Bayesian Deep Learning

We empirically investigate how well popular approximate inference algorithms for Bayesian Neural Networks (BNNs) respect the theoretical properties of Bayesian belief updating. We find strong evidence on synthetic regression and real-world image classification tasks that common BNN algorithms such as variational inference, Laplace approximation, SWAG, and SGLD fail to update in a consistent manner, forget about old data under sequential updates, and violate the predictive coherence properties that would be expected of Bayesian methods. These observed behaviors imply that care should be taken when treating BNNs as true Bayesian models, particularly when using them beyond static prediction settings, such as for active, continual, or transfer learning.

We revisit the Bayesian Black–Litterman (BL) portfolio model and remove its reliance on subjective investor views. Classical BL requires an investor “view”: a forecast vector $q$ and its uncertainty matrix $\Omega$ that describe how much a chosen portfolio should outperform the market. Our key idea is to treat $(q,\Omega)$ as latent variables and learn them from market data within a single Bayesian network. Consequently, the resulting posterior estimation admits closed-form expression, enabling fast inference and stable portfolio weights. Building on these, we propose two mechanisms to capture how features interact with returns: shared-latent parametrization and feature-influenced views; both recover classical BL and Markowitz portfolios as special cases. Empirically, on 30-year Dow-Jones and 20-year sector-ETF data, we improve Sharpe ratios by 50\% and cut turnover by 55\% relative to Markowitz and the index baselines. This work turns BL into a fully data-driven, view-free, and coherent Bayesian framework for portfolio optimization.

Test-time adaptation (TTA) addresses the machine learning challenge of adapting models to unlabeled test data from shifting distributions in dynamic environments. A key issue in this online setting arises from using unsupervised learning techniques, which introduce explicit gradient noise that degrades model weights. To invest in weight degradation, we propose a Bayesian weight enhancement framework, which generalizes existing weight-based TTA methods that effectively mitigate the issue. Our framework enables robust adaptation to distribution shifts by accounting for diverse weights by modeling weight distributions. Building on our framework, we identify a key limitation in existing methods: their neglect of time-varying covariance reflects the influence of the gradient noise. To address this gap, we propose a novel steady-state adaptation (SSA) algorithm that balances covariance dynamics during adaptation. SSA is derived through the solution of a stochastic differential equation for the TTA process and online inference. The resulting algorithm incorporates a covariance-aware learning rate adjustment mechanism. Through extensive experiments, we demonstrate that SSA consistently improves state-of-the-art methods in various TTA scenarios, datasets, and model architectures, establishing its effectiveness in instability and adaptability.

Decision making under uncertainty is challenging as the data-generating process (DGP) is often unknown. Bayesian inference proceeds by estimating the DGP through posterior beliefs on the model’s parameters. However, minimising the expected risk under these beliefs can lead to suboptimal decisions due to model uncertainty or limited, noisy observations. To address this, we introduce Distributionally Robust Optimisation with Bayesian Ambiguity Sets (DRO-BAS) which hedges against model uncertainty by optimising the worst-case risk over a posterior-informed ambiguity set. We provide two such sets, based on the posterior expectation (DRO-BAS(PE)) or the posterior predictive (DRO-BAS(PP)) and prove that both admit, under conditions, strong dual formulations leading to efficient single-stage stochastic programs which are solved with a sample average approximation. For DRO-BAS(PE), this covers all conjugate exponential family members while for DRO-BAS(PP) this is shown under conditions on the predictive's moment generating function. Our DRO-BAS formulations outperform existing Bayesian DRO on the Newsvendor problem and achieve faster solve times with comparable robustness on the Portfolio problem.

