backprop
Heterogeneous automatic differentation https://backprop.jle.im
Version on this page:  0.2.6.2 
LTS Haskell 13.18:  0.2.6.2 
Stackage Nightly 20190426:  0.2.6.2 
Latest on Hackage:  0.2.6.2 
Module documentation for 0.2.6.2
backprop
Automatic heterogeneous backpropagation.
Write your functions to compute your result, and the library will automatically generate functions to compute your gradient.
Differs from ad by offering full heterogeneity – each intermediate step and the resulting value can have different types (matrices, vectors, scalars, lists, etc.).
Useful for applications in differentiable programming and deep learning for creating and training numerical models, especially as described in this blog post on a purely functional typed approach to trainable models. Overall, intended for the implementation of gradient descent and other numeric optimization techniques. Comparable to the python library autograd.
Currently up on hackage, with haddock documentation! However, a proper library introduction and usage tutorial is available here. See also my introductory blog post. You can also find help or support on the gitter channel.
If you want to provide backprop for users of your library, see this guide to equipping your library with backprop.
MNIST Digit Classifier Example
My blog post introduces the concepts in this library in the context of training a handwritten digit classifier. I recommend reading that first.
There are some literate haskell examples in the source, though (rendered as pdf here), which can be built (if stack is installed) using:
$ ./Build.hs exe
There is a followup tutorial on using the library with more advanced types, with extensible neural networks a la this blog post, available as literate haskell and also rendered as a PDF.
Brief example
(This is a really brief version of the documentation walkthrough and my blog post)
The quick example below describes the running of a neural network with one
hidden layer to calculate its squared error with respect to target targ
,
which is parameterized by two weight matrices and two bias vectors.
Vector/matrix types are from the hmatrix package.
Let’s make a data type to store our parameters, with convenient accessors using lens:
import Numeric.LinearAlgebra.Static.Backprop
data Network = Net { _weight1 :: L 20 100
, _bias1 :: R 20
, _weight2 :: L 5 20
, _bias2 :: R 5
}
makeLenses ''Network
(R n
is an nlength vector, L m n
is an mbyn matrix, etc., #>
is
matrixvector multiplication)
“Running” a network on an input vector might look like this:
runNet net x = z
where
y = logistic $ (net ^^. weight1) #> x + (net ^^. bias1)
z = logistic $ (net ^^. weight2) #> y + (net ^^. bias2)
logistic :: Floating a => a > a
logistic x = 1 / (1 + exp (x))
And that’s it! neuralNet
is now backpropagatable!
We can “run” it using evalBP
:
evalBP2 runNet :: Network > R 100 > R 5
If we write a function to compute errors:
squaredError target output = error `dot` error
where
error = target  output
we can “test” our networks:
netError target input net = squaredError (auto target)
(runNet net (auto input))
This can be run, again:
evalBP (netError myTarget myVector) :: Network > Double
Now, we just wrote a normal function to compute the error of our network. With the backprop library, we now also have a way to compute the gradient, as well!
gradBP (netError myTarget myVector) :: Network > Network
Now, we can perform gradient descent!
gradDescent
:: R 100
> R 5
> Network
> Network
gradDescent x targ n0 = n0  0.1 * gradient
where
gradient = gradBP (netError targ x) n0
Ta dah! We were able to compute the gradient of our error function, just by only saying how to compute the error itself.
For a more fleshed out example, see the documentaiton, my blog post and the MNIST tutorial (also rendered as a pdf)
Benchmarks and Performance
Here are some basic benchmarks comparing the library’s automatic differentiation process to “manual” differentiation by hand. When using the MNIST tutorial as an example:
Here we compare:
 “Manual” differentiation of a 784 x 300 x 100 x 10 fullyconnected feedforward ANN.
 Automatic differentiation using backprop and the lensbased accessor interface
 Automatic differentiation using backprop and the “higherkinded data”based pattern matching interface
 A hybrid approach that manually provides gradients for individual layers but uses automatic differentiation for chaining the layers together.
We can see that simply running the network and functions (using evalBP
)
incurs virtually zero overhead. This means that library authors could actually
export only backproplifted functions, and users would be able to use them
without losing any performance.
As for computing gradients, there exists some associated overhead, from three main sources. Of these, the building of the computational graph and the Wengert Tape wind up being negligible. For more information, see a detailed look at performance, overhead, and optimization techniques in the documentation.
Note that the manual and hybrid modes almost overlap in the range of their random variances.
Comparisons
backprop can be compared and contrasted to many other similar libraries with some overlap:

The ad library (and variants like diffhask) support automatic differentiation, but only for homogeneous/monomorphic situations. All values in a computation must be of the same type — so, your computation might be the manipulation of
Double
s through aDouble > Double
function.backprop allows you to mix matrices, vectors, doubles, integers, and even keyvalue maps as a part of your computation, and they will all be backpropagated properly with the help of the
Backprop
typeclass. 
The autograd library is a very close equivalent to backprop, implemented in Python for Python applications. The difference between backprop and autograd is mostly the difference between Haskell and Python — static types with type inference, purity, etc.

