Training generative models to sample from unnormalized density functions is an important and challenging task in machine learning. Traditional training methods often rely on the reverse Kullback-Leibler (KL) divergence due to its tractability. However, the mode-seeking behavior of reverse KL hinders effective approximation of multi-modal target distributions. To address this, we propose to minimize the reverse KL along diffusion trajectories of both model and target densities. We refer to this objective as the reverse diffusive KL divergence, which allows the model to capture multiple modes. Leveraging this objective, we train neural samplers that can efficiently generate samples from the target distribution in one step. We demonstrate that our method enhances sampling performance across various Boltzmann distributions, including both synthetic multi-modal densities and n-body particle systems.

We propose a novel Stochastic Differential Equation (SDE) framework to address the problem of learning uncertainty-aware representations for graph-structured data. While Graph Neural Ordinary Differential Equations (GNODEs) have shown promise in learning node representations, they lack the ability to quantify uncertainty. To address this, we introduce Latent Graph Neural Stochastic Differential Equations (LGNSDE), which enhance GNODE by embedding randomness through a Bayesian prior-posterior mechanism for epistemic uncertainty and Brownian motion for aleatoric uncertainty. By leveraging the existence and uniqueness of solutions to graph-based SDEs, we prove that the variance of the latent space bounds the variance of model outputs, thereby providing theoretically sensible guarantees for the uncertainty estimates. Furthermore, we show mathematically that LGNSDEs are robust to small perturbations in the input, maintaining stability over time. Empirical results across several benchmarks demonstrate that our framework is competitive in out-of-distribution detection, robustness to noise perturbations, and active learning, underscoring the ability of LGNSDEs to quantify uncertainty reliably.

Contrastive losses have been extensively used as a tool for multimodal representation learning. However, it has been empirically observed that their use is not effective to learn an aligned representation space. In this paper, we argue that this phenomenon is caused by the presence of modality-specific information in the representation space. Although some of the most widely used contrastive losses maximize the mutual information between representations of both modalities, they are not designed to remove the modality-specific information. We give a theoretical description of this problem through the lens of the Information Bottleneck Principle. We also empirically analyze how different hyperparameters affect the emergence of this phenomenon in a controlled experimental setup. Finally, we propose a regularization term in the loss function that is derived by means of a variational approximation and aims to increase the representational alignment. We analyze in a set of controlled experiments and real-world applications the advantages of including this regularization term.

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.

Recent research has focused on designing neural samplers that amortize the process of sampling from unnormalized densities. However, despite significant advancements, they still fall short of the state-of-the-art MCMC approach, Parallel Tempering (PT), when it comes to the efficiency of target evaluations. On the other hand, unlike a well-trained neural sampler, PT yields only dependent samples and needs to be rerun---at considerable computational cost---whenever new samples are required. To address these weaknesses, we propose the Progressive Tempering Sampler with Diffusion (PTSD), which trains diffusion models sequentially across temperatures, leveraging the advantages of PT to improve the training of neural samplers. We also introduce a novel method to combine high-temperature diffusion models to generate approximate lower-temperature samples, which are minimally refined using MCMC and used to train the next diffusion model. PTSD enables efficient reuse of sample information across temperature levels while generating well-mixed, uncorrelated samples. Our method significantly improves target evaluation efficiency, outperforming diffusion-based neural samplers.

Proteolysis-targeting chimeras (PROTACs) are a groundbreaking technology for targeted protein degradation, but designing effective linkers that connect two molecular fragments to form a drug-candidate PROTAC molecule remains a key challenge. While diffusion models show promise in molecular generation, current diffusion models for PROTAC linker design are typically trained on small molecule datasets, introducing distribution mismatches in the chemical space between small molecules and target PROTACs. Direct fine-tuning on limited PROTAC datasets often results in overfitting and poor generalization. In this work, we propose DAD-PROTAC, a domain-adapted diffusion model for PROTAC linker design, which addresses this distribution mismatch in chemical space through density ratio estimation to bridge the gap between small-molecule and PROTAC domains. By decomposing the target score estimator into a pre-trained score function and a lightweight score correction term, DAD-PROTAC achieves efficient fine-tuning without full retraining. Experimental results demonstrate its superior ability to generate high-quality PROTAC linkers.

The prevalence of temporal networks in real-world applications, like financial transaction networks for loan approval prediction, poses significant challenges for ensuring fairness across different groups. These dynamic systems increasingly rely on Temporal Graph Neural Networks (TGNNs) to model evolving interactions between users over time, but TGNNs can inadvertently produce unfair outcomes across different demographic groups. In this work, we are the first to investigate group fairness on temporal graphs and propose a novel real-time framework for monitoring and improving group fairness in TGNNs. We begin by incorporating a fixed fairness regularization term into the TGNN framework, named FTGNN-R, which operates in real-time but exhibits several critical limitations. To address this, we propose FTGNN-M, a new monitoring-based approach that assesses fairness on the fly, without relying on unseen test data. By conducting a sensitivity analysis, FTGNN-M further identifies the specific channels of node embeddings responsible for unfairness and adaptively adjusts the corresponding subset of model parameters. This approach enables a trade-off between fairness and utility in dynamic settings. FTGNN-M offers theoretical guarantees for both fairness assessment and fairness promotion. Extensive experiments on five temporal transaction network datasets demonstrate the effectiveness of our proposed FTGNN-M model in terms of both utility and fairness metrics.

We present Free energy Estimators with Adaptive Transport (FEAT), a novel framework for free energy estimation---a critical challenge across scientific domains. FEAT leverages learned transports implemented via stochastic interpolants and provides consistent, minimum-variance estimators based on escorted Jarzynski equality and controlled Crooks theorem, alongside variational upper and lower bounds on free energy differences. Unifying equilibrium and non-equilibrium methods under a single theoretical framework, FEAT establishes a principled foundation for neural free energy calculations. Experimental validation on toy examples, molecular simulations, and quantum field theory demonstrates promising improvements over existing learning-based methods. Our PyTorch implementation is available at https://github.com/jiajunhe98/FEAT.

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/.

As is well known, both sampling from the posterior and computing the mean of the posterior in Gaussian process regression reduces to solving a large linear system of equations. We study the use of stochastic gradient descent for solving this linear system, and show that when done right---by which we mean using specific insights from the optimisation and kernel communities---stochastic gradient descent is highly effective. To that end, we introduce a particularly simple stochastic dual descent algorithm, explain its design in an intuitive manner and illustrate the design choices through a series of ablation studies. Further experiments demonstrate that our new method is highly competitive. In particular, our evaluations on the UCI regression tasks and on Bayesian optimisation set our approach apart from preconditioned conjugate gradients and variational Gaussian process approximations. Moreover, our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.

Retrosynthesis is the task of planning a series of chemical reactions to create a desired molecule from simpler, buyable molecules. While previous works have proposed algorithms to find optimal solutions for a range of metrics (e.g. shortest, lowest-cost), these works generally overlook the fact that we have imperfect knowledge of the space of possible reactions, meaning plans created by algorithms may not work in a laboratory. In this paper we propose a novel formulation of retrosynthesis in terms of stochastic processes to account for this uncertainty. We then propose a novel greedy algorithm called retro-fallback which maximizes the probability that at least one synthesis plan can be executed in the lab. Using in-silico benchmarks we demonstrate that retro-fallback generally produces better sets of synthesis plans than the popular MCTS and retro* algorithms.