Recent theoretical and empirical successes in deep learning, including the celebrated neural scaling laws, are punctuated by the observation that many objects of interest tend to exhibit some form of heavy-tailed or power law behavior. In particular, the prevalence of heavy-tailed spectral densities in Jacobians, Hessians, and weight matrices has led to the introduction of the concept of *heavy-tailed mechanistic universality* (HT-MU). Multiple lines of empirical evidence suggest a robust correlation between heavy-tailed metrics and model performance, indicating that HT-MU may be a fundamental aspect of deep learning efficacy. Here, we propose a general family of random matrix models---the *high-temperature Marchenko-Pastur (HTMP) ensemble*---to explore attributes that give rise to heavy-tailed behavior in trained neural networks. Under this model, spectral densities with power laws on (upper and lower) tails arise through a combination of three independent factors (complex correlation structures in the data; reduced temperatures during training; and reduced eigenvector entropy), appearing as an implicit bias in the model structure, and they can be controlled with an "eigenvalue repulsion'' parameter. Implications of our model on other appearances of heavy tails, including neural scaling laws, optimizer trajectories, and the five-plus-one phases of neural network training, are discussed.

We introduce Flat Hilbert Bayesian Inference (FHBI), an algorithm designed to enhance generalization in Bayesian inference. Our approach involves an iterative two-step procedure with an adversarial functional perturbation step and a functional descent step within the reproducing kernel Hilbert spaces. This methodology is supported by a theoretical analysis that extends previous findings on generalization ability from finite-dimensional Euclidean spaces to infinite-dimensional functional spaces. To evaluate the effectiveness of FHBI, we conduct comprehensive comparisons against nine baseline methods on the VTAB-1K benchmark, which encompasses 19 diverse datasets across various domains with diverse semantics. Empirical results demonstrate that FHBI consistently outperforms the baselines by notable margins, highlighting its practical efficacy.

Offline reinforcement learning (RL) is crucial when online exploration is costly or unsafe but often struggles with high epistemic uncertainty due to limited data. Existing methods rely on fixed conservative policies, restricting adaptivity and generalization. To address this, we propose Reflect-then-Plan (RefPlan), a novel _doubly Bayesian_ offline model-based (MB) planning approach. RefPlan unifies uncertainty modeling and MB planning by recasting planning as Bayesian posterior estimation. At deployment, it updates a belief over environment dynamics using real-time observations, incorporating uncertainty into MB planning via marginalization. Empirical results on standard benchmarks show that RefPlan significantly improves the performance of conservative offline RL policies. In particular, RefPlan maintains robust performance under high epistemic uncertainty and limited data, while demonstrating resilience to changing environment dynamics, improving the flexibility, generalizability, and robustness of offline-learned policies.

One-Shot Federated Learning (FL) is a recent paradigm that enables multiple clients to cooperatively learn a global model in a single round of communication with a central server. In this paper, we analyze the One-Shot FL problem through the lens of Bayesian inference and propose FedBEns, an algorithm that leverages the inherent multimodality of local loss functions to find better global models.Our algorithm leverages a mixture of Laplace approximations for the clients' local posteriors, which the server then aggregates to infer the global model. We conduct extensive experiments on various datasets, demonstrating that the proposed method outperforms competing baselines that typically rely on unimodal approximations of the local losses.

Autoregressive and recurrent networks have achieved remarkable progress across various fields, from weather forecasting to molecular generation and Large Language Models. Despite their strong predictive capabilities, these models lack a rigorous framework for addressing uncertainty, which is key in scientific applications such as PDE solving, molecular generation and machine l earning Force Fields. To address this shortcoming we present BARNN: a variational Bayesian Autoregressive and Recurrent Neural Network. BARNNs aim to provide a principled way to turn any autoregressive or recurrent model into its Bayesian version. BARNN is based on the variational dropout method, allowing to apply it to large recurrent neural networks as well. We also introduce a temporal version of the “Variational Mixtures of Posteriors” prior (tVAMP-prior) to make Bayesian inference efficient and well-calibrated. Extensive experiments on PDE modelling and molecular generation demonstrate that BARNN not only achieves comparable or superior accuracy compared to existing methods, but also excels in uncertainty quantification and modelling long-range dependencies.