There is a link between backprop and deep learning/neural network libraries like tensorflow, caffe, and theano, which all all support some form of heterogeneous automatic differentiation. Haskell libraries doing similar things include grenade.
These are all frameworks for working with neural networks or other gradientbased optimizations — they include things like builtin optimizers, methods to automate training data, builtin models to use out of the box. backprop could be used as a part of such a framework, like I described in my A Purely Functional Typed Approach to Trainable Models blog series; however, the backprop library itself does not provide any built in models or optimizers or automated data processing pipelines.
See documentation for a more detailed look.
Todo

Benchmark against competing backpropagation libraries like ad, and autodifferentiating tensor libraries like grenade

Write tests!

Explore opportunities for parallelization. There are some naive ways of directly parallelizing right now, but potential overhead should be investigated.

Some open questions:
a. Is it possible to support constructors with existential types?
b. How to support “monadic” operations that depend on results of previous operations? (
ApBP
already exists for situations that don’t)c. What needs to be done to allow us to automatically do second, thirdorder differentiation, as well? This might be useful for certain ODE solvers which rely on second order gradients and hessians.
Changes
Changelog
Version 0.2.6.2
April 7, 2019
https://github.com/mstksg/backprop/releases/tag/v0.2.6.2
 Fix a numerical bug that would occur when an input is used directly as the
result of a computation. (For example,
gradBP id
orgradBP2 const
).  Some internal changes to strictness which offers some performance boosts in computation of gradients.
Version 0.2.6.1
August 6, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.6.1
 Removed redundant constraints from
Floating
instance ofOp
.  Fixed lower bound for vinyl dependency.
Version 0.2.6.0
August 6, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.6.0
 Dropped
Expr
instance ofBackprop
. I don’t think anyone was actually using this. If you need this, please useNumeric.Backprop.Num
instead!  Removed Rec reexports.
 Compatibility with vinyl0.9, using the Data.Vinyl.Recursive interface. This requires some minor reshuffling of constraints but should not affect any monomorphic usage.
Version 0.2.5.0
June 19, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.5.0