We investigate the problem of explainability for machine learning models, focusing on Feature Attribution Methods (FAMs) that evaluate feature importance through perturbation tests. Despite their utility, FAMs struggle to distinguish the contributions of different features, when their prediction changes are similar after perturbation. To enhance FAMs’ discriminative power, we introduce Feature Attribution with Necessity and Sufficiency (FANS), which find a neighborhood of the input such that perturbing samples within this neighborhood have a high Probability of being Necessity and Sufficiency (PNS) cause for the change in predictions, and use this PNS as the importance of the feature. Specifically, FANS compute this PNS via a heuristic strategy for estimating the neighborhood and a perturbation test involving two stages (factual and interventional) for counterfactual reasoning. To generate counterfactual samples, we use a resampling-based approach on the observed samples to approximate the required conditional distribution. We demonstrate that FANS outperforms existing attribution methods on six benchmarks. Please refer to the source code via https://github.com/DMIRLAB-Group/FANS.

The inadequate mixing of conventional Markov Chain Monte Carlo (MCMC) methods for multi-modal distributions presents a significant challenge in practical applications such as Bayesian inference and molecular dynamics. Addressing this, we propose Diffusive Gibbs Sampling (DiGS), an innovative family of sampling methods designed for effective sampling from distributions characterized by distant and disconnected modes. DiGS integrates recent developments in diffusion models, leveraging Gaussian convolution to create an auxiliary noisy distribution that bridges isolated modes in the original space and applying Gibbs sampling to alternately draw samples from both spaces. A novel Metropolis-within-Gibbs scheme is proposed to enhance mixing in the denoising sampling step. DiGS exhibits a better mixing property for sampling multi-modal distributions than state-of-the-art methods such as parallel tempering, attaining substantially improved performance across various tasks, including mixtures of Gaussians, Bayesian neural networks and molecular dynamics.

Research into optimisation for deep learning is characterised by a tension between the computational efficiency of first-order, gradient-based methods (such as SGD and Adam) and the theoretical efficiency of second-order, curvature-based methods (such as quasi-Newton methods and K-FAC). Noting that second-order methods often only function effectively with the addition of stabilising heuristics (such as Levenberg-Marquardt damping), we ask how much these (as opposed to the second-order curvature model) contribute to second-order algorithms’ performance. We thus study AdamQLR: an optimiser combining damping and learning rate selection techniques from K-FAC (Martens & Grosse, 2015) with the update directions proposed by Adam, inspired by considering Adam through a second-order lens. We evaluate AdamQLR on a range of regression and classification tasks at various scales and hyperparameter tuning methodologies, concluding K-FAC’s adaptive heuristics are of variable standalone general effectiveness, and finding an untuned AdamQLR setting can achieve comparable performance vs runtime to tuned benchmarks.

In the current landscape of deep learning research, there is a predominant emphasis on achieving high predictive accuracy in supervised tasks involving large image and language datasets. However, a broader perspective reveals a multitude of overlooked metrics, tasks, and data types, such as uncertainty, active and continual learning, and scientific data, that demand attention. Bayesian deep learning (BDL) constitutes a promising avenue, offering advantages across these diverse settings. This paper posits that BDL can elevate the capabilities of deep learning. It revisits the strengths of BDL, acknowledges existing challenges, and highlights some exciting research avenues aimed at addressing these obstacles. Looking ahead, the discussion focuses on possible ways to combine large-scale foundation models with BDL to unlock their full potential.

Relative entropy coding (REC) algorithms encode a random sample following a target distribution $Q$, using a coding distribution $P$ shared between the sender and receiver. Sadly, general REC algorithms suffer from prohibitive encoding times, at least on the order of $2^{D_{\text{KL}}[Q||P]}$, and faster algorithms are limited to very specific settings. This work addresses this issue by introducing a REC scheme utilizing space partitioning to reduce runtime in practical scenarios. We provide theoretical analyses of our method and demonstrate its effectiveness with both toy examples and practical applications. Notably, our method successfully handles REC tasks with $D_{\text{KL}}[Q||P]$ about three times greater than what previous methods can manage, and reduces the bitrate by approximately 5-15\% in VAE-based lossless compression on MNIST and INR-based lossy compression on CIFAR-10, compared to previous methods, significantly improving the practicality of REC for neural compression.

Correctly capturing the symmetry transformations of data can lead to efficient models with strong generalization capabilities, though methods incorporating symmetries often require prior knowledge. While recent advancements have been made in learning those symmetries directly from the dataset, most of this work has focused on the discriminative setting. In this paper, we take inspiration from group theoretic ideas to construct a generative model that explicitly aims to capture the data's approximate symmetries. This results in a model that, given a prespecified broad set of possible symmetries, learns to what extent, if at all, those symmetries are actually present. Our model can be seen as a generative process for data augmentation. We provide a simple algorithm for learning our generative model and empirically demonstrate its ability to capture symmetries under affine and color transformations, in an interpretable way. Combining our symmetry model with standard generative models results in higher marginal test-log-likelihoods and improved data efficiency.

Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community. This paper focuses on iterative methods, which use linear system solvers, like conjugate gradients, alternating projections or stochastic gradient descent, to construct an estimate of the marginal likelihood gradient. We discuss three key improvements which are applicable across solvers: (i) a pathwise gradient estimator, which reduces the required number of solver iterations and amortises the computational cost of making predictions, (ii) warm starting linear system solvers with the solution from the previous step, which leads to faster solver convergence at the cost of negligible bias, (iii) early stopping linear system solvers after a limited computational budget, which synergises with warm starting, allowing solver progress to accumulate over multiple marginal likelihood steps. These techniques provide speed-ups of up to $72\times$ when solving to tolerance, and decrease the average residual norm by up to $7\times$ when stopping early.

Modelling partial differential equations (PDEs) is of crucial importance in science and engineering, and it includes tasks ranging from forecasting to inverse problems, such as data assimilation. However, most previous numerical and machine learning approaches that target forecasting cannot be applied out-of-the-box for data assimilation. Recently, diffusion models have emerged as a powerful tool for conditional generation, being able to flexibly incorporate observations without retraining. In this work, we perform a comparative study of score-based diffusion models for forecasting and assimilation of sparse observations. In particular, we focus on diffusion models that are either trained in a conditional manner, or conditioned after unconditional training. We address the shortcomings of existing models by proposing 1) an autoregressive sampling approach, that significantly improves performance in forecasting, 2) a new training strategy for conditional score-based models that achieves stable performance over a range of history lengths, and 3) a hybrid model which employs flexible pre-training conditioning on initial conditions and flexible post-training conditioning to handle data assimilation. We empirically show that these modifications are crucial for successfully tackling the combination of forecasting and data assimilation, a task commonly encountered in real-world scenarios.

Relative entropy coding (REC) algorithms encode a sample from a target distribution $Q$ using a proposal distribution $P$ using as few bits as possible. Unlike entropy coding, REC does not assume discrete distributions and require quantisation. As such, it can be naturally integrated into communication pipelines such as learnt compression and differentially private federated learning. Unfortunately, despite their practical benefits, REC algorithms have not seen widespread application, due to their prohibitively slow runtimes or restrictive assumptions. In this paper, we make progress towards addressing these issues. We introduce Greedy Rejection Coding (GRC), which generalises the rejection sampling-based algorithm of Harsha et al. (2007) to arbitrary probability spaces and partitioning schemes. We first show that GRC terminates almost surely and returns unbiased samples from $Q$, and then focus on two variants of GRC, namely GRCS and GRCD. We show that for continuous $Q$ and $P$ over $\mathbb{R}$ with unimodal $dQ/dP$, the expected runtime of GRCS is upper bounded by $\beta D_{KL}(Q||P) + \mathcal{O}(1)$ where $\beta \approx 4.82$, and its expected codelength is optimal. This makes GRCS the first REC algorithm with guaranteed optimal runtime for this class of distributions, up to the multiplicative constant $\beta$. This significantly improves upon the previous state-of-the-art method, A* coding (Flamich et al., 2022). Under the same assumptions, we experimentally observe and conjecture that the expected runtime and codelength of GRCD are upper bounded by $D_{KL}(Q||P) + \mathcal{O}(1)$. Finally, we evaluate GRC in a compression pipeline with variational autoencoders on MNIST, and show that a modified training objective and a codelength-compression method can further improve compression efficiency.

