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Probabilistic Methods · Bayesian Models and Methods

Kouroche Bouchiat, Alexander Immer, Hugo Yèche, Gunnar Ratsch, Vincent Fortuin

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.

Probabilistic Methods

JIAN XU, Delu Zeng, John Paisley

Deep Gaussian processes (DGPs) provide a robust paradigm in Bayesian deep learning. In DGPs, a set of sparse integration locations called inducing points are selected to approximate the posterior distribution of the model. This is done to reduce computational complexity and improve model efficiency. However, inferring the posterior distribution of inducing points is not straightforward. Traditional variational inference techniques methods to approximate the posterior often leads to significant bias. To address this issue, we propose an alternative named Denoising Diffusion Variational Inference (DDVI) that utilizes a denoising diffusion stochastic differential equation (SDE) for generating posterior samples of inducing variables. We refer to the score matching method in the denoising diffusion model to approximate challenging score functions using a neural network. Furthermore, by combining classical mathematical theory of SDE with the minimization of KL divergence between the approximate and true processes, we propose a novel explicit variational lower bound for the marginal likelihood function of DGP. Through extensive experiments on various datasets and comparisons with baseline methods, we empirically demonstrate the effectiveness of the DDVI method in posterior inference of inducing points for DGP models.

Applications · Health

Isabel Chien, Wessel Bruinsma, Javier Gonzalez, Richard E Turner

In drug development, early phase dose-finding clinical trials are carried out to identify an optimal dose to administer to patients in larger confirmatory clinical trials. Standard trial procedures do not optimize for participant benefit and do not consider participant heterogeneity, despite consequences to participants' health and downstream impacts to under-represented population subgroups. Many novel drugs also do not obey parametric modelling assumptions made in common dose-finding procedures. We present Safe Allocation for Exploration of Treatments SAFE-T, a procedure for adaptive dose-finding that adheres to safety constraints, improves utility for heterogeneous participants, and works well with small sample sizes. SAFE-T flexibly learns non-parametric multi-output Gaussian process models for dose toxicity and efficacy, using Bayesian optimization, and provides accurate final dose recommendations. We provide theoretical guarantees for the satisfaction of safety constraints. Using a comprehensive set of realistic synthetic scenarios, we demonstrate empirically that SAFE-T generally outperforms comparable methods and maintains performance across variations in sample size and subgroup distribution. Finally, we extend SAFE-T to a new adaptive setting, demonstrating its potential to improve traditional clinical trial procedures.

Probabilistic Methods

Louis Sharrock, Jack Simons, Song Liu, Mark Beaumont

We introduce Sequential Neural Posterior Score Estimation (SNPSE), a score-based method for Bayesian inference in simulator-based models. Our method, inspired by the remarkable success of score-based methods in generative modelling, leverages conditional score-based diffusion models to generate samples from the posterior distribution of interest. The model is trained using an objective function which directly estimates the score of the posterior. We embed the model into a sequential training procedure, which guides simulations using the current approximation of the posterior at the observation of interest, thereby reducing the simulation cost. We also introduce several alternative sequential approaches, and discuss their relative merits. We then validate our method, as well as its amortised, non-sequential, variant on several numerical examples, demonstrating comparable or superior performance to existing state-of-the-art methods such as Sequential Neural Posterior Estimation (SNPE).

Probabilistic Methods

Manuel Gloeckler, Michael Deistler, Christian Weilbach, Frank Wood, Jakob Macke

Amortized Bayesian inference trains neural networks to solve stochastic inference problems using model simulations, thereby making it possible to rapidly perform Bayesian inference for any newly observed data. However, current simulation-based amortized inference methods are simulation-hungry and inflexible: They require the specification of a fixed parametric prior, simulator, and inference tasks ahead of time. Here, we present a new amortized inference method---the Simformer---which overcomes these limitations. By training a probabilistic diffusion model with transformer architectures, the Simformer outperforms current state-of-the-art amortized inference approaches on benchmark tasks and is substantially more flexible: It can be applied to models with function-valued parameters, it can handle inference scenarios with missing or unstructured data, and it can sample arbitrary conditionals of the joint distribution of parameters and data, including both posterior and likelihood. We showcase the performance and flexibility of the Simformer on simulators from ecology, epidemiology, and neuroscience, and demonstrate that it opens up new possibilities and application domains for amortized Bayesian inference on simulation-based models.

