Getting your computer to tell you the gradient of a function, without resorting to finite difference approximation, or coding an analytic derivative by hand. We usually mean this in the sense of automatic forward or reverse mode differentiation, which is not, as such, a symbolic technique, but symbolic differentiation gets an incidental look-in, and these ideas do of course relate.

Infinitesimal/Taylor series formulations, the related dual number formulations, and even fancier hyperdual formulations. Reverse-mode, a.k.a. Backpropagation, versus forward-mode etc. Computational complexity of all the above.

There is a beautiful explanation of reverse-mode the basics by Sanjeev Arora and Tengyu Ma.

You might want to do this for ODE quadrature, or sensitivity analysis, or for optimisation, either batch or SGD, especially in neural networks, matrix factorisations, variational approximation etc. This is not news these days, but it took a stunningly long time to become common since its inception in the… 1970s? See, e.g. Justin Domke, who claimed automatic differentiation to be the most criminally underused tool in the machine learning toolbox. (That escalated quickly.) See also a timely update by Tim Viera.

Related: symbolic mathematical calculators.

There are many ways you can do automatic differentiation, and I won’t attempt to comprehensively introduce the various approaches here. This is a well-ploughed field. There is much of good material out there already with fancy diagrams and the like. Symbolic, numeric, dual/forward, backwards mode… Notably, you don’t have to choose between them - e.g. you can use forward differentiation to calculate an expedient step in the middle of backward differentiation, for example.

To do: investigate unorthodox methods such as Benoît Pasquier’s F-1 Method. (source)

This package implements the F-1 algorithm described

…It allows for efficient quasi-auto-differentiation of an objective function defined implicitly by the solution of a steady-state problem.

See, e.g.
Mike Innes’ hand-on introduction,
or his terse, opinionated introductory paper, Innes (2018).
There is a well-establish terminoligy for sensititvity analysis discussing *adjoints*, e.g.
Steven Johnson’s class notes, and his references
(Johnson 2012; Errico 1997; Cao et al. 2003).

## Software

### jax

`jax`

(python) is a successor to classic python
`autograd`

.

JAX is Autograd and XLA, brought together for high-performance machine learning research.

With its updated version of Autograd, JAX can automatically differentiate native Python and NumPy functions. It can differentiate through loops, branches, recursion, and closures, and it can take derivatives of derivatives of derivatives. It supports reverse-mode differentiation (a.k.a. backpropagation) via grad as well as forward-mode differentiation, and the two can be composed arbitrarily to any order.

What’s new is that JAX uses XLA to compile and run your NumPy programs on GPUs and TPUs. Compilation happens under the hood by default, with library calls getting just-in-time compiled and executed. But JAX also lets you just-in-time compile your own Python functions into XLA-optimized kernels using a one-function API, jit. Compilation and automatic differentiation can be composed arbitrarily, so you can express sophisticated algorithms and get maximal performance without leaving Python.

Dig a little deeper, and you'll see that JAX is really an extensible system for composable function transformations. Both grad and jit are instances of such transformations. Another is vmap for automatic vectorization, with more to come.

This is a research project, not an official Google product. Expect bugs and sharp edges. Please help by trying it out, reporting bugs, and letting us know what you think!

AFAICT the conda installation mode is

`conda install -c conda-forge jaxlib`

### Python autograd

can automatically differentiate native Python and Numpy code. It can handle a large subset of Python’s features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives. It uses reverse-mode differentiation (a.k.a. backpropagation), which means it can efficiently take gradients of scalar-valued functions with respect to array-valued arguments. The main intended application is gradient-based optimization.

This is the most pythonic of the choices here; not as fast as tensorflow but simple to use and can differentiate more general things than Tensorflow.

autograd-forward will mingle forward-mode differentiation in to calculate Jacobian-vector products and Hessian-vector products for scalar-valued loss functions, which is useful for classic optimization. AFAICT there are no guarantees about computational efficiency for these.

### Theano

Mentioned for historical accuracy.

Theano, (python) supports autodiff as a basic feature and had a massive user base, although it is now discontinued in favour of the next two…

### Tensorflow

See Tensorflow. FYI there is an interesting discussion of its workings in the tensorflow jacobians ticket request

### Pytorch

See `pytorch`

.

Another neural-net style thing like tensorflow,
but with dynamic graph construction as in
`autograd`

.

### Julia

Julia has an embarrassment of different methods of autodiff (Homoiconicity and introspection makes this comparatively easy.) and it’s not always clear the comparative selling points of each.