Many common types of data can be represented as functions that map coordinates to signal values, such as pixel locations to RGB values in the case of an image. Based on this view, data can be compressed by overfitting a compact neural network to its functional representation and then encoding the network weights. However, most current solutions for this are inefficient, as quantization to low-bit precision substantially degrades the reconstruction quality. To address this issue, we propose overfitting variational Bayesian neural networks to the data and compressing an approximate posterior weight sample using relative entropy coding instead of quantizing and entropy coding it. This strategy enables direct optimization of the rate-distortion performance by minimizing the $\beta$-ELBO, and target different rate-distortion trade-offs for a given network architecture by adjusting $\beta$. Moreover, we introduce an iterative algorithm for learning prior weight distributions and employ a progressive refinement process for the variational posterior that significantly enhances performance. Experiments show that our method achieves strong performance on image and audio compression while retaining simplicity.

Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian~optimization~task.

Coupling normalizing flows allow for fast sampling and density evaluation, making them the tool of choice for probabilistic modeling of physical systems. However, the standard coupling architecture precludes endowing flows that operate on the Cartesian coordinates of atoms with the SE(3) and permutation invariances of physical systems. This work proposes a coupling flow that preserves SE(3) and permutation equivariance by performing coordinate splits along additional augmented dimensions. At each layer, the flow maps atoms' positions into learned SE(3) invariant bases, where we apply standard flow transformations, such as monotonic rational-quadratic splines, before returning to the original basis. Crucially, our flow preserves fast sampling and density evaluation, and may be used to produce unbiased estimates of expectations with respect to the target distribution via importance sampling. When trained on the DW4, LJ13, and QM9-positional datasets, our flow is competitive with equivariant continuous normalizing flows and diffusion models, while allowing sampling more than an order of magnitude faster. Moreover, to the best of our knowledge, we are the first to learn the full Boltzmann distribution of alanine dipeptide by only modeling the Cartesian positions of its atoms. Lastly, we demonstrate that our flow can be trained to approximately sample from the Boltzmann distribution of the DW4 and LJ13 particle systems using only their energy functions.

The Tanimoto coefficient is commonly used to measure the similarity between molecules represented as discrete fingerprints, either as a distance metric or a positive definite kernel. While many kernel methods can be accelerated using random feature approximations, at present there is a lack of such approximations for the Tanimoto kernel. In this paper we propose two kinds of novel random features to allow this kernel to scale to large datasets, and in the process discover a novel extension of the kernel to real-valued vectors. We theoretically characterize these random features, and provide error bounds on the spectral norm of the Gram matrix. Experimentally, we show that these random features are effective at approximating the Tanimoto coefficient of real-world datasets and are useful for molecular property prediction and optimization tasks. Future updates to this work will be available at http://arxiv.org/abs/2306.14809.

Normalizing flows are a popular class of models for approximating probability distributions. However, their invertible nature limits their ability to model target distributions whose support have a complex topological structure, such as Boltzmann distributions. Several procedures have been proposed to solve this problem but many of them sacrifice invertibility and, thereby, tractability of the log-likelihood as well as other desirable properties. To address these limitations, we introduce a base distribution for normalizing flows based on learned rejection sampling, allowing the resulting normalizing flow to model complicated distributions without giving up bijectivity. Furthermore, we develop suitable learning algorithms using both maximizing the log-likelihood and the optimization of the Kullback-Leibler divergence, and apply them to various sample problems, i.e. approximating 2D densities, density estimation of tabular data, image generation, and modeling Boltzmann distributions. In these experiments our method is competitive with or outperforms the baselines.

The linearised Laplace method for estimating model uncertainty has received renewed attention in the Bayesian deep learning community. The method provides reliable error bars and admits a closed-form expression for the model evidence, allowing for scalable selection of model hyperparameters. In this work, we examine the assumptions behind this method, particularly in conjunction with model selection. We show that these interact poorly with some now-standard tools of deep learning–stochastic approximation methods and normalisation layers–and make recommendations for how to better adapt this classic method to the modern setting. We provide theoretical support for our recommendations and validate them empirically on MLPs, classic CNNs, residual networks with and without normalisation layers, generative autoencoders and transformers.

Relative entropy coding (REC) algorithms encode a sample from a target distribution Q using a proposal distribution P, such that the expected codelength is O(KL[Q || P]). REC can be seamlessly integrated with existing learned compression models since, unlike entropy coding, it does not assume discrete Q or P, and does not require quantisation. However, general REC algorithms require an intractable $\Omega$(exp(KL[Q || P])) runtime. We introduce AS* and AD* coding, two REC algorithms based on A* sampling. We prove that, for continuous distributions over the reals, if the density ratio is unimodal, AS* has O(D$\infty$[Q || P]) expected runtime, where D$\infty$[Q || P] is the Renyi $\infty$-divergence. We provide experimental evidence that AD* also has O(D$\infty$[Q || P]) expected runtime. We prove that AS* and AD* achieve an expected codelength of O(KL[Q || P]). Further, we introduce DAD*, an approximate algorithm based on AD* which retains its favourable runtime and has bias similar to that of alternative methods. Focusing on VAEs, we propose the IsoKL VAE (IKVAE), which can be used with DAD* to further improve compression efficiency. We evaluate A* coding with (IK)VAEs on MNIST, showing that it can losslessly compress images near the theoretically optimal limit.

Perceived signals in real-world scenarios are usually high-dimensional and noisy, and finding and using their representation that contains essential and sufficient information required by downstream decision-making tasks will help improve computational efficiency and generalization ability in the tasks. In this paper, we focus on partially observable environments and propose to learn a minimal set of state representations that capture sufficient information for decision-making, termed Action-Sufficient state Representations (ASRs). We build a generative environment model for the structural relationships among variables in the system and present a principled way to characterize ASRs based on structural constraints and the goal of maximizing cumulative reward in policy learning. We then develop a structured sequential Variational Auto-Encoder to estimate the environment model and extract ASRs. Our empirical results on CarRacing and VizDoom demonstrate a clear advantage of learning and using ASRs for policy learning. Moreover, the estimated environment model and ASRs allow learning behaviors from imagined outcomes in the compact latent space to improve sample efficiency.

Variational Autoencoders (VAEs) have recently been highly successful at imputing and acquiring heterogeneous missing data. However, within this specific application domain, existing VAE methods are restricted by using only one layer of latent variables and strictly Gaussian posterior approximations. To address these limitations, we present HH-VAEM, a Hierarchical VAE model for mixed-type incomplete data that uses Hamiltonian Monte Carlo with automatic hyper-parameter tuning for improved approximate inference. Our experiments show that HH-VAEM outperforms existing baselines in the tasks of missing data imputation and supervised learning with missing features. Finally, we also present a sampling-based approach for efficiently computing the information gain when missing features are to be acquired with HH-VAEM. Our experiments show that this sampling-based approach is superior to alternatives based on Gaussian approximations.