Since typecombinators has been unmaintained for over two years, and is no longer compatible with modern GHC, the library internals was rewritten to be built on the typelevel combinators in the vinyl library instead. The main external API change is basically
Every
is replaced withAllConstrained
, andKnown Length
is replaced withRecApplicative
.To most users, this should make no difference APIwise. The only users affected should be those using the “N” family of functions (
backpropN
), who have to pass in heterogeneous lists. Heterogeneous lists now must be passed in using vinyl syntax and operators instead of the previous typecombinators interface. 
bpOp
added, to allow for nonrankN storage of backpropagatable functions in containers without impredicative types. 
Benchmarks use microlens and microlensth instead of lens.
Version 0.2.4.0
May 28, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.4.0
NOTE Major breaking changes to Explicit modules, and some reshuffling of typeclass constraints on various nonexplicit functions that should only affect polymorphic usage.
 Huge improvements in performance! Around 2040% reduction in
runtimes/overheads, with savings higher for large matrix situations or
situations with expensive
add
.  However, this restructuring required major reshuffling of constraints on
Backprop
/Num
for most functions. These are potentially breaking changes for polymorphic code, but monomorphic code should remain unchanged. However, code using the Explicit interfaces is most likely broken unfortunately. Fixes just include adding or droppingzeroFunc
s to the appropriate functions.  Added warnings to Explicit modules that the API is “semistable”.
overVar
and%~~
, for modifying fields. Essentially a wrapper over aviewVar
andsetVar
. Argument order in the
backpropWith
family of functions changed again; breaking change for those using anybackpropWith
function. However, the new order is much more usable.  Changes to the argument order in the
backprop
family of functions in the Explicit interfaces now reverted back to previous order, from v0.2.0 and before. Should be an “unbreaking” change, but will break code written in v0.2.3 style.  Bechmarks now include HKD access and a “hybrid” approach. Documentation updated to reflect results.
 Documentation updated to include a new “unital” law for
one
, namelyone = gradBP id
.  Fixity declarations for
^^?
,^^?!
, and<$>
.  Added
fmap . const
and<$
to Prelude modules. Backprop
instances forExpr
from simplereflect Added
zeroVecNum
andoneVecNum
to Numeric.Backprop.Class, which is potentially more efficient thanzeroVec
andoneVec
if the items are instances ofNum
and the vectors are larger. Also addedNumVec
newtype wrapper givingBackprop
instances to vectors usingzeroVecNum
andoneVecNum
instead ofzeroVec
andoneVec
. Build.hs
build script now also builds profiling results
Version 0.2.3.0
May 25, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.3.0
 Argument order in
backpropWith
family of functions switched around to allow for final gradient to be given afterthefact. Breaking change for anyone using anybackpropWith
function.  As a consequence of the previous change,
backprop
family of functions in Explicit interfaces also all changed argument order. Breaking change only for those using the Explicit interfaces.  Explicit
collectVar
no longer needs aZeroFunc
for the container, and so all versions ofcollectVar
and functions that use it (fmap
,liftA2
,liftA3
,traverse
,mapAccumL
,mapAccumR
) no longer requireBackprop
orNum
instances for the final returned container type. This enables a lot more flexibility in container types. Breaking change only for those using the Explicit interfaces. BV
pattern synonym added to Numeric.Backprop, abstracting over application ofsplitBV
andjoinBV
.foldr
andfoldl'
added to Prelude modules, for convenience.round
andfromIntegral'
(“unround”) added to Prelude modules.
Version 0.2.2.0
May 12, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.2.0
evalBP0
added, for convenience for noargument values that need to be evaluated without backpropagation.splitBV
andjoinBV
for “higherkinded data” styleBVar
manipulation, via theBVGroup
helper typeclass.toList
,mapAccumL
, andmapAccumR
for Prelude.Backprop modulesBackprop
instance forBVar
 COMPLETE pragmas for
T2
andT3
 Unexported
gzero
,gadd
, andgone
from Numeric.Backprop.Class  Many, many more instances of
Backprop
Backprop
instance forProxy
made nonstrict foradd
 Swapped type variable order for a few library functions, which might potentially be breaking changes.
Internal
 Fixed documentation for Num and Explicit Prelude modules, and rewrote normal and Num Prelude modules in terms of canonical Prelude definitions
 Switched to
errorWithoutStackTrace
wherever appropriate (in Internal module)
Version 0.2.1.0
May 8, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.1.0
 Added
ABP
newtype wrapper to Numeric.Backprop.Class (reexported from Numeric.Backprop and Numeric.Backprop.Explicit) to give freeBackprop
instances for Applicative actions.  Added
NumBP
newtype wrapper to Numeric.Backprop.Class (reexported in the same places asABP
) to give freeBackprop
instances forNum
instances.  Added
^^?!
(unsafe access) to Numeric.Backprop and Numeric.Backprop.Num. Backprop
instance forNatural
from Numeric.Natural. Should actually be safe, unlike itsNum
instance!zfFunctor
andofFunctor
for instances ofFunctor
for Numeric.Backprop.Explicit.realToFrac
andfromIntegral
to Prelude modulesT2
andT3
patterns for Numeric.Backprop, for conveniently constructing and deconstructing tuples.
Version 0.2.0.0
May 1, 2018
https://github.com/mstksg/backprop/releases/tag/v0.2.0.0
 Added
Backprop
class in Numeric.Backprop.Class, which is a typeclass specifically for “backpropagatable” values. This will replaceNum
.  API of Numeric.Backprop completely rewritten to require values be
instances of
Backprop
instead ofNum
. This closes some outstanding issues with the reliance ofNum
