For a real function, automatic differentiation is such a standard algorithm used to efficiently compute its gradient, that it is integrated in various neural network frameworks. However, despite the recent advances in using complex functions in machine learning and the well-established usefulness of automatic differentiation, the support of automatic differentiation for complex functions is not as well-established and widespread as for real functions. In this work we propose an efficient and seamless scheme to implement automatic differentiation for complex functions, which is a compatible generalization of the current scheme for real functions. This scheme can significantly simplify the implementation of neural networks which use complex numbers.
We show that reverse mode automatic differentiation and symbolic differentiation are equivalent in the sense that they both perform the same operations when computing derivatives. This is in stark contrast to the common claim that they are substantially different. The difference is often illustrated by claiming that symbolic differentiation suffers from "expression swell" whereas automatic differentiation does not. Here, we show that this statement is not true. "Expression swell" refers to the phenomenon of a much larger representation of the derivative as opposed to the representation of the …
Shanghai Jiao Tong University（上海交通大学）；Shanghai Qi Zhi Institute（上海期智研究院）；Alibaba Group（阿里巴巴集团）；Microsoft Research（微软研究院）；Institute for AI Industry Research (AIR), Tsinghua University（清华大学人工智能产业研究院（AIR））
An activation function is an element-wise mathematical function and plays a crucial role in deep neural networks (DNN). Many novel and sophisticated activation functions have been proposed to improve the DNN accuracy but also consume massive memory in the training process with back-propagation. In this study, we propose the nested forward automatic differentiation (Forward-AD), specifically for the element-wise activation function for memory-efficient DNN training. We deploy nested Forward-AD in two widely-used deep learning frameworks, TensorFlow and PyTorch, which support the static and dyna…
National University of Ireland Maynooth（爱尔兰国立大学梅努斯分校）
Automatic differentiation---the mechanical transformation of numeric computer programs to calculate derivatives efficiently and accurately---dates to the origin of the computer age. Reverse mode automatic differentiation both antedates and generalizes the method of backwards propagation of errors used in machine learning. Despite this, practitioners in a variety of fields, including machine learning, have been little influenced by automatic differentiation, and make scant use of available tools. Here we review the technique of automatic differentiation, describe its two main modes, and explain…
Automatic differentiation (AD) is an important family of algorithms which enables derivative based optimization. We show that AD can be simply implemented with effects and handlers by doing so in the Frank language. By considering how our implementation behaves in Frank's operational semantics, we show how our code performs the dynamic creation of programs during evaluation.
Toulouse School of Economics（图卢兹经济学院）；Univ. Toulouse（图卢兹大学）；IRIT, CNRS（法国国家科学研究中心信息与自动化研究所）
Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning. In this work we articulate the relationships between differentiation of programs as implemented in practice and differentiation of nonsmooth functions. To this end we provide a simple class of functions, a nonsmooth calculus, and show how they apply to stochastic approximation methods. We also evidence the issue of artificial critical points created by algorithmic differentiation and show how usual methods avoid these points with probability one.
We decompose reverse-mode automatic differentiation into (forward-mode) linearization followed by transposition. Doing so isolates the essential difference between forward- and reverse-mode AD, and simplifies their joint implementation. In particular, once forward-mode AD rules are defined for every primitive operation in a source language, only linear primitives require an additional transposition rule in order to arrive at a complete reverse-mode AD implementation. This is how reverse-mode AD is written in JAX and Dex.
We present the classical coordinate-free formalism for forward and backward mode ad in the real and complex setting. We show how to formally derive the forward and backward formulae for a number of matrix functions starting from basic principles.
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…
Huawei Research Centre（华为研究中心）；University of Birmingham（伯明翰大学）；University College London（伦敦大学学院）
We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness. To give an efficient yet principled implementation of the AD algorithm we define a sound and complete representation of hierarchical string diagrams as a class of hierarchical hypergraphs we call hypernets.
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Mount Holyoke College（曼荷莲学院）
We develop nested automatic differentiation (AD) algorithms for exact inference and learning in integer latent variable models. Recently, Winner, Sujono, and Sheldon showed how to reduce marginalization in a class of integer latent variable models to evaluating a probability generating function which contains many levels of nested high-order derivatives. We contribute faster and more stable AD algorithms for this challenging problem and a novel algorithm to compute exact gradients for learning. These contributions lead to significantly faster and more accurate learning algorithms, and are the …
In this note, we report the back propagation formula for complex valued singular value decompositions (SVD). This formula is an important ingredient for a complete automatic differentiation(AD) infrastructure in terms of complex numbers, and it is also the key to understand and utilize AD in tensor networks.
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.