A growing number of people are seeking healthcare advice online. Usually, they diagnose their medical conditions based on the symptoms they are experiencing, which is also known as self-diagnosis. From the machine learning perspective, online disease diagnosis is a sequential feature (symptom) selection and classification problem. Reinforcement learning (RL) methods are the standard approaches to this type of tasks. Generally, they perform well when the feature space is small, but frequently become inefficient in tasks with a large number of features, such as the self-diagnosis. To address the challenge, we propose a non-RL Bipartite Scalable framework for Online Disease diAgnosis, called BSODA. BSODA is composed of two cooperative branches that handle symptom-inquiry and disease-diagnosis, respectively. The inquiry branch determines which symptom to collect next by an information-theoretic reward. We employ a Product-of-Experts encoder to significantly improve the handling of partial observations of a large number of features. Besides, we propose several approximation methods to substantially reduce the computational cost of the reward to a level that is acceptable for online services. Additionally, we leverage the diagnosis model to estimate the reward more precisely. For the diagnosis branch, we use a knowledge-guided self-attention model to perform predictions. In particular, BSODA determines when to stop inquiry and output predictions using both the inquiry and diagnosis models. We demonstrate that BSODA outperforms the state-of-the-art methods on several public datasets. Moreover, we propose a novel evaluation method to test the transferability of symptom checking methods from synthetic to real-world tasks. Compared to existing RL baselines, BSODA is more effectively scalable to large search spaces.

Online education platforms enable teachers to share a large number of educational resources such as questions to form exercises and quizzes for students. With large volumes of available questions, it is important to have an automated way to quantify their properties and intelligently select them for students, enabling effective and personalized learning experiences. In this work, we propose a framework for mining insights from educational questions at scale. We utilize the state-of-the-art Bayesian deep learning method, in particular partial variational auto-encoders (p-VAE), to analyze real students' answers to a large collection of questions. Based on p-VAE, we propose two novel metrics that quantify question quality and difficulty, respectively, and a personalized strategy to adaptively select questions for students. We apply our proposed framework to a real-world dataset with tens of thousands of questions and tens of millions of answers from an online education platform. Our framework not only demonstrates promising results in terms of statistical metrics but also obtains highly consistent results with domain experts' evaluation.

Specifying a Bayesian prior is notoriously difficult for complex models such as neural networks. Reasoning about parameters is made challenging by the high-dimensionality and over-parameterization of the space. Priors that seem benign and uninformative can have unintuitive and detrimental effects on a model’s predictions. For this reason, we propose predictive complexity priors: a functional prior that is defined by comparing the model’s predictions to those of a reference model. Although originally defined on the model outputs, we transfer the prior to the model parameters via a change of variables. The traditional Bayesian workflow can then proceed as usual. We apply our predictive complexity prior to high-dimensional regression, reasoning over neural network depth, and sharing of statistical strength for few-shot learning.

Sliced Stein discrepancy (SSD) and its kernelized variants have demonstrated promising successes in goodness-of-fit tests and model learning in high dimensions. Despite the theoretical elegance, their empirical performance depends crucially on the search of the optimal slicing directions to discriminate between two distributions. Unfortunately, previous gradient-based optimisation approach returns sub-optimal results for the slicing directions: it is computationally expensive, sensitive to initialization, and it lacks theoretical guarantee for convergence. We address these issues in two steps. First, we show in theory that the requirement of using optimal slicing directions in the kernelized version of SSD can be relaxed, validating the resulting discrepancy with finite random slicing directions. Second, given that good slicing directions are crucial for practical performance, we propose a fast algorithm for finding good slicing directions based on ideas of active sub-space construction and spectral decomposition. Experiments in goodness-of-fit tests and model learning show that our approach achieves both the best performance and the fastest convergence. Especially, we demonstrate 14-80x speed-up in goodness-of-fit tests when compared with the gradient-based approach.

Hamiltonian Monte Carlo (HMC) is one of the most successful sampling methods in machine learning. However, its performance is significantly affected by the choice of hyperparameter values. Existing approaches for optimizing the HMC hyperparameters either optimize a proxy for mixing speed or consider the HMC chain as an implicit variational distribution and optimize a tractable lower bound that can be very loose in practice. Instead, we propose to optimize an objective that quantifies directly the speed of convergence to the target distribution. Our objective can be easily optimized using stochastic gradient descent. We evaluate our proposed method and compare to baselines on a variety of problems including sampling from synthetic 2D distributions, reconstructing sparse signals, learning deep latent variable models and sampling molecular configurations from the Boltzmann distribution of a 22 atom molecule. We find that our method is competitive with or improves upon alternative baselines in all these experiments.

The Bayesian paradigm has the potential to solve core issues of deep neural networks such as poor calibration and data inefficiency. Alas, scaling Bayesian inference to large weight spaces often requires restrictive approximations. In this work, we show that it suffices to perform inference over a small subset of model weights in order to obtain accurate predictive posteriors. The other weights are kept as point estimates. This subnetwork inference framework enables us to use expressive, otherwise intractable, posterior approximations over such subsets. In particular, we implement subnetwork linearized Laplace as a simple, scalable Bayesian deep learning method: We first obtain a MAP estimate of all weights and then infer a full-covariance Gaussian posterior over a subnetwork using the linearized Laplace approximation. We propose a subnetwork selection strategy that aims to maximally preserve the model’s predictive uncertainty. Empirically, our approach compares favorably to ensembles and less expressive posterior approximations over full networks.

Bayesian inference in the space of functions has been an important topic for Bayesian modeling in the past. In this paper, we propose a new solution to this problem called Functional Variational Inference (FVI). In FVI, we minimize a divergence in function space between the variational distribution and the posterior process. This is done by using as functional variational family a new class of flexible distributions called Stochastic Process Generators (SPGs), which are cleverly designed so that the functional ELBO can be estimated efficiently using analytic solutions and mini-batch sampling. FVI can be applied to stochastic process priors when random function samples from those priors are available. Our experiments show that FVI consistently outperforms weight-space and function space VI methods on several tasks, which validates the effectiveness of our approach.

Optimization in the latent space of variational autoencoders is a promising approach to generate high-dimensional discrete objects that maximize an expensive black-box property (e.g., drug-likeness in molecular generation, function approximation with arithmetic expressions). However, existing methods lack robustness as they may decide to explore areas of the latent space for which no data was available during training and where the decoder can be unreliable, leading to the generation of unrealistic or invalid objects. We propose to leverage the epistemic uncertainty of the decoder to guide the optimization process. This is not trivial though, as a naive estimation of uncertainty in the high-dimensional and structured settings we consider would result in high estimator variance. To solve this problem, we introduce an importance sampling-based estimator that provides more robust estimates of epistemic uncertainty. Our uncertainty-guided optimization approach does not require modifications of the model architecture nor the training process. It produces samples with a better trade-off between black-box objective and validity of the generated samples, sometimes improving both simultaneously. We illustrate these advantages across several experimental settings in digit generation, arithmetic expression approximation and molecule generation for drug design.

Great computational effort is invested in generating equilibrium states for molecular systems using, for example, Markov chain Monte Carlo. We present a probabilistic model that generates statistically independent samples for molecules from their graph representations. Our model learns a low-dimensional manifold that preserves the geometry of local atomic neighborhoods through a principled learning representation that is based on Euclidean distance geometry. In a new benchmark for molecular conformation generation, we show experimentally that our generative model achieves state-of-the-art accuracy. Finally, we show how to use our model as a proposal distribution in an importance sampling scheme to compute molecular properties.

Automating molecular design using deep reinforcement learning (RL) holds the promise of accelerating the discovery of new chemical compounds. Existing approaches work with molecular graphs and thus ignore the location of atoms in space, which restricts them to 1) generating single organic molecules and 2) heuristic reward functions. To address this, we present a novel RL formulation for molecular design in Cartesian coordinates, thereby extending the class of molecules that can be built. Our reward function is directly based on fundamental physical properties such as the energy, which we approximate via fast quantum-chemical methods. To enable progress towards de-novo molecular design, we introduce MolGym, an RL environment comprising several challenging molecular design tasks along with baselines. In our experiments, we show that our agent can efficiently learn to solve these tasks from scratch by working in a translation and rotation invariant state-action space.

