We introduce a deep learning accelerated methodology to solve PDE-based Bayesian inverse problems with guaranteed accuracy. This is motivated by the ill-posed problem of inferring a spatio-temporal heat-flux parameter known as the Biot number given temperature data, however the methodology is generalisable to other settings. To accelerate Bayesian inference, we develop a novel training scheme that uses data to adaptively train a neural-network surrogate simulating the parametric forward model. By simultaneously identifying an approximate posterior distribution over the Biot number, and weighting a physics-informed training loss according to this, our approach approximates forward and inverse solution together without any need for external solves. Using a random Chebyshev series, we outline how to approximate a Gaussian process prior, and using the surrogate we apply Hamiltonian Monte Carlo (HMC) to sample from the posterior distribution. We derive convergence of the surrogate posterior to the true posterior distribution in the Hellinger metric as our adaptive loss approaches zero. Additionally, we describe how this surrogate-accelerated HMC approach can be combined with traditional PDE solvers in a delayed-acceptance scheme to a-priori control the posterior accuracy. This overcomes a major limitation of deep learning-based surrogate approaches, which do not achieve guaranteed accuracy a-priori due to their non-convex training. Biot number calculations are involved in turbo-machinery design, which is safety critical and highly regulated, therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high dimensions whilst retaining the convergence guarantees of a traditional PDE solver, and without the burden of evaluating this solver for proposals that are likely to be rejected. Numerical results are given using real and simulated data.

Latent Gaussian models have a rich history in statistics and machine learning, with applications ranging from factor analysis to compressed sensing to time series analysis. The classical method for maximizing the likelihood of these models is the expectation-maximization (EM) algorithm. For problems with high-dimensional latent variables and large datasets, EM scales poorly because it needs to invert as many large covariance matrices as the number of data points. We introduce probabilistic unrolling, a method that combines Monte Carlo sampling with iterative linear solvers to circumvent matrix inversion. Our theoretical analyses reveal that unrolling and backpropagation through the iterations of the solver can accelerate gradient estimation for maximum likelihood estimation. In experiments on simulated and real data, we demonstrate that probabilistic unrolling learns latent Gaussian models up to an order of magnitude faster than gradient EM, with minimal losses in model performance.

With only observational data on two variables, and without other assumptions, it is not possible to infer which one causes the other. Much of the causal literature has focused on guaranteeing identifiability of causal direction in statistical models for datasets where strong assumptions hold, such as additive noise or restrictions on parameter count. These methods are then subsequently tested on realistic datasets, most of which violate their assumptions. Building on previous attempts, we show how to use causal assumptions within the Bayesian framework. This allows us to specify models with realistic assumptions, while also encoding independent causal mechanisms, leading to an asymmetry between the causal directions. Identifying causal direction then becomes a Bayesian model selection problem. We analyse why Bayesian model selection works for known identifiable cases and flexible model classes, while also providing correctness guarantees about its behaviour. To demonstrate our approach, we construct a Bayesian non-parametric model that can flexibly model the joint. We then outperform previous methods on a wide range of benchmark datasets with varying data generating assumptions showing the usefulness of our method.

Boosting is a commonly used technique to enhance the performance of a set of base models by combining them into a strong ensemble model. Though widely adopted, boosting is typically used in supervised learning where the data is labeled accurately. However, in weakly supervised learning, where most of the data is labeled through weak and noisy sources, it remains nontrivial to design effective boosting approaches. In this work, we show that the standard implementation of the convex combination of base learners can hardly work due to the presence of noisy labels. Instead, we propose $\textit{LocalBoost}$, a novel framework for weakly-supervised boosting. LocalBoost iteratively boosts the ensemble model from two dimensions, i.e., intra-source and inter-source. The intra-source boosting introduces locality to the base learners and enables each base learner to focus on a particular feature regime by training new base learners on granularity-varying error regions. For the inter-source boosting, we leverage a conditional function to indicate the weak source where the sample is more likely to appear. To account for the weak labels, we further design an estimate-then-modify approach to compute the model weights. Experiments on seven datasets show that our method significantly outperforms vanilla boosting methods and other weakly-supervised methods.

Deep Ensembles (DEs) demonstrate improved accuracy, calibration and robustness to perturbations over single neural networks partly due to their functional diversity. Particle-based variational inference (ParVI) methods enhance diversity by formalizing a repulsion term based on a network similarity kernel. However, weight-space repulsion is inefficient due to over-parameterization, while direct function-space repulsion has been found to produce little improvement over DEs. To sidestep these difficulties, we propose First-order Repulsive Deep Ensemble (FoRDE), an ensemble learning method based on ParVI, which performs repulsion in the space of first-order input gradients. As input gradients uniquely characterize a function up to translation and are much smaller in dimension than the weights, this method guarantees that ensemble members are functionally different. Intuitively, diversifying the input gradients encourages each network to learn different features, which is expected to improve the robustness of an ensemble. Experiments on image classification datasets show that FoRDE significantly outperforms the gold-standard DEs and other ensemble methods in accuracy and calibration under covariate shift due to input perturbations.