The juliadiff project produces ForwardDiff.jl and ReverseDiff.jl which do what I would expect, namely autodiff in forward and reverse mode respectively. ForwardDiff claims to be very advanced. ReverseDiff works but is abandoned.

ForwardDiff implements methods to take derivatives, gradients, Jacobians, Hessians, and higher-order derivatives of native Julia functions

In my casual tests it seems to be a slow for my purposes,
due to constantly needing to create
a new closure with a single argument it and differentiate it *all* the time.
Or
maybe I’m doing it wrong, and the compiler will deal with this if I set it up
right?
Or maybe most people are not solving my kind of problems,
e.g. finding many
different optima in similar sub problems.
I suspect this difficulty would vanish
if you were solving one big expensive optimisation with many steps, as with
neural networks.
**update**: I has doing it wrong. This gets faster if you avoid type ambiguity
by, e.g setting up your problem in a function to avoid type ambiguities. I’m not
sure if there is any remaining overhead in this closure-based system, but it’s
not so bad.
Other needful optimisations might be covered by ForwradDiff2 which rolls in a bunch of commonly needed optimisations.

In forward mode (desirable when, e.g. I have few parameters with respect to
which I must differentiate), when do I use
DualNumbers.jl?
Probably never; it seems to be deprecated in favour of
a similar system
in ForwardDiff.jl.
But ForwardDiff is well supported.
It seems to be fast for functions with low-dimensional arguments.
It is not clearly documented how one would provide custom derivatives, but apparently you can still
use method extensions for Dual types,
of which there is an example in the issue tracker.
The recommended way
is extending `DiffRules.jl`

which is a little circuitous if you are building
custom functions to interpolate.
It does not seem to support Wirtinger derivatives yet.

Related to this forward differential formalism is Luis Benet and David P. Sanders’ TaylorSeries.jl, which is satisfyingly explicit, and seems to generalise in several unusual directions.

TaylorSeries.jl is an implementation of high-order automatic differentiation, as presented in the book by W. Tucker (2011). The general idea is the following.

The Taylor series expansion of an analytical function \(f(t)\) with

oneindependent variable \(t\) around \(t_0\) can be written as\[ f(t) = f_0 + f_1 (t-t_0) + f_2 (t-t_0)^2 + \cdots + f_k (t-t_0)^k + \cdots, \] where \(f_0=f(t_0)\), and the Taylor coefficients \(f_k = f_k(t_0)\) are the \(k\)th

normalized derivativesat \(t_0\):\[ f_k = \frac{1}{k!} \frac{{\rm d}^k f} {{\rm d} t^k}(t_0). \]

Thus, computing the high-order derivatives of \(f(t)\) is equivalent to computing its Taylor expansion.… Arithmetic operations involving Taylor series can be expressed as operations on the coefficients.

It has a number of functional-approximation analysis tricks. 🏗

HyperDualNumbers,
promises cheap 2nd order derivatives by generalizing Dual Numbers to HyperDuals.
(ForwardDiff claims to support Hessians by Dual Duals, which are supposed to be
the same as HyperDuals.)
I am curious which is the faster way of generating Hessians out of
`ForwardDiff`

’s Dual-of-Dual and `HyperDualNumbers`

.
`HyperDualNumbers`

has some very nice tricks.
Look at the `HyperDualNumbers`

homepage example, where we are
evaluating derivatives of `f`

at `x`

by evaluating it at
`hyper(x, 1.0, 1.0, 0.0)`

.

```
> f(x) = ℯ^x / (sqrt(sin(x)^3 + cos(x)^3))
> t0 = Hyper(1.5, 1.0, 1.0, 0.0)
> y = f(t0)
4.497780053946162 + 4.053427893898621ϵ1 +
4.053427893898621ϵ2 + 9.463073681596601ϵ1ϵ2
```

The first term is the function value, the coefficients of both ϵ1 and ϵ2 (which correspond to the second and third arguments of hyper) are equal to the first derivative, and the coefficient of ϵ1ϵ2 is the second derivative.

*Really* nice. However, AFAICT this method does not actually get you a Hessian,
except in a trivial sense, because it only seems
to return the right answer for scalar functions of scalar arguments.
This is amazing, if you can reduce your function to scalar parameters,
in the sense of having a diagonal Hessian.
But that skips lots of interesting cases.
One useful case it does not skip, if that is so,
is *diagonal* preconditioning of tricky optimisations.