We study the problem of learning the topology of a directed Gaussian Graphical Model under the equal-variance assumption, where the graph has $n$ nodes and maximum in-degree $d$. Prior work has established that $O(d \log n)$ samples are sufficient for this task. However, an important factor that is often overlooked in these analyses is the dependence on the condition number of the covariance matrix of the model. Indeed, all algorithms from prior work require a number of samples that grows polynomially with this condition number. In many cases this is unsatisfactory, since the condition number could grow polynomially with $n$, rendering these prior approaches impractical in high-dimensional settings. In this work, we provide an algorithm that recovers the underlying graph and prove that the number of samples required is independent of the condition number. Furthermore, we establish lower bounds that nearly match the upper bound up to a $d$-factor, thus providing an almost tight characterization of the true sample complexity of the problem. Moreover, under a further assumption that all the variances of the variables are bounded, we design a polynomial-time algorithm that recovers the underlying graph, at the cost of an additional polynomial dependence of the sample complexity on $d$. We complement our theoretical findings with simulations on synthetic datasets that confirm our predictions.

Auction plays a crucial role in many modern trading environments, including online advertising and public resource allocation. As the number of competing bidders increases, learning Bayesian Nash Equilibrium (BNE) in auctions faces significant scalability challenges. Existing methods often experience slow convergence in large-scale auctions. For example, in a classic symmetric auction setting, the convergence rate depends on the number of bidders quadratically. To address this issue, we propose the *Approximate Best Response Gradient* method, a new approach for learning BNE efficiently in auction games. We leverage an analytic solution for gradient estimation to enable efficient gradient computation during optimization. Moreover, we introduce the *Best Response Distance* objective, which serves as an upper bound of approximation quality to BNE. By optimizing the new objective, our method is proven to achieve a local convergence rate independent of bidder numbers and circumvent the traditional quadratic complexity in the classic symmetric setting. Extensive experiments across various auction formats demonstrate that our approach accelerates convergence and enhances learning efficiency in complex auction settings.

Personalized federated learning (PFL) based on Bayesian approach tackle the challenges from statistical heterogeneity of client data by computing a personalized posterior distribution over the parameters of each client's local model and constructing a global distribution by aggregating the parameters of these personalized posteriors. However, the heuristic aggregation methods introduce strong biases and result in global models with poor generalization. We thus propose a novel hierarchical Bayesian inference framework for PFL by specifying a conjugate hyper-prior over the parameters of the personalized posteriors. This allows us to jointly compute a global posterior distribution for aggregation and the personalized ones at local level. This hierarchical Bayesian inference framework achieves elegant balance between local personalization and global model robustness. Extensive empirical study shows that by effectively sharing the heterogeneous statistical strength across the local models while retaining their distinctive characteristics, our framework yields state-of-the-art performance. We also show that existing Bayesian PFLs are special cases of our framework.

Scaling has been a major driver of recent advancements in deep learning. Numerous empirical studies have found that scaling laws often follow the power-law and proposed several variants of power-law functions to predict the scaling behavior at larger scales. However, existing methods mostly rely on point estimation and do not quantify uncertainty, which is crucial for real-world applications involving decision-making problems such as determining the expected performance improvements achievable by investing additional computational resources. In this work, we explore a Bayesian framework based on Prior-data Fitted Networks (PFNs) for neural scaling law extrapolation. Specifically, we design a prior distribution that enables the sampling of infinitely many synthetic functions resembling real-world neural scaling laws, allowing our PFN to meta-learn the extrapolation. We validate the effectiveness of our approach on real-world neural scaling laws, comparing it against both the existing point estimation methods and Bayesian approaches. Our method demonstrates superior performance, particularly in data-limited scenarios such as Bayesian active learning, underscoring its potential for reliable, uncertainty-aware extrapolation in practical applications.

The rapid proliferation of large language models (LLMs), such as GPT-4 and Gemini, underscores the intense demand for resources during their training processes, posing significant challenges due to substantial computational and environmental costs. In this paper, we introduce a novel checkpoint merging strategy aimed at making efficient use of intermediate checkpoints during LLM pretraining. This method utilizes intermediate checkpoints with shared training trajectories, and is rooted in an extensive search space exploration for the best merging weight via Bayesian optimization. Through various experiments, we demonstrate that: (1) Our proposed methodology exhibits the capacity to augment pretraining, presenting an opportunity akin to obtaining substantial benefits at minimal cost; (2) Our proposed methodology, despite requiring a given held-out dataset, still demonstrates robust generalization capabilities across diverse domains, a pivotal aspect in pretraining.