Modern systems-on-chip (SoCs) include not only general-purpose CPUs but also specialized hardware accelerators. Typically, there are three coherence model choices to integrate an accelerator with the memory hierarchy: no coherence, coherent with the last-level cache (LLC), and private cache based full coherence. However, there has been very limited research on finding which coherence models are optimal for the accelerators of a complex many-accelerator SoC. This paper focuses on determining a cost-aware coherence interface for an SoC and its target application: find the best coherence models for the accelerators that optimize their power and performance, considering both workload characteristics and system-level contention. A novel comprehensive methodology is proposed that uses Bayesian optimization to efficiently find the cost-aware coherence interfaces for SoCs that are modeled using the gem5-Aladdin architectural simulator. For a complete analysis, gem5-Aladdin is extended to support LLC coherence in addition to already-supported no coherence and full coherence. For a heterogeneous SoC targeting applications with varying amount of accelerator-level parallelism, the proposed framework rapidly finds cost-aware coherence interfaces that show significant performance and power benefits over the other commonly-used coherence interfaces.

Existing methods for estimating uncertainty in deep learning tend to require multiple forward passes, making them unsuitable for applications where computational resources are limited. To solve this, we perform probabilistic reasoning over the depth of neural networks. Different depths correspond to subnetworks which share weights and whose predictions are combined via marginalisation, yielding model uncertainty. By exploiting the sequential structure of feed-forward networks, we are able to both evaluate our training objective and make predictions with a single forward pass. We validate our approach on real-world regression and image classification tasks. Our approach provides uncertainty calibration, robustness to dataset shift, and accuracies competitive with more computationally expensive baselines.

When designing new molecules with particular properties, it is not only important what to make but crucially how to make it. These instructions form a synthesis directed acyclic graph (DAG), describing how a large vocabulary of simple building blocks can be recursively combined through chemical reactions to create more complicated molecules of interest. In contrast, many current deep generative models for molecules ignore synthesizability. We therefore propose a deep generative model that better represents the real world process, by directly outputting molecule synthesis DAGs. We argue that this provides sensible inductive biases, ensuring that our model searches over the same chemical space that chemists would also have access to, as well as interoperability. We show that our approach is able to model chemical space well, producing a wide range of diverse molecules, and allows for unconstrained optimization of an inherently constrained problem: maximize certain chemical properties such that discovered molecules are synthesizable.

Variational Autoencoders (VAEs) have seen widespread use in learned image compression. They are used to learn expressive latent representations on which downstream compression methods can operate with high efficiency. Recently proposed 'bits-back' methods can indirectly encode the latent representation of images with codelength close to the relative entropy between the latent posterior and the prior. However, due to the underlying algorithm, these methods can only be used for lossless compression, and they only achieve their nominal efficiency when compressing multiple images simultaneously; they are inefficient for compressing single images. As an alternative, we propose a novel method, Relative Entropy Coding (REC), that can directly encode the latent representation with codelength close to the relative entropy for single images, supported by our empirical results obtained on the Cifar10, ImageNet32 and Kodak datasets. Moreover, unlike previous bits-back methods, REC is immediately applicable to lossy compression, where it is competitive with the state-of-the-art on the Kodak dataset.

Deep generative models often perform poorly in real-world applications due to the heterogeneity of natural data sets. Heterogeneity arises from data containing different types of features (categorical, ordinal, continuous, etc.) and features of the same type having different marginal distributions. We propose an extension of variational autoencoders (VAEs) called VAEM to handle such heterogeneous data. VAEM is a deep generative model that is trained in a two stage manner, such that the first stage provides a more uniform representation of the data to the second stage, thereby sidestepping the problems caused by heterogeneous data. We provide extensions of VAEM to handle partially observed data, and demonstrate its performance in data generation, missing data prediction and sequential feature selection tasks. Our results show that VAEM broadens the range of real-world applications where deep generative models can be successfully deployed.

Many important problems in science and engineering, such as drug design, involve optimizing an expensive black-box objective function over a complex, high-dimensional, and structured input space. Although machine learning techniques have shown promise in solving such problems, existing approaches substantially lack sample efficiency. We introduce an improved method for efficient black-box optimization, which performs the optimization in the low-dimensional, continuous latent manifold learned by a deep generative model. In contrast to previous approaches, we actively steer the generative model to maintain a latent manifold that is highly useful for efficiently optimizing the objective. We achieve this by periodically retraining the generative model on the data points queried along the optimization trajectory, as well as weighting those data points according to their objective function value. This weighted retraining can be easily implemented on top of existing methods, and is empirically shown to significantly improve their efficiency and performance on synthetic and real-world optimization problems.

Generative models can be used for a wide range of tasks, and have the appealing ability to learn from both labelled and unlabelled data. In contrast, discriminative models cannot learn from unlabelled data, but tend to outperform their generative counterparts in supervised tasks. We develop a framework to jointly train deep generative and discriminative models, enjoying the benefits of both. The framework allows models to learn from labelled and unlabelled data, as well as naturally account for uncertainty in predictive distributions, providing the first Bayesian approach to semi-supervised learning with deep generative models. We demonstrate that our blended discriminative and generative models outperform purely generative models in both predictive performance and uncertainty calibration in a number of semi-supervised learning tasks.

We introduce the implicit processes (IPs), a stochastic process that places implicitly defined multivariate distributions over any finite collections of random variables. IPs are therefore highly flexible implicit priors over functions, with examples including data simulators, Bayesian neural networks and non-linear transformations of stochastic processes. A novel and efficient approximate inference algorithm for IPs, namely the variational implicit processes (VIPs), is derived using generalised wake-sleep updates. This method returns simple update equations and allows scalable hyper-parameter learning with stochastic optimization. Experiments show that VIPs return better uncertainty estimates and lower errors over existing inference methods for challenging models such as Bayesian neural networks, and Gaussian processes.

Many real-life decision making situations allow further relevant information to be acquired at a specific cost, for example, in assessing the health status of a patient we may decide to take additional measurements such as diagnostic tests or imaging scans before making a final assessment. Acquiring more relevant information enables better decision making, but may be costly. How can we trade off the desire to make good decisions by acquiring further information with the cost of performing that acquisition? To this end, we propose a principled framework, named EDDI (Efficient Dynamic Discovery of high-value Information), based on the theory of Bayesian experimental design. In EDDI, we propose a novel partial variational autoencoder (Partial VAE) to predict missing data entries problematically given any subset of the observed ones, and combine it with an acquisition function that maximizes expected information gain on a set of target variables. We show cost reduction at the same decision quality and improved decision quality at the same cost in multiple machine learning benchmarks and two real-world health-care applications.

Dropout regularization of deep neural networks has been a mysterious yet effective tool to prevent overfitting. Explanations for its success range from the prevention of "co-adapted" weights to it being a form of cheap Bayesian inference. We propose a novel framework for understanding multiplicative noise in neural networks, considering continuous distributions as well as Bernoulli noise (i.e. dropout). We show that multiplicative noise induces structured shrinkage priors on a network’s weights. We derive the equivalence through reparametrization properties of scale mixtures and without invoking any approximations. Given the equivalence, we then show that dropout’s Monte Carlo training objective approximates marginal MAP estimation. We leverage these insights to propose a novel shrinkage framework for resnets, terming the prior ’automatic depth determination’ as it is the natural analog of automatic relevance determination for network depth. Lastly, we investigate two inference strategies that improve upon the aforementioned MAP approximation in regression benchmarks.

