We address the solution of large-scale Bayesian optimal experimental design (OED) problems governed by partial differential equations (PDEs) with infinite-dimensional parameter fields. The OED problem seeks to find sensor locations that maximize the expected information gain (EIG) in the solution of the underlying Bayesian inverse problem. Computation of the EIG is usually prohibitive for PDE-based OED problems. To make the evaluation of the EIG tractable, we approximate the (PDE-based) parameter-to-observable map with a derivative-informed projected neural network (DIPNet) surrogate, which exploits the geometry, smoothness, and intrinsic low-dimensionality of the map using a small and dimension-independent number of PDE solves. The surrogate is then deployed within a greedy algorithm-based solution of the OED problem such that no further PDE solves are required. We analyze the EIG approximation error in terms of the generalization error of the DIPNet, and demonstrate the efficiency and accuracy of the method via numerical experiments involving inverse scattering and inverse reactive transport.
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We study a new two-time-scale stochastic gradient method for solving optimization problems, where the gradients are computed with the aid of an auxiliary variable under samples generated by time-varying Markov random processes parameterized by the underlying optimization variable. These time-varying samples make gradient directions in our update biased and dependent, which can potentially lead to the divergence of the iterates. In our two-time-scale approach, one scale is to estimate the true gradient from these samples, which is then used to update the estimate of the optimal solution. While these two iterates are implemented simultaneously, the former is updated "faster" (using bigger step sizes) than the latter (using smaller step sizes). Our first contribution is to characterize the finite-time complexity of the proposed two-time-scale stochastic gradient method. In particular, we provide explicit formulas for the convergence rates of this method under different structural assumptions, namely, strong convexity, convexity, the Polyak-Lojasiewicz condition, and general non-convexity. We apply our framework to two problems in control and reinforcement learning. First, we look at the standard online actor-critic algorithm over finite state and action spaces and derive a convergence rate of O(k^(-2/5)), which recovers the best known rate derived specifically for this problem. Second, we study an online actor-critic algorithm for the linear-quadratic regulator and show that a convergence rate of O(k^(-2/3)) is achieved. This is the first time such a result is known in the literature. Finally, we support our theoretical analysis with numerical simulations where the convergence rates are visualized.
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural architecture to learn solution operators of PDEs with (possibly stochastic) forcing from partially observed data. The proposed Neural SPDE model provides an extension to two popular classes of physics-inspired architectures. On the one hand, it extends Neural CDEs and variants -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information arriving irregularly in time and observed at arbitrary spatial resolutions. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can parameterize solution operators of SPDEs depending simultaneously on the initial condition and a realization of the driving noise. By performing operations in the spectral domain, we show how a Neural SPDE can be evaluated in two ways, either by calling an ODE solver (emulating a spectral Galerkin scheme), or by solving a fixed point problem. Experiments on various semilinear SPDEs, including the stochastic Navier-Stokes equations, demonstrate how the Neural SPDE model is capable of learning complex spatiotemporal dynamics in a resolution-invariant way, with better accuracy and lighter training data requirements compared to alternative models, and up to 3 orders of magnitude faster than traditional solvers.
Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. The novelty, in this paper, lies in letting the discretization parameter (time step-size) vary from layer to layer, which needs to be learned, in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional-DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. Stability of some of the existing continuous DNNs such as Fractional-DNN is also studied. The proposed approach is applied to an ill-posed 3D-Maxwell's equation.
We investigate the feature compression of high-dimensional ridge regression using the optimal subsampling technique. Specifically, based on the basic framework of random sampling algorithm on feature for ridge regression and the A-optimal design criterion, we first obtain a set of optimal subsampling probabilities. Considering that the obtained probabilities are uneconomical, we then propose the nearly optimal ones. With these probabilities, a two step iterative algorithm is established which has lower computational cost and higher accuracy. We provide theoretical analysis and numerical experiments to support the proposed methods. Numerical results demonstrate the decent performance of our methods.
We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.
In this paper, we present a scalable deep learning approach to solve opinion dynamics stochastic optimal control problems with mean field term coupling in the dynamics and cost function. Our approach relies on the probabilistic representation of the solution of the Hamilton-Jacobi-Bellman partial differential equation. Grounded on the nonlinear version of the Feynman-Kac lemma, the solutions of the Hamilton-Jacobi-Bellman partial differential equation are linked to the solution of Forward-Backward Stochastic Differential Equations. These equations can be solved numerically using a novel deep neural network with architecture tailored to the problem in consideration. The resulting algorithm is tested on a polarized opinion consensus experiment. The large-scale (10K) agents experiment validates the scalability and generalizability of our algorithm. The proposed framework opens up the possibility for future applications on extremely large-scale problems.
We introduce a novel methodology for particle filtering in dynamical systems where the evolution of the signal of interest is described by a SDE and observations are collected instantaneously at prescribed time instants. The new approach includes the discretisation of the SDE and the design of efficient particle filters for the resulting discrete-time state-space model. The discretisation scheme converges with weak order 1 and it is devised to create a sequential dependence structure along the coordinates of the discrete-time state vector. We introduce a class of space-sequential particle filters that exploits this structure to improve performance when the system dimension is large. This is numerically illustrated by a set of computer simulations for a stochastic Lorenz 96 system with additive noise. The new space-sequential particle filters attain approximately constant estimation errors as the dimension of the Lorenz 96 system is increased, with a computational cost that increases polynomially, rather than exponentially, with the system dimension. Besides the new numerical scheme and particle filters, we provide in this paper a general framework for discrete-time filtering in continuous-time dynamical systems described by a SDE and instantaneous observations. Provided that the SDE is discretised using a weakly-convergent scheme, we prove that the marginal posterior laws of the resulting discrete-time state-space model converge to the posterior marginal posterior laws of the original continuous-time state-space model under a suitably defined metric. This result is general and not restricted to the numerical scheme or particle filters specifically studied in this manuscript.
Policy gradient (PG) estimation becomes a challenge when we are not allowed to sample with the target policy but only have access to a dataset generated by some unknown behavior policy. Conventional methods for off-policy PG estimation often suffer from either significant bias or exponentially large variance. In this paper, we propose the double Fitted PG estimation (FPG) algorithm. FPG can work with an arbitrary policy parameterization, assuming access to a Bellman-complete value function class. In the case of linear value function approximation, we provide a tight finite-sample upper bound on policy gradient estimation error, that is governed by the amount of distribution mismatch measured in feature space. We also establish the asymptotic normality of FPG estimation error with a precise covariance characterization, which is further shown to be statistically optimal with a matching Cramer-Rao lower bound. Empirically, we evaluate the performance of FPG on both policy gradient estimation and policy optimization, using either softmax tabular or ReLU policy networks. Under various metrics, our results show that FPG significantly outperforms existing off-policy PG estimation methods based on importance sampling and variance reduction techniques.
Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving l_1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
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