Pro tip: the actual manual is the walk-through which is not linked from the purported manual.

Another curiosity: Benoît Pasquier’s (n.d.) (F-1 Method) Dual Matrix Tools and Hyper Dual Matrix Tools. which extend this to certain implicit derivatives arising in something or other.

How about
`Zygote.jl`

then?
That’s an alternative AD library from the creators of the aforementioned
`Flux`

.
It usually operates in
reverse mode
and does some zany compilation tricks to get extra fast.
It also has forward mode.
Has many
fancy features including
compiling to Google Cloud TPUs.
Hessian support is “somewhat”.
Flux itself does not yet default to Zygote,
using its own specialised reverse-mode autodiff
`Tracker`

,
but promises to switch transparently to Zygote in the future.
In the interim Zygote is still attractive has many luxurious options,
such as defining optimised custom derivatives easily, as well as weird quirks
such as occasionally bizarre error messages and failures to notice source code
updates.

One could roll one’s own autodiff system using the basic diff definitions in
`DiffRules`

. There is also the very
fancy planned Capstan, which aims to
use a tape system to inject forward and reverse mode differentiation into even
hostile code, and do much more besides.
However it also doesn’t work yet, and depends upon Julia features that also
don’t work yet, so don’t hold your breath. (Or: help them out!)

See also `XGrad`

which does symbolic
differentiation. It prefers to have access to the source code as text rather
than as an AST.
So I think that makes it similar to Zygote, but with worse PR?

### algopy

allows you to differentiate functions implemented as computer programs by using Algorithmic Differentiation (AD) techniques in the forward and reverse mode. The forward mode propagates univariate Taylor polynomials of arbitrary order. Hence it is also possible to use AlgoPy to evaluate higher-order derivative tensors.

Speciality of AlgoPy is the possibility to differentiate functions that contain matrix functions as +,-,*,/, dot, solve, qr, eigh, cholesky.

Looks sophisticated, and indeed supports differentiation elegantly; but not so actively maintained, and the source code is hard to find.

### Casadi

A classic is CasADi (Python, C++, MATLAB)

a symbolic framework for numeric optimization implementing automatic differentiation in forward and reverse modes on sparse matrix-valued computational graphs. It supports self-contained C-code generation and interfaces state-of-the-art codes such as SUNDIALS, IPOPT etc. It can be used from C++, Python or Matlab

…CasADi is an open-source tool, written in self-contained C++ code, depending only on the C++ Standard Library. It is developed by Joel Andersson and Joris Gillis at the Optimization in Engineering Center, OPTEC of the K.U. Leuven under supervision of Moritz Diehl. CasADi is distributed under the LGPL license, meaning the code can be used royalty-free even in commercial applications.

Documentation is minimal; probably should read the source or the published papers to understand how well this will fit your needs and, e.g. which arithmetic operations it supports.

It might be worth it for such features as graceful support for 100-fold nonlinear composition, for example. But the price you pay is a weird DSL that you must learn to use it.

### aDOL

Another classic.
`ADOL-C`

is a popular
C++ differentiation library with python binding.
Looks clunky from python but tenable from c++.

### ad

`ad`

, which is based off
uncertainties (and therefore python)
also does it.

### ceres solver

ceres-solver, (C++), the google least squares solver, seems to be pretty good at this although mostly focussed on least-squares losses.

### Misc

Symbolic math packages such as Sympy, MAPLE and Mathematica can all do actual symbolic differentiation, which is different again, but sometimes leads to the same thing. I haven’t tried Sympy or MAPLE, but Mathematica’s support for matrix calculus is weak.

`autodiff`

, which is usually referred to as`audi`

for the sake of clarity, offers light automatic differentiation for MATLAB. I think MATLAB now has a whole deep learning toolkit built in which surely supports something natively in this domain.

Baydin, Atilim Gunes, and Barak A. Pearlmutter. 2014. “Automatic Differentiation of Algorithms for Machine Learning,” April. http://arxiv.org/abs/1404.7456.

Baydin, Atilim Gunes, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. 2015. “Automatic Differentiation in Machine Learning: A Survey,” February. http://arxiv.org/abs/1502.05767.

Baydin, Atılım Güneş, Barak A. Pearlmutter, and Jeffrey Mark Siskind. 2016. “Tricks from Deep Learning,” November. http://arxiv.org/abs/1611.03777.

Cao, Y., S. Li, L. Petzold, and R. Serban. 2003. “Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution.” *SIAM Journal on Scientific Computing* 24 (3): 1076–89. https://doi.org/10.1137/S1064827501380630.