Imperial College London（帝国理工学院）；Argonne National Laboratory（阿贡国家实验室）
Stencil loops are a common motif in computations including convolutional neural networks, structured-mesh solvers for partial differential equations, and image processing. Stencil loops are easy to parallelise, and their fast execution is aided by compilers, libraries, and domain-specific languages. Reverse-mode automatic differentiation, also known as algorithmic differentiation, autodiff, adjoint differentiation, or back-propagation, is sometimes used to obtain gradients of programs that contain stencil loops. Unfortunately, conventional automatic differentiation results in a memory access p…
Banded matrices can be used as precision matrices in several models including linear state-space models, some Gaussian processes, and Gaussian Markov random fields. The aim of the paper is to make modern inference methods (such as variational inference or gradient-based sampling) available for Gaussian models with banded precision. We show that this can efficiently be achieved by equipping an automatic differentiation framework, such as TensorFlow or PyTorch, with some linear algebra operators dedicated to banded matrices. This paper studies the algorithmic aspects of the required operators, d…
Technical University of Munich（慕尼黑工业大学）；Imperial College London（帝国理工学院）；University of Oxford（牛津大学）；OpenMined；IIIT Hyderabad（海得拉巴国际信息技术学院）
We introduce Tritium, an automatic differentiation-based sensitivity analysis framework for differentially private (DP) machine learning (ML). Optimal noise calibration in this setting requires efficient Jacobian matrix computations and tight bounds on the L2-sensitivity. Our framework achieves these objectives by relying on a functional analysis-based method for sensitivity tracking, which we briefly outline. This approach interoperates naturally and seamlessly with static graph-based automatic differentiation, which enables order-of-magnitude improvements in compilation times compared to pre…
The need to efficiently calculate first- and higher-order derivatives of increasingly complex models expressed in Python has stressed or exceeded the capabilities of available tools. In this work, we explore techniques from the field of automatic differentiation (AD) that can give researchers expressive power, performance and strong usability. These include source-code transformation (SCT), flexible gradient surgery, efficient in-place array operations, higher-order derivatives as well as mixing of forward and reverse mode AD. We implement and demonstrate these ideas in the Tangent software li…
Toulouse School of Economics, University of Toulouse Capitole（图卢兹经济学院，图卢兹第一大学）；IRIT, CNRS, Université de Toulouse, ANITI（图卢兹信息学研究所，法国国家科学研究中心，图卢兹大学，图卢兹人工智能与自然智能研究所）
Differentiation along algorithms, i.e., piggyback propagation of derivatives, is now routinely used to differentiate iterative solvers in differentiable programming. Asymptotics is well understood for many smooth problems but the nondifferentiable case is hardly considered. Is there a limiting object for nonsmooth piggyback automatic differentiation (AD)? Does it have any variational meaning and can it be used effectively in machine learning? Is there a connection with classical derivative? All these questions are addressed under appropriate nonexpansivity conditions in the framework of conser…
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）
Automatic differentiation plays a prominent role in scientific computing and in modern machine learning, often in the context of powerful programming systems. The relation of the various embodiments of automatic differentiation to the mathematical notion of derivative is not always entirely clear---discrepancies can arise, sometimes inadvertently. In order to study automatic differentiation in such programming contexts, we define a small but expressive programming language that includes a construct for reverse-mode differentiation. We give operational and denotational semantics for this langua…
The successes of deep learning, variational inference, and many other fields have been aided by specialized implementations of reverse-mode automatic differentiation (AD) to compute gradients of mega-dimensional objectives. The AD techniques underlying these tools were designed to compute exact gradients to numerical precision, but modern machine learning models are almost always trained with stochastic gradient descent. Why spend computation and memory on exact (minibatch) gradients only to use them for stochastic optimization? We develop a general framework and approach for randomized automa…
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）
Toulouse School of Economics（图卢兹经济学院）；Université Toulouse 1 Capitole（图卢兹第一大学）；IRIT（图卢兹计算机研究所）；Université de Toulouse（图卢兹大学）；IRT Saint Exupery（圣埃克苏佩里技术研究院）
Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible calculus. We introduce generalized derivatives called conservative fields for which we develop a calculus and provide representation formulas. Functions having a conservative field are called path differentiable: convex, concave, Clarke regular and any semialgebraic Lipschitz continuous functions are path differentiable. Using Whitney stratification techniques for semialgebraic and definable sets, our model provides variational formulas for nonsmooth automatic differentiation oracles, as for instance th…
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…
University of Oxford（牛津大学）；National University of Ireland Maynooth（爱尔兰国立大学梅努斯分校）；Massachusetts Institute of Technology（麻省理工学院）；Purdue University（普渡大学）
Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. Automatic differentiation (AD), also called algorithmic differentiation or simply "autodiff", is a family of techniques similar to but more general than backpropagation for efficiently and accurately evaluating derivatives of numeric functions expressed as computer programs. AD is a small but established field with applications in areas including computational fluid dynamics, atmospheric sciences, and engineering design optimization. Until very recently, the fields of machine learning and AD have lar…
