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

The numerical solution of nonlinear equation systems is often achieved by so-called quasi-Newton methods. They preserve the rapid local convergence of Newton’s method at a significantly reduced cost per step by successively approximating the system Jacobian though low-rank updates. We analyze two variants of the recently proposed adjoint Broyden update, which for the first time combines the classical least change property with heredity on affine systems. However, the new update does require, the evaluation of so-called adjoint vectors, namely products of the transposed Jacobian with certain dual direction vectors. The resulting quasi-Newton method is linear contravariant in the sense of Deuflhard (Newton methods for nonlinear equations. Springer, Heidelberg, 2006) and it is shown here to be locally and q-superlinearly convergent. Our numerical results on a range of test problems demonstrate that the new method usually outperforms Newton’s and Broyden’s method in terms of runtime and iterations count, respectively.

On the local convergence of adjoint Broyden methods

This article presents a new area of application for Automatic Differentiation (AD): Computing parametric sensitivities for optimization problems. For an optimization problem containing parameters which are not among the optimization variables, the term parametric sensitivity refers to the derivative of an optimal solution with respect to the parameters. We treat non-linear finite- and infinite-dimensional optimization problems, in particular optimal control problems involving ordinary differential equations with control and state constraints, and compute their parametric sensitivities using AD. Particular attention is given to the generation of second-order derivatives required in the process. Copyright © 2003 John Wiley & Sons, Ltd.

Parametric sensitivities for optimal control problems using automatic differentiation

In contrast to integration, the differentiation of a function is an ill-conditioned process, if only an oracle is available for its pointwise evaluation. That is, unrelated small variations in the value of the composite function are allowed at nearly identical arguments. In contrast, we show here that, if the function is defined by an evaluation procedure as a composition of arithmetic operations and elementary functions, then automatic, or algorithmic differentiation is backward stable in the sense of Wilkinson. More specifically, the derivative values obtained are exact for a perturbation of the elementary components at the level of the machine precision. We also provide a forward error analysis for both the forward and reverse mode. The theoretical analysis is confirmed by numerical experiments.

On the numerical stability of algorithmic differentiation

Computer-aided design (CAD) tools are extensively used to design industrial components, however, contrary to e.g. computational fluid dynamics (CFD) solvers, shape sensitivities for gradient-based ...

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Algorithmic differentiation of the Open CASCADE Technology CAD kernel and its coupling with an adjoint CFD solver

Using the basic reverse mode of automatic differentiation, the memory needed for the computation of discrete adjoints is proportional to the number of operations performed. This behavior is frequently not acceptable, especially for large-scale problems that involve a kind of time-stepping procedure. Therefore, we integrate a checkpointing mechanism into ADOL-C, a tool for the automatic differentiation of C and C++ programs. This checkpointing procedure is optimal for a given number of checkpoints in the sense that it yields the minimal number of recomputations. The resulting effects on the run-time behavior are illustrated by means of the derivative computation for an ODE-based optimization problem.

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Andrea Walther

Papers

On the local convergence of adjoint Broyden methods

Parametric sensitivities for optimal control problems using automatic differentiation

On the numerical stability of algorithmic differentiation

Algorithmic differentiation of the Open CASCADE Technology CAD kernel and its coupling with an adjoint CFD solver

Optimal checkpointing for time-stepping procedures in ADOL-C