In mathematics and computer algebra, automatic differentiation (AD) is a set of techniques to evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.), elementary functions (exp, log, sin, cos, etc.) and control flow statements. AD takes source code of a function as input and produces source code of the derived function. By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.
This paper presents AD techniques available in ROOT, supported by Cling, to produce derivatives of arbitrary C/C++ functions through implementing source code transformation and employing the chain rule of differential calculus in both forward mode and reverse mode. We explain its current integration for gradient computation in TFormula. We demonstrate the correctness and performance improvements in ROOT's fitting algorithms.
Expression templates are a well-known set of techniques for improving the efficiency of operator overloading-based forward mode automatic differentiation schemes in the C++ programming language by translating the differentiation from individual operators to whole expressions. However standard expression template approaches result in a large amount of duplicate computation, particularly for large expression trees, degrading their performance. In this paper we describe several techniques for improving the efficiency of expression templates and their implementation in the automatic differentiatio…
We give a gentle introduction to using various software tools for automatic differentiation (AD). Ready-to-use examples are discussed, and links to further information are presented. Our target audience includes all those who are looking for a straightforward way to get started using the available AD technology. The document is dynamic in the sense that its content will be updated as the AD software evolves.
We discuss the role of automatic differentiation tools in optimization software. We emphasize issues that are important to large-scale optimization and that have proved useful in the installation of nonlinear solvers in the NEOS Server. Our discussion centers on the computation of the gradient and Hessian matrix for partially separable functions and shows that the gradient and Hessian matrix can be computed with guaranteed bounds in time and memory requirements
Automatic differentiation is a set of techniques to efficiently and accurately compute the derivative of a function represented by a computer program. Existing C++ libraries for automatic differentiation (e.g. Adept, Stan Math Library), however, exhibit large memory consumptions and runtime performance issues. This paper introduces FastAD, a new C++ template library for automatic differentiation, that overcomes all of these challenges in existing libraries by using vectorization, simpler memory management using a fully expression-template-based design, and other compile-time optimizations to r…
We discuss the calculation of the derivatives of ODE systems with the automatic differentiation tool ADiMat. Using the well-known Lotka-Volterra equations and the ode23 ODE solver as examples we show the analytic derivatives and detail how to differentiate a top-level function that calls ode23 somewhere with ADiMat. This involves the manual construction of substitution function to propagate the derivatives in forward and reverse mode. We also show how to use the reverse mode code to evaluate the Hessian in forward-over-reverse mode.
Argonne National Laboratory（阿贡国家实验室）；Inria Sophia-Antipolis（法国国家信息与自动化研究所索菲亚-安提波利斯研究中心）
This paper presents our work toward correct and efficient automatic differentiation of OpenMP parallel worksharing loops in forward and reverse mode. Automatic differentiation is a method to obtain gradients of numerical programs, which are crucial in optimization, uncertainty quantification, and machine learning. The computational cost to compute gradients is a common bottleneck in practice. For applications that are parallelized for multicore CPUs or GPUs using OpenMP, one also wishes to compute the gradients in parallel. We propose a framework to reason about the correctness of the generate…
Automatic differentiation (AD) is an essential primitive for machine learning programming systems. Tangent is a new library that performs AD using source code transformation (SCT) in Python. It takes numeric functions written in a syntactic subset of Python and NumPy as input, and generates new Python functions which calculate a derivative. This approach to automatic differentiation is different from existing packages popular in machine learning, such as TensorFlow and Autograd. Advantages are that Tangent generates gradient code in Python which is readable by the user, easy to understand and …
Machine learning and neural network models in particular have been improving the state of the art performance on many artificial intelligence related tasks. Neural network models are typically implemented using frameworks that perform gradient based optimization methods to fit a model to a dataset. These frameworks use a technique of calculating derivatives called automatic differentiation (AD) which removes the burden of performing derivative calculations from the model designer. In this report we describe AD, its motivations, and different implementation approaches. We briefly describe dataf…
Automatic differentiation is a technique which allows a programmer to define a numerical computation via compositions of a broad range of numeric and computational primitives and have the underlying system support the computation of partial derivatives of the result with respect to any of its inputs, without making any finite difference approximations, and without manipulating large symbolic expressions representing the computation. This note describes a novel approach to reverse mode automatic differentiation using constraint logic programmming, specifically, the constraint handling rules (CH…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the calculation of derivatives and is preferable to more traditional methods, especially when differentiating complex algorithms and mathematical functions. The implementation of automatic differentiation however requires some care to insure efficiency. Modern differentiation packages deploy a broad range of computational techniques to improve applicability, run time, …
In this paper we compare several Python tools for automatic differentiation. In order to assess the difference in performance and precision, the problem of finding the optimal geometrical structure of the cluster with identical atoms is used as follows. First, we compare performance of calculating gradients for the objective function. We showed that the PyADOL-C and PyCppAD tools have much better performance for big clusters than the other ones. Second, we assess precision of these two tools by calculating the difference between the obtained at the optimal configuration gradient norms. We conc…
In this review we present hyper-dual numbers as a tool for the automatic differentiation of computer programs via operator overloading. We start with a motivational introduction into the ideas of algorithmic differentiation. Then we illuminate the concepts behind operator overloading and dual numbers. Afterwards, we present hyper-dual numbers (and vectors) as an extension of dual numbers for the computation of the Jacobian and the Hessian matrices of a computer program. We review a mathematical theorem that proves the correctness of the derivative information that is obtained from hyper-dual n…
Sylvester, Lyapunov, and algebraic Riccati equations are the bread and butter of control theorists. They are used to compute infinite-horizon Gramians, solve optimal control problems in continuous or discrete time, and design observers. While popular numerical computing frameworks (e.g., scipy) provide efficient solvers for these equations, these solvers are still largely missing from most automatic differentiation libraries. Here, we derive the forward and reverse-mode derivatives of the solutions to all three types of equations, and showcase their application on an inverse control problem.