, and allows backpropagation to work with nonNum instances like variablelength vectors, matrices, lists, tuples, etc. (including types from accelerate)  Numeric.Backprop.Num and Prelude.Backprop.Num modules added, providing
the old interface that uses
Num
instances instead ofBackprop
instances, for those who wish to avoid writing orphan instances when working with external types.  Numeric.Backprop.Explicit and Prelude.Backprop.Explicit modules added,
providing an interface that allows users to manually specify how zeroing,
addition, and oneing works on a pervalue basis. Useful for those who
wish to avoid writing orphan instances of
Backprop
for types with noNum
instances, or if you are mixing and matching styles. backpropWith
variants added, allowing you to specify a “final gradient”, instead of assuming it to be 1. Added
auto
, a shorter alias forconstVar
inspired by the ad library.  Numeric.Backprop.Tuple module removed. I couldn’t find a significant
reason to keep it now that
Num
is no longer required for backpropagation.
Version 0.1.5.2
Apr 26, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.5.2
 Added
coerceVar
to Numeric.Backprop  Added
Random
instaces for all tuple types. Same as forBinary
, this does incur a random and time dependency only from the tuple types. Again, because these packages are a part of GHC’s boot libraries, this is hopefully not too bad.
Version 0.1.5.1
Apr 9, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.5.1
 Fixed
NFData
instance forT
; before, was shallow.  Added
Typeable
instances for all tuple types, and forBVar
.  Added
Eq
,Ord
,Show
, etc. instances forT
.  Added
Binary
instances for all tuple types. Note that this does incur a binary dependency only because of the tuple types; however, this will hopefully be not too much of an issue because binary is a GHC library anyway.
Version 0.1.5.0
Mar 30, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.5.0
T
added to Numeric.Backprop.Tuple: basically anHList
with aNum
instance.Eq
andOrd
instances forBVar
. Is this sound?
Internal
 Refactored
Monoid
instances in Numeric.Backprop.Tuple
Version 0.1.4.0
Mar 25, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.4.0
isoVar
,isoVar2
,isoVar3
, andisoVarN
: convenient aliases for applying isomorphisms toBVar
s. Helpful for use with constructors and deconstructors.opIso2
andopIso3
added to Numeric.Backprop.Op, for convenience.T0
(Unit with numeric instances) added to Numeric.Backprop.Tuple.
Internal
 Completely decoupled the internal implementation from
Num
, which showed some performance benefits. Mostly just to make the code slightly cleaner, and to prepare for some day potentially decoupling the external API fromNum
as well.
Version 0.1.3.0
Feb 12, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.3.0
 Preulude.Backprop module added with lifted versions of several Prelude and base functions.
liftOpX
family of operators now have a more logical ordering for type variables. This change breaks backwardscompatibility.opIsoN
added to export list of Numeric.BackpropnoGrad
andnoGrad1
added to Numeric.Backprop.Op, for functions with no defined gradient.
Internal
 Completely decoupled the internal implementation from
Num
, which showed some performance benefits.
Version 0.1.2.0
Feb 7, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.2.0
 Added currying and uncurrying functions for tuples in Numeric.Backprop.Tuple.
opIsoN
, for isomorphisms between a tuple of values and a value. (Internal) AD engine now using
Any
from ghcprim instead ofSome I
from typecombinators
Version 0.1.1.0
Feb 6, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.1.0
 Added canonical strict tuple types with
Num
instances, in the module Numeric.Backprop.Tuple. This is meant to be a bandaid for the problem of orphan instances and potential mismatched tuple types.  Fixed bug in
collectVar
that occurs if container sizes change
Internal
 Internal tweaks to the underlying automatic differentiation types that
decouple backpropagation from
Num
, internally.Num
is now just used externally as a part of the API, which might someday be made optional.
Version 0.1.0.0
Feb 5, 2018
https://github.com/mstksg/backprop/releases/tag/v0.1.0.0
 First nonalpha release.
 More or less complete redesign of library. The entire API is completely
changed, and there is no backwards compatibility!
 Everything is now “implicit” style, and there is no more
BP
monad.  Accessing items in
BVar
s is now lens, prism, and traversal based, instead of iso and genericsbased. Op
is no longer monadic Mono modules are removed.
 Implicit modules are removed, since they are the default
 Iso module is removed, since
Iso
s no longer play major role in the implementation of the library.
 Everything is now “implicit” style, and there is no more
 Removed dependency on ad and adbased ops, which had been pulling in the vast majority of dependencies.
 Moved from .cabal file to hpack system.
Version 0.0.3.0
Alpha
https://github.com/mstksg/backprop/releases/tag/v0.0.3.0

Removed samples as registered executables in the cabal file, to reduce dependences to a bare minimum. For convenience, build script now also compiles the samples into the local directory if stack is installed.

Added experimental (unsafe) combinators for working with GADTs with existential types,
withGADT
, to Numeric.Backprop module. 
Fixed broken links in changelog.
Version 0.0.2.0
Alpha
https://github.com/mstksg/backprop/releases/tag/v0.0.2.0

Added optimized numeric
Op
s, and rewriteNum
/Fractional
/Floating
instances in terms of them. 
Removed all traces of
Summer
/Unity
from the library, eliminating a whole swath of “explicitSummer”/“explicitUnity” versions of functions. As a consequence, the library now only works withNum
instances. The API, however, is now much more simple. 
Benchmark suite added for MNIST example.
Version 0.0.1.0
Alpha
https://github.com/mstksg/backprop/releases/tag/v0.0.1.0
 Initial prerelease, as a request for comments. API is in a usable form and everything is fully documented, but there are definitely some things left to be done. (See README.md)