Parametric verification of linear temporal properties for stochastic models can be expressed as computing the satisfaction probability of a certain property as a function of the parameters of the model. Smoothed model checking (smMC) aims at inferring the satisfaction function over the entire parameter space from a limited set of observations obtained via simulation. As observations are costly and noisy, smMC is framed as a Bayesian inference problem so that the estimates have an additional quantification of the uncertainty. In smMC the authors use Gaussian Processes (GP), inferred by means of the Expectation Propagation algorithm. This approach provides accurate reconstructions with statistically sound quantification of the uncertainty. However, it inherits the well-known scalability issues of GP. In this paper, we exploit recent advances in probabilistic machine learning to push this limitation forward, making Bayesian inference of smMC scalable to larger datasets and enabling its application to models with high dimensional parameter spaces. We propose Stochastic Variational Smoothed Model Checking (SV-smMC), a solution that exploits stochastic variational inference (SVI) to approximate the posterior distribution of the smMC problem. The strength and flexibility of SVI make SV-smMC applicable to two alternative probabilistic models: Gaussian Processes (GP) and Bayesian Neural Networks (BNN). The core ingredient of SVI is a stochastic gradient-based optimization that makes inference easily parallelizable and that enables GPU acceleration. In this paper, we compare the performances of smMC against those of SV-smMC by looking at the scalability, the computational efficiency and the accuracy of the reconstructed satisfaction function.
相關內容
Cross-validation is the standard approach for tuning parameter selection in many non-parametric regression problems. However its use is less common in change-point regression, perhaps as its prediction error-based criterion may appear to permit small spurious changes and hence be less well-suited to estimation of the number and location of change-points. We show that in fact the problems of cross-validation with squared error loss are more severe and can lead to systematic under- or over-estimation of the number of change-points, and highly suboptimal estimation of the mean function in simple settings where changes are easily detectable. We propose two simple approaches to remedy these issues, the first involving the use of absolute error rather than squared error loss, and the second involving modifying the holdout sets used. For the latter, we provide conditions that permit consistent estimation of the number of change-points for a general change-point estimation procedure. We show these conditions are satisfied for optimal partitioning using new results on its performance when supplied with the incorrect number of change-points. Numerical experiments show that the absolute error approach in particular is competitive with common change-point methods using classical tuning parameter choices when error distributions are well-specified, but can substantially outperform these in misspecified models. An implementation of our methodology is available in the R package crossvalidationCP on CRAN.
This paper studies delayed stochastic algorithms for weakly convex optimization in a distributed network with workers connected to a master node. More specifically, we consider a structured stochastic weakly convex objective function which is the composition of a convex function and a smooth nonconvex function. Recently, Xu et al. 2022 showed that an inertial stochastic subgradient method converges at a rate of $\mathcal{O}(\tau/\sqrt{K})$, which suffers a significant penalty from the maximum information delay $\tau$. To alleviate this issue, we propose a new delayed stochastic prox-linear ($\texttt{DSPL}$) method in which the master performs the proximal update of the parameters and the workers only need to linearly approximate the inner smooth function. Somewhat surprisingly, we show that the delays only affect the high order term in the complexity rate and hence, are negligible after a certain number of $\texttt{DSPL}$ iterations. Moreover, to further improve the empirical performance, we propose a delayed extrapolated prox-linear ($\texttt{DSEPL}$) method which employs Polyak-type momentum to speed up the algorithm convergence. Building on the tools for analyzing $\texttt{DSPL}$, we also develop improved analysis of delayed stochastic subgradient method ($\texttt{DSGD}$). In particular, for general weakly convex problems, we show that convergence of $\texttt{DSGD}$ only depends on the expected delay.
Generative Adversarial Networks (GANs) have shown immense potential in fields far from physics, such as in text and image generation. Here we use GANs to learn a prototypical stochastic process on a lattice. By suitably adding noise to the original data we succeed in bringing both the Generator and the Discriminator loss functions close to their ideal value. However, as typical for adversarial approaches, oscillations persist. This undermines model selection and the quality of the generated trajectory. We demonstrate that a suitable multi-model procedure where stochastic trajectories are advanced at each step upon randomly selecting a Generator leads to a remarkable increase in accuracy. Based on the reported findings GANs appears as a promising tool to tackle complex statistical dynamics.
Dimensionality reduction (DR) algorithms compress high-dimensional data into a lower dimensional representation while preserving important features of the data. DR is a critical step in many analysis pipelines as it enables visualisation, noise reduction and efficient downstream processing of the data. In this work, we introduce the ProbDR variational framework, which interprets a wide range of classical DR algorithms as probabilistic inference algorithms in this framework. ProbDR encompasses PCA, CMDS, LLE, LE, MVU, diffusion maps, kPCA, Isomap, (t-)SNE, and UMAP. In our framework, a low-dimensional latent variable is used to construct a covariance, precision, or a graph Laplacian matrix, which can be used as part of a generative model for the data. Inference is done by optimizing an evidence lower bound. We demonstrate the internal consistency of our framework and show that it enables the use of probabilistic programming languages (PPLs) for DR. Additionally, we illustrate that the framework facilitates reasoning about unseen data and argue that our generative models approximate Gaussian processes (GPs) on manifolds. By providing a unified view of DR, our framework facilitates communication, reasoning about uncertainties, model composition, and extensions, particularly when domain knowledge is present.