Probabilistic Methods

JIAN XU, Delu Zeng, John Paisley

Deep Gaussian processes (DGPs) provide a robust paradigm in Bayesian deep learning. In DGPs, a set of sparse integration locations called inducing points are selected to approximate the posterior distribution of the model. This is done to reduce computational complexity and improve model efficiency. However, inferring the posterior distribution of inducing points is not straightforward. Traditional variational inference techniques methods to approximate the posterior often leads to significant bias. To address this issue, we propose an alternative named Denoising Diffusion Variational Inference (DDVI) that utilizes a denoising diffusion stochastic differential equation (SDE) for generating posterior samples of inducing variables. We refer to the score matching method in the denoising diffusion model to approximate challenging score functions using a neural network. Furthermore, by combining classical mathematical theory of SDE with the minimization of KL divergence between the approximate and true processes, we propose a novel explicit variational lower bound for the marginal likelihood function of DGP. Through extensive experiments on various datasets and comparisons with baseline methods, we empirically demonstrate the effectiveness of the DDVI method in posterior inference of inducing points for DGP models.

Optimization · Optimization and Learning under Uncertainty

Yucen Li, Tim G. J. Rudner, Andrew Gordon Wilson

Bayesian optimization is a highly efficient approach to optimizing objective functions which are expensive to query. These objectives are typically represented by Gaussian process (GP) surrogate models which are easy to optimize and support exact inference. While standard GP surrogates have been well-established in Bayesian optimization, Bayesian neural networks (BNNs) have recently become practical function approximators, with many benefits over standard GPs such as the ability to naturally handle non-stationarity and learn representations for high-dimensional data. In this paper, we study BNNs as alternatives to standard GP surrogates for optimization. We consider a variety of approximate inference procedures for finite-width BNNs, including high-quality Hamiltonian Monte Carlo, low-cost stochastic MCMC, and heuristics such as deep ensembles. We also consider infinite-width BNNs, linearized Laplace approximations, and partially stochastic models such as deep kernel learning. We evaluate this collection of surrogate models on diverse problems with varying dimensionality, number of objectives, non-stationarity, and discrete and continuous inputs. We find: (i) the ranking of methods is highly problem dependent, suggesting the need for tailored inductive biases; (ii) HMC is the most successful approximate inference procedure for fully stochastic BNNs; (iii) full stochasticity may be unnecessary as deep kernel learning is relatively competitive; (iv) deep ensembles perform relatively poorly; (v) infinite-width BNNs are particularly promising, especially in high dimensions.

Probabilistic Methods · Bayesian Models and Methods

Edward Milsom, Ben Anson, Laurence Aitchison

Standard infinite-width limits of neural networks sacrifice the ability for intermediate layers to learn representations from data. Recent work (“A theory of representation learning gives a deep generalisation of kernel methods”, Yang et al. 2023) modified the Neural Network Gaussian Process (NNGP) limit of Bayesian neural networks so that representation learning is retained. Furthermore, they found that applying this modified limit to a deep Gaussian process gives a practical learning algorithm which they dubbed the “deep kernel machine” (DKM). However, they only considered the simplest possible setting: regression in small, fully connected networks with e.g. 10 input features. Here, we introduce convolutional deep kernel machines. This required us to develop a novel inter-domain inducing point approximation, as well as introducing and experimentally assessing a number of techniques not previously seen in DKMs, including analogues to batch normalisation, different likelihoods, and different types of top-layer. The resulting model trains in roughly 77 GPU hours, achieving around 99\% test accuracy on MNIST, 72\% on CIFAR-100, and 92.7\% on CIFAR-10, which is SOTA for kernel methods.

