We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covariance matrices can become too large to be stored. By leveraging results about sequential disintegrations of Gaussian measures, we are able to introduce an implicit representation of posterior covariance matrices that reduces the memory footprint by only storing low rank intermediate matrices, while allowing individual elements to be accessed on-the-fly without needing to build full posterior covariance matrices. Moreover, it allows for fast sequential inclusion of new observations. These features are crucial when considering sequential experimental design tasks. We demonstrate our approach by computing sequential data collection plans for excursion set recovery for a gravimetric inverse problem, where the goal is to provide fine resolution estimates of high density regions inside the Stromboli volcano, Italy. Sequential data collection plans are computed by extending the weighted integrated variance reduction (wIVR) criterion to inverse problems. Our results show that this criterion is able to significantly reduce the uncertainty on the excursion volume, reaching close to minimal levels of residual uncertainty. Overall, our techniques allow the advantages of probabilistic models to be brought to bear on large-scale inverse problems arising in the natural sciences.
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There are proposals that extend the classical generalized additive models (GAMs) to accommodate high-dimensional data ($p>>n$) using group sparse regularization. However, the sparse regularization may induce excess shrinkage when estimating smoothing functions, damaging predictive performance. Moreover, most of these GAMs consider an "all-in-all-out" approach for functional selection, rendering them difficult to answer if nonlinear effects are necessary. While some Bayesian models can address these shortcomings, using Markov chain Monte Carlo algorithms for model fitting creates a new challenge, scalability. Hence, we propose Bayesian hierarchical generalized additive models as a solution: we consider the smoothing penalty for proper shrinkage of curve interpolation and separation of smoothing function linear and nonlinear spaces. A novel spike-and-slab spline prior is proposed to select components of smoothing functions. Two scalable and deterministic algorithms, EM-Coordinate Descent and EM-Iterative Weighted Least Squares, are developed for different utilities. Simulation studies and metabolomics data analyses demonstrate improved predictive or computational performance against state-of-the-art models, mgcv, COSSO and sparse Bayesian GAM. The software implementation of the proposed models is freely available via an R package BHAM.
The nature of available economic data has changed fundamentally in the last decade due to the economy's digitisation. With the prevalence of often black box data-driven machine learning methods, there is a necessity to develop interpretable machine learning methods that can conduct econometric inference, helping policymakers leverage the new nature of economic data. We therefore present a novel Variational Bayesian Inference approach to incorporate a time-varying parameter auto-regressive model which is scalable for big data. Our model is applied to a large blockchain dataset containing prices, transactions of individual actors, analyzing transactional flows and price movements on a very granular level. The model is extendable to any dataset which can be modelled as a dynamical system. We further improve the simple state-space modelling by introducing non-linearities in the forward model with the help of machine learning architectures.
We present a framework for transfer learning based on modular variational Gaussian processes (GP). We develop a module-based method that having a dictionary of well fitted GPs, one could build ensemble GP models without revisiting any data. Each model is characterised by its hyperparameters, pseudo-inputs and their corresponding posterior densities. Our method avoids undesired data centralisation, reduces rising computational costs and allows the transfer of learned uncertainty metrics after training. We exploit the augmentation of high-dimensional integral operators based on the Kullback-Leibler divergence between stochastic processes to introduce an efficient lower bound under all the sparse variational GPs, with different complexity and even likelihood distribution. The method is also valid for multi-output GPs, learning correlations a posteriori between independent modules. Extensive results illustrate the usability of our framework in large-scale and multi-task experiments, also compared with the exact inference methods in the literature.
The stochastic multi-armed bandit (MAB) problem is a common model for sequential decision problems. In the standard setup, a decision maker has to choose at every instant between several competing arms, each of them provides a scalar random variable, referred to as a "reward." Nearly all research on this topic considers the total cumulative reward as the criterion of interest. This work focuses on other natural objectives that cannot be cast as a sum over rewards, but rather more involved functions of the reward stream. Unlike the case of cumulative criteria, in the problems we study here the oracle policy, that knows the problem parameters a priori and is used to "center" the regret, is not trivial. We provide a systematic approach to such problems, and derive general conditions under which the oracle policy is sufficiently tractable to facilitate the design of optimism-based (upper confidence bound) learning policies. These conditions elucidate an interesting interplay between the arm reward distributions and the performance metric. Our main findings are illustrated for several commonly used objectives such as conditional value-at-risk, mean-variance trade-offs, Sharpe-ratio, and more.
