetic data and real-world applications on human activity recognition, electric motor condition monitoring, and eye-state recognition from electroencephalography (EEG) measurements illustrates the versatility of the approach and points to the potential of EMCs to efficiently track, model, and understand live, real-world processes.

As the significance of understanding the cause-and-effect relationships among variables increases in the development of modern systems and algorithms, learning causality from observational data has become a preferred and efficient approach over conducting randomized control trials. However, purely observational data could be insufficient to reconstruct the true causal graph. Consequently, many researchers tried to utilise some form of prior knowledge to improve causal discovery process. In this context, the impressive capabilities of large language models (LLMs) have emerged as a promising alternative to the costly acquisition of prior expert knowledge. In this work, we further explore the potential of using LLMs to enhance causal discovery approaches, particularly focusing on score-based methods, and we propose a general framework to utilise the capacity of not only one but multiple LLMs to augment the discovery process.

This paper presents a novel approach to binary classification using dynamic logistic ensemble models. The proposed method addresses the challenges posed by datasets containing inherent internal clusters that lack explicit feature-based separations. By extending traditional logistic regression, we develop an algorithm that automatically partitions the dataset into multiple subsets, constructing an ensemble of logistic models to enhance classification accuracy. A key innovation in this work is the recursive probability calculation, derived through algebraic manipulation and mathematical induction, which enables scalable and efficient model construction. Compared to traditional ensemble methods such as Bagging and Boosting, our approach maintains interpretability while offering competitive performance. Furthermore, we systematically employ maximum likelihood and cost functions to facilitate the analytical derivation of recursive gradients as functions of ensemble depth. The effectiveness of the proposed approach is validated on a custom dataset created by introducing noise and shifting data to simulate group structures, resulting in significant performance improvements with layers. Implemented in Python, this work balances computational efficiency with theoretical rigor, providing a robust and interpretable solution for complex classification tasks with broad implications for machine learning applications. Code at https://github.com/ensemble-art/Dynamic-Logistic-Ensembles

The primary challenge of multi-label active learning, differing it from multi-class active learning, lies in assessing the informativeness of an indefinite number of labels while also accounting for the inherited label correlation. Existing studies either require substantial computational resources to leverage correlations or fail to fully explore label dependencies. Additionally, real-world scenarios often require addressing intrinsic biases stemming from imbalanced data distributions. In this paper, we propose a new multi-label active learning strategy to address both challenges. Our method incorporates progressively updated positive and negative correlation matrices to capture co-occurrence and disjoint relationships within the label space of annotated samples, enabling a holistic assessment of uncertainty rather than treating labels as isolated elements. Furthermore, alongside diversity, our model employs ensemble pseudo labeling and beta scoring rules to address data imbalances. Extensive experiments on four realistic datasets demonstrate that our strategy consistently achieves more reliable and superior performance, compared to several established methods.