Carpenter, Bob, Matthew D. Hoffman, Marcus Brubaker, Daniel Lee, Peter Li, and Michael Betancourt. 2015. “The Stan Math Library: Reverse-Mode Automatic Differentiation in C++.” *arXiv Preprint arXiv:1509.07164*. http://arxiv.org/abs/1509.07164.

Errico, Ronald M. 1997. “What Is an Adjoint Model?” *Bulletin of the American Meteorological Society* 78 (11): 2577–92. https://doi.org/10.1175/1520-0477(1997)078<2577:WIAAM>2.0.CO;2.

Fike, Jeffrey, and Juan Alonso. 2011. “The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations.” In *49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition*. Orlando, Florida: American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2011-886.

Fischer, Keno, and Elliot Saba. 2018. “Automatic Full Compilation of Julia Programs and ML Models to Cloud TPUs,” October. http://arxiv.org/abs/1810.09868.

Giles, Mike B. 2008. “Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation.” In *Advances in Automatic Differentiation*, edited by Christian H. Bischof, H. Martin Bücker, Paul Hovland, Uwe Naumann, and Jean Utke, 64:35–44. Berlin, Heidelberg: Springer Berlin Heidelberg. http://eprints.maths.ox.ac.uk/1079/.

Gower, R. M., and A. L. Gower. 2016. “Higher-Order Reverse Automatic Differentiation with Emphasis on the Third-Order.” *Mathematical Programming* 155 (1-2): 81–103. https://doi.org/10.1007/s10107-014-0827-4.

Griewank, Andreas, and Andrea Walther. 2008. *Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation*. 2nd ed. Philadelphia, PA: Society for Industrial and Applied Mathematics.

Haro, A. 2008. “Automatic Differentiation Methods in Computational Dynamical Systems: Invariant Manifolds and Normal Forms of Vector Fields at Fixed Points.” *IMA Note*. http://www.maia.ub.es/~alex/admcds/admcds.pdf.

Innes, Michael. 2018. “Don’t Unroll Adjoint: Differentiating SSA-Form Programs,” October. http://arxiv.org/abs/1810.07951.

Johnson, Steven G. 2012. “Notes on Adjoint Methods for 18.335,” 6.

Laue, Soeren, Matthias Mitterreiter, and Joachim Giesen. 2018. “Computing Higher Order Derivatives of Matrix and Tensor Expressions.” In *Advances in Neural Information Processing Systems 31*, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, 2750–9. Curran Associates, Inc. http://papers.nips.cc/paper/7540-computing-higher-order-derivatives-of-matrix-and-tensor-expressions.pdf.

Maclaurin, Dougal, David K. Duvenaud, and Ryan P. Adams. 2015. “Gradient-Based Hyperparameter Optimization Through Reversible Learning.” In *ICML*, 2113–22. http://www.jmlr.org/proceedings/papers/v37/maclaurin15.pdf.

Neidinger, R. 2010. “Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming.” *SIAM Review* 52 (3): 545–63. https://doi.org/10.1137/080743627.

Neuenhofen, Martin. 2018. “Review of Theory and Implementation of Hyper-Dual Numbers for First and Second Order Automatic Differentiation,” January. http://arxiv.org/abs/1801.03614.

Pasquier, B, and F Primeau. n.d. “The F-1 Algorithm for Efficient Computation of the Hessian Matrix of an Objective Function Defined Implicitly by the Solution of a Steady-State Problem.” *SIAM Journal on Scientific Computing*, 10. https://www.bpasquier.com/publication/pasquier_primeau_sisc_2019/.

Rall, Louis B. 1981. *Automatic Differentiation: Techniques and Applications*. Lecture Notes in Computer Science 120. Berlin ; New York: Springer-Verlag.

Revels, Jarrett, Miles Lubin, and Theodore Papamarkou. 2016. “Forward-Mode Automatic Differentiation in Julia,” July. http://arxiv.org/abs/1607.07892.

Rumelhart, David E., Geoffrey E. Hinton, and Ronald J. Williams. 1986. “Learning Representations by Back-Propagating Errors.” *Nature* 323 (6088): 533–36. https://doi.org/10.1038/323533a0.

Tucker, Warwick. 2011. *Validated Numerics: A Short Introduction to Rigorous Computations*. Princeton: Princeton University Press. http://public.eblib.com/choice/publicfullrecord.aspx?p=683309.