Bayesian optimisation in the latent space of a VAE is a powerful framework for optimisation tasks over complex structured domains, such as the space of valid molecules. However, existing approaches tightly couple the surrogate and generative models, which can lead to suboptimal performance when the latent space is not tailored to specific tasks, which in turn has led to the proposal of increasingly sophisticated algorithms. In this work, we explore a new direction, instead proposing a decoupled approach that trains a generative model and a GP surrogate separately, then combines them via a simple yet principled Bayesian update rule. This separation allows each component to focus on its strengths— structure generation from the VAE and predictive modelling by the GP. We show that our decoupled approach improves our ability to identify high-potential candidates in molecular optimisation problems under constrained evaluation budgets.

As machine learning-based prediction systems are increasingly used in high-stakes situations, it is important to understand how such predictive models will perform upon deployment. Distribution-free uncertainty quantification techniques such as conformal prediction provide guarantees about the loss black-box models will incur even when the details of the models are hidden. However, such methods are based on frequentist probability, which unduly limits their applicability. We revisit the central aspects of conformal prediction from a Bayesian perspective and thereby illuminate the shortcomings of frequentist guarantees. We propose a practical alternative based on Bayesian quadrature that provides interpretable guarantees and offers a richer representation of the likely range of losses to be observed at test time.

Applying Gaussian processes (GPs) to very large datasets remains a challenge due to limited computational scalability. Matrix structures, such as the Kronecker product, can accelerate operations significantly, but their application commonly entails approximations or unrealistic assumptions. In particular, the most common path to creating a Kronecker-structured kernel matrix is by evaluating a product kernel on gridded inputs that can be expressed as a Cartesian product. However, this structure is lost if any observation is missing, breaking the Cartesian product structure, which frequently occurs in real-world data such as time series. To address this limitation, we propose leveraging latent Kronecker structure, by expressing the kernel matrix of observed values as the projection of a latent Kronecker product. In combination with iterative linear system solvers and pathwise conditioning, our method facilitates inference of exact GPs while requiring substantially fewer computational resources than standard iterative methods. We demonstrate that our method outperforms state-of-the-art sparse and variational GPs on real-world datasets with up to five million examples, including robotics, automated machine learning, and climate applications.

Data assimilation for nonlinear state space models (SSMs) is inherently challenging due to non-Gaussian posteriors. We propose Deep Bayesian Filtering (DBF), a novel approach to data assimilation in nonlinear SSMs. DBF introduces latent variables $h_t$ in addition to physical variables $z_t$, ensuring Gaussian posteriors by (i) constraining state transitions in the latent space to be linear and (ii) learning a Gaussian inverse observation operator $r(h_t|o_t)$. This structured posterior design enables analytical recursive computation, avoiding the accumulation of Monte Carlo sampling errors over time steps. DBF optimizes these operators and other latent SSM parameters by maximizing the evidence lower bound. Experiments demonstrate that DBF outperforms existing methods in scenarios with highly non-Gaussian posteriors.

We propose *LEAPS*, an algorithm to sample from discrete distributions known up to normalization by learning a rate matrix of a continuous-time Markov chain (CTMC). LEAPS can be seen as a continuous-time formulation of annealed importance sampling and sequential Monte Carlo methods, extended so that the variance of the importance weights is offset by the inclusion of the CTMC. To derive these importance weights, we introduce a set of Radon-Nikodym derivatives of CTMCs over their path measures. Because the computation of these weights is intractable with standard neural network parameterizations of rate matrices, we devise a new compact representation for rate matrices via what we call \textit{locally equivariant} functions. To parameterize them, we introduce a family of locally equivariant multilayer perceptrons, attention layers, and convolutional networks, and provide an approach to make deep networks that preserve the local equivariance. This property allows us to propose a scalable training algorithm for the rate matrix such that the variance of the importance weights associated to the CTMC are minimal. We demonstrate the efficacy of LEAPS on problems in statistical physics. We provide code in https://github.com/malbergo/leaps/.

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.