Probabilistic Methods · Bayesian Models and Methods

Panagiotis Dimitrakopoulos, Giorgos Sfikas, Christophoros Nikou

Bayesian inference is the standard for providing full predictive distributions with well calibrated uncertainty estimates.However, scaling to a modern, overparameterized deep learning setting typically comes at the cost of severe and restrictive approximations, sacrificing model predictive strength.With our approach, we factor model parameters as a function of deterministic and probabilistic components;the model is solved by combining maximum a posteriori estimation of the former,with inference over a low-dimensional, Implicit Neural Representation of the latter.This results in a solution that combines both predictive accuracy and good calibration,as it entails inducing stochasticity over the full set of model weights while being comparatively cheap to compute.Experimentally, our approach compares favorably to the state of the art,including much more expensive methods as well as less expressive posterior approximations over full network parameters.

Probabilistic Methods · Graphical Models

Jean-Pierre Falet, Hae Beom Lee, Nikolay Malkin, Chen Sun, Dragos Secrieru, Dinghuai Zhang, Guillaume Lajoie, Yoshua Bengio

We present a new algorithm for amortized inference in sparse probabilistic graphical models (PGMs), which we call $\Delta$-amortized inference ($\Delta$-AI). Our approach is based on the observation that when the sampling of variables in a PGM is seen as a sequence of actions taken by an agent, sparsity of the PGM enables local credit assignment in the agent's policy learning objective. This yields a local constraint that can be turned into a local loss in the style of generative flow networks (GFlowNets) that enables off-policy training but avoids the need to instantiate all the random variables for each parameter update, thus speeding up training considerably. The $\Delta$-AI objective matches the conditional distribution of a variable given its Markov blanket in a tractable learned sampler, which has the structure of a Bayesian network, with the same conditional distribution under the target PGM. As such, the trained sampler recovers marginals and conditional distributions of interest and enables inference of partial subsets of variables. We illustrate $\Delta$-AI's effectiveness for sampling from synthetic PGMs and training latent variable models with sparse factor structure. Code: https://github.com/GFNOrg/Delta-AI.

Probabilistic Methods · Bayesian Models and Methods

Soham Gadgil, Ian Covert, Su-In Lee

Dynamic feature selection, where we sequentially query features to make accurate predictions with a minimal budget, is a promising paradigm to reduce feature acquisition costs and provide transparency into the prediction process. The problem is challenging, however, as it requires both making predictions with arbitrary feature sets and learning a policy to identify the most valuable selections. Here, we take an information-theoretic perspective and prioritize features based on their mutual information with the response variable. The main challenge is implementing this policy, and we design a new approach that estimates the mutual information in a discriminative rather than a generative fashion. Building on our learning approach, we introduce several further improvements: allowing variable feature budgets across samples, enabling non-uniform costs between features, incorporating prior information, and exploring modern architectures to handle partial input information. We find that our method provides consistent gains over recent state-of-the-art methods across a variety of datasets.

Reinforcement Learning

Mahdi Kallel, Debabrota Basu, Riad Akrour, Carlo D'Eramo

Deterministic policies are often preferred over stochastic ones when implemented on physical systems. They can prevent erratic and harmful behaviors while being easier to implement and interpret. However, in practice, exploration is largely performed by stochastic policies.First-order Bayesian Optimization (BO) methods offer a principled way of performing exploration using deterministic policies. This is done through a learned probabilistic model of the objective function and its gradient. Nonetheless, such approaches treat policy search as a black-box problem, and thus, neglect the reinforcement learning nature of the problem. In this work, we leverage the performance difference lemma to introduce a novel mean function for the probabilistic model. This results in augmenting BO methods with the action-value function. Hence, we call our method Augmented Bayesian Search (ABS).Interestingly, this new mean function enhances the posterior gradient with the deterministic policy gradient, effectively bridging the gap between BO and policy gradient methods. The resulting algorithm combines the convenience of the direct policy search with the scalability of reinforcement learning.We validate ABS on high-dimensional locomotion problems and demonstrate competitive performance compared to existing direct policy search schemes.