Deep generative models are able to suggest new organic molecules by generating strings, trees, and graphs representing their structure. While such models allow one to generate molecules with desirable properties, they give no guarantees that the molecules can actually be synthesized in practice. We propose a new molecule generation model, mirroring a more realistic real-world process, where (a) reactants are selected, and (b) combined to form more complex molecules. More specifically, our generative model proposes a bag of initial reactants (selected from a pool of commercially-available molecules) and uses a reaction model to predict how they react together to generate new molecules. We first show that the model can generate diverse, valid and unique molecules due to the useful inductive biases of modeling reactions. Furthermore, our model allows chemists to interrogate not only the properties of the generated molecules but also the feasibility of the synthesis routes. We conclude by using our model to solve retrosynthesis problems, predicting a set of reactants that can produce a target product.

In this paper, we address the ice-start problem, i.e., the challenge of deploying machine learning models when only a little or no training data is initially available, and acquiring each feature element of data is associated with costs. This setting is representative of the real-world machine learning applications. For instance, in the health care domain, obtaining every single measurement comes with a cost. We propose Icebreaker, a principled framework for elementwise training data acquisition. Icebreaker introduces a full Bayesian Deep Latent Gaussian Model (BELGAM) with a novel inference method, which combines recent advances in amortized inference and stochastic gradient MCMC to enable fast and accurate posterior inference. By utilizing BELGAM’s ability to fully quantify model uncertainty, we also propose two information acquisition functions for imputation and active prediction problems. We demonstrate that BELGAM performs significantly better than previous variational autoencoder (VAE) based models, when the data set size is small, using both machine learning benchmarks and real world recommender systems and health-care applications. Moreover, Icebreaker not only demonstrates improved performance compared to baselines, but it is also capable of achieving better test performance with less training data available.

Posterior sampling for reinforcement learning (PSRL) is an effective method for balancing exploration and exploitation in reinforcement learning. Randomised value functions (RVF) can be viewed as a promising approach to scaling PSRL. However, we show that most contemporary algorithms combining RVF with neural network function approximation do not possess the properties which make PSRL effective, and provably fail in sparse reward problems. Moreover, we find that propagation of uncertainty, a property of PSRL previously thought important for exploration, does not preclude this failure. We use these insights to design Successor Uncertainties (SU), a cheap and easy to implement RVF algorithm that retains key properties of PSRL. SU is highly effective on hard tabular exploration benchmarks. Furthermore, on the Atari 2600 domain, it surpasses human performance on 38 of 49 games tested (achieving a median human normalised score of 2.09), and outperforms its closest RVF competitor, Bootstrapped DQN, on 36 of those.

Leveraging the wealth of unlabeled data produced in recent years provides great potential for improving supervised models. When the cost of acquiring labels is high, probabilistic active learning methods can be used to greedily select the most informative data points to be labeled. However, for many large-scale problems standard greedy procedures become computationally infeasible and suffer from negligible model change. In this paper, we introduce a novel Bayesian batch active learning approach that mitigates these issues. Our approach is motivated by approximating the complete data posterior of the model parameters. While naive batch construction methods result in correlated queries, our algorithm produces diverse batches that enable efficient active learning at scale. We derive interpretable closed-form solutions akin to existing active learning procedures for linear models, and generalize to arbitrary models using random projections. We demonstrate the benefits of our approach on several large-scale regression and classification tasks.

Bayesian neural networks with latent variables are scalable and flexible probabilistic models: they account for uncertainty in the estimation of the network weights and, by making use of latent variables, can capture complex noise patterns in the data. Using these models we show how to perform and utilize a decomposition of uncertainty in aleatoric and epistemic components for decision making purposes. This allows us to successfully identify informative points for active learning of functions with heteroscedastic and bimodal noise. Using the decomposition we further define a novel risk-sensitive criterion for reinforcement learningto identify policies that balance expected cost, model-bias and noise aversion.

Deep Gaussian Processes (DGPs) are hierarchical generalizations of Gaussian Processes that combine well calibrated uncertainty estimates with the high flexibility of multilayer models. One of the biggest challenges with these models is that exact inference is intractable. The current state-of-the-art inference method, Variational Inference (VI), employs a Gaussian approximation to the posterior distribution. This can be a potentially poor unimodal approximation of the generally multimodal posterior. In this work, we provide evidence for the non-Gaussian nature of the posterior and we apply the Stochastic Gradient Hamiltonian Monte Carlo method to generate samples. To efficiently optimize the hyperparameters, we introduce the Moving Window MCEM algorithm. This results in significantly better predictions at a lower computational cost than its VI counterpart. Thus our method establishes a new state-of-the-art for inference in DGPs.

Chemical space is so large that brute force searches for new interesting molecules are infeasible. High-throughput virtual screening via computer cluster simulations can speed up the discovery process by collecting very large amounts of data in parallel, e.g., up to hundreds or thousands of parallel measurements. Bayesian optimization (BO) can produce additional acceleration by sequentially identifying the most useful simulations or experiments to be performed next. However, current BO methods cannot scale to the large numbers of parallel measurements and the massive libraries of molecules currently used in high-throughput screening. Here, we propose a scalable solution based on a parallel and distributed implementation of Thompson sampling (PDTS). We show that, in small scale problems, PDTS performs similarly as parallel expected improvement (EI), a batch version of the most widely used BO heuristic. Additionally, in settings where parallel EI does not scale, PDTS outperforms other scalable baselines such as a greedy search, $\epsilon$-greedy approaches and a random search method. These results show that PDTS is a successful solution for large-scale parallel BO.

This paper proposes a general method for improving the structure and quality of sequences generated by a recurrent neural network (RNN), while maintaining information originally learned from data, as well as sample diversity. An RNN is first pre-trained on data using maximum likelihood estimation (MLE), and the probability distribution over the next token in the sequence learned by this model is treated as a prior policy. Another RNN is then trained using reinforcement learning (RL) to generate higher-quality outputs that account for domain-specific incentives while retaining proximity to the prior policy of the MLE RNN. To formalize this objective, we derive novel off-policy RL methods for RNNs from KL-control. The effectiveness of the approach is demonstrated on two applications; 1) generating novel musical melodies, and 2) computational molecular generation. For both problems, we show that the proposed method improves the desired properties and structure of the generated sequences, while maintaining information learned from data.

Deep generative models have been wildly successful at learning coherent latent representations for continuous data such as natural images, artwork, and audio. However, generative modeling of discrete data such as arithmetic expressions and molecular structures still poses significant challenges. Crucially, state-of-the-art methods often produce outputs that are not valid. We make the key observation that frequently, discrete data can be represented as a parse tree from a context-free grammar. We propose a variational autoencoder which directly encodes from and decodes to these parse trees, ensuring the generated outputs are always syntactically valid. Surprisingly, we show that not only does our model more often generate valid outputs, it also learns a more coherent latent space in which nearby points decode to similar discrete outputs. We demonstrate the effectiveness of our learned models by showing their improved performance in Bayesian optimization for symbolic regression and molecule generation.

In this paper we propose using machine learning to improve the design of deep neural network hardware accelerators. We show how to adapt multi-objective Bayesian optimization to overcome a challenging design problem: optimizing deep neural network hardware accelerators for both accuracy and energy efficiency. DNN accelerators exhibit all aspects of a challenging optimization space: the landscape is rough, evaluating designs is expensive, the objectives compete with each other, and both design spaces (algorithmic and microarchitectural) are unwieldy. With multi-objective Bayesian optimization, the design space exploration is made tractable and the design points found vastly outperform traditional methods across all metrics of interest.

Variational methods have been recently considered for scaling the training process of Gaussian process classifiers to large datasets. As an alternative, we describe here how to train these classifiers efficiently using expectation propagation (EP). The proposed EP method allows to train Gaussian process classifiers on very large datasets, with millions of instances, that were out of the reach of previous implementations of EP. More precisely, it can be used for (i) training in a distributed fashion where the data instances are sent to different nodes in which the required computations are carried out, and for (ii) maximizing an estimate of the marginal likelihood using a stochastic approximation of the gradient. Several experiments involving large datasets show that the method described is competitive with the variational approach.