In this paper, we study sampling from a posterior derived from a neural network. We propose a new probabilistic model consisting of adding noise at every pre- and post-activation in the network, arguing that the resulting posterior can be sampled using an efficient Gibbs sampler. The Gibbs sampler attains similar performances as the state-of-the-art Monte Carlo Markov chain methods, such as the Hamiltonian Monte Carlo or the Metropolis adjusted Langevin algorithm, both on real and synthetic data. By framing our analysis in the teacher-student setting, we introduce a thermalization criterion that allows us to detect when an algorithm, when run on data with synthetic labels, fails to sample from the posterior. The criterion is based on the fact that in the teacher-student setting we can initialize an algorithm directly at equilibrium.

We address the tasks of Bayesian state estimation and forecasting for a model-free process in an unsupervised learning setup. In the article, we propose DANSE -- a Data-driven Nonlinear State Estimation method. DANSE provides a closed-form posterior of the state of the model-free process, given linear measurements of the state. In addition, it provides a closed-form posterior for forecasting. A data-driven recurrent neural network (RNN) is used in DANSE to provide the parameters of a prior of the state. The prior depends on the past measurements as input, and then we find the closed-form posterior of the state using the current measurement as input. The data-driven RNN captures the underlying non-linear dynamics of the model-free process. The training of DANSE, mainly learning the parameters of the RNN, is executed using an unsupervised learning approach. In unsupervised learning, we have access to a training dataset comprising only a set of measurement data trajectories, but we do not have any access to the state trajectories. Therefore, DANSE does not have access to state information in the training data and can not use supervised learning. Using simulated linear and non-linear process models (Lorenz attractor and Chen attractor), we evaluate the unsupervised learning-based DANSE. We show that the proposed DANSE, without knowledge of the process model and without supervised learning, provides a competitive performance against model-driven methods, such as the Kalman filter (KF), extended KF (EKF), unscented KF (UKF), and a recently proposed hybrid method called KalmanNet.

Models of many real-life applications, such as queuing models of communication networks or computing systems, have a countably infinite state-space. Algorithmic and learning procedures that have been developed to produce optimal policies mainly focus on finite state settings, and do not directly apply to these models. To overcome this lacuna, in this work we study the problem of optimal control of a family of discrete-time countable state-space Markov Decision Processes (MDPs) governed by an unknown parameter $\theta\in\Theta$, and defined on a countably-infinite state space $\mathcal X=\mathbb{Z}_+^d$, with finite action space $\mathcal A$, and an unbounded cost function. We take a Bayesian perspective with the random unknown parameter $\boldsymbol{\theta}^*$ generated via a given fixed prior distribution on $\Theta$. To optimally control the unknown MDP, we propose an algorithm based on Thompson sampling with dynamically-sized episodes: at the beginning of each episode, the posterior distribution formed via Bayes' rule is used to produce a parameter estimate, which then decides the policy applied during the episode. To ensure the stability of the Markov chain obtained by following the policy chosen for each parameter, we impose ergodicity assumptions. From this condition and using the solution of the average cost Bellman equation, we establish an $\tilde O(\sqrt{|\mathcal A|T})$ upper bound on the Bayesian regret of our algorithm, where $T$ is the time-horizon. Finally, to elucidate the applicability of our algorithm, we consider two different queuing models with unknown dynamics, and show that our algorithm can be applied to develop approximately optimal control algorithms.

We propose a unified approach to obtain structured sparse optimal paths in the latent space of a variational autoencoder (VAE) using dynamic programming and Gumbel propagation. We solve the classical optimal path problem by a probability softening solution, called the stochastic optimal path, and transform a wide range of DP problems into directed acyclic graphs in which all possible paths follow a Gibbs distribution. We show the equivalence of the Gibbs distribution to a message-passing algorithm by the properties of the Gumbel distribution and give all the ingredients required for variational Bayesian inference. Our approach obtaining latent optimal paths enables end-to-end training for generative tasks in which models rely on the information of unobserved structural features. We validate the behavior of our approach and showcase its applicability in two real-world applications: text-to-speech and singing voice synthesis.

Commonly used objective functions (losses) for a supervised optimization of discriminative neural network classifiers were either distribution-based or metric-based. The distribution-based losses could compromise the generalization or cause classification biases towards the dominant classes of an imbalanced class-sample distribution. The metric-based losses could make the network model independent of any distribution and thus improve its generalization. However, they could still be biased towards the dominant classes and could suffer from discrepancies when a class was absent in both the reference (ground truth) and the predicted labels. In this paper, we proposed a novel optimization process which not only tackled the unbalancedness of the class-sample distribution of the training samples but also provided a mechanism to tackle errors in the reference labels of the training samples. This was achieved by proposing a novel algorithm to find candidate classification labels of the training samples from their prior probabilities and the currently estimated posteriors on the network and a novel objective function for the optimizations. The algorithm was the result of casting the generalized Kelly criterion for optimal betting into a multiclass classification problem. The proposed objective function was the expected free energy of a prospective active inference and could incorporate the candidate labels, the original reference labels, and the priors of the training samples while still being distribution-based. The incorporation of the priors into the optimization not only helped to tackle errors in the reference labels but also allowed to reduce classification biases towards the dominant classes by focusing the attention of the neural network on important but minority foreground classes.