In this article, we describe an automatic differentiation module of PyTorch — a library designed to enable rapid research on machine learning models. It builds upon a few projects, most notably Lua Torch, Chainer, and HIPS Autograd [4], and provides a high performance environment with easy access to automatic differentiation of models executed on different devices (CPU and GPU). To make prototyping easier, PyTorch does not follow the symbolic approach used in many other deep learning frameworks, but focuses on differentiation of purely imperative programs, with a focus on extensibility and low overhead. Note that this preprint is a draft of certain sections from an upcoming paper covering all PyTorch features.

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Automatic differentiation in PyTorch

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

The brms package implements Bayesian multilevel models in R using the probabilistic programming language Stan. A wide range of distributions and link functions are supported, allowing users to fit - among others - linear, robust linear, binomial, Poisson, survival, ordinal, zero-inflated, hurdle, and even non-linear models all in a multilevel context. Further modeling options include autocorrelation of the response variable, user defined covariance structures, censored data, as well as meta-analytic standard errors. Prior specifications are flexible and explicitly encourage users to apply prior distributions that actually reflect their beliefs. In addition, model fit can easily be assessed and compared with the Watanabe-Akaike information criterion and leave-one-out cross-validation.

brms: An R Package for Bayesian Multilevel Models Using Stan

Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.

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Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

We consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models. Such models, implemented as computer codes, are often generic in the sense that by a suitable choice of some of the model's input parameters the code can be used to predict the behaviour of the system in a variety of specific applications. However, in any specific application the values of necessary parameters may be unknown. In this case, physical observations of the system in the specific context are used to learn about the unknown parameters. The process of fitting the model to the observed data by adjusting the parameters is known as calibration. Calibration is typically effected by ad hoc fitting, and after calibration the model is used, with the fitted input values, to predict the future behaviour of the system. We present a Bayesian calibration technique which improves on this traditional approach in two respects. First, the predictions allow for all sources of uncertainty, including the remaining uncertainty over the fitted parameters. Second, they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best-fitting parameter values. The method is illustrated by using data from a nuclear radiation release at Tomsk, and from a more complex simulated nuclear accident exercise.

https://www.jstor.org/stable/pdfplus/2680584.pdf

Bayesian Calibration of computer models

This article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure. The natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation. However, with the approach presented in this article, much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate Taylor series of a suitable degree. It is applicable for arbitrary orders of derivatives. Also it is possible to calculate derivatives only in some directions instead of the full derivative tensor. Explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given.

/pdf/evaluating-higher-derivative-tensors-by-forward-propagation-eqdvlfcz6c.pdf

Evaluating higher derivative tensors by forward propagation of univariate Taylor series

This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results.

Automatic differentiation of explicit Runge-Kutta methods for optimal control

The computation of a sparse Hessian matrix H using automatic differentiation (AD) can be made efficient using the following four-step procedure: (1) Determine the sparsity structure of H, (2) obtain a seed matrix S that defines a column partition of H using a specialized coloring on the adjacency graph of H, (3) compute the compressed Hessian matrix B ≡ HS, and (4) recover the numerical values of the entries of H from B.

The coloring variant used in the second step depends on whether the recovery in the fourth step is direct or indirect: a direct method uses star coloring and an indirect method uses acyclic coloring. In an earlier work, we had designed and implemented effective heuristic algorithms for these two NP-hard coloring problems. Recently, we integrated part of the developed software with the AD tool ADOL-C, which has recently acquired a sparsity detection capability. In this paper, we provide a detailed description and analysis of the recovery algorithms and experimentally demonstrate the efficacy of the coloring techniques in the overall process of computing the Hessian of a given function using ADOL-C as an example of an AD tool. We also present new analytical results on star and acyclic coloring of chordal graphs. The experimental results show that sparsity exploitation via coloring yields enormous savings in runtime and makes the computation of Hessians of very large size feasible. The results also show that evaluating a Hessian via an indirect method is often faster than a direct evaluation. This speedup is achieved without compromising numerical accuracy.

Efficient Computation of Sparse Hessians Using Coloring and Automatic Differentiation

The proceedings represent the state of knowledge in the area of algorithmic differentiation (AD). The 31 contributed papers presented at the AD2012 conference cover the application of AD to many areas in science and engineering as well as aspects of AD theory and its implementation in tools. For all papers the referees, selected from the program committee and the greater community, as well as the editors have emphasized accessibility of the presented ideas also to non-AD experts. In the AD tools arena new implementations are introduced covering, for example, Java and graphical modeling environments or join the set of existing tools for Fortran. New developments in AD algorithms target the efficiency of matrix-operation derivatives, detection and exploitation of sparsity, partial separability, the treatment of nonsmooth functions, and other high-level mathematical aspects of the numerical computations to be differentiated. Applications stem from the Earth sciences, nuclear engineering, fluid dynamics, and chemistry, to name just a few. In many cases the applications in a given area of science or engineering share characteristics that require specific approaches to enable AD capabilities or provide an opportunity for efficiency gains in the derivative computation. The description of these characteristics and of the techniques for successfully using AD should make the proceedings a valuable source of information for users of AD tools.

Recent Advances in Algorithmic Differentiation

The computation of derivatives for optimizing time-dependent flow problems is often based on the integration of the adjoint differential equation. For this purpose, the knowledge of the complete forward solution is required. In the area of flow control, especially for three-dimensional problems, it may be impossible to keep track of the full forward solution due to the lack of storage capacities. One usual method to overcome this problem is checkpointing. Using a checkpointing strategy, only parts of the forward solution are kept in the main memory and additional time steps have to be performed. If one extends this approach such that some checkpoints can be stored on disc too, one can reduce the number of additional time steps. On the other hand, one has to take the access cost to one checkpoint into account. On parallel machines, one may use parallel I/O facilities to store and retrieve checkpoints. In these cases, the access cost of the checkpoints is also no longer negligible. Therefore, in this paper, the write and read counts for each checkpoint in a binomial checkpointing approach are examined. This way, the overall access cost to checkpoints is minimized. Numerical results illustrate the derived checkpointing approaches. They also show that checkpointing techniques may reduce the overall computing time despite the required recalculations.

Andrea Walther

Papers

Evaluating higher derivative tensors by forward propagation of univariate Taylor series

Automatic differentiation of explicit Runge-Kutta methods for optimal control

Efficient Computation of Sparse Hessians Using Coloring and Automatic Differentiation

Recent Advances in Algorithmic Differentiation

MultiStage Approaches for Optimal Offline Checkpointing