Deep Learning · Sequential Models, Time series

Konstantin Hess, Valentyn Melnychuk, Dennis Frauen, Stefan Feuerriegel

Treatment effect estimation in continuous time is crucial for personalized medicine. However, existing methods for this task are limited to point estimates of the potential outcomes, whereas uncertainty estimates have been ignored. Needless to say, uncertainty quantification is crucial for reliable decision-making in medical applications. To fill this gap, we propose a novel Bayesian neural controlled differential equation (BNCDE) for treatment effect estimation in continuous time. In our BNCDE, the time dimension is modeled through a coupled system of neural controlled differential equations and neural stochastic differential equations, where the neural stochastic differential equations allow for tractable variational Bayesian inference. Thereby, for an assigned sequence of treatments, our BNCDE provides meaningful posterior predictive distributions of the potential outcomes. To the best of our knowledge, ours is the first tailored neural method to provide uncertainty estimates of treatment effects in continuous time. As such, our method is of direct practical value for promoting reliable decision-making in medicine.

James Harrison, John Willes, Jasper Snoek

We introduce a deterministic variational formulation for training Bayesian last layer neural networks. This yields a sampling-free, single-pass model and loss that effectively improves uncertainty estimation. Our variational Bayesian last layer (VBLL) can be trained and evaluated with only quadratic complexity in last layer width, and is thus (nearly) computationally free to add to standard architectures. We experimentally investigate VBLLs, and show that they improve predictive accuracy, calibration, and out of distribution detection over baselines across both regression and classification. Finally, we investigate combining VBLL layers with variational Bayesian feature learning, yielding a lower variance collapsed variational inference method for Bayesian neural networks.

Probabilistic Methods · Variational Inference

Junbo Li, Zichen Miao, Qiang Qiu, Ruqi Zhang

Bayesian neural networks (BNNs) offer uncertainty quantification but come with the downside of substantially increased training and inference costs. Sparse BNNs have been investigated for efficient inference, typically by either slowly introducing sparsity throughout the training or by post-training compression of dense BNNs. The dilemma of how to cut down massive training costs remains, particularly given the requirement to learn about the uncertainty. To solve this challenge, we introduce Sparse Subspace Variational Inference (SSVI), the first fully sparse BNN framework that maintains a consistently sparse Bayesian model throughout the training and inference phases. Starting from a randomly initialized low-dimensional sparse subspace, our approach alternately optimizes the sparse subspace basis selection and its associated parameters. While basis selection is characterized as a non-differentiable problem, we approximate the optimal solution with a removal-and-addition strategy, guided by novel criteria based on weight distribution statistics. Our extensive experiments show that SSVI sets new benchmarks in crafting sparse BNNs, achieving, for instance, a 10-20× compression in model size with under 3\% performance drop, and up to 20× FLOPs reduction during training. Remarkably, SSVI also demonstrates enhanced robustness to hyperparameters, reducing the need for intricate tuning in VI and occasionally even surpassing VI-trained dense BNNs.

Deep Learning · Large Language Models

Adam Yang, Maxime Robeyns, Xi Wang, Laurence Aitchison

Parameter-efficient fine-tuning (PEFT) has emerged as a new paradigm for cost-efficient fine-tuning of large language models (LLMs), with low-rank adaptation (LoRA) being a widely adopted choice. However, fine-tuned LLMs often become overconfident especially when fine-tuned on small datasets. Bayesian methods, with their inherent ability to estimate uncertainty, serve as potent tools to mitigate overconfidence and enhance calibration. In this work, we introduce Laplace-LoRA, a straightforward yet effective Bayesian method, which applies the Laplace approximation to the LoRA parameters and, considerably boosts the calibration of fine-tuned LLMs.