CASUS - Center for Advanced System Understanding（先进系统理解中心）；Helmholtz-Zentrum Dresden-Rossendorf e.V. (HZDR)（亥姆霍兹德累斯顿罗森多夫研究中心）
We present a novel class of approximations for variational losses, being applicable for the training of physics-informed neural nets (PINNs). The loss formulation reflects classic Sobolev space theory for partial differential equations and their weak formulations. The loss computation rests on an extension of Gauss-Legendre cubatures, we term Sobolev cubatures, replacing automatic differentiation (A.D.). We prove the runtime complexity of training the resulting Soblev-PINNs (SC-PINNs) to be less than required by PINNs relying on A.D. On top of one-to-two order of magnitude speed-up the SC-PINN…
Frankfurt Institute for Advanced Studies（法兰克福高等研究院）；Xidian-FIAS International Joint Research Center（西电-法兰克福高等研究院国际联合研究中心）；McGill University（麦吉尔大学）；Stony Brook University（石溪大学）
Reconstructing spectral functions from Euclidean Green's functions is an important inverse problem in physics. The prior knowledge for specific physical systems routinely offers essential regularization schemes for solving the ill-posed problem approximately. Aiming at this point, we propose an automatic differentiation framework as a generic tool for the reconstruction from observable data. We represent the spectra by neural networks and set chi-square as loss function to optimize the parameters with backward automatic differentiation unsupervisedly. In the training process, there is no expli…
In min-min optimization or max-min optimization, one has to compute the gradient of a function defined as a minimum. In most cases, the minimum has no closed-form, and an approximation is obtained via an iterative algorithm. There are two usual ways of estimating the gradient of the function: using either an analytic formula obtained by assuming exactness of the approximation, or automatic differentiation through the algorithm. In this paper, we study the asymptotic error made by these estimators as a function of the optimization error. We find that the error of the automatic estimator is clos…
National University of Singapore（新加坡国立大学）；Tsinghua University（清华大学）
The performance of deep neural networks is well-known to be sensitive to the setting of their hyperparameters. Recent advances in reverse-mode automatic differentiation allow for optimizing hyperparameters with gradients. The standard way of computing these gradients involves a forward and backward pass of computations. However, the backward pass usually needs to consume unaffordable memory to store all the intermediate variables to exactly reverse the forward training procedure. In this work we propose a simple but effective method, DrMAD, to distill the knowledge of the forward pass into a s…
Data Science Institute（数据科学研究所）；Columbia University（哥伦比亚大学）；Princeton University（普林斯顿大学）
Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient …
We review the current state of automatic differentiation (AD) for array programming in machine learning (ML), including the different approaches such as operator overloading (OO) and source transformation (ST) used for AD, graph-based intermediate representations for programs, and source languages. Based on these insights, we introduce a new graph-based intermediate representation (IR) which specifically aims to efficiently support fully-general AD for array programming. Unlike existing dataflow programming representations in ML frameworks, our IR naturally supports function calls, higher-orde…
Modelers use automatic differentiation (AD) of computation graphs to implement complex Deep Learning models without defining gradient computations. Stochastic AD extends AD to stochastic computation graphs with sampling steps, which arise when modelers handle the intractable expectations common in Reinforcement Learning and Variational Inference. However, current methods for stochastic AD are limited: They are either only applicable to continuous random variables and differentiable functions, or can only use simple but high variance score-function estimators. To overcome these limitations, we …
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…
We study two practically important cases of model based clustering using Gaussian Mixture Models: (1) when there is misspecification and (2) on high dimensional data, in the light of recent advances in Gradient Descent (GD) based optimization using Automatic Differentiation (AD). Our simulation studies show that EM has better clustering performance, measured by Adjusted Rand Index, compared to GD in cases of misspecification, whereas on high dimensional data GD outperforms EM. We observe that both with EM and GD there are many solutions with high likelihood but poor cluster interpretation. To …
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）
DIKU, U. Copenhagen（哥本哈根大学DIKU）；U. Edinburgh（爱丁堡大学）
We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars, vectors, matrices or tensors represented as multi-dimensional arrays. We show that both symbolic and automatic differentiation can be performed using a differential calculus for generating linear functions representing Fréchet derivatives based on rules for primitive, constant, linear and bilinear functions as well as their sequential and parallel composition. Lin…
Tampere University（坦佩雷大学）；HIAB（HIAB公司）；Aalto University（阿尔托大学）；TU Darmstadt（达姆施塔特工业大学）
The ability of Gaussian processes (GPs) to predict the behavior of dynamical systems as a more sample-efficient alternative to parametric models seems promising for real-world robotics research. However, the computational complexity of GPs has made policy search a highly time and memory consuming process that has not been able to scale to larger problems. In this work, we develop a policy optimization method by leveraging fast predictive sampling methods to process batches of trajectories in every forward pass, and compute gradient updates over policy parameters by automatic differentiation of…