Imagine we show an image to a person and ask her/him to decide whether the scene in the image is warm or not warm, and whether it is easy or not to spot a squirrel in the image. For exactly the same image, the answers to those questions are likely to differ from person to person. This is because the task is inherently ambiguous. Such an ambiguous, therefore challenging, task is pushing the boundary of computer vision in showing what can and can not be learned from visual data. Crowdsourcing has been invaluable for collecting annotations. This is particularly so for a task that goes beyond a clear-cut dichotomy as multiple human judgments per image are needed to reach a consensus. This paper makes conceptual and technical contributions. On the conceptual side, we define disagreements among annotators as privileged information about the data instance. On the technical side, we propose a framework to incorporate annotation disagreements into the classifiers. The proposed framework is simple, relatively fast, and outperforms classifiers that do not take into account the disagreements, especially if tested on high confidence annotations.

Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are nonparametric probabilistic models and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. This paper develops a new approximate Bayesian learning scheme that enables DGPs to be applied to a range of medium to large scale regression problems for the first time. The new method uses an approximate Expectation Propagation procedure and a novel and efficient extension of the probabilistic backpropagation algorithm for learning. We evaluate the new method for non-linear regression on eleven real-world datasets, showing that it always outperforms GP regression and is almost always better than state-of-the-art deterministic and sampling-based approximate inference methods for Bayesian neural networks. As a by-product, this work provides a comprehensive analysis of six approximate Bayesian methods for training neural networks.

We present \small PESMO, a Bayesian method for identifying the Pareto set of multi-objective optimization problems, when the functions are expensive to evaluate. \small PESMO chooses the evaluation points to maximally reduce the entropy of the posterior distribution over the Pareto set. The \small PESMO acquisition function is decomposed as a sum of objective-specific acquisition functions, which makes it possible to use the algorithm in \emphdecoupled scenarios in which the objectives can be evaluated separately and perhaps with different costs. This decoupling capability is useful to identify difficult objectives that require more evaluations. \small PESMO also offers gains in efficiency, as its cost scales linearly with the number of objectives, in comparison to the exponential cost of other methods. We compare \small PESMO with other methods on synthetic and real-world problems. The results show that \small PESMO produces better recommendations with a smaller number of evaluations, and that a decoupled evaluation can lead to improvements in performance, particularly when the number of objectives is large.

Black-box alpha (BB-α) is a new approximate inference method based on the minimization of α-divergences. BB-αscales to large datasets because it can be implemented using stochastic gradient descent. BB-αcan be applied to complex probabilistic models with little effort since it only requires as input the likelihood function and its gradients. These gradients can be easily obtained using automatic differentiation. By changing the divergence parameter α, the method is able to interpolate between variational Bayes (VB) (α→0) and an algorithm similar to expectation propagation (EP) (α= 1). Experiments on probit regression and neural network regression and classification problems show that BB-αwith non-standard settings of α, such as α= 0.5, usually produces better predictions than with α→0 (VB) or α= 1 (EP).

The continued success of Deep Neural Networks (DNNs) in classification tasks has sparked a trend of accelerating their execution with specialized hardware. While published designs easily give an order of magnitude improvement over general-purpose hardware, few look beyond an initial implementation. This paper presents Minerva, a highly automated co-design approach across the algorithm, architecture, and circuit levels to optimize DNN hardware accelerators. Compared to an established fixed-point accelerator baseline, we show that fine-grained, heterogeneous data type optimization reduces power by 1.5×, aggressive, in-line predication and pruning of small activity values further reduces power by 2.0×, and active hardware fault detection coupled with domain-aware error mitigation eliminates an additional 2.7× through lowering SRAM voltages. Across five datasets, these optimizations provide a collective average of 8.1× power reduction over an accelerator baseline without compromising DNN model accuracy. Minerva enables highly accurate, ultra-low power DNN accelerators (in the range of tens of milliwatts), making it feasible to deploy DNNs in power-constrained IoT and mobile devices.

We present an information-theoretic framework for solving global black-box optimization problems that also have black-box constraints. Of particular interest to us is to efficiently solve problems with decoupled constraints, in which subsets of the objective and constraint functions may be evaluated independently. For example, when the objective is evaluated on a CPU and the constraints are evaluated independently on a GPU. These problems require an acquisition function that can be separated into the contributions of the individual function evaluations. We develop one such acquisition function and call it Predictive Entropy Search with Constraints (PESC). PESC is an approximation to the expected information gain criterion and it compares favorably to alternative approaches based on improvement in several synthetic and real- world problems. In addition to this, we consider problems with a mix of functions that are fast and slow to evaluate. These problems require balancing the amount of time spent in the meta- computation of PESC and in the actual evaluation of the target objective. We take a bounded rationality approach and develop a partial update for PESC which trades off accuracy against speed. We then propose a method for adaptively switching between the partial and full updates for PESC. This allows us to interpolate between versions of PESC that are efficient in terms of function evaluations and those that are efficient in terms of wall-clock time. Overall, we demonstrate that PESC is an effective algorithm that provides a promising direction towards a unified solution for constrained Bayesian optimization.

Multi-task feature selection methods often make the hypothesis that learning tasks share relevant and irrelevant features. However, this hypothesis may be too restrictive in practice. For example, there may be a few tasks with specific relevant and irrelevant features (outlier tasks). Similarly, a few of the features may be relevant for only some of the tasks (outlier features). To account for this, we propose a model for multi-task feature selection based on a robust prior distribution that introduces a set of binary latent variables to identify outlier tasks and outlier features. Expectation propagation can be used for efficient approximate inference under the proposed prior. Several experiments show that a model based on the new robust prior provides better predictive performance than other benchmark methods.

Unknown constraints arise in many types of expensive black-box optimization problems. Several methods have been proposed recently for performing Bayesian optimization with constraints, based on the expected improvement (EI) heuristic. However, EI can lead to pathologies when used with constraints. For example, in the case of decoupled constraints—i.e., when one can independently evaluate the objective or the constraints—EI can encounter a pathology that prevents exploration. Additionally, computing EI requires a current best solution, which may not exist if none of the data collected so far satisfy the constraints. By contrast, information-based approaches do not suffer from these failure modes. In this paper, we present a new information-based method called Predictive Entropy Search with Constraints (PESC). We analyze the performance of PESC and show that it compares favorably to EI-based approaches on synthetic and benchmark problems, as well as several real-world examples. We demonstrate that PESC is an effective algorithm that provides a promising direction towards a unified solution for constrained Bayesian optimization.

Large multilayer neural networks trained with backpropagation have recently achieved state-of-the-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a large number of hyperparameters to the data, lack of calibrated probabilistic predictions, and a tendency to overfit the training data. In principle, the Bayesian approach to learning neural networks does not have these problems. However, existing Bayesian techniques lack scalability to large dataset and network sizes. In this work we present a novel scalable method for learning Bayesian neural networks, called probabilistic backpropagation (PBP). Similar to classical backpropagation, PBP works by computing a forward propagation of probabilities through the network and then doing a backward computation of gradients. A series of experiments on ten real-world datasets show that PBP is significantly faster than other techniques, while offering competitive predictive abilities. Our experiments also show that PBP provides accurate estimates of the posterior variance on the network weights.