Probabilistic Methods · Variational Inference

Jiajun He, Gergely Flamich, Zongyu Guo, José Miguel Hernández Lobato

COMpression with Bayesian Implicit NEural Representations (COMBINER) is a recent data compression method that addresses a key inefficiency of previous Implicit Neural Representation (INR)-based approaches: it avoids quantization and enables direct optimization of the rate-distortion performance. However, COMBINER still has significant limitations: 1) it uses factorized priors and posterior approximations that lack flexibility; 2) it cannot effectively adapt to local deviations from global patterns in the data; and 3) its performance can be susceptible to modeling choices and the variational parameters' initializations. Our proposed method, Robust and Enhanced COMBINER (RECOMBINER), addresses these issues by 1) enriching the variational approximation while retaining a low computational cost via a linear reparameterization of the INR weights, 2) augmenting our INRs with learnable positional encodings that enable them to adapt to local details and 3) splitting high-resolution data into patches to increase robustness and utilizing expressive hierarchical priors to capture dependency across patches. We conduct extensive experiments across several data modalities, showcasing that RECOMBINER achieves competitive results with the best INR-based methods and even outperforms autoencoder-based codecs on low-resolution images at low bitrates. Our PyTorch implementation is available at https://github.com/cambridge-mlg/RECOMBINER/.

Optimization · Zero-order and Black-box Optimization

Scott Sussex, Pier Giuseppe Sessa, Anastasia Makarova, Andreas Krause

In Causal Bayesian Optimization (CBO), an agent intervenes on a structural causal model with known graph but unknown mechanisms to maximize a downstream reward variable. In this paper, we consider the generalization where other agents or external events also intervene on the system, which is key for enabling adaptiveness to non-stationarities such as weather changes, market forces, or adversaries. We formalize this generalization of CBO as Adversarial Causal Bayesian Optimization (ACBO) and introduce the first algorithm for ACBO with bounded regret: Causal Bayesian Optimization with Multiplicative Weights (CBO-MW). Our approach combines a classical online learning strategy with causal modeling of the rewards. To achieve this, it computes optimistic counterfactual reward estimates by propagating uncertainty through the causal graph. We derive regret bounds for CBO-MW that naturally depend on graph-related quantities. We further propose a scalable implementation for the case of combinatorial interventions and submodular rewards. Empirically, CBO-MW outperforms non-causal and non-adversarial Bayesian optimization methods on synthetic environments and environments based on real-word data. Our experiments include a realistic demonstration of how CBO-MW can be used to learn users' demand patterns in a shared mobility system and reposition vehicles in strategic areas.

Shikai Fang, Xin Yu, Zheng Wang, Shibo Li, Mike Kirby, Shandian Zhe

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at {https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition}.

Applications · Neuroscience, Cognitive Science

Ganchao Wei

Modern neural recording techniques allow neuroscientists to obtain spiking activity of multiple neurons from different brain regions over long time periods, which requires new statistical methods to be developed for understanding structure of the large-scale data. In this paper, we develop a bi-clustering method to cluster the neural spiking activity spatially and temporally, according to their low-dimensional latent structures. The spatial (neuron) clusters are defined by the latent trajectories within each neural population, while the temporal (state) clusters are defined by (populationally) synchronous local linear dynamics shared with different periods. To flexibly extract the bi-clustering structure, we build the model non-parametrically, and develop an efficient Markov chain Monte Carlo (MCMC) algorithm to sample the posterior distributions of model parameters. Validating our proposed MCMC algorithm through simulations, we find the method can recover unknown parameters and true bi-clustering structures successfully. We then apply the proposed bi-clustering method to multi-regional neural recordings under different experiment settings, where we find that simultaneously considering latent trajectories and spatial-temporal clustering structures can provide us with a more accurate and interpretable result. Overall, the proposed method provides scientific insights for large-scale (counting) time series with elongated recording periods, and it can potentially have application beyond neuroscience.