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Maintained by [email protected]
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Documentation and Walkthrough

Automatic heterogeneous back-propagation.

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 follow-up 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 n-length vector, L m n is an m-by-n matrix, etc., #> is matrix-vector multiplication)

“Running” a network on an input vector might look like this:

runNet net x = z
    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
    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!

    :: R 100
    -> R 5
    -> Network
    -> Network
gradDescent x targ n0 = n0 - 0.1 * gradient
    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:

  1. “Manual” differentiation of a 784 x 300 x 100 x 10 fully-connected feed-forward ANN.
  2. Automatic differentiation using backprop and the lens-based accessor interface
  3. Automatic differentiation using backprop and the “higher-kinded data”-based pattern matching interface
  4. 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 backprop-lifted 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.


backprop can be compared and contrasted to many other similar libraries with some overlap:

  1. 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 Doubles through a Double -> Double function.

    backprop allows you to mix matrices, vectors, doubles, integers, and even key-value maps as a part of your computation, and they will all be backpropagated properly with the help of the Backprop typeclass.

  2. 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.

  3. 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 gradient-based optimizations — they include things like built-in optimizers, methods to automate training data, built-in 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.


  1. Benchmark against competing back-propagation libraries like ad, and auto-differentiating tensor libraries like grenade

  2. Write tests!

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

  4. 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, third-order differentiation, as well? This might be useful for certain ODE solvers which rely on second order gradients and hessians.




July 23, 2023

  • vinyl-0.14 compatibility (@msakai), which can be disabled via cabal flag
  • GHC 9.0+ compatibility (@msakai)

Thanks to all generous contributors and commenters!


June 30, 2020

  • Compatibility with ghc-8.10.1 (@tonyday567)


August 13, 2019

  • Add Backprop instances for the various vinyl types.
  • Rewrite many Backprop instances over newtypes to coerce instead of go through Generics


April 7, 2019

  • Fix a numerical bug that would occur when an input is used directly as the result of a computation. (For example, gradBP id or gradBP2 const).
  • Some internal changes to strictness which offers some performance boosts in computation of gradients.


August 6, 2018

  • Removed redundant constraints from Floating instance of Op.
  • Fixed lower bound for vinyl dependency.


August 6, 2018

  • Dropped Expr instance of Backprop. I don’t think anyone was actually using this. If you need this, please use Numeric.Backprop.Num instead!
  • Removed Rec re-exports.
  • Compatibility with vinyl-0.9, using the Data.Vinyl.Recursive interface. This requires some minor reshuffling of constraints but should not affect any monomorphic usage.


June 19, 2018

  • Since type-combinators 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 type-level combinators in the vinyl library instead. The main external API change is basically Every is replaced with AllConstrained, and Known Length is replaced with RecApplicative.

    To most users, this should make no difference API-wise. 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 type-combinators interface.

  • bpOp added, to allow for non-rank-N storage of backpropagatable functions in containers without impredicative types.

  • