From 48bfb84b913d3f958351a8cbdbe3e3f4447bdff1 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 1 Dec 2023 15:21:57 +0100 Subject: [PATCH] added articlels --- doc/Articles/Autodiff/17-468.bbl | 1475 ++++++++ doc/Articles/Autodiff/17-468.tex | 694 ++++ .../{AutodiffArticle => Autodiff/adiff.tar} | Bin .../figures/approx-error/approx-error.tex | 2189 +++++++++++ .../backprop/backprop-eps-converted-to.pdf | Bin 0 -> 11125 bytes .../Autodiff/figures/backprop/backprop.eps | 650 ++++ .../figures/comp-graph/comp-graph.tex | 43 + .../differentiation/differentiation.tex | 79 + .../Autodiff/figures/helmholtz/helmholtz.tex | 598 +++ doc/Articles/Autodiff/jmlr2e_mod.sty | 353 ++ doc/Articles/CNN.pdf | Bin 0 -> 58577 bytes doc/Articles/Fig1.pdf | Bin 0 -> 527629 bytes doc/Articles/bibliography.bib | 3268 +++++++++++++++++ doc/Articles/fig_qst_beryllium.pdf | Bin 0 -> 362862 bytes doc/Articles/fig_qst_rydberg.pdf | Bin 0 -> 362847 bytes doc/Articles/fig_supervised.pdf | Bin 0 -> 157934 bytes doc/Articles/fig_vmc.pdf | Bin 0 -> 326313 bytes doc/Articles/main.bbl | 2688 ++++++++++++++ doc/Articles/main.tex | 734 ++++ doc/Articles/mlquantum.tar | Bin 0 -> 4362240 bytes doc/Articles/rnn.pdf | Bin 0 -> 2124405 bytes 21 files changed, 12771 insertions(+) create mode 100644 doc/Articles/Autodiff/17-468.bbl create mode 100644 doc/Articles/Autodiff/17-468.tex rename doc/Articles/{AutodiffArticle => Autodiff/adiff.tar} (100%) create mode 100644 doc/Articles/Autodiff/figures/approx-error/approx-error.tex create mode 100644 doc/Articles/Autodiff/figures/backprop/backprop-eps-converted-to.pdf create mode 100644 doc/Articles/Autodiff/figures/backprop/backprop.eps create mode 100644 doc/Articles/Autodiff/figures/comp-graph/comp-graph.tex create mode 100644 doc/Articles/Autodiff/figures/differentiation/differentiation.tex create mode 100644 doc/Articles/Autodiff/figures/helmholtz/helmholtz.tex create mode 100644 doc/Articles/Autodiff/jmlr2e_mod.sty create mode 100644 doc/Articles/CNN.pdf create mode 100644 doc/Articles/Fig1.pdf create mode 100644 doc/Articles/bibliography.bib create mode 100644 doc/Articles/fig_qst_beryllium.pdf create mode 100644 doc/Articles/fig_qst_rydberg.pdf create mode 100644 doc/Articles/fig_supervised.pdf create mode 100644 doc/Articles/fig_vmc.pdf create mode 100644 doc/Articles/main.bbl create mode 100644 doc/Articles/main.tex create mode 100644 doc/Articles/mlquantum.tar create mode 100644 doc/Articles/rnn.pdf diff --git a/doc/Articles/Autodiff/17-468.bbl b/doc/Articles/Autodiff/17-468.bbl new file mode 100644 index 000000000..c63d3c847 --- /dev/null +++ b/doc/Articles/Autodiff/17-468.bbl @@ -0,0 +1,1475 @@ +\begin{thebibliography}{208} +\providecommand{\natexlab}[1]{#1} +\providecommand{\url}[1]{\texttt{#1}} +\expandafter\ifx\csname urlstyle\endcsname\relax + \providecommand{\doi}[1]{doi: #1}\else + \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi + +\bibitem[Abadi et~al.(2016)Abadi, Agarwal, Barham, Brevdo, Chen, Citro, + Corrado, Davis, Dean, Devin, et~al.]{abadi2016tensorflow} +Mart{\'\i}n Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, + Craig Citro, Greg~S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et~al. +\newblock {TensorFlow}: Large-scale machine learning on heterogeneous + distributed systems. +\newblock \emph{arXiv preprint arXiv:1603.04467}, 2016. + +\bibitem[Adamson and Winant(1969)]{Adamson1969} +D.~S. 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Byrd, Peihuang Lu, and Jorge Nocedal. +\newblock Algorithm 778: {L-BFGS-B}: {Fortran} subroutines for large-scale + bound-constrained optimization. +\newblock \emph{ACM Transactions on Mathematical Software (TOMS)}, 23\penalty0 + (4):\penalty0 550--60, 1997. +\newblock \doi{10.1145/279232.279236}. + +\end{thebibliography} diff --git a/doc/Articles/Autodiff/17-468.tex b/doc/Articles/Autodiff/17-468.tex new file mode 100644 index 000000000..6a62db0d2 --- /dev/null +++ b/doc/Articles/Autodiff/17-468.tex @@ -0,0 +1,694 @@ +\documentclass[twoside,11pt]{article} + +% Any additional packages needed should be included after jmlr2e. +% Note that jmlr2e.sty includes epsfig, amssymb, natbib and graphicx, +% and defines many common macros, such as 'proof' and 'example'. +% +% It also sets the bibliographystyle to plainnat; for more information on +% natbib citation styles, see the natbib documentation, a copy of which +% is archived at http://www.jmlr.org/format/natbib.pdf + +\PassOptionsToPackage{hyphens}{url} +\usepackage{jmlr2e_mod} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage[inline]{enumitem} +\usepackage{paralist} +\usepackage{adjustbox} +\usepackage{tikz} +\usetikzlibrary{decorations.pathmorphing} +\usetikzlibrary{arrows} +\usetikzlibrary{positioning} +\usepackage{standalone} +\usepackage{rotating} +\usepackage{tabularx} +\usepackage{booktabs} +\usepackage{multirow} +\usepackage{todonotes} +\usepackage{microtype} +\hyphenation{DiffSharp} +\hyphenation{TensorFlow} +\hyphenation{auto-diff} +\hyphenation{ADIFOR} + +% Definitions of handy macros can go here +\newcommand{\dataset}{{\cal D}} +\newcommand{\fracpartial}[2]{\frac{\partial #1}{\partial #2}} +\DeclareMathOperator*{\argmin}{arg\,min} +\newcommand{\Del}{\mathrm{\Delta}} +\newcommand{\RedMark}{{\color[rgb]{1,0,0}$X$}} +\newcommand{\GreenMark}{{\color[rgb]{0,0.69,0}$Y$}} +\newcommand{\BlueMark}{{\color[rgb]{0,0,1}$Z$}} + +% Heading arguments are {volume}{year}{pages}{date submitted}{date published}{paper id}{author-full-names} +% \jmlrheading{1}{2000}{1-48}{4/00}{10/00}{meila00a}{Marina Meil\u{a} and Michael I. Jordan} + +\jmlrheading{18}{2018}{1-43}{8/17}{2/18}{17-468}{Atılım Güneş Baydin, Barak~A.~Pearlmutter, Alexey~Andreyevich~Radul, and Jeffrey~Mark~Siskind} + +% Short headings should be running head and authors last names + +\ShortHeadings{Automatic Differentiation in Machine Learning: a Survey}{Baydin, Pearlmutter, Radul, and Siskind} +\firstpageno{1} + +\begin{document} + +\title{Automatic Differentiation\\in Machine Learning: a Survey} + +\author{\name Atılım Güneş Baydin \email gunes@robots.ox.ac.uk \\ + \addr Department of Engineering Science\\ + University of Oxford\\ + Oxford OX1 3PJ, United Kingdom + \AND + \name Barak~A.~Pearlmutter \email barak@pearlmutter.net \\ + \addr Department of Computer Science\\ + National University of Ireland Maynooth\\ + Maynooth, Co. Kildare, Ireland + \AND + \name Alexey~Andreyevich~Radul \email axch@mit.edu \\ + \addr Department of Brain and Cognitive Sciences\\ + Massachusetts Institute of Technology\\ + Cambridge, MA 02139, United States + \AND + \name Jeffrey~Mark~Siskind \email qobi@purdue.edu \\ + \addr School of Electrical and Computer Engineering\\ + Purdue University\\ + West Lafayette, IN 47907, United States} + +\editor{Léon Bottou} + +\maketitle + +\begin{abstract}% <- trailing '%' for backward compatibility of .sty file +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 largely been unaware of each other and, in some cases, have independently discovered each other's results. Despite its relevance, general-purpose AD has been missing from the machine learning toolbox, a situation slowly changing with its ongoing adoption under the names ``dynamic computational graphs'' and ``differentiable programming''. We survey the intersection of AD and machine learning, cover applications where AD has direct relevance, and address the main implementation techniques. By precisely defining the main differentiation techniques and their interrelationships, we aim to bring clarity to the usage of the terms ``autodiff'', ``automatic differentiation'', and ``symbolic differentiation'' as these are encountered more and more in machine learning settings. + +\end{abstract} + +\begin{keywords} + Backpropagation, Differentiable Programming +\end{keywords} + +\section{Introduction} + +Methods for the computation of derivatives in computer programs can be classified into four categories: +\begin{inparaenum}[(1)] + \item manually working out derivatives and coding them; + \item \emph{numerical differentiation} using finite difference approximations; + \item \emph{symbolic differentiation} using expression manipulation in computer algebra systems such as Mathematica, Maxima, and Maple; and + \item \emph{automatic differentiation}, also called \emph{algorithmic differentiation}, which is the subject matter of this paper. +\end{inparaenum} + +Conventionally, many methods in machine learning have required the evaluation of derivatives and most of the traditional learning algorithms have relied on the computation of gradients and Hessians of an objective function \citep{Sra2011}. When introducing new models, machine learning researchers have spent considerable effort on the manual derivation of analytical derivatives to subsequently plug these into standard optimization procedures such as L-BFGS \citep{Zhu1997} or stochastic gradient descent \citep{Bottou1998}. Manual differentiation is time consuming and prone to error. Of the other alternatives, numerical differentiation is simple to implement but can be highly inaccurate due to round-off and truncation errors \citep{Jerrell1997}; more importantly, it scales poorly for gradients, rendering it inappropriate for machine learning where gradients with respect to millions of parameters are commonly needed. Symbolic differentiation addresses the weaknesses of both the manual and numerical methods, but often results in complex and cryptic expressions plagued with the problem of ``expression swell'' \citep{Corliss1988}. Furthermore, manual and symbolic methods require models to be defined as closed-form expressions, ruling out or severely limiting algorithmic control flow and expressivity. + +We are concerned with the powerful fourth technique, automatic differentiation (AD). AD performs a non-standard interpretation of a given computer program by replacing the domain of the variables to incorporate derivative values and redefining the semantics of the operators to propagate derivatives per the chain rule of differential calculus. Despite its widespread use in other fields, general-purpose AD has been underused by the machine learning community until very recently.\footnote{See, e.g., \url{https://justindomke.wordpress.com/2009/02/17/automatic-differentiation-the-most-criminally-underused-tool-in-the-potential-machine-learning-toolbox/}} Following the emergence of deep learning \citep{lecun2015deep,goodfellow2016deep} as the state-of-the-art in many machine learning tasks and the modern workflow based on rapid prototyping and code reuse in frameworks such as Theano \citep{Bastien2012}, Torch \citep{collobert2011torch7}, and TensorFlow \citep{abadi2016tensorflow}, the situation is slowly changing where projects such as autograd\footnote{\url{https://github.com/HIPS/autograd}} \citep{maclaurin2016modeling}, Chainer\footnote{\url{https://chainer.org/}} \citep{tokui2015chainer}, and PyTorch\footnote{\url{http://pytorch.org/}} \citep{paszke2017automatic} are leading the way in bringing general-purpose AD to the mainstream. + +The term ``automatic'' in AD can be a source of confusion, causing machine learning practitioners to put the label ``automatic differentiation'', or just ``autodiff'', on any method or tool that does not involve manual differentiation, without giving due attention to the underlying mechanism. We would like to stress that AD as a technical term refers to a specific family of techniques that compute derivatives through accumulation of values during code execution to generate numerical derivative evaluations rather than derivative expressions. This allows accurate evaluation of derivatives at machine precision with only a small constant factor of overhead and ideal asymptotic efficiency. In contrast with the effort involved in arranging code as closed-form expressions under the syntactic and semantic constraints of symbolic differentiation, AD can be applied to regular code with minimal change, allowing branching, loops, and recursion. Because of this generality, AD has been applied to computer simulations in industry and academia and found applications in fields including engineering design optimization \citep{forth2002aerofoil,casanova2002application}, computational fluid dynamics \citep{Muller2005,thomas2006using,Bischof2006}, physical modeling \citep{Ekstrom2010}, optimal control \citep{Walther2007}, structural mechanics \citep{haase2002optimal}, atmospheric sciences \citep{Carmichael1997,Charpentier2000}, and computational finance \citep{Bischof2002,Capriotti2011}. + +In machine learning, a specialized counterpart of AD known as the backpropagation algorithm has been the mainstay for training neural networks, with a colorful history of having been reinvented at various times by independent researchers \citep{Griewank2012,schmidhuber2015deep}. It has been one of the most studied and used training algorithms since the day it became popular mainly through the work of \citet{rumelhart1986learning}. In simplest terms, backpropagation models learning as gradient descent in neural network weight space, looking for the minima of an objective function. The required gradient is obtained by the backward propagation of the sensitivity of the objective value at the output (Figure~\ref{FigureBackpropagation}), utilizing the chain rule to compute partial derivatives of the objective with respect to each weight. The resulting algorithm is essentially equivalent to transforming the network evaluation function composed with the objective function under reverse mode AD, which, as we shall see, actually generalizes the backpropagation idea. Thus, a modest understanding of the mathematics underlying backpropagation provides one with sufficient background for grasping AD techniques. + +\begin{figure} + \centering + \trimbox{0cm -0.4cm}{\resizebox{0.75\textwidth}{!}{\includegraphics{figures/backprop/backprop}}} + \caption{Overview of backpropagation. (a) Training inputs $x_i$ are fed forward, generating corresponding activations $y_i$. An error $E$ between the actual output $y_3$ and the target output $t$ is computed. (b) The error adjoint is propagated backward, giving the gradient with respect to the weights $\nabla_{w_i}E = \left(\frac{\partial E}{\partial w_1},\dots,\frac{\partial E}{\partial w_6}\right)$, which is subsequently used in a gradient-descent procedure. The gradient with respect to inputs $\nabla_{x_i}E$ can be also computed in the same backward pass.} + \label{FigureBackpropagation} +\end{figure} + +In this paper we review AD from a machine learning perspective, covering its origins, applications in machine learning, and methods of implementation. Along the way, we also aim to dispel some misconceptions that we believe have impeded wider recognition of AD by the machine learning community. In Section~\ref{SectionWhatADIsNot} we start by explicating how AD differs from numerical and symbolic differentiation. Section~\ref{SectionPreliminaries} gives an introduction to the AD technique and its forward and reverse accumulation modes. Section~\ref{SectionDerivativesAndMachineLearning} discusses the role of derivatives in machine learning and examines cases where AD has relevance. Section~\ref{SectionImplementations} covers various implementation approaches and general-purpose AD tools, followed by Section~\ref{SectionConclusions} where we discuss future directions. + +\section{What AD Is Not} +\label{SectionWhatADIsNot} + +Without proper introduction, one might assume that AD is either a type of numerical or symbolic differentiation. Confusion can arise because AD does in fact provide numerical values of derivatives (as opposed to derivative expressions) and it does so by using symbolic rules of differentiation (but keeping track of derivative values as opposed to the resulting expressions), giving it a two-sided nature that is partly symbolic and partly numerical \citep{Griewank2003}. We start by emphasizing how AD is different from, and in several aspects superior to, these two commonly encountered techniques of computing derivatives. + +\begin{figure*} + \centering + \trimbox{0cm -0.4cm}{\resizebox{\textwidth}{!}{\small\input{figures/differentiation/differentiation.tex}}} + \caption{The range of approaches for differentiating mathematical expressions and computer code, looking at the example of a truncated logistic map (upper left). Symbolic differentiation (center right) gives exact results but requires closed-form input and suffers from expression swell; numerical differentiation (lower right) has problems of accuracy due to round-off and truncation errors; automatic differentiation (lower left) is as accurate as symbolic differentiation with only a constant factor of overhead and support for control flow.} + \label{FigureDifferentiation} +\end{figure*} + +\subsection{AD Is Not Numerical Differentiation} + +Numerical differentiation is the finite difference approximation of derivatives using values of the original function evaluated at some sample points \citep{Burden2001} (Figure~\ref{FigureDifferentiation}, lower right). In its simplest form, it is based on the limit definition of a derivative. For example, for a multivariate function $f:\mathbb{R}^n \to \mathbb{R}$, one can approximate the gradient $\nabla f=\left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right)$ using +\begin{equation} + \label{EquationForwardDifference} + \frac{\partial f(\mathbf{x})}{\partial x_i} \approx \frac{f(\mathbf{x} + h \mathbf{e}_i) - f(\mathbf{x})}{h}\;, +\end{equation} +where $\mathbf{e}_i$ is the $i$-th unit vector and $h > 0$ is a small step size. This has the advantage of being uncomplicated to implement, but the disadvantages of performing $O(n)$ evaluations of $f$ for a gradient in $n$ dimensions and requiring careful consideration in selecting the step size $h$. + +Numerical approximations of derivatives are inherently ill-conditioned and unstable,\footnote{Using the limit definition of the derivative for finite difference approximation commits both cardinal sins of numerical analysis: \emph{``thou shalt not add small numbers to big numbers''}, and \emph{``thou shalt not subtract numbers which are approximately equal''}.} with the exception of complex variable methods that are applicable to a limited set of holomorphic functions \citep{Fornberg1981}. This is due to the introduction of truncation\footnote{Truncation error is the error of approximation, or inaccuracy, one gets from $h$ not actually being zero. It is proportional to a power of $h$.} and round-off\footnote{Round-off error is the inaccuracy one gets from valuable low-order bits of the final answer having to compete for machine-word space with high-order bits of $f(\mathbf{x} + h \mathbf{e}_i)$ and $f(\mathbf{x})$ (Eq.~\ref{EquationForwardDifference}), which the computer has to store just until they cancel in the subtraction at the end. Round-off error is inversely proportional to a power of $h$.} errors inflicted by the limited precision of computations and the chosen value of the step size $h$. Truncation error tends to zero as $h \to 0$. However, as $h$ is decreased, round-off error increases and becomes dominant (Figure~\ref{FigureApproximationError}). + +\begin{figure*} + \centering + \resizebox{0.82\textwidth}{!}{\small\input{figures/approx-error/approx-error.tex}} + \caption{Error in the forward (Eq.~\ref{EquationForwardDifference}) and center difference (Eq.~\ref{EquationCenterDifference}) approximations as a function of step size $h$, for the derivative of the truncated logistic map ${f(x)=64x(1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2}$. Plotted errors are computed using ${E_{\mathrm{forward}}(h,x_0)=\left|\frac{f(x_0+h)-f(x_0)}{h} - \frac{d}{dx}f(x)\big|_{x_0}\right|}$ and ${E_{\mathrm{center}}(h,x_0)=\left|\frac{f(x_0+h)-f(x_0-h)}{2h} - \frac{d}{dx}f(x)\big|_{x_0}\right|}$ at $x_0=0.2$\;.} + \label{FigureApproximationError} +\end{figure*} + +Various techniques have been developed to mitigate approximation errors in numerical differentiation, such as using a center difference approximation +\begin{equation} + \label{EquationCenterDifference} + \frac{\partial f(\mathbf{x})}{\partial x_i} = \frac{f(\mathbf{x} + h \mathbf{e}_i) - f(\mathbf{x} - h \mathbf{e}_i)}{2h} + O(h^{2})\;, +\end{equation} +where the first-order errors cancel and one effectively moves the truncation error from first-order to second-order in $h$.\footnote{This does not avoid either of the cardinal sins, and is still highly inaccurate due to truncation.} For the one-dimensional case, it is just as costly to compute the forward difference (Eq.~\ref{EquationForwardDifference}) and the center difference (Eq.~\ref{EquationCenterDifference}), requiring only two evaluations of $f$. However, with increasing dimensionality, a trade-off between accuracy and performance is faced, where computing a Jacobian matrix of a function $f: \mathbb{R}^n \to \mathbb{R}^m$ requires $2mn$ evaluations. + +Other techniques for improving numerical differentiation, including higher-order finite differences, Richardson extrapolation to the limit \citep{Brezinski1991}, and differential quadrature methods using weighted sums \citep{Bert1996}, have increased computational complexity, do not completely eliminate approximation errors, and remain highly susceptible to floating point truncation. + +The $O(n)$ complexity of numerical differentiation for a gradient in $n$ dimensions is the main obstacle to its usefulness in machine learning, where $n$ can be as large as millions or billions in state-of-the-art deep learning models \citep{shazeer2017outrageously}. In contrast, approximation errors would be tolerated in a deep learning setting thanks to the well-documented error resiliency of neural network architectures \citep{gupta2015deep}. + +\subsection{AD Is Not Symbolic Differentiation} + +Symbolic differentiation is the automatic manipulation of expressions for obtaining derivative expressions \citep{Grabmeier2003} (Figure~\ref{FigureDifferentiation}, center right), carried out by applying transformations representing rules of differentiation such as +\begin{equation} +\begin{aligned} +\frac{d}{dx} \left(f(x) + g(x)\right) &\leadsto \frac{d}{dx} f(x) + \frac{d}{dx} g(x)\\ +\frac{d}{dx} \left(f(x)\,g(x)\right) &\leadsto \left(\frac{d}{dx} f(x)\right) g(x) + f(x) \left(\frac{d}{dx} g(x)\right)\; . +\end{aligned} +\label{EquationMultiplicationRule} +\end{equation} + +When formulae are represented as data structures, symbolically differentiating an expression tree is a perfectly mechanistic process, considered subject to mechanical automation even at the very inception of calculus \citep{Leibniz1685}. This is realized in modern computer algebra systems such as Mathematica, Maxima, and Maple and machine learning frameworks such as Theano. + +In optimization, symbolic derivatives can give valuable insight into the structure of the problem domain and, in some cases, produce analytical solutions of extrema (e.g., solving for $\frac{d}{dx}f(x)=0$) that can eliminate the need for derivative calculation altogether. On the other hand, symbolic derivatives do not lend themselves to efficient runtime calculation of derivative values, as they can get exponentially larger than the expression whose derivative they represent. + +Consider a function $h(x)=f(x)g(x)$ and the multiplication rule in Eq.~\ref{EquationMultiplicationRule}. Since $h$ is a product, $h(x)$ and $\frac{d}{dx}h(x)$ have some common components, namely $f(x)$ and $g(x)$. Note also that on the right hand side, $f(x)$ and $\frac{d}{dx}f(x)$ appear separately. If we just proceeded to symbolically differentiate $f(x)$ and plugged its derivative into the appropriate place, we would have nested duplications of any computation that appears in common between $f(x)$ and $\frac{d}{dx}f(x)$. Hence, careless symbolic differentiation can easily produce exponentially large symbolic expressions which take correspondingly long to evaluate. This problem is known as \emph{expression swell} (Table~\ref{TableExpressionSwell}). + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \caption{Iterations of the logistic map $l_{n+1}=4l_n (1-l_n)$, $l_1=x$ and the corresponding derivatives of $l_n$ with respect to $x$, illustrating expression swell.} + \label{TableExpressionSwell} + {\small + \begin{tabularx}{\columnwidth}{@{}lp{2.8cm}XX@{}} + \toprule + $n$ & $l_n$ & $\frac{d}{dx}l_n$ & $\frac{d}{dx}l_n$ (Simplified form)\\ + \addlinespace + \midrule + 1 & $x$ & $1$ & $1$\\ + \addlinespace + 2 & $4x(1 - x)$ & $4(1 - x) -4x$ & $4 - 8x$\\ + \addlinespace + 3 & $16x(1 - x)(1 - 2 x)^2$ & $16(1 - x)(1 - 2 x)^2 - 16x(1 - 2 x)^2 - 64x(1 - x)(1 - 2 x)$ & $16 (1 - 10 x + 24 x^2 - 16 x^3)$\\ + \addlinespace + 4 & $64x(1 - x)(1 - 2 x)^2$ $(1 - 8 x + 8 x^2)^2$ & $128x(1 - x)(-8 + 16 x)(1 - 2 x)^2 (1 - 8 x + 8 x^2) + 64 (1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 64x(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 256x(1 - x)(1 - 2 x)(1 - 8 x + 8 x^2)^2$ & $64 (1 - 42 x + 504 x^2 - 2640 x^3 + 7040 x^4 - 9984 x^5 + 7168 x^6 - 2048 x^7)$\\ + \bottomrule + \end{tabularx}} +\end{table} + +When we are concerned with the accurate numerical evaluation of derivatives and not so much with their actual symbolic form, it is in principle possible to significantly simplify computations by storing only the values of intermediate sub-expressions in memory. Moreover, for further efficiency, we can interleave as much as possible the differentiation and simplification steps. This interleaving idea forms the basis of AD and provides an account of its simplest form: \emph{apply symbolic differentiation at the elementary operation level and keep intermediate numerical results, in lockstep with the evaluation of the main function.} This is AD in the forward accumulation mode, which we shall introduce in the following section. + + +\section{AD and Its Main Modes} +\label{SectionPreliminaries} + +AD can be thought of as performing a non-standard interpretation of a computer program where this interpretation involves augmenting the standard computation with the calculation of various derivatives. All numerical computations are ultimately compositions of a finite set of elementary operations for which derivatives are known \citep{Verma2000,Griewank2008}, and combining the derivatives of the constituent operations through the chain rule gives the derivative of the overall composition. Usually these elementary operations include the binary arithmetic operations, the unary sign switch, and transcendental functions such as the exponential, the logarithm, and the trigonometric functions. + +On the left hand side of Table~\ref{TableForwardADExample} we see the representation of the computation $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ as an \emph{evaluation trace} of elementary operations---also called a Wengert list \citep{Wengert1964}. We adopt the three-part notation used by \citet{Griewank2008}, where a function $f: \mathbb{R}^n \to \mathbb{R}^m$ is constructed using intermediate variables $v_i$ such that +\begin{compactitem} + \item variables $v_{i-n} = x_i,\;i = 1, \dotsc, n$ are the input variables, + \item variables $v_i\;i = 1, \dotsc, l$ are the working (intermediate) variables, and + \item variables $y_{m-i} = v_{l-i},\;i = m - 1, \dotsc, 0$ are the output variables. +\end{compactitem} +Figure~\ref{FigureComputationalGraph} shows the given trace of elementary operations represented as a computational graph \citep{Bauer1974}, useful in visualizing dependency relations between intermediate variables. + +\begin{figure} + \centering + \trimbox{0cm -0.5cm}{\resizebox{0.8\textwidth}{!}{\normalsize\input{figures/comp-graph/comp-graph.tex}}} + \caption{Computational graph of the example $f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$. See the primal trace in Tables \ref{TableForwardADExample} or \ref{TableReverseADExample} for the definitions of the intermediate variables $v_{-1} \dots v_5$\;.} + \label{FigureComputationalGraph} +\end{figure} + +Evaluation traces form the basis of the AD techniques. An important point to note here is that AD can differentiate not only closed-form expressions in the classical sense, but also algorithms making use of control flow such as branching, loops, recursion, and procedure calls, giving it an important advantage over symbolic differentiation which severely limits such expressivity. This is thanks to the fact that any numeric code will eventually result in a numeric evaluation trace with particular values of the input, intermediate, and output variables, which are the only things one needs to know for computing derivatives using chain rule composition, regardless of the specific control flow path that was taken during execution. Another way of expressing this is that AD is blind with respect to any operation, including control flow statements, which do not directly alter numeric values. + +\subsection{Forward Mode} + +AD in forward accumulation mode\footnote{Also called \emph{tangent linear} mode.} is the conceptually most simple type. Consider the evaluation trace of the function $f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ given on the left-hand side in Table~\ref{TableForwardADExample} and in graph form in Figure~\ref{FigureComputationalGraph}. For computing the derivative of $f$ with respect to $x_1$, we start by associating with each intermediate variable $v_i$ a derivative +\begin{equation*} + \dot{v}_i = \frac{\partial v_i}{\partial x_1}\; . +\end{equation*} + +Applying the chain rule to each elementary operation in the forward primal trace, we generate the corresponding tangent (derivative) trace, given on the right-hand side in Table~\ref{TableForwardADExample}. Evaluating the primals $v_i$ in lockstep with their corresponding tangents $\dot{v}_i$ gives us the required derivative in the final variable $\dot{v}_5=\frac{\partial y}{\partial x_1}$\;. + +%\footnote{In an implementation, the order in which $v_i$ and its corresponding $\dot{v}_i$ are computed can make a difference.} + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \caption{Forward mode AD example, with $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ evaluated at $(x_1, x_2) = (2, 5)$ and setting $\dot{x}_1 = 1$ to compute $\frac{\partial y}{\partial x_1}$. The original forward evaluation of the primals on the left is augmented by the tangent operations on the right, where each line complements the original directly to its left.} + \label{TableForwardADExample} + \begin{minipage}[c]{0.47\textwidth} + {\footnotesize + \begin{tabularx}{\textwidth}{p{0.2mm}p{2mm}p{16mm}X} + \toprule + \multicolumn{4}{l}{Forward Primal Trace}\\ + \multirow{9}{2mm}{\begin{tikzpicture}\draw[->,>=triangle 60,thick](0,0)--(0,-3.8);\end{tikzpicture}} & $v_{-1}$ & $=x_1$ & $=2$\\ + & $v_0$ & $=x_2$ & $=5$\\ + \cmidrule{2-4} + & $v_1$ & $=\ln{v_{-1}}$ & $=\ln{2}$\\ + & $v_2$ & $=v_{-1} \times v_0$ & $=2 \times 5$\\ + & $v_3$ & $=\sin{v_0}$ & $=\sin{5}$\\ + & $v_4$ & $=v_1+v_2$ & $=0.693+10$\\ + & $v_5$ & $=v_4-v_3$ & $=10.693+0.959$\\ + \cmidrule{2-4} + & $y$ & $=v_5$ & $=11.652$\\ + \bottomrule + \end{tabularx}} + \end{minipage} + \begin{minipage}[c]{0.52\textwidth} + \setlength{\fboxsep}{0pt}\colorbox{gray!20} + {\footnotesize + \begin{tabularx}{\textwidth}{p{0.2mm}p{2mm}p{28mm}X} + \toprule + \multicolumn{4}{l}{Forward Tangent (Derivative) Trace}\\ + \multirow{9}{2mm}{\begin{tikzpicture}\draw[->,>=triangle 60,thick](0,0)--(0,-3.8);\end{tikzpicture}} & $\dot{v}_{-1}$ & $=\dot{x}_1$ & $=1$\\ + & $\dot{v}_0$ & $=\dot{x}_2$ & $=0$\\ + \cmidrule{2-4} + & $\dot{v}_1$ & $=\dot{v}_{-1}/v_{-1}$ & $=1/2$\\ + & $\dot{v}_2$ & $=\dot{v}_{-1} \times v_0 + \dot{v}_0 \times v_{-1}$ & $=1 \times 5 + 0 \times 2$\\ + & $\dot{v}_3$ & $=\dot{v}_0 \times \cos{v_0}$ & $=0 \times \cos{5}$\\ + & $\dot{v}_4$ & $=\dot{v}_1+\dot{v}_2$ & $=0.5+5$\\ + & $\dot{v}_5$ & $=\dot{v}_4-\dot{v}_3$ & $=5.5-0$\\ + \cmidrule{2-4} + & \boldmath$\dot{y}$ & \boldmath$=\dot{v}_5$ & \boldmath$=5.5$\\ + \bottomrule + \end{tabularx}} + \end{minipage} +\end{table} + +This generalizes naturally to computing the Jacobian of a function $f : \mathbb{R}^n \to \mathbb{R}^m$ with $n$ independent (input) variables $x_i$ and $m$ dependent (output) variables $y_j$. In this case, each forward pass of AD is initialized by setting only one of the variables $\dot{x}_i=1$ and setting the rest to zero (in other words, setting $\dot{\mathbf{x}} = \mathbf{e}_i$, where $\mathbf{e}_i$ is the $i$-th unit vector). A run of the code with specific input values $\mathbf{x}=\mathbf{a}$ then computes +\begin{equation*} +\dot{y}_j = \left.\frac{\partial y_j}{\partial x_i}\right|_{\mathbf{x}=\mathbf{a}},\;j = 1, \dotsc, m\;, +\end{equation*} +giving us one column of the Jacobian matrix +\begin{equation*} +\mathbf{J}_f = \left. \begin{bmatrix} + \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ + \vdots & \ddots & \vdots \\ + \frac{\partial y_m}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_n} + \end{bmatrix} \right|_{\mathbf{x}\; = \; \mathbf{a}} +\end{equation*} +evaluated at point $\mathbf{a}$. Thus, the full Jacobian can be computed in $n$ evaluations. + +Furthermore, forward mode AD provides a very efficient and matrix-free way of computing Jacobian--vector products +\begin{equation} + \mathbf{J}_f\,\mathbf{r} = \begin{bmatrix} + \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ + \vdots & \ddots & \vdots \\ + \frac{\partial y_m}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_n} + \end{bmatrix} + \begin{bmatrix} + r_1 \\ + \vdots \\ + r_n + \end{bmatrix}\; , +\label{EquationJacobianVectorProduct} +\end{equation} +simply by initializing with $\dot{\mathbf{x}}=\mathbf{r}$. Thus, we can compute the Jacobian--vector product in just one forward pass. As a special case, when $f: \mathbb{R}^n \to \mathbb{R}$, we can obtain the directional derivative along a given vector $\mathbf{r}$ as a linear combination of the partial derivatives +\begin{equation*} + \nabla f \cdot \mathbf{r} +\end{equation*} +by starting the AD computation with the values $\dot{\mathbf{x}}=\mathbf{r}$. + +Forward mode AD is efficient and straightforward for functions $f: \mathbb{R} \to \mathbb{R}^m$, as all the derivatives $\frac{d y_i}{d x}$ can be computed with just one forward pass. Conversely, in the other extreme of $f: \mathbb{R}^n \to \mathbb{R}$, forward mode AD requires $n$ evaluations to compute the gradient +\begin{equation*} + \nabla f = \left( \frac{\partial y}{\partial x_1}, \dots, \frac{\partial y}{\partial x_n}\right)\; , +\end{equation*} +which also corresponds to a $1 \times n$ Jacobian matrix that is built one column at a time with the forward mode in $n$ evaluations. + +In general, for cases $f: \mathbb{R}^n \to \mathbb{R}^m$ where $n \gg m$, a different technique is often preferred. +We will describe AD in \emph{reverse accumulation mode} in Section~\ref{sec:reverse-mode}. + +\subsubsection{Dual Numbers} +\label{SectionDualNumbers} +Mathematically, forward mode AD (represented by the left- and right-hand sides in Table~\ref{TableForwardADExample}) can be viewed as evaluating a function using dual numbers,\footnote{First introduced by \citet{Clifford1873}, with important uses in linear algebra and physics.} which can be defined as truncated Taylor series of the form +\begin{equation*} + v + \dot{v}\epsilon \;, +\end{equation*} +where $v, \dot{v} \in \mathbb{R}$ and $\epsilon$ is a nilpotent number such +that $\epsilon^2 = 0$ and $\epsilon \neq 0$. Observe, for example, that +\begin{align*} + (v + \dot{v}\epsilon) + (u + \dot{u}\epsilon) &= (v + u) + (\dot{v} + \dot{u})\epsilon\\ + (v + \dot{v}\epsilon)(u + \dot{u}\epsilon) &= (vu) + (v\dot{u} + \dot{v}u)\epsilon\;, +\end{align*} +in which the coefficients of $\epsilon$ conveniently mirror symbolic differentiation rules (e.g., Eq.~\ref{EquationMultiplicationRule}). We can utilize this by setting up a regime where +\begin{equation}\label{EquationDualRule} + f(v + \dot{v}\epsilon) = f(v) + f'(v)\dot{v}\epsilon +\end{equation} +and using dual numbers as data structures for carrying the tangent value together with the primal.\footnote{Just as the complex number written $x + y i$ is represented in the computer as a pair in memory $(x, y)$ whose two slots are reals, the dual number written $x + \dot{x}\epsilon$ is represented as the pair $(x, \dot{x})$. Such pairs are sometimes called Argand pairs \citep[][p107 Eqs.~(157) and (158)]{Hamilton1837}.} The chain rule works as expected on this representation: two applications of Eq.~\ref{EquationDualRule} give +\begin{align*} + f(g(v + \dot{v}\epsilon)) &= f(g(v) + g'(v)\dot{v}\epsilon)\\ + &= f(g(v)) + f'(g(v))g'(v)\dot{v}\epsilon\;. +\end{align*} +The coefficient of $\epsilon$ on the right-hand side is exactly the derivative of the composition of $f$ and $g$. This means that since we implement elementary operations to respect the invariant Eq.~\ref{EquationDualRule}, all compositions of them will also do so. This, in turn, means that we can extract the derivative of a function by interpreting any non-dual number $v$ as $v + 0 \epsilon$ and evaluating the function in this non-standard way on an initial input with a coefficient $1$ for $\epsilon$: +\begin{align*} +\left.\frac{df(x)}{dx}\right|_{x=v} = \textrm{epsilon-coefficient}(\textrm{dual-version}(f)(v + 1\epsilon))\;. +\end{align*} + +This also extends to arbitrary program constructs, since dual numbers, as data types, can be contained in any data structure. As long as a dual number remains in a data structure with no arithmetic operations being performed on it, it will just remain a dual number; and if it is taken out of the data structure and operated on again, then the differentiation will continue. + +In practice, a function $f$ coded in a programming language of choice would be fed into an AD tool, which would then augment it with corresponding extra code to handle the dual operations so that the function and its derivative are simultaneously computed. This can be implemented through calls to a specific library, in the form of source code transformation where a given source code will be automatically modified, or through operator overloading, making the process transparent to the user. We discuss these implementation techniques in Section~\ref{SectionImplementations}. + +\subsection{Reverse Mode} +\label{sec:reverse-mode} + +AD in the reverse accumulation mode\footnote{Also called \emph{adjoint} or \emph{cotangent linear} mode.} corresponds to a generalized backpropagation algorithm, in that it propagates derivatives backward from a given output. This is done by complementing each intermediate variable $v_i$ with an adjoint +\begin{equation*} + \bar{v}_i = \frac{\partial y_j}{\partial v_i}\; , +\end{equation*} +which represents the sensitivity of a considered output $y_j$ with respect to changes in $v_i$. In the case of backpropagation, $y$ would be a scalar corresponding to the error $E$ (Figure~\ref{FigureBackpropagation}). + +In reverse mode AD, derivatives are computed in the second phase of a two-phase process. In the first phase, the original function code is run \emph{forward}, populating intermediate variables $v_i$ and recording the dependencies in the computational graph through a bookkeeping procedure. In the second phase, derivatives are calculated by propagating adjoints $\bar{v}_i$ in \emph{reverse}, from the outputs to the inputs. + +Returning to the example $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$, in Table~\ref{TableReverseADExample} we see the adjoint statements on the right-hand side, corresponding to each original elementary operation on the left-hand side. In simple terms, we are interested in computing the contribution $\bar{v}_i = \frac{\partial y}{\partial v_i}$ of the change in each variable $v_i$ to the change in the output $y$. Taking the variable $v_0$ as an example, we see in Figure~\ref{FigureComputationalGraph} that the only way it can affect $y$ is through affecting $v_2$ and $v_3$, so its contribution to the change in $y$ is given by +\begin{align*} + \frac{\partial y}{\partial v_0} &= \frac{\partial y}{\partial v_2}\frac{\partial v_2}{\partial v_0} + \frac{\partial y}{\partial v_3}\frac{\partial v_3}{\partial v_0}& + \text{or}&& + \bar{v}_0 &= \bar{v}_2\frac{\partial v_2}{\partial v_0} + \bar{v}_3\frac{\partial v_3}{\partial v_0}\;. +\end{align*} + +In Table~\ref{TableReverseADExample}, this contribution is computed in two incremental steps +\begin{align*} + \bar{v}_0 &= \bar{v}_3\frac{\partial v_3}{\partial v_0} & + \text{and} && + \bar{v}_0 &= \bar{v}_0 + \bar{v}_2\frac{\partial v_2}{\partial v_0}\;, +\end{align*} +lined up with the lines in the forward trace from which these expressions originate. + +After the forward pass on the left-hand side, we run the reverse pass of the adjoints on the right-hand side, starting with $\bar{v}_5 = \bar{y} = \frac{\partial y}{\partial y} = 1$. In the end we get the derivatives $\frac{\partial y}{\partial x_1} = \bar{x}_1$ and $\frac{\partial y}{\partial x_2} = \bar{x}_2$ in just one reverse pass. + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \caption{Reverse mode AD example, with $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ evaluated at $(x_1, x_2) = (2, 5)$. After the forward evaluation of the primals on the left, the adjoint operations on the right are evaluated in reverse (cf.\ Figure~\ref{FigureBackpropagation}). Note that both $\frac{\partial y}{\partial x_1}$ and $\frac{\partial y}{\partial x_2}$ are computed in the same reverse pass, starting from the adjoint $\bar{v}_5 = \bar{y} = \frac{\partial y}{\partial y} = 1$.} + \label{TableReverseADExample} + \begin{minipage}[t]{0.41\textwidth} + {\footnotesize + \begin{tabularx}{\textwidth}[t]{p{0.5mm}p{0.8mm}p{18mm}@{}X} + \toprule + \multicolumn{4}{l}{Forward Primal Trace}\\ + \multirow{9}{1mm}{\begin{tikzpicture}\draw[->,>=triangle 60,thick](0,0)--(0,-5.4);\end{tikzpicture}} & $v_{-1}$ & $=x_1$ & $=2$\\ + & $v_0$ & $=x_2$ & $=5$\\ + \cmidrule{2-4} + & $v_1$ & $=\ln{v_{-1}}$ & $=\ln{2}$\vspace{0.25mm}\\ + & $v_2$ & $=v_{-1} \times v_0$ & $=2 \times 5$\vspace{0.25mm}\\ + &\vspace{0.25mm}\\ + & $v_3$ & $=\sin{v_0}$ & $=\sin{5}$\vspace{0.25mm}\\ + & $v_4$ & $=v_1+v_2$ & $=0.693+10$\vspace{0.25mm}\\ + &\vspace{0.25mm}\\ + & $v_5$ & $=v_4-v_3$ & $=10.693+0.959$\vspace{0.25mm}\\ + &\vspace{0.25mm}\\ + \cmidrule{2-4} + & $y$ & $=v_5$ & $=11.652$\\ + \bottomrule + \end{tabularx}}\vspace{1mm} + \end{minipage} + \begin{minipage}[t]{0.58\textwidth} + \setlength{\fboxsep}{0pt}\colorbox{gray!20} + {\footnotesize + \begin{tabularx}{\textwidth}[t]{p{0.5mm}p{1mm}p{23mm}@{\hspace{1mm}}p{24mm}@{}X} + \toprule + \multicolumn{5}{l}{Reverse Adjoint (Derivative) Trace}\\ + \multirow{9}{1mm}{\begin{tikzpicture}\draw[<-,>=triangle 60,thick](0,0)--(0,-5.4);\end{tikzpicture}} & \boldmath$\bar{x}_1$ & \boldmath$=\bar{v}_{-1}$ & & \boldmath$=5.5$\\ + & \boldmath$\bar{x}_2$ & \boldmath$=\bar{v}_0$ & & \boldmath$=1.716$\\ + \cmidrule{2-5} + & $\bar{v}_{-1}$ & $=\bar{v}_{-1} + \bar{v}_1 \frac{\partial v_1}{\partial v_{-1}}$ & $=\bar{v}_{-1} + \bar{v}_1 / v_{-1}$ & $=5.5$\\ + & $\bar{v}_0$ & $=\bar{v}_0 + \bar{v}_2 \frac{\partial v_2}{\partial v_0}$ & $=\bar{v}_0 + \bar{v}_2 \times v_{-1}$ & $=1.716$\\ + & $\bar{v}_{-1}$ & $=\bar{v}_2 \frac{\partial v_2}{\partial v_{-1}}$ & $=\bar{v}_2 \times v_0$ & $=5$\\ + & $\bar{v}_0$ & $=\bar{v}_3 \frac{\partial v_3}{\partial v_0}$ & $=\bar{v}_3 \times \cos{v_0}$ & $=-0.284$\\ + & $\bar{v}_2$ & $=\bar{v}_4 \frac{\partial v_4}{\partial v_2}$ & $=\bar{v}_4 \times 1$ & $=1$\\ + & $\bar{v}_1$ & $=\bar{v}_4 \frac{\partial v_4}{\partial v_1}$ & $=\bar{v}_4 \times 1$ & $=1$\\ + & $\bar{v}_3$ & $=\bar{v}_5 \frac{\partial v_5}{\partial v_3}$ & $=\bar{v}_5 \times (-1)$ & $=-1$\\ + & $\bar{v}_4$ & $=\bar{v}_5 \frac{\partial v_5}{\partial v_4}$ & $=\bar{v}_5 \times 1$ & $=1$\\ + \cmidrule{2-5} + & $\bar{v}_5$ & $=\bar{y}$ & $=1$\\ + \bottomrule + \end{tabularx}} + \end{minipage} +\end{table} + +Compared with the straightforwardness of forward accumulation mode, reverse mode AD can, at first, appear somewhat ``mysterious'' \citep{Dennis1996}. \citet{Griewank2008} argue that this is in part because of the common acquaintance with the chain rule as a mechanistic procedure propagating derivatives forward. + +An important advantage of the reverse mode is that it is significantly less costly to evaluate (in terms of operation count) than the forward mode for functions with a large number of inputs. In the extreme case of $f: \mathbb{R}^n \to \mathbb{R}$, only one application of the reverse mode is sufficient to compute the full gradient $\nabla f = \left(\frac{\partial y}{\partial x_1},\dots,\frac{\partial y}{\partial x_n}\right)$, compared with the $n$ passes of the forward mode needed for populating the same. Because machine learning practice principally involves the gradient of a scalar-valued objective with respect to a large number of parameters, this establishes the reverse mode, as opposed to the forward mode, as the mainstay technique in the form of the backpropagation algorithm. + +In general, for a function $f: \mathbb{R}^n \to \mathbb{R}^m$, if we denote the operation count to evaluate the original function by $\textrm{ops}(f)$, the time it takes to calculate the $m \times n$ Jacobian by the forward mode is $n\;c\;\textrm{ops}(f)$, whereas the same computation can be done via reverse mode in $m\;c\;\textrm{ops}(f)$, where $c$ is a constant guaranteed to be $c<6$ and typically $c \sim [2,3]$ \citep{Griewank2008}. That is to say, reverse mode AD performs better when $m \ll n$. + +Similar to the matrix-free computation of Jacobian--vector products with forward mode (Eq.~\ref{EquationJacobianVectorProduct}), reverse mode can be used for computing the transposed Jacobian--vector product +\begin{equation*} + \mathbf{J}^{\intercal}_f\,\mathbf{r} = \begin{bmatrix} + \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_1} \\ + \vdots & \ddots & \vdots \\ + \frac{\partial y_1}{\partial x_n} & \cdots & \frac{\partial y_m}{\partial x_n} + \end{bmatrix} + \begin{bmatrix} + r_1 \\ + \vdots \\ + r_m + \end{bmatrix}\;, +\end{equation*} +by initializing the reverse phase with $\bar{\mathbf{y}}=\mathbf{r}$. + +The advantages of reverse mode AD, however, come with the cost of increased storage requirements growing (in the worst case) in proportion to the number of operations in the evaluated function. It is an active area of research to improve storage requirements in implementations by using advanced methods such as checkpointing strategies and data-flow analysis \citep{Dauvergne2006,siskind2017divide}. + +\subsection{Origins of AD and Backpropagation} + +Ideas underlying AD date back to the 1950s \citep{Nolan1953,Beda1959}. Forward mode AD as a general method for evaluating partial derivatives was essentially discovered by \citet{Wengert1964}. It was followed by a period of relatively low activity, until interest in the field was revived in the 1980s mostly through the work of \citet{Griewank1989}, also supported by improvements in modern programming languages and the feasibility of an efficient reverse mode AD. + +Reverse mode AD and backpropagation have an intertwined history. The essence of the reverse mode, cast in a continuous-time formalism, is the Pontryagin maximum principle \citep{Rozonoer-Pontryagin-1959a, Boltyanskii-Gamkrelidze-Pontryagin-1960a}. This method was understood in the control theory community \citep{Bryson-1962a, Bryson-Ho-1969a} and cast in more formal terms with discrete-time variables topologically sorted in terms of dependency by \citet{Werbos-1974a}. Prior to Werbos, the work by \citet{linnainmaa1970representation,linnainmaa1976taylor} is often cited as the first published description of the reverse mode. \citet{Speelpenning80} subsequently introduced reverse mode AD as we know it, in the sense that he gave the first implementation that was actually automatic, accepting a specification of a computational process written in a general-purpose programming language and automatically performing the reverse mode transformation. + +Incidentally, \citet{Hecht1989} cites the work of \citet{Bryson-Ho-1969a} and \citet{Werbos-1974a} as the two earliest known instances of backpropagation. Within the machine learning community, the method has been reinvented several times, such as by \citet{Parker1985}, until it was eventually brought to fame by \citet{rumelhart1986learning} and the Parallel Distributed Processing (PDP) group. The PDP group became aware of Parker's work only after their own discovery; similarly, Werbos' work was not appreciated until it was found by Parker \citep{Hecht1989}. This tells us an interesting story of two highly interconnected research communities that have somehow also managed to stay detached during this foundational period. + +For a thorough review of the development of AD, we advise readers to refer to \citet{Rall2006}. Interested readers are highly recommended to read \citet{Griewank2012} for an investigation of the origins of the reverse mode and \citet{schmidhuber2015deep} for the same for backpropagation. + +\section{AD and Machine Learning} +\label{SectionDerivativesAndMachineLearning} + +In the following, we examine the main uses of derivatives in machine learning and report on a selection of works where general-purpose AD, as opposed to just backpropagation, has been successfully applied in a machine learning context. Areas where AD has seen use include optimization, neural networks, computer vision, natural language processing, and probabilistic inference. + +\subsection{Gradient-Based Optimization} + +Gradient-based optimization is one of the pillars of machine learning \citep{bottou2016optimization}. Given an objective function $f: \mathbb{R}^n \to \mathbb{R}$, classical gradient descent has the goal of finding (local) minima $\mathbf{w}^* = \argmin_{\mathbf{w}} f(\mathbf{w})$ via updates of the form $\Del \mathbf{w} = -\eta \nabla f$, where $\eta>0$ is a step size. Gradient-based methods make use of the fact that $f$ decreases steepest if one goes in the direction of the negative gradient. The convergence rate of gradient-based methods is usually improved by adaptive step-size techniques that adjust the step size $\eta$ on every iteration \citep{duchi2011adaptive,schaul2013no,kingma2015adam}. + +As we have seen, for large $n$, reverse mode AD provides a highly efficient method for computing gradients.\footnote{See \url{http://DiffSharp.github.io/DiffSharp/examples-gradientdescent.html} for an example of a general-purpose AD-based gradient descent routine using DiffSharp.} Figure~\ref{FigureHelmholtz} and Table~\ref{TableHelmholtz} demonstrate how gradient computation scales differently for forward and reverse mode AD and numerical differentiation, looking at the Helmholtz free energy function that has been used in AD literature for benchmarking gradient calculations \citep{Griewank1989,Griewank2008,griewank2012numerical}. + +\begin{figure} + \centering + \trimbox{0cm -1cm}{\resizebox{0.65\textwidth}{!}{\normalsize\input{figures/helmholtz/helmholtz.tex}}} + \caption{Evaluation time of the Helmholtz free energy function of a mixed fluid, based on the Peng-Robinson equation of state \citep{Peng1976}, ${f(\mathbf{x}) = R \, T \sum_{i = 0}^{n} \log \frac{x_i}{1 - \mathbf{b^T} \mathbf{x}} - \frac{\mathbf{x^T} \mathbf{A} \mathbf{x}}{\sqrt{8} \mathbf{b^T} \mathbf{x}} \log \frac{1 + (1 + \sqrt{2}) \mathbf{b^T} \mathbf{x}}{1 + (1 - \sqrt{2}) \mathbf{b^T} \mathbf{x}}}$, where $R$ is the universal gas constant, $T$ is the absolute temperature, $\mathbf{b} \in \mathbb{R}^n$ is a vector of constants, $\mathbf{A} \in \mathbb{R}^{n \times n}$ is a symmetric matrix of constants, and $\mathbf{x} \in \mathbb{R}^n$ is the vector of independent variables describing the system. The plots show the evaluation time of $f$ and the gradient $\nabla f$ with numerical differentiation (central difference), forward mode AD, and reverse mode AD, as a function of the number of variables $n$. Reported times are relative to the evaluation time of $f$ with $n=1$. The lower plot uses logarithmic scale for illustrating the behavior for small $n$. Numerical results are given in Table~\ref{TableHelmholtz}. (Code: {\small\url{http://DiffSharp.github.io/DiffSharp/misc/Benchmarks-h-grad-v0.5.7.fsx}})} + \label{FigureHelmholtz} +\end{figure} + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \setlength{\tabcolsep}{1.85mm} + \caption{Evaluation times of the Helmholtz free energy function and its gradient (Figure~\ref{FigureHelmholtz}). Times are given relative to that of the original function with both (1) $n=1$ and (2) $n$ corresponding to each column. (For instance, reverse mode AD with $n=43$ takes approximately twice the time to evaluate relative to the original function with $n=43$.) Times are measured by averaging a thousand runs on a machine with Intel Core i7-4785T 2.20 GHz CPU and 16 GB RAM, using DiffSharp 0.5.7. The evaluation time for the original function with $n=1$ is 0.0023 ms.} + \label{TableHelmholtz} + {\small + \begin{tabularx}{\columnwidth}{@{}p{42.8mm}rrrrrrrr@{}} + \toprule + & \multicolumn{4}{l}{$n$, number of variables}\\ + \cmidrule(l){2-9} + & 1 & 8 & 15 & 22 & 29 & 36 & 43 & 50 \\ + \midrule + $f$, original\\ + \hspace{2mm} Relative $n=1$ & 1 & 5.12 & 14.51 & 29.11 & 52.58 & 84.00 & 127.33 & 174.44 \\ + $\nabla f$, numerical diff.\\ + \hspace{2mm} Relative $n=1$ & 1.08 & 35.55 & 176.79 & 499.43 & 1045.29 & 1986.70 & 3269.36 & 4995.96 \\ + \hspace{2mm} Relative $n$ in column & 1.08 & 6.93 & 12.17 & 17.15 & 19.87 & 23.64 & 25.67 & 28.63 \\ + $\nabla f$, forward AD\\ + \hspace{2mm} Relative $n=1$ & 1.34 & 13.69 & 51.54 & 132.33 & 251.32 & 469.84 & 815.55 & 1342.07\\ + \hspace{2mm} Relative $n$ in column & 1.34 & 2.66 & 3.55 & 4.54 & 4.77 & 5.59 & 6.40 & 7.69 \\ + $\nabla f$, reverse AD\\ + \hspace{2mm} Relative $n=1$ & 1.52 & 11.12 & 31.37 & 67.27 & 113.99 & 174.62 & 254.15 & 342.33 \\ + \hspace{2mm} Relative $n$ in column & 1.52 & 2.16 & 2.16 & 2.31 & 2.16 & 2.07 & 1.99 & 1.96 \\ + \bottomrule + \end{tabularx}} +\end{table} + +%Second order descent methods, Newton, Gauss-Newton, Levenberg-Marquardt +Second-order methods based on Newton's method make use of both the gradient $\nabla f$ and the Hessian $\mathbf{H}_f$, working via updates of the form $\Del \mathbf{w} = -\eta\,\mathbf{H}^{-1}_f \nabla f$ and providing significantly faster convergence \citep{Press2007}. AD provides a way of automatically computing the exact Hessian, enabling succinct and convenient general-purpose implementations.\footnote{See \url{http://DiffSharp.github.io/DiffSharp/examples-newtonsmethod.html} for an implementation of Newton's method with the full Hessian.} Newton's method converges in fewer iterations, but this comes at the cost of having to compute $\mathbf{H}_f$ in each iteration. In large-scale problems, the Hessian is usually replaced by a numerical approximation using first-order updates from gradient evaluations, giving rise to quasi-Newton methods. A highly popular such method is the BFGS\footnote{After Broyden–Fletcher–Goldfarb–Shanno, who independently discovered the method in the 1970s.} algorithm, together with its limited-memory variant L-BFGS \citep{Dennis1996}. On the other hand, Hessians arising in large-scale applications are typically sparse. This sparsity along with symmetry can be readily exploited by AD techniques such as computational graph elimination \citep{Dixon1991}, partial separability \citep{Gay1996}, and matrix coloring and compression \citep{Gebremedhin2009}. + +In many cases one does not need the full Hessian but only a Hessian--vector product $\mathbf{H} \mathbf{v}$, which can be computed efficiently using a reverse-on-forward configuration of AD by applying the reverse mode to take the gradient of code produced by the forward mode.\footnote{\citet{Christianson2012ALN} demonstrates that the second derivative can be computed with the same arithmetic operation sequence using forward-on-reverse, reverse-on-forward, and reverse-on-reverse. The taping overheads of these methods may differ in implementation-dependent ways.} Given the function $f:\mathbb{R}^n\to\mathbb{R}$, the evaluation point $\mathbf{x}$, and the vector $\mathbf{v}$, one can accomplish this by first computing the directional derivative $\nabla f \cdot \mathbf{v}$ through the forward mode via setting $\dot{\mathbf{x}}=\mathbf{v}$ and then applying the reverse mode on this result to get $\nabla^2 f \cdot \mathbf{v}=\mathbf{H}_f \mathbf{v}$ \citep{Pearlmutter1994}. This computes $\mathbf{H} \mathbf{v}$ with $O(n)$ complexity, even though $\mathbf{H}$ is a $n \times n$ matrix. Availability of robust AD tools may make more sophisticated optimization methods applicable to large-scale machine-learning problems. For instance, when fast stochastic Hessian--vector products are available, these can be used as the basis of stochastic Newton's methods \citep{agarwal2016second}, which have the potential to endow stochastic optimization with quadratic convergence. + +Another approach for improving the rate of convergence of gradient-based methods is to use gain adaptation methods such as stochastic meta-descent (SMD) \citep{Schraudolph1999}, where stochastic sampling is introduced to avoid local minima and reduce the computational expense. An example using SMD with AD Hessian--vector products is given by \citet{Vishwanathan2006} on conditional random fields (CRF). Similarly, \citet{Schraudolph2003} use Hessian--vector products in their model combining conjugate gradient techniques with stochastic gradient descent. + +\subsection{Neural Networks, Deep Learning, Differentiable Programming} + +Training of a neural network is an optimization problem with respect to its set of weights, which can in principle be addressed by using any method ranging from evolutionary algorithms \citep{such2017deep} to gradient-based methods such as BFGS \citep{Apostolopoulou2009} or the mainstay stochastic gradient descent \citep{bottou2010large} and its many variants \citep{kingma2015adam,tieleman2012lecture,duchi2011adaptive}. As we have seen, the backpropagation algorithm is only a special case of AD: by applying reverse mode AD to an objective function evaluating a network's error as a function of its weights, we can readily compute the partial derivatives needed for performing weight updates.\footnote{See \url{http://DiffSharp.github.io/DiffSharp/examples-neuralnetworks.html} for an implementation of backpropagation with reverse mode AD.} + +The LUSH system \citep{LUSH2002}, and its predecessor SN \citep{bottou-lecun-88}, were the first production systems that targeted efficient neural network simulation while incorporating both a general-purpose programming language and AD. Modern deep learning frameworks provide differentiation capability in one way or another, but the underlying mechanism is not always made clear and confusion abounds regarding the use of the terms ``autodiff'', ``automatic differentiation'', and ``symbolic differentiation'', which are sometimes even used interchangeably. In mainstream frameworks including Theano\footnote{Theano is a computational graph optimizer and compiler with GPU support and it currently handles derivatives in a highly optimized form of symbolic differentiation. The result can be interpreted as a hybrid of symbolic differentiation and reverse mode AD, but Theano does not use the general-purpose reverse accumulation as we describe in this paper. (Personal communication with the authors.)} \citep{Bastien2012}, TensorFlow \citep{abadi2016tensorflow}, Caffe \citep{jia2014caffe}, and CNTK \citep{seide2016cntk} the user first constructs a model as a computational graph using a domain-specific mini language, which then gets interpreted by the framework during execution. This approach has the advantage of enabling optimizations of the computational graph structure (e.g., as in Theano), but the disadvantages of having limited and unintuitive control flow and being difficult to debug. In contrast, the lineage of recent frameworks led by autograd \citep{maclaurin2016modeling}, Chainer \citep{tokui2015chainer}, and PyTorch \citep{paszke2017automatic} provide truly general-purpose reverse mode AD of the type we outline in Section~\ref{SectionPreliminaries}, where the user directly uses the host programming language to define the model as a regular program of the forward computation. This eliminates the need for an interpreter, allows arbitrary control flow statements, and makes debugging simple and intuitive. + +Simultaneously with the ongoing adoption of general-purpose AD in machine learning, we are witnessing a modeling-centric terminology emerge within the deep learning community. The terms \emph{define-and-run} and \emph{static computational graph} refer to Theano-like systems where a model is constructed, before execution, as a computational graph structure, which later gets executed with different inputs while remaining fixed. In contrast, the terms \emph{define-by-run} and \emph{dynamic computational graph} refer to the general-purpose AD capability available in newer PyTorch-like systems where a model is a regular program in the host programming language, whose execution dynamically constructs a computational graph on-the-fly that can freely change in each iteration.\footnote{Note that the terms ``static'' and ``dynamic'' here are used in the sense of having a fixed versus non-fixed computational graph topology and not in the sense of data flow architectures.} + +\emph{Differentiable programming}\footnote{A term advocated by Christopher Olah (\url{http://colah.github.io/posts/2015-09-NN-Types-FP/}), David Dalrymple (\url{https://www.edge.org/response-detail/26794}), and Yann LeCun (\url{https://www.facebook.com/yann.lecun/posts/10155003011462143}) from a deep learning point of view. Note the difference from \emph{differential} dynamic programming \citep{jacobson1970differential} in optimal control.} is another emerging term referring to the realization that deep learning practice essentially amounts to writing program templates of potential solutions to a problem, which are constructed as differentiable directed graphs assembled from functional blocks whose parameters are learned from examples using gradient-based optimization. Expressed in this paradigm, neural networks are just a class of parameterized differentiable programs composed of building blocks such as feed-forward, convolutional, and recurrent elements. We are increasingly seeing these traditional building blocks freely composed in arbitrary algorithmic structures using control flow, as well as the introduction of novel differentiable architectures such as the neural Turing machine \citep{graves2014neural}, a range of controller--interface abstractions \citep{graves2016hybrid,zaremba2016learning,joulin2015inferring,sukhbaatar2015end}, and differentiable versions of data structures such as stacks, queues, deques \citep{grefenstette2015learning}. Availability of general-purpose AD greatly simplifies the implementation of such architectures by enabling their expression as regular programs that rely on the differentiation infrastructure. Although the differentiable programming perspective on deep learning is new, we note that programming with differentiable functions and having differentiation as a language infrastructure has been the main research subject of the AD community for many decades and realized in a wide range of systems and languages as we shall see in Section~\ref{SectionImplementations}. + +There are instances in neural network literature---albeit few---where explicit reference has been made to AD for computing error gradients, such as \citet{Eriksson1998} using AD for large-scale feed-forward networks, and the work by \citet{Yang2008}, where the authors use AD to train a neural-network-based proportional-integral-derivative (PID) controller. Similarly, \citet{Rollins2009} uses reverse mode AD in conjunction with neural networks for the problem of optimal feedback control. Another example is given for continuous time recurrent neural networks (CTRNN) by \citet{AlSeyab2008}, where the authors apply AD for the training of CTRNNs predicting dynamic behavior of nonlinear processes in real time and report significantly reduced training time compared with other methods. + + +\subsection{Computer Vision} + +Since the influential work by \citet{krizhevsky2012imagenet}, computer vision has been dominated by deep learning, specifically, variations of convolutional neural networks \citep{lecun1998gradient}. These models are trained end-to-end, meaning that a mapping from raw input data to corresponding outputs is learned, automatically discovering the representations needed for feature detection in a process called representation learning \citep{bengio2013representation}. + +Besides deep learning, an interesting area where AD can be applied to computer vision problems is inverse graphics \citep{horn1977understanding,hinton1997generative}---or analysis-by-synthesis \citep{yildirim2015efficient}---where vision is seen as the inference of parameters for a generative model of a scene. Using gradient-based optimization in inverse graphics requires propagating derivatives through whole image synthesis pipelines including the renderer. \citet{eslami2016attend} use numerical differentiation for this purpose. \citet{loper2014opendr} implement the Open Differentiable Renderer (OpenDR), which is a scene renderer that also supplies derivatives of the image pixels with respect to scene parameters, and demonstrate it in the task of fitting an articulated and deformable 3D model of the human body to image and range data from a Kinect device. Similarly, \citet{kulkarni2015picture} implement a differentiable approximate renderer for the task of inference in probabilistic programs describing scenes. + +\citet{srajer2016benchmark} investigate the use of AD for three tasks in computer vision and machine learning, namely bundle adjustment \citep{triggs1999bundle}, Gaussian mixture model fitting, and hand tracking \citep{taylor2014user}, and provide a comprehensive benchmark of various AD tools for the computation of derivatives in these tasks. + +\citet{Pock2007} make use of AD in addressing the problems of denoising, segmentation, and recovery of information from stereoscopic image pairs, and note the usefulness of AD in identifying sparsity patterns in large Jacobian and Hessian matrices. In another study, \citet{Grabner2008} use reverse mode AD for GPU-accelerated medical 2D/3D registration, a task involving the alignment of data from different sources such as X-ray images or computed tomography. The authors report a six-fold increase in speed compared with numerical differentiation using center difference (cf.\ our benchmark with the Helmholtz function, Figure~\ref{FigureHelmholtz} and Table~\ref{TableHelmholtz}). + +\citet{Barrett2013} present a use of general-purpose AD for the task of video event detection using hidden Markov models (HMMs) and \citet{Dalal2005} object detectors, performing training on a corpus of pre-tracked video using an adaptive step size gradient descent with reverse mode AD. Initially implemented with the R6RS-AD package\footnote{\url{https://github.com/qobi/R6RS-AD}} which provides forward and reverse mode AD in Scheme, the resulting gradient code was later ported to C and highly optimized.\footnote{Personal communication.} + + +\subsection{Natural Language Processing} + +Natural language processing (NLP) constitutes one of the areas where rapid progress is being made by applying deep learning techniques \citep{goldberg2016primer}, with applications in tasks including machine translation \citep{bahdanau2014neural}, language modeling \citep{mikolov2010recurrent}, dependency parsing \citep{chen2014fast}, and question answering \citep{pmlr-v48-kumar16}. Besides deep learning approaches, statistical models in NLP are commonly trained using general purpose or specialized gradient-based methods and mostly remain expensive to train. Improvements in training time can be realized by using online or distributed training algorithms \citep{Gimpel2010}. An example using stochastic gradient descent for NLP is given by \citet{Finkel2008} optimizing conditional random field parsers through an objective function. Related with the work on video event detection in the previous section, \citet{Yu2013} report their work on sentence tracking, representing an instance of grounded language learning paired with computer vision, where the system learns word meanings from short video clips paired with descriptive sentences. The method uses HMMs to represent changes in video frames and meanings of different parts of speech. This work is implemented in C and computes the required gradients using AD through the ADOL-C tool.\footnote{An implementation of the sentence tracker applied to video search using sentence-based queries can be accessed online: \url{http://upplysingaoflun.ecn.purdue.edu/~qobi/cccp/sentence-tracker-video-retrieval.html}} + +\subsection{Probabilistic Modeling and Inference} + +Inference in probabilistic models can be static, such as compiling a given model to Bayesian networks and using algorithms such as belief propagation for inference; or they can be dynamic, executing a model forward many times and computing statistics on observed values to infer posterior distributions. Markov chain Monte Carlo (MCMC) \citep{Neal1993} methods are often used for dynamic inference, such as the Metropolis--Hastings algorithm based on random sampling \citep{Chib1995}. \citet{Meyer2003} give an example of how AD can be used to speed up Bayesian posterior inference in MCMC, with an application in stochastic volatility. Amortized inference \citep{gershman2014amortized,stuhlmuller2013learning} techniques based on deep learning \citep{le2016inference,ritchie2016deep} work by training neural networks for performing approximate inference in generative models defined as probabilistic programs \citep{gordon2014probabilistic}. + +When model parameters are continuous, the Hamiltonian---or, hybrid---Monte Carlo (HMC) algorithm provides improved convergence characteristics avoiding the slow exploration of random sampling, by simulating Hamiltonian dynamics through auxiliary ``momentum variables'' \citep{Duane1987}. The advantages of HMC come at the cost of requiring gradient evaluations of complicated probability models. AD is highly suitable here for complementing probabilistic modeling, because it relieves the user from the manual derivation of gradients for each model.\footnote{See \url{http://diffsharp.github.io/DiffSharp/examples-hamiltonianmontecarlo.html} for an implementation of HMC with reverse mode AD.} For instance, the probabilistic programming language Stan \citep{carpenter2016stan} implements automatic Bayesian inference based on HMC and the No-U-Turn sampler (NUTS) \citep{Hoffman2014} and uses reverse mode AD for the calculation of gradients for both HMC and NUTS \citep{carpenter2015stan}. Similarly, \citet{Wingate2011} demonstrate the use of AD as a non-standard interpretation of probabilistic programs enabling efficient inference algorithms. \citet{kucukelbir2017automatic} present an AD-based method for deriving variational inference (VI) algorithms. + +PyMC3 \citep{salvatier2016probabilistic} allows fitting of Bayesian models using MCMC and VI, for which it uses gradients supplied by Theano. Edward \citep{tran2016edward} is a library for deep probabilistic modeling, inference, and criticism \citep{tran2017deep} that supports VI using TensorFlow. Availability of general-purpose AD in this area has enabled new libraries such as Pyro\footnote{\url{http://pyro.ai/}} and ProbTorch \citep{siddharth2017learning} for deep \emph{universal} probabilistic programming with support for recursion and control flow, relying, in both instances, on VI using gradients supplied by PyTorch's reverse mode AD infrastructure. + +When working with probabilistic models, one often needs to backpropagate derivatives through sampling operations of random variables in order to achieve stochastic optimization of model parameters. The score-function estimator, or REINFORCE \citep{williams1992simple}, method provides a generally applicable unbiased gradient estimate, albeit with high variance. When working with continuous random variables, one can substitute a random variable by a deterministic and differentiable transformation of a simpler random variable, a method known as the ``reparameterization trick'' \citep{williams1992simple,kingma2014auto,rezende2014stochastic}. For discrete variables, the REBAR \citep{tucker2017rebar} method provides a lower-variance unbiased gradient estimator by using continuous relaxation. A generalization of REBAR called RELAX \citep{grathwohl2017backpropagation} works by learning a free-form control variate parameterized by a neural network and is applicable in both discrete and continuous settings. + +\section{Implementations} +\label{SectionImplementations} + +It is useful to have an understanding of the different ways in which AD can be implemented. Here we cover major implementation strategies and provide a survey of existing tools. + +A principal consideration in any AD implementation is the performance overhead introduced by the AD arithmetic and bookkeeping. In terms of computational complexity, AD guarantees that the amount of arithmetic goes up by no more than a small constant factor \citep{Griewank2008}. On the other hand, managing this arithmetic can introduce a significant overhead if done carelessly. For instance, naïvely allocating data structures for holding dual numbers will involve memory access and allocation for every arithmetic operation, which are usually more expensive than arithmetic operations on modern computers.\footnote{The implementation of forward mode in Julia \citep{revels2016forward} attempts to avoid this, and some current compilers can avoid this expense by unboxing dual numbers \citep{leroy1997effectiveness, jones1993glasgow, jones1991unboxed, siskind2016efficient}. This method is also used to reduce the memory-access overhead in the implementations of forward mode in Stalingrad and the Haskell \emph{ad} library.} Likewise, using operator overloading may introduce method dispatches with attendant costs, which, compared to raw numerical computation of the original function, can easily amount to a slowdown of an order of magnitude.\footnote{Flow analysis \citep{shivers1991control} and/or partial evaluation \citep{jones1993partial}, together with tag stripping \citep{appel1989runtime, peterson1989untagged}, can remove this method dispatch. These, together with unboxing, can often make it possible to completely eliminate the memory access, memory allocation, memory reclamation, and method dispatch overhead of dual numbers \citep{siskind2016efficient}.} + +Another major issue is the risk of hitting a class of bugs called ``perturbation confusion'' \citep{SiskindPearlmutter2005a,manzyuk2012confusion}. This essentially means that if two ongoing differentiations affect the same piece of code, the two formal epsilons they introduce (Section~\ref{SectionDualNumbers}) need to be kept distinct. It is very easy to have bugs---particularly in performance-oriented AD implementations---that confuse these in various ways. Such situations can also arise when AD is nested, that is, derivatives are computed for functions that internally compute derivatives. + +Translation of mathematics into computer code often requires attention to numeric issues. For instance, the mathematical expressions $\log(1 + x)$ or $\sqrt{x^2+y^2+z^2}$ or $\tan^{-1}(y/x)$ should not be naïvely translated, but rather expressed as \texttt{log1p(x)}, \texttt{hypot(x,hypot(y,z))}, and \texttt{atan2(y,x)}. In machine learning, the most prominent example of this is probably the so-called log-sum-exp trick to improve the numerics of calculations of the form $\log\sum_i \exp x_i$. AD is not immune to such numeric considerations. For example, code calculating $E=\sum_i E_i$, processed by AD, will calculate $\nabla_w E=\sum_i\nabla_w E_i$. If the system is seeking a local minimum of $E$ then $\nabla_w E = \sum_i\nabla_w E_i \rightarrow_t 0$, and naïvely adding a set of large numbers whose sum is near zero is numerically fraught. This is to say that AD is not immune to the perils of floating point arithmetic, and can sometimes introduce numeric issues which were not present in the primal calculation. Issues of numeric analysis are outside our present scope, but there is a robust literature on the numerics of AD (e.g., \citet{griewank2012numerical}) involving using subgradients to allow optimization to proceed despite non-differentiability of the objective, appropriate subgradients and approximations for functions like $\lvert \cdot \rvert$ and $\lVert\cdot\rVert_2$ and $\sqrt{\cdot}$ near zero, and a spate of related issues. + +One should also be cautious about approximated functions and AD \citep{sirkes-tziperman-1997a}. In this case, if one has a procedure \emph{approximating} an ideal function, AD always gives the derivative of the procedure that was actually programmed, which may not be a good approximation of the derivative of the ideal function that the procedure was approximating. For instance, consider $e^x$ computed by a piecewise-rational approximation routine. Using AD on this routine would produce an approximated derivative in which each piece of the piecewise formula will get differentiated. Even if this would remain an approximation of the derivative of $e^x$, we know that $\frac{de^x}{dx} = e^x$ and the original approximation itself was already a better approximation for the derivative of $e^x$.\footnote{In modern systems this is not an issue, because $e^x$ is a primitive implemented in hardware.} Users of AD implementations must be therefore cautious to \emph{approximate the derivative, not differentiate the approximation}. This would require explicitly approximating a known derivative, in cases where a mathematical function can only be computed approximately but has a well-defined mathematical derivative. + +We note that there are similarities as well as differences between machine learning workloads and those studied in the traditional AD literature \citep{baydin2016tricks}. Deep learning systems are generally compute-bound and spend a considerable amount of computation time in highly-optimized numerical kernels for matrix operations \citep{hadjis2015caffe,chetlur2014cudnn}. This is a situation which is arguably amenable to operator-overloading-based AD implementations on high-level operations, as is commonly found in current machine learning frameworks. In contrast, numerical simulation workloads in traditional AD applications can be bandwidth-bound, making source code transformation and compiler optimization approaches more relevant. Another difference worth noting is that whereas high numerical precision is desirable in traditional application domains of AD such as computational fluid dynamics \citep{cohen2009fast}, in deep learning lower-precision is sufficient and even desirable in improving computational efficiency, thanks to the error resiliency of neural networks \citep{gupta2015deep,courbariaux2015binaryconnect}. + +There are instances in recent literature where implementation-related experience from the AD field has been put to use in machine learning settings. One particular area of recent interest is implicit and iterative AD techniques \citep{Griewank2008}, which has found use in work incorporating constrained optimization within deep learning \citep{amos2017optnet} and probabilistic graphical models and neural networks \citep{johnson2016composing}. Another example is checkpointing strategies \citep{Dauvergne2006,siskind2017divide}, which allow balancing of application-specific trade-offs between time and space complexities of reverse mode AD by not storing the full tape of intermediate variables in memory and reconstructing these as needed by re-running parts of the forward computation from intermediate checkpoints. This is highly relevant in deep learning workloads running on GPUs with limited memory budgets. A recent example in this area is the work by \citet{gruslys2016memory}, where the authors construct a checkpointing variety of the backpropagation through time (BPTT) algorithm for recurrent neural networks and demonstrate it saving up to 95\% memory usage at the cost of a 33\% increase in computation time in one instance. + +In Table~\ref{TableADImplementations} we present a review of notable general-purpose AD implementations.\footnote{Also see the website \url{http://www.autodiff.org/} for a list of tools maintained by the AD community.} A thorough taxonomy of implementation techniques was introduced by \citet{Juedes1991}, which was later revisited by \citet{Bischof2008} and simplified into \emph{elemental}, \emph{operator overloading}, \emph{compiler-based}, and \emph{hybrid} methods. We adopt a similar classification for the following part of this section. + +% Remarks concerning further implementation methods in \citet{Gay2006}. + +\addtolength{\tabcolsep}{-3pt} +\begin{sidewaystable} + \centering + \renewcommand{\arraystretch}{1.5} + \caption{Survey of AD implementations. Tools developed primarily for machine learning are highlighted in bold.} + \label{TableADImplementations} + {\tiny + \begin{tabularx}{\textwidth}{@{}p{12mm}p{20mm}p{5mm}p{6mm}p{62mm}p{38mm}p{65mm}@{}} + \toprule + Language & Tool & Type & Mode & Institution / Project & Reference & URL\\ + \midrule + AMPL & AMPL & INT & F, R & Bell Laboratories & \citet{Fourer2002} & \tiny\url{http://www.ampl.com/}\\ + C, C++ & ADIC & ST & F, R & Argonne National Laboratory & \citet{Bischof1997} & \tiny\url{http://www.mcs.anl.gov/research/projects/adic/}\\ + & ADOL-C & OO & F, R & Computational Infrastructure for Operations Research & \citet{Walther2012} & \tiny\url{https://projects.coin-or.org/ADOL-C}\\ + C++ & Ceres Solver & LIB & F & Google & & \tiny\url{http://ceres-solver.org/}\\ + & CppAD & OO & F, R & Computational Infrastructure for Operations Research & \citet{Bell2008} & \tiny\url{http://www.coin-or.org/CppAD/}\\ + & FADBAD++ & OO & F, R & Technical University of Denmark & \citet{Bendtsen1996} & \tiny\url{http://www.fadbad.com/fadbad.html}\\ + & Mxyzptlk & OO & F & Fermi National Accelerator Laboratory & \citet{Ostiguy2007} & \\ + C\# & AutoDiff & LIB & R & George Mason Univ., Dept. of Computer Science & \citet{Shtof2013} & \tiny\url{http://autodiff.codeplex.com/}\\ + F\#, C\# & \textbf{DiffSharp} & OO & F, R & Maynooth University, Microsoft Research Cambridge & \citet{baydin2016diffsharp} & \tiny\url{http://diffsharp.github.io}\\ + Fortran & ADIFOR & ST & F, R & Argonne National Laboratory & \citet{Bischof1996} & \tiny\url{http://www.mcs.anl.gov/research/projects/adifor/}\\ + & NAGWare & COM & F, R & Numerical Algorithms Group & \citet{Naumann2005} & \tiny\url{http://www.nag.co.uk/nagware/Research/ad_overview.asp}\\ + & TAMC & ST & R & Max Planck Institute for Meteorology & \citet{Giering1998} & \tiny\url{http://autodiff.com/tamc/}\\ + Fortran, C & COSY & INT & F & Michigan State Univ., Biomedical and Physical Sci. & \citet{Berz1996} & \tiny\url{http://www.bt.pa.msu.edu/index_cosy.htm}\\ + & Tapenade & ST & F, R & INRIA Sophia-Antipolis & \citet{Hascoet2013} & \tiny\url{http://www-sop.inria.fr/tropics/tapenade.html}\\ + Haskell & ad & OO & F, R & Haskell package & & \tiny\url{http://hackage.haskell.org/package/ad}\\ + Java & ADiJaC & ST & F, R & University Politehnica of Bucharest & \citet{slusanschi2016adijac} & \tiny\url{http://adijac.cs.pub.ro}\\ + & Deriva & LIB & R & Java \& Clojure library & & \tiny\url{https://github.com/lambder/Deriva}\\ + Julia & JuliaDiff & OO & F, R & Julia packages & \citet{RevelsLubinPapamarkou2016} & \tiny\url{http://www.juliadiff.org/}\\ + Lua & \textbf{torch-autograd} & OO & R & Twitter Cortex & & \tiny\url{https://github.com/twitter/torch-autograd}\\ + MATLAB & ADiMat & ST & F, R & Technical University of Darmstadt, Scientific Comp. & \citet{Willkomm2013} & \tiny\url{http://adimat.sc.informatik.tu-darmstadt.de/}\\ + & INTLab & OO & F & Hamburg Univ. of Technology, Inst. for Reliable Comp. & \citet{Rump1999} & \tiny\url{http://www.ti3.tu-harburg.de/rump/intlab/}\\ + & TOMLAB/MAD & OO & F & Cranfield University \& Tomlab Optimization Inc. & \citet{Forth2006} & \tiny\url{http://tomlab.biz/products/mad}\\ + Python & ad & OO & R & Python package & & \tiny\url{https://pypi.python.org/pypi/ad}\\ + & \textbf{autograd} & OO & F, R & Harvard Intelligent Probabilistic Systems Group & \citet{maclaurin2016modeling} & \tiny\url{https://github.com/HIPS/autograd}\\ + & \textbf{Chainer} & OO & R & Preferred Networks & \citet{tokui2015chainer} & \tiny\url{https://chainer.org/}\\ + & \textbf{PyTorch} & OO & R & PyTorch core team & \citet{paszke2017automatic} & \tiny\url{http://pytorch.org/}\\ + & \textbf{Tangent} & ST & F, R & Google Brain & \citet{van2017tangent} & \tiny\url{https://github.com/google/tangent}\\ + Scheme & R6RS-AD & OO & F, R & Purdue Univ., School of Electrical and Computer Eng. & & \tiny\url{https://github.com/qobi/R6RS-AD}\\ + & Scmutils & OO & F & MIT Computer Science and Artificial Intelligence Lab. & \citet{Sussman2001} & \tiny\url{http://groups.csail.mit.edu/mac/users/gjs/6946/refman.txt}\\ + & Stalingrad & COM & F, R & Purdue Univ., School of Electrical and Computer Eng. & \citet{pearlmutter2008reverse} & \tiny\url{http://www.bcl.hamilton.ie/~qobi/stalingrad/}\\ + % Odyssee reverse mode source transformation (predecessor of Tapenade) + % TAMC reverse mode source transformation + \bottomrule + \addlinespace + \multicolumn{7}{l}{F: Forward, R: Reverse; COM: Compiler, INT: Interpreter, LIB: Library, OO: Operator overloading, ST: Source transformation} + \end{tabularx} + } +\end{sidewaystable} +\addtolength{\tabcolsep}{3pt} + +\subsection{Elemental Libraries} + +These implementations form the most basic category and work by replacing mathematical operations with calls to an AD-enabled library. Methods exposed by the library are then used in function definitions, meaning that the decomposition of any function into elementary operations is done manually when writing the code. + +The approach has been utilized since the early days of AD, with prototypical examples being the WCOMP and UCOMP packages of \citet{Lawson1971}, the APL package of \citet{Neidinger1989}, and the work by \citet{Hinkins1994}. Likewise, \citet{Hill1992} formulate their implementation of AD in MATLAB using elemental methods. + +Elemental libraries still constitute the simplest strategy to implement AD for languages without operator overloading. + +\subsection{Compilers and Source Code Transformation} + +These implementations provide extensions to programming languages that automate the decomposition of algorithms into AD-enabled elementary operations. They are typically executed as preprocessors\footnote{Preprocessors transform program source code before it is given as an input to a compiler.} to transform the input in the extended language into the original language. + +Classical instances of source code transformation include the Fortran preprocessors GRESS \citep{Horwedel1988} and PADRE2 \citep{Kubo1990}, which transform AD-enabled variants of Fortran into standard Fortran 77 before compiling. Similarly, the ADIFOR tool \citep{Bischof1996}, given a Fortran source code, generates an augmented code in which all specified partial derivatives are computed in addition to the original result. For procedures coded in ANSI C, the ADIC tool \citep{Bischof1997} implements AD as a source code transformation after the specification of dependent and independent variables. A recent and popular tool also utilizing this approach is Tapenade \citep{Pascual2008,Hascoet2013}, implementing forward and reverse mode AD for Fortran and C programs. Tapenade itself is implemented in Java and can be run locally or as an online service.\footnote{\url{http://www-tapenade.inria.fr:8080/tapenade/index.jsp}} + +In addition to language extensions through source code transformation, there are implementations introducing new languages with tightly integrated AD capabilities through special-purpose compilers or interpreters. Some of the earliest AD tools such as SLANG \citep{Adamson1969} and PROSE \citep{Pfeiffer1987} belong to this category. The NAGWare Fortran 95 compiler \citep{Naumann2005} is a more recent example, where the use of AD-related extensions triggers automatic generation of derivative code at compile time. + +As an example of interpreter-based implementation, the algebraic modeling language AMPL \citep{Fourer2002} enables objectives and constraints to be expressed in mathematical notation, from which the system deduces active variables and arranges the necessary AD computations. Other examples in this category include the FM/FAD package \citep{Mazourik1991}, based on the Algol-like DIFALG language, and the object-oriented COSY language \citep{Berz1996} similar to Pascal. + +The Stalingrad compiler \citep{pearlmutter2008reverse,Siskind2008}, working on the Scheme-based AD-aware VLAD language, also falls under this category. The newer DVL compiler\footnote{\url{https://github.com/axch/dysvunctional-language}} is based on Stalingrad and uses a reimplementation of portions of the VLAD language. + +Motivated by machine learning applications, the Tangent library \citep{van2017tangent} implements AD using source code transformation, and accepts numeric functions written in a syntactic subset of Python and Numpy. + +\subsection{Operator Overloading} + +In modern programming languages with polymorphic features, operator overloading provides the most straightforward way of implementing AD, exploiting the capability of redefining elementary operation semantics. + +A popular tool implemented with operator overloading in C++ is ADOL-C \citep{Walther2012}. ADOL-C requires the use of AD-enabled types for variables, and records arithmetic operations on variables in tape data structures, which can subsequently be ``played back'' during reverse mode AD computations. The Mxyzptlk package \citep{Michelotti1990} is another example for C++ capable of computing arbitrary-order partial derivatives via forward propagation. The FADBAD++ library \citep{Bendtsen1996} implements AD for C++ using templates and operator overloading. For Python, the \emph{ad} package\footnote{\url{http://pythonhosted.org/ad/}} uses operator overloading to compute first- and second-order derivatives, while the newer autograd package\footnote{\url{https://github.com/HIPS/autograd}} provides forward and reverse mode AD with support for higher-order derivatives. + +For functional languages, examples include R6RS-AD\footnote{\url{https://github.com/NUIM-BCL/R6RS-AD}} and the AD routines within the Scmutils library\footnote{\url{http://groups.csail.mit.edu/mac/users/gjs/6946/refman.txt}} for Scheme, the \emph{ad} library\footnote{\url{http://hackage.haskell.org/package/ad}} for Haskell, and DiffSharp\footnote{\url{http://diffsharp.github.io}} for F\# and C\#. + +%\subsection{Parallelization} +% Discuss advantages of AD for parallelization. +%Parallel implementation of Hessian calculations, automated generation of parallel code \citep{Bucker2008} +%\citet{Bischof2008} give an example of parallel reverse mode AD for a plasma simulation code + +\section{Conclusions} +\label{SectionConclusions} +Backpropagation and gradient-based optimization are behind virtually all recent successes in machine learning, yielding state-of-the-art results in computer vision, speech recognition and synthesis, and machine translation. We expect these techniques to remain at the core of machine learning for the foreseeable future. Research in the field involves a rapid prototyping and development cycle for testing new models and ideas, using a collection of increasingly higher-quality machine learning frameworks. These frameworks are in the process of transition from coarse-grained (module level) backpropagation towards fine-grained, general-purpose AD, allowing models to be implemented as regular programs in general-purpose programming languages with differentiation as an integral part of the infrastructure. We strongly believe that general-purpose AD is the future of gradient-based machine learning and we expect it to become an indispensable tool in the machine learning toolbox. + +It is an exciting time for working at the intersection of AD and machine learning, and there are many opportunities for bringing advanced techniques and expertise from AD literature to bear on machine learning problems. Techniques that have been developed by the AD community such as tape reduction and elimination \citep{naumann2004optimal}, fixed-point iterations \citep{christianson1994reverse}, utilizing sparsity by matrix coloring \citep{Gebremedhin2009,gebremedhin2013colpack}, and reverse AD checkpointing \citep{Dauvergne2006} are just a few examples that can find potential use in machine learning for increasing performance, improving convergence of optimization, using hardware more efficiently, and even enabling new types of machine learning models to be implemented. Similarly, exciting new AD modes like direct propagation of the inverse Jacobian \citep{srinivasan-todorov-2015a} have emerged from the machine learning community, but have yet to be examined and formalized by the AD community. + +An important direction for future work is to make use of nested AD techniques in machine learning, allowing differentiation to be nested arbitrarily deep with referential transparency \citep{Siskind2008b,pearlmutter2008reverse}. Nested AD is highly relevant in hyperparameter optimization as it can effortlessly provide exact hypergradients, that is, derivatives of a training objective with respect to the hyperparameters of an optimization routine \citep{Maclaurin2015,baydin2017online}. Potential applications include Bayesian model selection \citep{Rasmussen2006} and gradient-based tuning of Hamiltonian Monte Carlo step sizes and mass matrices \citep{Salimans2014}. Besides hyperparameters, models internally using higher-order derivatives constitute a straightforward usage case for nested AD. The Riemannian manifold Langevin and Hamiltonian Monte Carlo methods \citep{Girolami2011} use higher-order derivative information to more closely track the information geometry of the sampled distribution for faster convergence and exploration. In neural networks, it is very natural to use nested derivatives in defining objective functions that take input transformations into account, such as the Tangent Prop method \citep{Simard1998} for imposing invariance under a set of chosen transformations. + +% Acknowledgements should go at the end, before appendices and references + +\acks{We thank the anonymous reviewers whose comments helped improve this manuscript. This work was supported, in part, by Science Foundation Ireland grant 09/IN.1/I2637, by the Army Research Laboratory, accomplished under Cooperative Agreement Number W911NF-10-2-0060, by the National Science Foundation under Grants 1522954-IIS and 1734938-IIS, and by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/Interior Business Center (DOI/IBC) contract number D17PC00341. Any opinions, findings, views, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views, official policies, or endorsements, either expressed or implied, of the sponsors. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes, notwithstanding any copyright notation herein. +} + +% Manual newpage inserted to improve layout of sample file - not +% needed in general before appendices/bibliography. +% +% \newpage +% +% \appendix +% \section*{Appendix A.} +% \label{app:theorem} + +% Note: in this sample, the section number is hard-coded in. Following +% proper LaTeX conventions, it should properly be coded as a reference: + +%In this appendix we prove the following theorem from +%Section~\ref{sec:textree-generalization}: + +\vskip 0.2in +\bibliography{17-468} + +\end{document} diff --git a/doc/Articles/AutodiffArticle b/doc/Articles/Autodiff/adiff.tar similarity index 100% rename from doc/Articles/AutodiffArticle rename to doc/Articles/Autodiff/adiff.tar diff --git a/doc/Articles/Autodiff/figures/approx-error/approx-error.tex b/doc/Articles/Autodiff/figures/approx-error/approx-error.tex new file mode 100644 index 000000000..3dec348fd --- /dev/null +++ b/doc/Articles/Autodiff/figures/approx-error/approx-error.tex @@ -0,0 +1,2189 @@ +% Created by tikzDevice version 0.8.1 on 2015-04-18 01:33:45 +% !TEX encoding = UTF-8 Unicode +\begin{tikzpicture}[x=1pt,y=1pt] +\definecolor{fillColor}{RGB}{255,255,255} +\path[use as bounding box,fill=fillColor,fill opacity=0.00] (0,0) rectangle (325.21,216.81); +\begin{scope} +\path[clip] ( 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b/doc/Articles/Autodiff/figures/differentiation/differentiation.tex @@ -0,0 +1,79 @@ +\documentclass[tikz]{standalone} +\usepackage{framed} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{xcolor} + +\usetikzlibrary{decorations.pathmorphing} +\usetikzlibrary{arrows} + +\begin{document} +\begin{tikzpicture}[ + pencildraw/.style={ + decorate, + decoration={random steps,segment length=2pt,amplitude=1pt} + }] + %\draw[help lines] (0,0) grid (20,10); + \tikzstyle{paperbox} = [pencildraw,draw,thick,fill=white,text width=7cm,inner sep=5mm] + \tikzstyle{codebox} = [draw,thick,text width=7cm,inner sep=5mm] + \tikzstyle{edge} = [->,>=triangle 60,thick] + + \node[paperbox] (a) at (-5.5,-12.5) { + $l_1=x$\\ + $l_{n+1}=4l_n(1-l_n)$\\ + \vspace{4mm} + $f(x)=l_4=64x(1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2$ + }; + \node[paperbox,fill=gray!20] (b) at (5.5,-12.5) {$f'(x)=128x(1 - x)(-8 + 16 x)(1 - 2 x)^2(1 - 8 x + 8 x^2) + 64 (1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 64x(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 256x(1 - x)(1 - 2 x)(1 - 8 x + 8 x^2)^2$}; + \node[codebox,fill=white] (c) at (-5.5,-18.5) {\texttt{\parbox{7cm}{ + f(x):\\ + \hphantom{tt} v = x\\ + \hphantom{tt} for i = 1 to 3\\ + \hphantom{tttt} v = 4*v*(1 - v)\\ + \hphantom{tt} return v\\ + \hphantom{t}\\ + \textrm{or, in closed-form,}\\ + \hphantom{t}\\ + f(x):\\ + \hphantom{tt}\parbox{6cm}{\hangindent=0.4cm \hangafter=1 return 64*x*(1-x)*((1-2*x)\^{}2)\\ *(1-8*x+8*x*x)\^{}2}\\ + }}}; + \node[codebox,black,fill=gray!20] (d) at (5.5,-18.5) {\color{black}\texttt{\parbox{7cm}{\textbf{ + f'(x):\\ + \hphantom{tt}\parbox{6cm}{\hangindent=0.4cm \hangafter=1 return 128*x*(1 - x)*(-8 + 16*x)\\ *((1 - 2*x)\^{}2)*(1 - 8*x + 8*x*x)\\+ 64*(1 - x)*((1 - 2*x)\^{}2)*((1 - 8*x + 8*x*x)\^{}2) - (64*x*(1 - 2*x)\^{}2)*(1 - 8*x + 8*x*x)\^{}2 - 256*x*(1 - x)*(1 - 2*x)*(1 - 8*x + 8*x*x)\^{}2}\\} + \flushright \color{gray} f'($\mathtt{x_0}$) $= f'(x_0)$\\\textrm{Exact} + }}}; + \node[codebox,black,fill=gray!20] (e) at (-5.5,-25.5) {\color{black}\texttt{\parbox{7cm}{\textbf{ + f'(x):\\ + \hphantom{tt} (v,dv) = (x,1)\\ + \hphantom{tt} for i = 1 to 3\\ + \hphantom{tttt} (v,dv) = (4*v*(1-v), 4*dv-8*v*dv)\\ + \hphantom{tt} return (v,dv)\\} + \flushright \color{gray} f'($\mathtt{x_0}$) $= f'(x_0)$\\\textrm{Exact} + }}}; + \node[codebox,black,fill=gray!20] (f) at (5.5,-25.5) {\color{black}\texttt{\parbox{7cm}{\textbf{ + f'(x):\\ + \hphantom{tt} h = 0.000001\\ + \hphantom{tt} return (f(x + h) - f(x)) / h\\} + \flushright \color{gray} f'($\mathtt{x_0}$) $\approx f'(x_0)$\\\textrm{Approximate} + }}}; + + \draw (a) edge [edge] (b); + \node[align=center,below] at (0,-12.5) {Manual\\Differentiation}; + + \draw (c) edge [edge] (d); + \node[align=center,below] at (0,-18.5) {Symbolic\\Differentiation\\of the Closed-form}; + + \draw (a) edge [edge] (c); + \node[align=left,right] at (-5.5,-15) {Coding}; + + \draw (b) edge [edge] (d); + \node[align=left,right] at (5.5,-15) {Coding}; + + \draw (c) edge [edge] (f); + 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scale= 1.00] at ( 39.60,206.57) {0}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,247.93) {1000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,289.28) {2000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,330.64) {3000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,371.99) {4000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,413.35) {5000}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,198.30) -- + (313.21,198.30) -- + (313.21,421.62) -- + ( 54.00,421.62) -- + ( 54.00,198.30); +\end{scope} +\begin{scope} +\path[clip] ( 54.00,198.30) rectangle (313.21,421.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] ( 63.60,206.62) -- + ( 97.89,208.04) -- + (132.18,213.88) -- + (166.46,227.23) -- + (200.75,249.80) -- + (235.04,288.73) -- + (269.33,341.78) -- + (303.61,413.18); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,203.80) -- + ( 66.42,206.62) -- + ( 63.60,209.44) -- + ( 60.78,206.62) -- + ( 63.60,203.80); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89,205.22) -- + (100.71,208.04) -- + ( 97.89,210.86) -- + ( 95.07,208.04) -- + ( 97.89,205.22); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,211.06) -- + (135.00,213.88) -- + (132.18,216.70) -- + (129.36,213.88) -- + (132.18,211.06); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,224.41) -- + (169.28,227.23) -- + (166.46,230.05) -- + (163.64,227.23) -- + (166.46,224.41); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,246.98) -- + (203.57,249.80) -- + (200.75,252.62) -- + (197.93,249.80) -- + (200.75,246.98); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,285.91) -- + (237.86,288.73) -- + (235.04,291.55) -- + (232.22,288.73) -- + (235.04,285.91); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,338.96) -- + (272.15,341.78) -- + (269.33,344.60) -- + (266.51,341.78) -- + (269.33,338.96); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,410.36) -- + (306.43,413.18) -- + (303.61,416.00) -- + (300.79,413.18) -- + (303.61,410.36); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] ( 63.60,206.63) -- + ( 97.89,207.14) -- + (132.18,208.70) -- + (166.46,212.04) -- + (200.75,216.96) -- + (235.04,226.00) -- + (269.33,240.30) -- + (303.61,262.07); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,210.13) -- + ( 66.63,204.88) -- + ( 60.57,204.88) -- + ( 63.60,210.13); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89,210.64) -- + (100.92,205.39) -- + ( 94.86,205.39) -- + ( 97.89,210.64); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,212.20) -- + (135.21,206.95) -- + (129.15,206.95) -- + (132.18,212.20); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,215.54) -- + (169.49,210.29) -- + (163.43,210.29) -- + (166.46,215.54); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,220.46) -- + (203.78,215.22) -- + (197.72,215.22) -- + (200.75,220.46); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,229.50) -- + (238.07,224.25) -- + (232.01,224.25) -- + (235.04,229.50); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,243.80) -- + (272.36,238.55) -- + (266.30,238.55) -- + (269.33,243.80); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,265.57) -- + (306.64,260.32) -- + (300.58,260.32) -- + (303.61,265.57); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt on 4pt off 3pt ,line join=round,line cap=round] ( 63.60,206.63) -- + ( 97.89,207.03) -- + (132.18,207.87) -- + (166.46,209.35) -- + (200.75,211.29) -- + (235.04,213.79) -- + (269.33,217.08) -- + (303.61,220.73); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,203.14) -- + ( 66.63,208.38) -- + ( 60.57,208.38) -- + ( 63.60,203.14); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89,203.53) -- + (100.92,208.78) -- + ( 94.86,208.78) -- + ( 97.89,203.53); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,204.37) -- + (135.21,209.62) -- + (129.15,209.62) -- + (132.18,204.37); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,205.85) -- + (169.49,211.10) -- + (163.43,211.10) -- + (166.46,205.85); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,207.79) -- + (203.78,213.03) -- + (197.72,213.03) -- + (200.75,207.79); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,210.29) -- + (238.07,215.54) -- + (232.01,215.54) -- + (235.04,210.29); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,213.58) -- + (272.36,218.83) -- + (266.30,218.83) -- + (269.33,213.58); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,217.23) -- + (306.64,222.48) -- + (300.58,222.48) -- + (303.61,217.23); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,411.28) rectangle (207.00,351.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 66.30,399.28) -- ( 84.30,399.28); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] ( 66.30,387.28) -- ( 84.30,387.28); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] ( 66.30,375.28) -- ( 84.30,375.28); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt on 4pt off 3pt ,line join=round,line cap=round] ( 66.30,363.28) -- ( 84.30,363.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 73.31,397.29) rectangle ( 77.29,401.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 75.30,384.46) -- + ( 78.12,387.28) -- + ( 75.30,390.10) -- + ( 72.48,387.28) -- + ( 75.30,384.46); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 75.30,378.78) -- + ( 78.33,373.53) -- + ( 72.27,373.53) -- + ( 75.30,378.78); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 75.30,359.78) -- + ( 78.33,365.03) -- + ( 72.27,365.03) -- + ( 75.30,359.78); + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,395.84) {$f$, original function}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,383.84) {$\nabla f$, numerical diff.}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,371.84) {$\nabla f$, forward AD}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,359.84) {$\nabla f$, reverse AD}; +\end{scope} +\begin{scope} +\path[clip] ( 54.00, 54.00) rectangle (313.21,185.10); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60, 58.86) -- + ( 97.89, 82.16) -- + (132.18, 96.98) -- + (166.46,106.90) -- + (200.75,115.33) -- + (235.04,122.01) -- + (269.33,127.93) -- + (303.61,132.42); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 61.61, 56.86) rectangle ( 65.59, 60.85); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 95.89, 80.16) rectangle ( 99.88, 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join=round,line cap=round] ( 58.70, 54.00) -- ( 58.70, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (107.68, 54.00) -- (107.68, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (156.67, 54.00) -- (156.67, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (205.65, 54.00) -- (205.65, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (254.63, 54.00) -- (254.63, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61, 54.00) -- (303.61, 48.00); + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 58.70, 32.40) {0}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (107.68, 32.40) {10}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (156.67, 32.40) {20}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (205.65, 32.40) {30}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (254.63, 32.40) {40}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (303.61, 32.40) {50}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 54.00) -- + (313.21, 54.00) -- + (313.21,185.10) -- + ( 54.00,185.10) -- + ( 54.00, 54.00); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,197.10); +\definecolor{drawColor}{RGB}{0,0,0} + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (183.61, 8.40) {$n$}; +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,433.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 58.86) -- ( 54.00,157.31); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 58.86) -- ( 48.00, 58.86); + 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west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 35.51,127.74) {2}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,150.56) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 35.51,160.56) {3}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 55.68) -- ( 54.00,185.04); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 55.68) -- ( 50.40, 55.68); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 57.35) -- ( 50.40, 57.35); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 68.73) -- ( 50.40, 68.73); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 74.51) -- ( 50.40, 74.51); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 78.61) -- ( 50.40, 78.61); + 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Smola (we should set up RCS or CVS) +% Last edited March 29, 2004 Erik G. Learned-Miller +% Last edited January 17, 2016 Charles Sutton +% Last edited January 9, 2017 Charles Sutton +% (We have now set up GIT, good thing that we waited for it to +% be invented.) +% +% The name of this file should follow the article document +% type, e.g. \documentstyle[jmlr]{article} + +% Copied and edited from similar file for Machine Learning Journal. +% Original Author: Jeff Schlimmer +% Edited by: Kevin Thompson, Martha Del Alto, Helen Stewart, Steve Minton \& Pandu Nayak. +% Last edited: Mon May 3 20:40:00 1993 by kthompso (Kevin Thompson) on muir + +\typeout{Document Style `jmlr' -- January 2016.} + +\newif\if@abbrvbib\@abbrvbibfalse +\DeclareOption{abbrvbib}{\@abbrvbibtrue} + +\newif\if@usehyper\@usehypertrue +\DeclareOption{nohyperref}{\@usehyperfalse} +\DeclareOption{hyperref}{\@usehypertrue} + +\DeclareOption*{\PackageWarning{jmlr}{Unknown ‘\CurrentOption’}} +\ProcessOptions\relax + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% REQUIRED PACKAGES +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{epsfig} +\RequirePackage{amssymb} +\RequirePackage{natbib} +\RequirePackage{graphicx} + +\if@usehyper +\RequirePackage[colorlinks=false,allbordercolors={1 1 1}]{hyperref} +\fi + +\if@abbrvbib +\bibliographystyle{abbrvnat} +\else +\bibliographystyle{plainnat} +\fi + +\bibpunct{(}{)}{;}{a}{,}{,} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% P A G E S I Z E +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Change the overall width of the page. If these parameters are +% changed, they will require corresponding changes in the +% maketitle section. +% +\renewcommand{\topfraction}{0.95} % let figure take up nearly whole page +\renewcommand{\textfraction}{0.05} % let figure take up nearly whole page + +% Specify the dimensions of each page + +\oddsidemargin .25in % Note \oddsidemargin = \evensidemargin +\evensidemargin .25in +\marginparwidth 0.07 true in +%\marginparwidth 0.75 true in +%\topmargin 0 true pt % Nominal distance from top of page to top of +%\topmargin 0.125in +\topmargin -0.5in +\addtolength{\headsep}{0.25in} +\textheight 8.5 true in % Height of text (including footnotes & figures) +\textwidth 6.0 true in % Width of text line. +\widowpenalty=10000 +\clubpenalty=10000 +\@twosidetrue \@mparswitchtrue \def\ds@draft{\overfullrule 5pt} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% S E C T I O N S +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Definitions for nicer (?) sections, etc., ideas from Pat Langley. +% Numbering for sections, etc. is taken care of automatically. + +\def\@startsiction#1#2#3#4#5#6{\if@noskipsec \leavevmode \fi + \par \@tempskipa #4\relax + \@afterindenttrue + \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi + \if@nobreak \everypar{}\else + \addpenalty{\@secpenalty}\addvspace{\@tempskipa}\fi \@ifstar + {\@ssect{#3}{#4}{#5}{#6}}{\@dblarg{\@sict{#1}{#2}{#3}{#4}{#5}{#6}}}} + +\def\@sict#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname}\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec.\hskip 0.1em} + {\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname\hskip 0.5em }\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +\def\thesection {\arabic{section}} +\def\thesubsection {\thesection.\arabic{subsection}} +\def\section{\@startsiction{section}{1}{\z@}{-0.24in}{0.10in} + {\large\bf\raggedright}} +\def\subsection{\@startsection{subsection}{2}{\z@}{-0.20in}{0.08in} + {\normalsize\bf\raggedright}} +\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-0.18in}{0.08in} + {\normalsize\sc\raggedright}} +\def\paragraph{\@startsiction{paragraph}{4}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} +\def\subparagraph{\@startsiction{subparagraph}{5}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% F O O T N O T E S +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Change the size of the footnote rule +% +% \renewcommand{\footnoterule}{\vspace{10pt}\hrule width 0mm} + +\long\def\@makefntext#1{\@setpar{\@@par\@tempdima \hsize + \advance\@tempdima-15pt\parshape \@ne 15pt \@tempdima}\par + \parindent 2em\noindent \hbox to \z@{\hss{\@thefnmark}. \hfil}#1} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% A B S T R A C T +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% use \begin{abstract} .. \end{abstract} for abstracts. +\renewenvironment{abstract} +{\centerline{\large\bf Abstract}\vspace{0.7ex}% + \bgroup\leftskip 20pt\rightskip 20pt\small\noindent\ignorespaces}% +{\par\egroup\vskip 0.25ex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% KEYWORDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% use \begin{keywords} .. \end{keywords} for keywordss. +\newenvironment{keywords} +{\bgroup\leftskip 20pt\rightskip 20pt \small\noindent{\bf Keywords:} }% +{\par\egroup\vskip 0.25ex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% FIRST PAGE, TITLE, AUTHOR +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Author information can be set in various styles: +% For several authors from the same institution: +% \author{Author 1 \and ... \and Author n \\ +% \addr{Address line} \\ ... \\ \addr{Address line}} +% if the names do not fit well on one line use +% Author 1 \\ {\bf Author 2} \\ ... \\ {\bf Author n} \\ +% To start a seperate ``row'' of authors use \AND, as in +% \author{Author 1 \\ \addr{Address line} \\ ... \\ \addr{Address line} +% \AND +% Author 2 \\ \addr{Address line} \\ ... \\ \addr{Address line} \And +% Author 3 \\ \addr{Address line} \\ ... \\ \addr{Address line}} + +% Title stuff, borrowed in part from aaai92.sty + +\newlength\aftertitskip \newlength\beforetitskip +\newlength\interauthorskip \newlength\aftermaketitskip + +%% Changeable parameters. +\setlength\aftertitskip{0.1in plus 0.2in minus 0.2in} +\setlength\beforetitskip{0.05in plus 0.08in minus 0.08in} +\setlength\interauthorskip{0.08in plus 0.1in minus 0.1in} +\setlength\aftermaketitskip{0.3in plus 0.1in minus 0.1in} + +%% overall definition of maketitle, @maketitle does the real work +\def\maketitle{\par + \begingroup + \def\thefootnote{\fnsymbol{footnote}} + \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} + \@maketitle \@thanks + \endgroup +\setcounter{footnote}{0} + \let\maketitle\relax \let\@maketitle\relax + \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax} + +\def\@startauthor{\noindent \normalsize\bf} +\def\@endauthor{} +\def\@starteditor{\noindent \small {\bf ~}} +\def\@endeditor{\normalsize} +\def\@maketitle{\vbox{\hsize\textwidth + \linewidth\hsize \vskip \beforetitskip + {\begin{center} \Large\bf \@title \par \end{center}} \vskip \aftertitskip + {\def\and{\unskip\enspace{\rm and}\enspace}% + \def\addr{\small\it}% + \def\email{\hfill\small\sc}% + \def\name{\normalsize\bf}% + \def\AND{\@endauthor\rm\hss \vskip \interauthorskip \@startauthor} + \@startauthor \@author \@endauthor} + + \vskip \aftermaketitskip + \noindent \@starteditor \@editor \@endeditor + \vskip \aftermaketitskip +}} + +\def\editor#1{\gdef\@editor{}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% +%%% Pagestyle +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% Defines the pagestyle for the title page. +%% Usage: \jmlrheading{1}{1993}{1-15}{8/93}{9/93}{14-115}{Jane Q. Public and A. U. Thor} +%% \jmlrheading{vol}{year}{pages}{Submitted date}{published date}{paper id}{authors} +%% +%% If your paper required revisions that were reviewed by the action editor, then indicate +%% this by, e.g. +%% \jmlrheading{1}{1993}{1-15}{8/93; Revised 10/93}{12/93}{14-115}{Jane Q. Public and A. U. Thor} + +\def\firstpageno#1{\setcounter{page}{#1}} + +\def\jmlrheading#1#2#3#4#5#6#7{\def\ps@jmlrtps{\let\@mkboth\@gobbletwo% +\def\@oddhead{\scriptsize }% +\def\@oddfoot{\parbox[t]{\textwidth}{\raggedright \scriptsize \hfill}}% + +\def\@evenhead{}\def\@evenfoot{}}% +\thispagestyle{jmlrtps}} + +%% Defines the pagestyle for the rest of the pages +%% Usage: \ShortHeadings{Minimizing Conflicts}{Minton et al} +%% \ShortHeadings{short title}{short authors} + +\def\ShortHeadings#1#2{\def\ps@jmlrps{\let\@mkboth\@gobbletwo% +\def\@oddhead{\hfill {\small\sc #1} \hfill}% +\def\@oddfoot{\hfill \small\rm \thepage \hfill}% +\def\@evenhead{\hfill {\small\sc #2} \hfill}% +\def\@evenfoot{\hfill \small\rm \thepage \hfill}}% +\pagestyle{jmlrps}} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% MISCELLANY +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Define macros for figure captions and table titles + +% Figurecaption prints the caption title flush left. +% \def\figurecaption#1#2{\noindent\hangindent 42pt +% \hbox to 36pt {\sl #1 \hfil} +% \ignorespaces #2} +% \def\figurecaption#1#2{\noindent\hangindent 46pt +% \hbox to 41pt {\small\sl #1 \hfil} +% \ignorespaces {\small #2}} +\def\figurecaption#1#2{\noindent\hangindent 40pt + \hbox to 36pt {\small\sl #1 \hfil} + \ignorespaces {\small #2}} +% Figurecenter prints the caption title centered. +\def\figurecenter#1#2{\centerline{{\sl #1} #2}} +\def\figurecenter#1#2{\centerline{{\small\sl #1} {\small #2}}} + +% +% Allow ``hanging indents'' in long captions +% +\long\def\@makecaption#1#2{ + \vskip 10pt + \setbox\@tempboxa\hbox{#1: #2} + \ifdim \wd\@tempboxa >\hsize % IF longer than one line: + \begin{list}{#1:}{ + \settowidth{\labelwidth}{#1:} + \setlength{\leftmargin}{\labelwidth} + \addtolength{\leftmargin}{\labelsep} + }\item #2 \end{list}\par % Output in quote mode + \else % ELSE center. + \hbox to\hsize{\hfil\box\@tempboxa\hfil} + \fi} + + +% Define strut macros for skipping spaces above and below text in a +% tabular environment. +\def\abovestrut#1{\rule[0in]{0in}{#1}\ignorespaces} +\def\belowstrut#1{\rule[-#1]{0in}{#1}\ignorespaces} + +% Acknowledgments +\long\def\acks#1{\vskip 0.3in\noindent{\large\bf Acknowledgments}\vskip 0.2in +\noindent #1} + +% Research Note +\long\def\researchnote#1{\noindent {\LARGE\it Research Note} #1} + +\renewcommand{\appendix}{\par + \setcounter{section}{0} + \setcounter{subsection}{0} + \def\thesection{\Alph{section}} +\def\section{\@ifnextchar*{\@startsiction{section}{1}{\z@}{-0.24in}{0.10in}% + {\large\bf\raggedright}}% +{\@startsiction{section}{1}{\z@}{-0.24in}{0.10in} + {\large\bf\raggedright Appendix\ }}}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PROOF, THEOREM, and FRIENDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\BlackBox}{\rule{1.5ex}{1.5ex}} % end of proof +\newenvironment{proof}{\par\noindent{\bf Proof\ }}{\hfill\BlackBox\\[2mm]} +\newtheorem{example}{Example} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{conjecture}[theorem]{Conjecture} +\newtheorem{axiom}[theorem]{Axiom} + + + diff --git a/doc/Articles/CNN.pdf b/doc/Articles/CNN.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cae8d4b6c612289bff7ab65fd7a161b245d31a6b GIT binary patch literal 58577 zcmeFa1z1)2zCXO_4(U?1bi<}W=|)ndLAtv^8tIY{0qK%P1f(0Jr3F#x?(TXwILcvTIrf1 z-oFoIlrpe1vNr~@fq!}kWE3?qw>Pi_|25aKHxM?^v(h&J^710u+1ncESRgt>tS5Gs z%&|NAAG#qjCtooZcmrfG$VB*y<4Y=B|0px_IGq@C`frX>B$tqHLj~< zpdRu9>+W{4loz0PD+S>=?Na(OjrhokNMmU+?&O)(spqEWXv4+E&__PEm4KFXPu(@K zi}O9}gL$X(RZsWR?dkPRzRT(M_KjKD9*$~jyz{t*m6;|qlZ_0%BNxT$;}vT!;3f{WQt;Nrc*8?Vr;|5jbW~0ZctcAQUhzt8qqp35aH*tixm;hL 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{\enquote {\bibinfo {title} {{NetKet: A Machine Learning + Toolkit for Many-Body Quantum Systems}},}\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} + {arXiv:1904.00031}} (\bibinfo {year} {2019})},\ \Eprint + {http://arxiv.org/abs/1904.00031} {arXiv:1904.00031 [quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {{Zeiler}}(2012)}]{2012arXiv1212.5701Z}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Matthew~D.}\ + \bibnamefont {{Zeiler}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{ADADELTA: An Adaptive Learning Rate Method}},}\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} + {arXiv:1212.5701}} (\bibinfo {year} {2012})},\ \Eprint + {http://arxiv.org/abs/1212.5701} {arXiv:1212.5701 [cs.LG]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Kandala}\ \emph {et~al.}(2017)\citenamefont + {Kandala}, \citenamefont {Mezzacapo}, \citenamefont {Temme}, 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\bibinfo {author} {\bibfnamefont {Caglar}\ + \bibnamefont {Gulcehre}}, \bibinfo {author} {\bibfnamefont {Dzmitry}\ + \bibnamefont {Bahdanau}}, \bibinfo {author} {\bibfnamefont {Fethi}\ + \bibnamefont {Bougares}}, \bibinfo {author} {\bibfnamefont {Holger}\ + \bibnamefont {Schwenk}}, \ and\ \bibinfo {author} {\bibfnamefont {Yoshua}\ + \bibnamefont {Bengio}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Learning phrase representations using {RNN} encoder{--}decoder for + statistical machine translation},}\ }in\ \href {\doibase 10.3115/v1/D14-1179} + {\emph {\bibinfo {booktitle} {Proceedings of the 2014 Conference on Empirical + Methods in Natural Language Processing ({EMNLP})}}}\ (\bibinfo {publisher} + {Association for Computational Linguistics},\ \bibinfo {address} {Doha, + Qatar},\ \bibinfo {year} {2014})\ pp.\ \bibinfo {pages} + {1724--1734}\BibitemShut {NoStop}% +\end{thebibliography}% diff --git a/doc/Articles/main.tex b/doc/Articles/main.tex new file mode 100644 index 000000000..4bde84092 --- /dev/null +++ b/doc/Articles/main.tex @@ -0,0 +1,734 @@ +\documentclass[twocolumn,english,reprint,superscriptaddress,longbibliography,pra]{revtex4-1} +\renewcommand{\familydefault}{\rmdefault} +\usepackage[T1]{fontenc} +\usepackage[latin9]{inputenc} +\setcounter{secnumdepth}{3} +\usepackage{color} +\usepackage{bm} +\usepackage{amstext} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{booktabs} +\usepackage{tabularx} +\usepackage{amsmath,mathrsfs} +\usepackage{enumitem} +\makeatletter +\usepackage{multibib} +\usepackage{listings} +\usepackage{mathtools} +\usepackage{braket} + + +\DeclareMathOperator*{\argmin}{argmin} + +\usepackage{babel} +\usepackage{bbold} +\usepackage{mathrsfs} +\usepackage{hyperref} +\hypersetup{ + linktocpage = true, + colorlinks, + citecolor=blue, + filecolor=black, + linkcolor=blue, + urlcolor=blue +} +\newcommand{\BeH}{BeH$_2$} +\newcommand{\tr}{\textrm{Tr}} +\newcommand{\FIG}{\textcolor{blue}{FIGURE\:}} +\newcommand{\REF}{[\textcolor{red}{ref}]} + + + +\usepackage{xcolor} +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} +\lstdefinestyle{mystyle}{ + backgroundcolor=\color{backcolour}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\tiny\color{codegray}, + stringstyle=\color{codepurple}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} +\lstset{style=mystyle} +\usepackage{xpatch} + +\makeatletter +\patchcmd{\@ssect@ltx} + {\addcontentsline{toc}{#1}{\protect\numberline{}#8}} + {} + {} + {} +\makeatother + + +\begin{document} + +\title{Neural networks in quantum many-body physics: a hands-on tutorial} + +\author{Juan Carrasquilla} +\affiliation{Vector Institute, MaRS Centre, Toronto, Ontario, M5G 1M1, Canada} +\author{Giacomo Torlai} +\email{gttorlai@amazon.com} +\thanks{\\This work has been done before Giacomo Torlai joined Amazon.} +\affiliation{AWS Center for Quantum Computing, Pasadena, CA 91125, USA} +\affiliation{Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010, USA} + + + +\begin{abstract} +Over the past years, machine learning has emerged as a powerful computational tool to tackle complex problems over a broad range of scientific disciplines. In particular, artificial neural networks have been successfully deployed to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built out of a large number of interacting particles. In this Article, we overview some applications of machine learning in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick-start for a newcomer to the field. We present supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and variational Monte Carlo with recurrent neural-networks for approximating the ground state of a many-body Hamiltonian. We briefly review the key ingredients of each algorithm and their corresponding neural-network implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions. +\end{abstract} + +\maketitle + +\section{Introduction} + +Quantum many-body physics refers to the mathematical framework to study the collective behavior of large numbers of interacting particles. The emerging cooperative phenomena that result from seemingly simple interactions can produce an astounding variety of phases of matter such as conventional metals and magnetically ordered states, as well as unanticipated states including high-temperature superconductivity, strange metals, and spin liquids~\cite{Xiao:803748}. In addition to naturally occurring quantum systems, many-body physics studies synthetic quantum matter, e.g., ultracold atoms, superconducting qubits, and trapped ions, which simultaneously reveal new phenomena in highly controlled laboratory settings and advances the development of quantum computers and other quantum information processing devices. + +In spite of the simplicity of the physical laws that govern such multi-particle quantum objects, the theoretical and experimental analysis of these systems confront us with complexities which are ultimately rooted in the "curse of dimensionality" associated with the exponential explosion of the size of the space where quantum many-body states live in. Traditionally, the study of many-body systems is performed with the help of tools designed to circumvent this dimensionality explosion and produce a succinct, low-dimensional description that captures the essential aspects of a quantum system. Such descriptions arise from the analysis of data generated in a wide range of theoretical, computational, and experimental devices. These include numerical simulations of model Hamiltonians based on quantum Monte Carlo or variational algorithms, but also experimental arrays of complex electronic-structure images obtained from spectroscopic imaging scanning tunnelling microscopy, or measurements of quantum states prepared on a physical quantum computing platform. + +Machine learning, already explored as a tool in several research areas in physics~\cite{RevModPhys.91.045002}, offers a set of alternative approaches to the study of quantum many-body systems in experiments and numerical simulations~\cite{doi:10.1080/23746149.2020.1797528,annurev-conmatphys-031119-050651}. The resurgence of activity at the intersection between physics and machine learning is in part due to a series of scientific breakthroughs in computer vision and natural language processing. Such progress has led to a burst of research where neural networks have been repurposed to tackle fundamental questions in condensed matter physics, quantum computing, statistical physics, and atomic, molecular and optical physics. Machine learning, and in particular deep neural networks, have been used to identify phases of matter in numerical simulations and experiments~\cite{carrasquilla2017nature, evert2017nature, torlai_learning_2016, leiwang2016, chng2017, broecker2017, eun-ah2017, dassarma2017, neupeurt2017, yi-ting2018, huembeli2018,PhysRevB.99.060404,PhysRevB.99.104410,Zhang_MLcuprates,Bohrdt2018,Rem2018,PhysRevLett.122.210503}, +to increase the performance of Monte Carlo simulations~\cite{huang2017,junwei2017, xiao_yan2017, inack2018, parolini2019, pilati2019, mcnaughton2020, albergo2019}, to accurately describe the state of classical~\cite{Wu_2019} and quantum systems~\cite{androsiuk1993,LAGARIS19971, Carleo_2017, zi2018, Di_Luo, pfau2019abinitio, hermann2019deep, PhysRevLett.122.250502, PhysRevLett.122.250501,PhysRevLett.122.250503,PhysRevB.99.214306, choo_fermionicnqs2020, PhysRevLett.124.020503, RNNWF_2020, roth2020iterative}, to develop novel quantum control strategies~\cite{PhysRevX.8.031086,PhysRevX.8.031084,PhysRevLett.122.020601,niu_universal_2019,2020arXiv201003655Y,coopmans2020}, to perform quantum tomography~\cite{torlai_Tomo,rocchetto,Torlai_latent,2018arXiv181206693Q,carrasquilla_povm,biamonte_qst,torlai_rydberg19,xin_local-measurement-based_2019,Sehayek2019,torlai_chemistry,PhysRevA.102.022412,NoriGAN,Tiunov:20,Cha2020,PhysRevA.102.042604,DeVlugt2020,2020arXiv200907601S,torlai_QPT,morawetz2020,Nori2020}, to accelerate density functional theory calculations~\cite{PhysRevLett.108.253002,doi:10.1063/1.4834075,PhysRevB.94.245129,brockherde_bypassing_2017,PhysRevA.100.022512,PhysRevLett.125.076402,PhysRevResearch.2.033388}, to develop and elucidate renormalization group analyses~\cite{2014arXiv1410.3831M,koch-janusz_mutual_2018,PhysRevE.97.053304,PhysRevLett.121.260601,PhysRevResearch.2.023369,2020arXiv201005703C}, to devise quantum error correction protocols~\cite{torlai_neural_2016,krastanov_deep_2017,Varsamopoulos_2017,Baireuther2018machinelearning,Chamberland_2018,Breuckmann2018scalableneural,Nautrup2019optimizingquantum,PhysRevLett.122.200501,PhysRevA.99.052351,Andreasson2019quantumerror,Evert2020QEC,Ni2020neuralnetwork,PhysRevResearch.2.033399,PhysRevResearch.2.023230}, among many other examples~\cite{PhysRevB.97.045153,Seif_2018,Melnikov1221,Dunjko_2018,PhysRevE.99.062106, 2019arXiv191211052C,PhysRevB.99.075113,PhysRevLett.124.010508, PhysRevX.10.011006,2020arXiv200600712H,2020arXiv200905580L,2020arXiv201014510L}. + +Such an explosion of activity indicates that machine learning techniques may soon become commonplace in quantum many-body physics research, both in experiments and numerical simulation. These clear trends call for the development of resources to stimulate researchers to familiarize with the wealth of concepts, intuition, algorithms, hardware, software, and research culture entailed by the adoption of machine learning and neural networks in physics research. Here, we take a step forward in this direction and develop a set of hands-on tutorials focused on a set of recent prototypical examples of applications of neural network technology to problems in statistical physics, condensed matter and quantum computing. + +\subsection*{Outline} +The Article is organized as follows. Starting with a preliminary discussion, we introduce in Sec~\ref{preliminariesA} some fundamental concepts in machine learning and neural networks. In Sec~\ref{preliminariesB} we present a concise description of the physical system studied in our numerical experiments, a two-dimensional array of interacting Rydberg atoms. In Sec~\ref{supervised} we discuss our first application, the classification of phases of matter with supervised machine learning of projective measurement data using a convolutional neural network, and demonstrate it on the quantum phase transition in the Rydberg atoms. In Sec~\ref{qst} we introduce quantum state tomography, and show how this problem can be phrased as an unsupervised machine learning task. Using the restricted Boltzmann machine, we show quantum tomography of the Rydberg ground states, as well as of the ground state of a small molecule from qubit measurement data. In Sec~\ref{vmc} we present the simulation of the ground state of a many-body Hamiltonian using variational Monte Carlo with a recurrent neural network wavefunction. For each of these applications, we also show the key components of the underlying software, with full code tutorials available in an external repository~\cite{coderepo}. + + +\section{Preliminaries} +\subsection{Machine learning with neural networks} +\label{preliminariesA} +Artificial intelligence, the scientific discipline that deals with the theory and development of computer programs with the ability to perform complex tasks, saw early success solving problems which are relatively straightforward to formalize in an abstract way. The solutions to this breed of problems are typically described by a list of very precise formal rules that computers can process efficiently. As remarkable example, computers have been beating humans at playing chess since 1997, due in part to the fact that chess involves a large set of formal rules. + +Modern machine learning, instead, deals with the challenge of automatizing the solution of real world tasks that may be easy for humans to process but that are hard to formally describe by simple rules. These techniques have spurred a recent revolution where algorithms trained using data have started to match humans' ability to recognize objects in an image, decipher speech or translate text to multiple languages, which are tasks that are difficult to formalize and articulate through simple rules. + +A key element behind these recent developments can be largely traced back to a series of breakthroughs in the development of powerful neural network models, where data is processed through the sequential combination of multiple nonlinear layers~\cite{Goodfellow-et-al-2016}. Such models solve a fundamental problem in learning real world tasks, namely the problem of automatically extracting knowledge from raw noisy data, rather than relying on hard-coded knowledge directly inscribed in the algorithms by a human. Neural networks automatize the construction of sets of increasingly complex representations of the data, which can be understood as the computational disentangling of complex concepts (e.g. an object in a cluttered image) out of simpler concepts (e.g. pixel values and basic shapes like edges). These representations, in turn, lead to solutions to learning tasks with unprecedented success. + +% ML tasks +\begin{figure*}[t] +\noindent \centering{}\includegraphics[width=2.05\columnwidth]{Fig1} +\caption{Rydberg atoms in a two-dimensional square array. ({\bf a}) Schematic representation of the phase diagram at a fixed value of the interaction $V=3$ MHz and $\Omega = 1$ MHz. At large and negative detuning, the system is in a disordered (paramagnetic) phase with all atoms in the ground state. At large and positive detuning, the atoms are found in a checkerboard pattern with N\'eel order. ({\bf b}) Snake-like geometry of the MPS path along the square lattice, used for the DMRG simulations. Ground state energy ({\bf c}) and the staggered magnetization (N\'eel order) ({\bf d}) as a function of the detuning $\delta$, for a $8\times8$ array ($V=3$ MHz). ({\bf e}) Absolute value of the average occupation number in momentum space $|n(\bm{k})|$ deep into the $Z_2$ ordered phase ($\delta=4$ MHz), showing a peak at $\bm{k}=(\pi,\pi)$, a signature of anti-ferromagnetic order. ({\bf f}) Energy gap $\Delta$ between the ground state and first excited state, detecting a quantum phase transition at detuning $\delta\approx1.3$.} +\label{Fig::1} +\end{figure*} + +For practical purposes, machine learning algorithms can be divided into the categories of supervised, unsupervised, and reinforcement learning, all of which have found applications to quantum many-body systems~\cite{doi:10.1080/23746149.2020.1797528}. While there is no formal difference between some of the algorithms in these categories when expressed in the language of probability~\cite{Goodfellow-et-al-2016,10.5555/1162264}, such a division is often used as a way to specify the details of the algorithms, the training setup, and the structure of the data sets involved. + +Supervised learning tasks aim at predicting a target output vector $\bm y$ associated with input vector $\bm x$, both of which can be discrete or continuous. The training data is thus a list of pairs of input/output tuples $\{ \bm x_i,\bm y_i\}_{i=1}^{M}$, where target output conveys that such a vector corresponds to the ideal output given the input vector~\cite{10.5555/1162264}. Starting with a training data set with $M$ entries, the learning algorithm outputs a function $\hat{\bm y} = f(\bm x)$ which estimates the output values for unseen input vectors $\bm x$. Examples of supervised learning include classification, where the objective is to assign each input vector to one of a set of discrete categories, and the task of +regression, where the output is a vector with continuous entries. Examples for classification and regression are respectively the problem of recognizing images of handwritten digits and the problem of determining the orbits of bodies around the sun from astronomical data. + +Unsupervised learning deals with the learning tasks where the training data is composed of a set of input vectors without a corresponding target output~\cite{10.5555/1162264}. These algorithms are typically used to discover hidden structure in the data sets. Examples of tasks in unsupervised learning problems include clustering, where the objective is to discover of groups of similar examples within the data, density estimation, where the objective is to estimate the underlying probability distribution associated with the data, as well as low-dimensional visualization of high-dimensional data algorithms, which depict complex data in two or three dimensions while trying to retain key spatial characteristics in the original data. + +Finally, reinforcement learning, although not discussed in this Article, develops algorithms dealing with the problem of discovering actions that maximize a numerical reward signal~\cite{10.5555/551283}. The learning algorithms are not necessarily directly exposed to examples of optimal actions. Instead, it must discover them by a process similar to a guided trial and error. Reinforcement learning augmented by deep neural networks has successfully learned policies from high-dimensional sensory input for game playing achieving human-level performance in several challenging games including Atari 2600~\cite{mnih_human-level_2015} as well as the board game Go~\cite{silver2016}. Likewise, reinforcement learning has been applied to the control of quantum systems~\cite{bukov2018,niu_universal_2019} as well as to the optimization of quantum error correction codes~\cite{Nautrup2019optimizingquantum,Andreasson2019quantumerror,Evert2020QEC}, one key ingredient in the development of fault-tolerant quantum computers. + + + +\subsection{Rydberg atoms in two dimensions} +\label{preliminariesB} +We demonstrate the machine learning algorithms discussed in this Article for a many-body system composed by interacting Rydberg atoms. Engineered arrays of cold Rydberg atoms are increasingly used for highly-controlled quantum simulations of strongly-interacting matter~\cite{Schauss1455,Endres2016,Labuhn,Bernien2017,keesling_quantum_2019,2020arXiv201212281E,2020arXiv201212268S}, as well as for quantum information processing~\cite{Levine2019}. We specifically consider a square array with linear dimension $L$ containing $N = L^2$ atoms. Each atom is described by a local Hilbert space spanned by the states $\{|g\rangle,|e\rangle\}$, referring respectively to the atomic ground and the highly-excited Rydberg states. The atoms are subject to a uniform laser drive with Rabi frequency $\Omega$ and detuning $\delta$, and they interact with one another via the Van der Waals potential $V(x)\approx r^{-6}$ at short distances. The resulting many-body Hamiltonian is +\begin{equation} +\hat{H} = -\Omega \sum_{\bm{r}}\hat{S}^x(\bm{r}) -\delta\sum_{\bm{r}}^N\hat{\Pi}(\bm{r}) +\frac{1}{2}\sum_{\bm{r},\bm{r^\prime}}V(\bm{r}-\bm{r^\prime})\hat{\Pi}(\bm{r}) \hat{\Pi}(\bm{r^\prime}) +\label{Eq::RydbergHamiltonian} +\end{equation} +where $\hat{\Pi}(\bm{r}) =|e\rangle\!\langle e|_{\bm{r}} $ is the projector onto the Rydberg state at position $\bm{r}$, $\hat{S}^x(\bm{r})=\frac{1}{2}\hat\sigma^x(\bm{r})$ are spin-$\frac{1}{2}$ operators, and $V(\bm{r}-\bm{r^\prime})=V_0/\|\bm{r}-\bm{r^\prime}\|^6$ is the Van der Waals potential between atoms at position $\bm{r}$ and $\bm{r^{\prime}}$. In the following, we assume $\Omega=1$ MHz. + +The phase diagram for the ground state of the Rydberg Hamiltonian is dictated by the mechanism of Rydberg blockade, a constraint that prevents two atoms at sufficiently small distances to be simultaneously excited to the Rydberg states. We can characterize the phase diagram in terms of the detuning $\delta$ and the interaction strength $V_0$. On the square lattice, several different orders have been detected by numerical simulations~\cite{Samajdar2020}. Here, we specifically focus on the $Z_2$ transition between a disordered phase at large and negative detuning, where all atoms are found in the ground state, and an ordered phase at large and positive detuning, where the system is found in one of the two symmetry-broken N\'eel states characterized by a checkerboard pattern in the atomic occupation number (Fig.~\ref{Fig::1}(a)). + +We perform numerical simulations of the ground state of Hamiltonian~(\ref{Eq::RydbergHamiltonian}) using the density matrix renormalization group (DMRG)~\cite{PhysRevLett.69.2863,PhysRevB.48.10345,SCHOLLWOCK201196} implemented using the software package ITensor~\cite{itensor}. We adopt a matrix product state (MPS) variational wavefunction $|\Psi\rangle$ with a snake-like geometry shown in Fig.~\ref{Fig::1}(b). We fix the interaction strength to $V_0=3$ MHz, and retain up to the third-nearest-neighbor interactions. For a several values of the detuning $\delta\in\{-5,5\}$ MHz, we run DMRG to find an approximation of the ground state, using a singular value decomposition cutoff of $10^{-10}$ and a target energy accuracy of $10^{-5}$. To certify convergence to the ground state, each run is repeated for different initialization of the starting MPS. + +We show the results of the simulations for a $8\times8$ array with open boundary conditions in Fig.~\ref{Fig::1}(c-f). We plot, as a function of the detuning, the ground state energy per site $E_0/N=\langle\Psi_0|\hat{H}|\Psi_0\rangle/N$, and the staggered magnetization $\langle\mathcal{N}\rangle=N^{-1}\sum_{\bm{r}}(-1)^{x+y}\langle\hat{S}^z(\bm{r})\rangle$, which can be used to detect N\'eel order. Whenever all atoms are in the ground state, $\langle\mathcal{N}\rangle\approx0$, while for an ordered state with a checkerboard pattern one has $\langle\mathcal{N}\rangle\approx0.5$. We also show the average occupation number in momentum space, +\begin{equation} +n(\bm{k})=\frac{1}{\sqrt{N}}\sum_{\bm{r}}e^{i\bm{k}\cdot\bm{r}}\langle\hat{n}(\bm{r})\rangle +\end{equation} +where $\hat{n}(\bm{r}) = \frac{1}{2}(1-2\hat{S}^z(\bm{r}))$. We observe a peak at $\linebreak\bm{k}=(\pi,\pi)$ for large detuning $\delta=4$ MHz (Fig.~\ref{Fig::1}(e)), and a featureless state at negative detuning (not shown). + +The two phases of the Rydberg atoms are separated by a second-order quantum phase transition at a critical point $\delta_c$. We can extract an approximation of $\delta_c$ by measuring the energy gap $\Delta = |E_0-E_1|$ between the ground state and the first excited state $|\Psi_1\rangle$. We compute $E_1$ by running DMRG on the Hamiltonian $\hat{H}^\prime=\hat{H}+\omega|\Psi\rangle\langle\Psi|$ where $\omega$ is an energy penalty. From the energy gap curve, we estimate the detuning where $\Delta\approx0$ to be $\delta_c\approx 1.3$ MHz. This approximate value will be sufficient for the purpose of this Article, though a more systematic scaling study with appropriate boundary conditions (to minimize finite-size effects) should be performed to accurately determine the critical point and critical exponents of the transition. + +Once we have solved for the ground states of the Rydberg Hamiltonian, the corresponding MPSs can be used to generate data to train the neural networks for the different applications. In this case, the data consists of projective measurements in the atomic occupation number basis $|\bm{\sigma}\rangle=|\sigma_1,\dots,\sigma_N\rangle$, where $\sigma_j=0$ and $\sigma_j=1$ refers respectively to the $j$-th atom being in the ground and Rydberg state. Given a wavefunction $|\Psi\rangle$, the probability to observe an atomic pattern $\bm{\sigma}$ following a measurement is simply given by the Born rule $P(\bm{\sigma})=|\langle\bm{\sigma}|\Psi\rangle|^2$. Because of the intrinsic one-dimensional geometry of an MPS, it is possible to efficiently sample the probability distribution $P(\bm{\sigma})$ by exploiting the chain rule of probabilities. Moreover, the sampling is exact in the sense that each sample is completely independent of one another~\cite{PhysRevB.85.165146}. %The sampling procedure consists of iteratively building one-site reduced density matrices conditional on the previous measurement outcomes~\cite{PhysRevB.85.165146}. + + + +%---------------------------------------------------------------------------------------- +%---------------------------------------------------------------------------------------- +% SUPERVISED LEARNING +%---------------------------------------------------------------------------------------- +%---------------------------------------------------------------------------------------- +%---------------------------------------------------------------------------------------- + + +\section{Learning a quantum phase transition} +\label{supervised} + +An important task in condensed matter and statistical physics is to characterize different phases of matter and the associated phase transitions between them. +Typically, phases of matter are described in terms of simple real-space patterns and their associated order parameters, which are theoretically understood using Landau symmetry-breaking paradigm~\cite{Xiao:803748}. While a wide array of theoretical and experimental tools to study interacting quantum systems have been constructed in relation to these patterns, there is an increasing set of states of matter whose theoretical and experimental understanding eludes the Landau symmetry-breaking paradigm. The characterization of these phases may rely on, e.g., out-of-equilibrium properties of the system as in the many-body localized phase~\cite{basko2006,RevModPhys.91.021001}, or on topological invariants in topological phases and spin liquids~\cite{Xiao:803748,savaryQuantumSpinLiquids2016, doi:10.1146/annurev-conmatphys-031218-013401}. + +Machine learning provides an alternative route to the characterization of phases of matter and their associated phase transitions in a semi-automated fashion without a direct use of manually specified real-space patterns and/or other signatures, provided that a sufficiently large training set is available. In its simplest form~\cite{carrasquilla2017nature}, given the existence of a classical or quantum phase transition between two phases in a physical system, one can use supervised learning to attempt to classify experimental or numerical snapshots of the phases of matter separated by the transition. This task can be achieved using most classification algorithms, e.g., those based on a neural network or a support vector machine~\cite{10.5555/1162264}, trained on snapshots of two phases of matter labelled according to the corresponding phase out of which the snapshot originated. Although here we only explore this simple strategy, we stress that machine learning approaches to studying phases and phase transitions have been significantly expanded and they no longer require the precise knowledge of the location of the critical point~\cite{evert2017nature,broecker2017b}, can be fully automatized, and can discover ordered phases~\cite{carrasquilla2017nature,leiwang2016,PhysRevE.96.022140}, topological phases~\cite{evert2017nature,rodriguez-nieva2019,PhysRevLett.125.127401}, and phases such as the many-body localized phase which is characterized by its dynamical properties~\cite{yi-ting2018,rao2018,PhysRevLett.120.257204}. + +The nature of the snapshots used to train the learning algorithms is vastly flexible, hence these strategies are of wide applicability, and can include numerically generated configurations visited during a classical or quantum Monte Carlo simulation of the physical system~\cite{carrasquilla2017nature,leiwang2016,evert2017nature,broecker2017,chng2017,PhysRevB.99.060404,PhysRevB.99.121104}, entanglement spectra~\cite{evert2017nature,yi-ting2018}, correlation matrices~\cite{PhysRevB.102.054512,PhysRevLett.125.170603}, tensors in an MPS~\cite{PhysRevLett.125.170603}, numerically generated projective measurements~\cite{berezutskii2020}, high-resolution real-space snapshots of complex many-body systems obtained with quantum gas microscopes for ultracold atoms~\cite{Bohrdt2018,PhysRevA.102.033326}, single-shot experimental momentum-space density images of ultracold quantum gases~\cite{Rem2018}, spectroscopic imaging scanning tunnelling microscopy data~\cite{Zhang_MLcuprates}, among many others. + +Below we explore learning a quantum phase in an array of interacting Rydberg atoms using projective measurements. As a classification algorithm, we make use a of a convolutional neural network~\cite{Goodfellow-et-al-2016} that readily takes advantage of the two-dimensional (2D) spatial arrangement of the the Rydberg atoms and their locality, as well as the approximate translation invariance of the system. + +\begin{figure}[t] +\noindent \centering{}\includegraphics[width=\columnwidth]{CNN.pdf} +\caption{A schematic representation of a convolutional neural network. The elements of the input $\mathsf{h}^{(0)}_{l,j,k}$ corresponds to the outcome of a projective measurement on the Rydberg system. The first operation is a convolutional layer with $ M_y \times M_x = 3 \times 3$ kernels with $I_{\text{out}} = 32$ output channels and $L_{\text{input}}=1$. This kernel is convolved with an input configuration with $N = 8 \times 8$ Rydberg atoms. Likewise, the second operation corresponds to a convolutional layer with $ M_y \times M_x = 3 \times 3$ kernels with $I_{\text{out}} = 32$ output channels and $L_{\text{input}}=32$. The output of the second convolutional layer is flattened and fed to an FC layer with a ReLU activation, followed by another FC layer with a softmax activation which produces the prediction outcome. } +\label{Fig::CNN} +\end{figure} + +\subsection{Convolutional neural networks and their training} +Convolutional neural networks (CNN) employ a mathematical operation called convolution to process information for data that has a natural grid-like topology~\cite{Goodfellow-et-al-2016}. A 2D convolutional layer implements the operation +\begin{align*} +\mathsf{h}^{(q)}_{i,j,k} & = F\left(\sum_{l,m_y,m_x} \mathsf{h}^{(q-1)}_{l,j+m_y,k+m_x} \mathsf{K}^{(q)}_{i,l,m_y,m_x} \right) \\ +&\coloneqq F\left( \mathsf{K}^{(q)} *\mathsf{h}^{(q-1)} \right) \nonumber %_{i,j,k} \nonumber +\end{align*} +where the trainable kernel $\mathsf{K}^{(q)}_{i,l,m_y,m_x}$ at layer $q$ specifies the connection strength between a unit in channel $i$ of the output and a unit in channel $l$ of the input, with a spatial offsets of $m_y$ rows (labeled y direction) and $m_x$ columns (labeled x direction) between the output and the input variables. The dimensions of the array $\mathsf{K}^{(q)}_{i,l,m,n}$ are $I_{\text{out}}$, $L_{\text{input}}$, $M_{y}$, $M_{x}$, which corresponds to the number of output channels, input channels, dimension of the filter along the vertical and horizontal directions, respectively. The activation at layer $q$ consists of elements $\mathsf{h}^{(q)}_{l,j,k}$, where $j$ and $k$ label vertical and horizontal directions, respectively, and $l$ specifies the channel. The activation units are labelled by $q$ where $q=0$ corresponds to the raw projective measurement data. Finally, the non-linear function $F(x)$, which in our examples is typically a rectified linear unit (ReLU) $F(x)=\text{max}(0,x)$, is applied element-wise to each of the components of its input. A convolutional neural network equipped with two convolutional layers is schematically shown in Fig.~\ref{Fig::CNN}. + + +Followed by the convolutional layers, a CNN typically processes information using sets of fully connected (FC) layers which implement a matrix-vector operation followed by a non-linearity $F$ as +\begin{equation} +\mathsf{h}^{(q)}_{i} = F\left(\sum_{l} \mathsf{h}^{(q-1)}_{l} \mathsf{K}^{(q)}_{i,l} + \mathsf{b}^{(q)}_{i}\right), +\end{equation} +where the trainable parameters of the FC layer are the kernel $\mathsf{K}^{(q)}_{i,l}$ and the bias vector $ \mathsf{b}^{(q)}_{i}$. +To feed the output of a convolutional layer $\mathsf{h}^{(q-1)}_{l,j,k}$ to an FC layer, the array is reshaped or ``flattened'' to $\mathsf{h^{\prime}}^{(q-1)}_{l}$ so that all the original components packed into a one-dimensional array with dimension $L_{\text{FC}}$. The last two layers of the CNN in Fig.~\ref{Fig::CNN} correspond to two fully connected layers with a ReLU and a softmax non-linearities, respectively. The softmax function $S$ is given by +\begin{equation} + \text{S}(\bm{v}) = \frac{\exp(\bm{v})}{\sum_i \exp(v_i)}. +\end{equation} +where $v_i$ are the components of a vector $\bm{v}$ and the $\text{exp}$ function acts element-wise on the components of the vector. We note that the input to the CNN and its trainable parameters are real, so that the outcome of the softmax layer can be interpreted as a probability distribution since $0 \le \text{S}(v_i)\le 1$ and $\sum_{i}\text{S}(v_i)=1$. + +Finally, we mention that we interpret our CNN as a model for the conditional probability of assigning a phase of matter $y=0,1$ to a projective measurement outcome $\bm{\sigma}=\mathsf{h}^{(0)}$, i.e. $P_{\bm{\theta}}(y|\bm{\sigma})$, where $\bm{\theta}$ encompasses all the trainable parameters of the CNN. The conditional is given by +\begin{widetext} +\begin{equation} + P_{\bm{\theta}}(y|\bm{\sigma}) = S\left( b^{(4)}_{y} + \sum_{m}\mathsf{K}^{(4)}_{y,m} F\left(b^{(3)}_{m} + \sum_{l} \mathsf{K}^{(3)}_{m,l}\,\text{Flatten}\left(F\left((\mathsf{K}^{(2)}*F\left(\mathsf{K}^{(1)}*\bm{\sigma} \right)\right)\right)_{l}\right) \right), + \label{Eq::CNNdist} +\end{equation} +\end{widetext} +where the function $\text{Flatten}()_{l}$ is the $l$-th component of a vector that arises from reshaping the incoming argument of the function to a one-dimensional array. + +%\subsection{Phase transition in the Rydberg array} + +To estimate the parameters of the CNN we use the maximum likelihood principle, where the parameters of a statistical model are selected by assigning high probability to the observed data. For a dataset with observations $\{\bm{\sigma}_n, y_n \}_{n=1}^{M}$, where $y_n=0,1$ label the phase of matter out of which a projective measurement $\bm{\sigma}_n$ was taken from, the likelihood assigned by the model to the dataset can be written as +\begin{equation} + p(\bm{y}|\bm{\theta}) = \prod_{n=1}^{M} + P_{\bm{\theta}}(y_n|\bm{\sigma}_n)^{y_n}(1-P_{\bm{\theta}}(y_n|\bm{\sigma}_n))^{1-y_n} +\end{equation} +where $\bm{y}=(y_1,...,y_{M})$. Instead of attempting to maximize the likelihood, it is convenient to define a loss function by taking the negative logarithm of the likelihood, which gives the cross-entropy +\begin{widetext} +\begin{equation} +E(\bm{\theta}) = -\ln( p(\bm{y}|\bm{\theta})) - = \sum_{n=1}^{M} \{ y_n \ln \left(P_{\bm{\theta}}(y_n|\bm{\sigma}_n))\right) + (1-y_n)\ln(1 - P_{\bm{\theta}}(y_n|\bm{\sigma}_n)) \}. +\label{Eq::NLL} +\end{equation} +\end{widetext} + +To train the model, we minimize $E(\bm{\theta})$ using gradient descent techniques~\cite{10.5555/1162264}. While it is possible to evaluate the gradients of $E(\bm{\theta})$ with respect to the parameters $\bm{\theta}$ in the CNN analytically using the chain rule, a more convenient and less error-prone approach is to use automatic differentiation (AD), which is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. A complete survery detailing AD can be found in Ref.~\cite{AD_2017}. + +In addition, instead of using the entire dataset in the calculation of $E(\bm{\theta})$ and its gradients, we use smaller batches of data of size $M_{\text{batch}}Pzwa^N=WIoy~c%kzVmuR6dVGtj4Y8%sw)kRVx zrTa;dp-PNt5H>RI7N+nsr1~cyCD`U^> zIxPHrkiHFVJWpX4M9B#4&uK;HUbm=tCNnSF>OhqxU+Irt*#A~DU&8f`$A=tRD|^XB z!3Pg9c&)0AsC4SqjRMo_yed9)s!gJLToIyXUJ&c53sIsNTN&9q*ct0v-n_Fmc#d*& zx$(^_J1Y+l*Ut}u>(qZ;f~@LlV+>+Z)Hk{PYiDfb00M5Nzq#RD+1TFN(azA=9>n$i z1959B2cX;@baRO^P~nlWk(s`jwF^j-75IRS6}U#8n~O&qg+<)X+UE9iHg+~hGnk8$4aCjK%FF}Y1<%dS%FM~j$)gQo5wkEhHu^usfLK)R%q)S=QT{F-#3E|v z@PDhWZ0!Kt+ze8NdaZRHQzU-KzZ-g4o!8m`l|axMQ9T%yV0f1$Y9Zuz`Qr8hGLcZkhjapS-Fn5Abl~CBVw6 zu3*5p>_2>8??E?znd_dD5jerLZpeNQJ8URJWsQvW-n4kNX1i*z{Ka2@f z;{<&-%8fVw7{q&9aw82e6rj?N zHXK|aU>^N^U}OEh3y&Rfy9x^aMbfAzx+`soEgY0%Hn0BFVo z0x35(SD~RK-YCIsGU)lh@bN$kv4fG2!Fh+lg0tP(SU#)pTJii(M_U8Gk z2^hrv%SgN+K((KCu}V5orIvan}!JR37Qun*${ELu)ZJ2YW?hJ8^4E8*8f@&gTJG%gh3h 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