In this paper several related estimation problems are addressed from a Bayesian point of view and optimal estimators are obtained for each of them when some natural loss functions are considered. Namely, we are interested in estimating a regression curve. Simultaneously, the estimation problems of a conditional distribution function, or a conditional density, or even the conditional distribution itself, are considered. All these problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, parametric and non-parametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function comonly used in estimating a real function of the unknown parameter. The cornerstone of the mentioned Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate these results.
We study the optimal variance reduction solutions for online controlled experiments by applying flexible machine learning tools to incorporate covariates that are independent from the treatment but have predictive power for the outcomes. Employing cross-fitting, we propose variance reduction procedures for both count metrics and ratio metrics in online experiments based on which the inference of the estimands are valid under mild convergence conditions. We also establish the asymptotic optimality of all these procedures under consistency condition of the machine learning estimators. In complement to the proposed nonlinear optimal procedure, a linear adjustment method for ratio metrics is also derived as a special case that is computationally efficient and can flexibly incorporate any pre-treatment covariates. Comprehensive simulation studies are performed and practical suggestions are given. When tested on real online experiment data from LinkedIn, the proposed optimal procedure for ratio metrics can reduce up to $80\%$ of variance compared to the standard difference-in-mean estimator and also further reduce up to $30\%$ of variance compared to the CUPED approach by going beyond linearity and incorporating a large number of extra covariates.
Refining low-resolution (LR) spatial fields with high-resolution (HR) information, often known as statistical downscaling, is challenging as the diversity of spatial datasets often prevents direct matching of observations. Yet, when LR samples are modeled as aggregate conditional means of HR samples with respect to a mediating variable that is globally observed, the recovery of the underlying fine-grained field can be framed as taking an "inverse" of the conditional expectation, namely a deconditioning problem. In this work, we propose a Bayesian formulation of deconditioning which naturally recovers the initial reproducing kernel Hilbert space formulation from Hsu and Ramos (2019). We extend deconditioning to a downscaling setup and devise efficient conditional mean embedding estimator for multiresolution data. By treating conditional expectations as inter-domain features of the underlying field, a posterior for the latent field can be established as a solution to the deconditioning problem. Furthermore, we show that this solution can be viewed as a two-staged vector-valued kernel ridge regressor and show that it has a minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate its proficiency in a synthetic and a real-world atmospheric field downscaling problem, showing substantial improvements over existing methods.
Dynamic topic models (DTMs) model the evolution of prevalent themes in literature, online media, and other forms of text over time. DTMs assume that word co-occurrence statistics change continuously and therefore impose continuous stochastic process priors on their model parameters. These dynamical priors make inference much harder than in regular topic models, and also limit scalability. In this paper, we present several new results around DTMs. First, we extend the class of tractable priors from Wiener processes to the generic class of Gaussian processes (GPs). This allows us to explore topics that develop smoothly over time, that have a long-term memory or are temporally concentrated (for event detection). Second, we show how to perform scalable approximate inference in these models based on ideas around stochastic variational inference and sparse Gaussian processes. This way we can train a rich family of DTMs to massive data. Our experiments on several large-scale datasets show that our generalized model allows us to find interesting patterns that were not accessible by previous approaches.
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.
We propose a new approach to inverse reinforcement learning (IRL) based on the deep Gaussian process (deep GP) model, which is capable of learning complicated reward structures with few demonstrations. Our model stacks multiple latent GP layers to learn abstract representations of the state feature space, which is linked to the demonstrations through the Maximum Entropy learning framework. Incorporating the IRL engine into the nonlinear latent structure renders existing deep GP inference approaches intractable. To tackle this, we develop a non-standard variational approximation framework which extends previous inference schemes. This allows for approximate Bayesian treatment of the feature space and guards against overfitting. Carrying out representation and inverse reinforcement learning simultaneously within our model outperforms state-of-the-art approaches, as we demonstrate with experiments on standard benchmarks ("object world","highway driving") and a new benchmark ("binary world").
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