We propose a universal method for the evaluation of generalized standard materials that greatly simplifies the material law implementation process. By means of automatic differentiation and a numerical integration scheme, AutoMat reduces the implementation effort to two potential functions. By moving AutoMat to the GPU, we close the performance gap to conventional evaluation routines and demonstrate in detail that the expression level reverse mode of automatic differentiation as well as its extension to second order derivatives can be applied inside CUDA kernels. We underline the effectiveness…
We present ForwardDiff, a Julia package for forward-mode automatic differentiation (AD) featuring performance competitive with low-level languages like C++. Unlike recently developed AD tools in other popular high-level languages such as Python and MATLAB, ForwardDiff takes advantage of just-in-time (JIT) compilation to transparently recompile AD-unaware user code, enabling efficient support for higher-order differentiation and differentiation using custom number types (including complex numbers). For gradient and Jacobian calculations, ForwardDiff provides a variant of vector-forward mode tha…
Most numerical solvers used to determine free variables of dynamical systems rely on first-order derivatives of the state of the system w.r.t. the free variables. The number of the free variables can be fairly large. One of the approaches of obtaining those derivatives is the integration of the derivatives simultaneously with the dynamical equations, which is best done with the automatic differentiation technique. Even though there exist many automatic differentiation tools, none have been found to be scalable and usable for practical purposes of dynamic systems modeling. Landau is a Turing in…
We show how forward-mode automatic differentiation (AD) can be employed within larger reverse-mode computations to dynamically differentiate broadcast operations in a GPU-friendly manner. Our technique fully exploits the broadcast Jacobian's inherent sparsity structure, and unlike a pure reverse-mode approach, this "mixed-mode" approach does not require a backwards pass over the broadcasted operation's subgraph, obviating the need for several reverse-mode-specific programmability restrictions on user-authored broadcast operations. Most notably, this approach allows broadcast fusion in primal c…
We present the results of our analysis of publication venues for papers on automatic differentiation (AD), covering academic journals and conference proceedings. Our data are collected from the AD publications database maintained by the autodiff.org community website. The database is purpose-built for the AD field and is expanding via submissions by AD researchers. Therefore, it provides a relatively noise-free list of publications relating to the field. However, it does include noise in the form of variant spellings of journal and conference names. We handle this by manually correcting and me…
SnadiOpt is a package that supports the use of the automatic differentiation package ADIFOR with the optimization package Snopt. Snopt is a general-purpose system for solving optimization problems with many variables and constraints. It minimizes a linear or nonlinear function subject to bounds on the variables and sparse linear or nonlinear constraints. It is suitable for large-scale linear and quadratic programming and for linearly constrained optimization, as well as for general nonlinear programs. The method used by Snopt requires the first derivatives of the objective and constraint funct…
LVMT, ENPC, Institut Polytechnique de Paris, Univ Gustave Eiffel（巴黎高等路桥学院，巴黎理工学院，古斯塔夫·埃菲尔大学，LVMT实验室）；BIFOLD – Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所（BIFOLD））；Machine Learning Group, Technical University of Berlin（柏林工业大学机器学习组）
A critical step in topology optimization (TO) is finding sensitivities. Manual derivation and implementation of the sensitivities can be quite laborious and error-prone, especially for non-trivial objectives, constraints and material models. An alternate approach is to utilize automatic differentiation (AD). While AD has been around for decades, and has also been applied in TO, wider adoption has largely been absent. In this educational paper, we aim to reintroduce AD for TO, and make it easily accessible through illustrative codes. In particular, we employ JAX, a high-performance Python libra…
In this paper we introduce DiffSharp, an automatic differentiation (AD) library designed with machine learning in mind. AD is a family of techniques that evaluate derivatives at machine precision with only a small constant factor of overhead, by systematically applying the chain rule of calculus at the elementary operator level. DiffSharp aims to make an extensive array of AD techniques available, in convenient form, to the machine learning community. These including arbitrary nesting of forward/reverse AD operations, AD with linear algebra primitives, and a functional API that emphasizes the …