Expectation propagation (EP) is a deterministic approximation algorithm that is often used to perform approximate Bayesian parameter learning. EP approximates the full intractable posterior distribution through a set of local-approximations that are iteratively refined for each datapoint. EP can offer analytic and computational advantages over other approximations, such as Variational Inference (VI), and is the method of choice for a number of models. The local nature of EP appears to make it an ideal candidate for performing Bayesian learning on large models in large-scale datasets settings. However, EP has a crucial limitation in this context: the number approximating factors needs to increase with the number of data-points, N, which often entails a prohibitively large memory overhead. This paper presents an extension to EP, called stochastic expectation propagation (SEP), that maintains a global posterior approximation (like VI) but updates it in a local way (like EP). Experiments on a number of canonical learning problems using synthetic and real-world datasets indicate that SEP performs almost as well as full EP, but reduces the memory consumption by a factor of N. SEP is therefore ideally suited to performing approximate Bayesian learning in the large model, large dataset setting.

Fully observed large binary matrices appear in a wide variety of contexts. To model them, probabilistic matrix factorization (PMF) methods are an attractive solution. However, current batch algorithms for PMF can be inefficient because they need to analyze the entire data matrix before producing any parameter updates. We derive an efficient stochastic inference algorithm for PMF models of fully observed binary matrices. Our method exhibits faster convergence rates than more expensive batch approaches and has better predictive performance than scalable alternatives. The proposed method includes new data subsampling strategies which produce large gains over standard uniform subsampling. We also address the task of automatically selecting the size of the minibatches of data used by our method. For this, we derive an algorithm that adjusts this hyper-parameter online.

We propose a probabilistic matrix factorization model for collaborative filtering that learns from data that is missing not at random(MNAR). Matrix factorization models exhibit state-of-the-art predictive performance in collaborative filtering. However, these models usually assume that the data is missing at random (MAR), and this is rarely the case. For example, the data is not MAR if users rate items they like more than ones they dislike. When the MAR assumption is incorrect, inferences are biased and predictive performance can suffer. Therefore, we model both the generative process for the data and the missing data mechanism. By learning these two models jointly we obtain improved performance over state-of-the-art methods when predicting the ratings and when modeling the data observation process. We present the first viable MF model for MNAR data. Our results are promising and we expect that further research on NMAR models will yield large gains in collaborative filtering.

We present a new matrix factorization model for rating data and a corresponding active learning strategy to address the cold-start problem. Cold-start is one of the most challenging tasks for recommender systems: what to recommend with new users or items for which one has little or no data. An approach is to use active learning to collect the most useful initial ratings. However, the performance of active learning depends strongly upon having accurate estimates of i) the uncertainty in model parameters and ii) the intrinsic noisiness of the data. To achieve these estimates we propose a heteroskedastic Bayesian model for ordinal matrix factorization. We also present a computationally efficient framework for Bayesian active learning with this type of complex probabilistic model. This algorithm successfully distinguishes between informative and noisy data points. Our model yields state-of-the-art predictive performance and, coupled with our active learning strategy, enables us to gain useful information in the cold-start setting from the very first active sample.

Copulas allow to learn marginal distributions separately from the multivariate dependence structure (copula) that links them together into a density function. Vine factorizations ease the learning of high-dimensional copulas by constructing a hierarchy of conditional bivariate copulas. However, to simplify inference, it is common to assume that each of these conditional bivariate copulas is independent from its conditioning variables. In this paper, we relax this assumption by discovering the latent functions that specify the shape of a conditional copula given its conditioning variables. We learn these functions by following a Bayesian approach based on sparse Gaussian processes with expectation propagation for scalable, approximate inference. Experiments on real-world datasets show that, when modeling all conditional dependencies, we obtain better estimates of the underlying copula of the data.

The accurate prediction of time-changing covariances is an important problem in the modeling of multivariate financial data. However, some of the most popular models suffer from a) overfitting problems and multiple local optima, b) failure to capture shifts in market conditions and c) large computational costs. To address these problems we introduce a novel dynamic model for time-changing covariances. Over-fitting and local optima are avoided by following a Bayesian approach instead of computing point estimates. Changes in market conditions are captured by assuming a diffusion process in parameter values, and finally computationally efficient and scalable inference is performed using particle filters. Experiments with financial data show excellent performance of the proposed method with respect to current standard models.

A probabilistic model based on the horseshoe prior is proposed for learning dependencies in the process of identifying relevant features for prediction. Exact inference is intractable in this model. However, expectation propagation offers an approximate alternative. Because the process of estimating feature selection dependencies may suffer from over-fitting in the model proposed, additional data from a multi-task learning scenario are considered for induction. The same model can be used in this setting with few modifications. Furthermore, the assumptions made are less restrictive than in other multi-task methods: The different tasks must share feature selection dependencies, but can have different relevant features and model coefficients. Experiments with real and synthetic data show that this model performs better than other multi-task alternatives from the literature. The experiments also show that the model is able to induce suitable feature selection dependencies for the problems considered, only from the training data.

The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be innacurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables. We evaluate the ability of our method to predict time-varying dependencies on several equities and currencies and observe consistent performance gains compared to static copula models and other time-varying copula methods.

We present a new model based on Gaussian processes (GPs) for learning pairwise preferences expressed by multiple users. Inference is simplified by using a \emph{preference kernel} for GPs which allows us to combine supervised GP learning of user preferences with unsupervised dimensionality reduction for multi-user systems. The model not only exploits collaborative information from the shared structure in user behavior, but may also incorporate user features if they are available. Approximate inference is implemented using a combination of expectation propagation and variational Bayes. Finally, we present an efficient active learning strategy for querying preferences. The proposed technique performs favorably on real-world data against state-of-the-art multi-user preference learning algorithms.

A new framework based on the theory of copulas is proposed to address semi-supervised domain adaptation problems. The presented method factorizes any multivariate density into a product of marginal distributions and bivariate copula functions. Therefore, changes in each of these factors can be detected and corrected to adapt a density model across different learning domains. Importantly, we introduce a novel vine copula model, which allows for this factorization in a non-parametric manner. Experimental results on regression problems with real-world data illustrate the efficacy of the proposed approach when compared to state-of-the-art techniques.

We conjecture that the distribution of the time-reversed residuals of a causal linear process is closer to a Gaussian than the distribution of the noise used to generate the process in the forward direction. This property is demonstrated for causal AR(1) processes assuming that all the cumulants of the distribution of the noise are defined. Based on this observation, it is possible to design a decision rule for detecting the direction of time series that can be described as linear processes: The correct direction (forward in time) is the one in which the residuals from a linear fit to the time series are less Gaussian. A series of experiments with simulated and real-world data illustrate the superior results of the proposed rule when compared with other state-of-the-art methods based on independence tests.

Multi-class Gaussian Process Classifiers (MGPCs) are often affected by over-fitting problems when labeling errors occur far from the decision boundaries. To prevent this, we investigate a robust MGPC (RMGPC) which considers labeling errors independently of their distance to the decision boundaries. Expectation propagation is used for approximate inference. Experiments with several datasets in which noise is injected in the class labels illustrate the benefits of RMGPC. This method performs better than other Gaussian process alternatives based on considering latent Gaussian noise or heavy-tailed processes. When no noise is injected in the labels, RMGPC still performs equal or better than the other methods. Finally, we show how RMGPC can be used for successfully identifying data instances which are difficult to classify accurately in practice.

We introduce a hierarchical Bayesian model for the discovery of putative regulators from gene expression data only. The hierarchy incorporates the knowledge that there are just a few regulators that by themselves only regulate a handful of genes. This is implemented through a so-called spike-and-slab prior, a mixture of Gaussians with different widths, with mixing weights from a hierarchical Bernoulli model. For efficient inference we implemented expectation propagation. Running the model on a malaria parasite data set, we found four genes with significant homology to transcription factors in an amoebe, one RNA regulator and three genes of unknown function (out of the top ten genes considered).