We provide the first convergence guarantee for full black-box variational inference (BBVI), also known as Monte Carlo variational inference. While preliminary investigations worked on simplified versions of BBVI (e.g., bounded domain, bounded support, only optimizing for the scale, and such), our setup does not need any such algorithmic modifications. Our results hold for log-smooth posterior densities with and without strong log-concavity and the location-scale variational family. Also, our analysis reveals that certain algorithm design choices commonly employed in practice, particularly, nonlinear parameterizations of the scale of the variational approximation, can result in suboptimal convergence rates. Fortunately, running BBVI with proximal stochastic gradient descent fixes these limitations, and thus achieves the strongest known convergence rate guarantees. We evaluate this theoretical insight by comparing proximal SGD against other standard implementations of BBVI on large-scale Bayesian inference problems.
Scientific machine learning has become an increasingly important tool in materials science and engineering. It is particularly well suited to tackle material problems involving many variables or to allow rapid construction of surrogates of material models, to name just a few. Mathematically, many problems in materials science and engineering can be cast as variational problems. However, handling of uncertainty, ever present in materials, in the context of variational formulations remains challenging for scientific machine learning. In this article, we propose a deep-learning-based numerical method for solving variational problems under uncertainty. Our approach seamlessly combines deep-learning approximation with Monte-Carlo sampling. The resulting numerical method is powerful yet remarkably simple. We assess its performance and accuracy on a number of variational problems.
Deep kernel processes are a recently introduced class of deep Bayesian models that have the flexibility of neural networks, but work entirely with Gram matrices. They operate by alternately sampling a Gram matrix from a distribution over positive semi-definite matrices, and applying a deterministic transformation. When the distribution is chosen to be Wishart, the model is called a deep Wishart process (DWP). This particular model is of interest because its prior is equivalent to a deep Gaussian process (DGP) prior, but at the same time it is invariant to rotational symmetries, leading to a simpler posterior distribution. Practical inference in the DWP was made possible in recent work ("A variational approximate posterior for the deep Wishart process" Ober and Aitchison 2021a) where the authors used a generalisation of the Bartlett decomposition of the Wishart distribution as the variational approximate posterior. However, predictive performance in that paper was less impressive than one might expect, with the DWP only beating a DGP on a few of the UCI datasets used for comparison. In this paper, we show that further generalising their distribution to allow linear combinations of rows and columns in the Bartlett decomposition results in better predictive performance, while incurring negligible additional computation cost.
We propose two market designs for the optimal day-ahead scheduling of energy exchanges within renewable energy communities. The first one implements a cooperative demand side management scheme inside a community where members objectives are coupled through grid tariffs, whereas the second allows in addition the valuation of excess generation in the community and on the retail market. Both designs are formulated as centralized optimization problems first, and as non cooperative games then. In the latter case, the existence and efficiency of the corresponding (Generalized) Nash Equilibria are rigorously studied and proven, and distributed implementations of iterative solution algorithms for finding these equilibria are proposed, with proofs of convergence. The models are tested on a use-case made by 55 members with PV generation, storage and flexible appliances, and compared with a benchmark situation where members act individually (situation without community). We compute the global REC costs and individual bills, inefficiencies of the decentralized models compared to the centralized optima, as well as technical indices such as self-consumption ratio, self-sufficiency ratio, and peak-to-average ratio.
The Stochastic Gradient Langevin Dynamics (SGLD) are popularly used to approximate Bayesian posterior distributions in statistical learning procedures with large-scale data. As opposed to many usual Markov chain Monte Carlo (MCMC) algorithms, SGLD is not stationary with respect to the posterior distribution; two sources of error appear: The first error is introduced by an Euler--Maruyama discretisation of a Langevin diffusion process, the second error comes from the data subsampling that enables its use in large-scale data settings. In this work, we consider an idealised version of SGLD to analyse the method's pure subsampling error that we then see as a best-case error for diffusion-based subsampling MCMC methods. Indeed, we introduce and study the Stochastic Gradient Langevin Diffusion (SGLDiff), a continuous-time Markov process that follows the Langevin diffusion corresponding to a data subset and switches this data subset after exponential waiting times. There, we show that the Wasserstein distance between the posterior and the limiting distribution of SGLDiff is bounded above by a fractional power of the mean waiting time. Importantly, this fractional power does not depend on the dimension of the state space. We bring our results into context with other analyses of SGLD.
The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning. Today's prevalent random field representations are restricted to unbounded domains or are too restrictive in terms of possible field properties. As a result, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields are especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. Unlike the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of a Helmholtz-like SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis (statFEM) and Gaussian process (GP) regression on geometrically complex domains. We use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the Helmholtz-like SPDE so that we can model on bounded domains and manifolds anisotropic, non-homogeneous random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The expressions for the mean vector and the precision matrix can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191