Sparser, Better, Faster, Stronger: Sparsity Detection for Efficient Automatic Differentiation
BIFOLD – Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）；Technical University of Berlin（柏林工业大学）；ENPC（巴黎高科路桥学校）；Institut Polytechnique de Paris（巴黎综合理工学院）；Univ Gustave Eiffel（古斯塔夫·埃菲尔大学）
The ocean general circulation model OPA is developed by the LODYC team at Paris VI university. OPA has recently undergone a major rewriting, migrating to FORTRAN95, and its adjoint code needs to be rebuilt. For earlier versions, the adjoint of OPA was written by hand at a high development cost. We use the Automatic Differentiation tool TAPENADE to build mechanicaly the tangent and adjoint codes of OPA. We validate the differentiated codes by comparison with divided differences, and also with an identical twin experiment. We apply state-of-the-art methods to improve the performance of the adjoi…
As computational challenges in optimization and statistical inference grow ever harder, algorithms that utilize derivatives are becoming increasingly more important. The implementation of the derivatives that make these algorithms so powerful, however, is a substantial user burden and the practicality of these algorithms depends critically on tools like automatic differentiation that remove the implementation burden entirely. The Stan Math Library is a C++, reverse-mode automatic differentiation library designed to be usable, extensive and extensible, efficient, scalable, stable, portable, and…
ADF95 is a tool to automatically calculate numerical first derivatives for any mathematical expression as a function of user defined independent variables. Accuracy of derivatives is achieved within machine precision. ADF95 may be applied to any FORTRAN 77/90/95 conforming code and requires minimal changes by the user. It provides a new derived data type that holds the value and derivatives and applies forward differencing by overloading all FORTRAN operators and intrinsic functions. An efficient indexing technique leads to a reduced memory usage and a substantially increased performance gain …
We present the new software OpDiLib, a universal add-on for classical operator overloading AD tools that enables the automatic differentiation (AD) of OpenMP parallelized code. With it, we establish support for OpenMP features in a reverse mode operator overloading AD tool to an extent that was previously only reported on in source transformation tools. We achieve this with an event-based implementation ansatz that is unprecedented in AD. Combined with modern OpenMP features around OMPT, we demonstrate how it can be used to achieve differentiation without any additional modifications of the so…
Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院马尔丘克计算数学研究所）；HSE University（高等经济学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）
In scientific computing and machine learning applications, matrices and more general multidimensional arrays (tensors) can often be approximated with the help of low-rank decompositions. Since matrices and tensors of fixed rank form smooth Riemannian manifolds, one of the popular tools for finding low-rank approximations is to use Riemannian optimization. Nevertheless, efficient implementation of Riemannian gradients and Hessians, required in Riemannian optimization algorithms, can be a nontrivial task in practice. Moreover, in some cases, analytic formulas are not even available. In this pape…
A C++ implementation of the discrete adjoint sensitivity analysis method for explicit adaptive Runge-Kutta methods enabled by automatic adjoint differentiation and SIMD vectorization
本文实现了一个面向自适应显式 Runge-Kutta 方法的 C++ 离散伴随灵敏度分析库，结合 AAD 与 SIMD 并行计算向量-Jacobian 乘积，性能略优于现有同类软件。
BIFOLD – Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）；LVMT, ENPC, Institut Polytechnique de Paris, Univ Gustave Eiffel（法国国立路桥学校，巴黎理工学院，古斯塔夫·埃菲尔大学）；Machine Learning Group, Technical University of Berlin（柏林工业大学机器学习组）
We propose an algorithm to compute the $C^\infty$-ring structure of arbitrary Weil algebra. It allows us to do some analysis with higher infinitesimals numerically and symbolically. To that end, we first give a brief description of the (Forward-mode) automatic differentiation (AD) in terms of $C^\infty$-rings. The notion of a $C^\infty$-ring was introduced by Lawvere and used as the fundamental building block of smooth infinitesimal analysis and synthetic differential geometry. We argue that interpreting AD in terms of $C^\infty$-rings gives us a unifying theoretical framework and modular ways…
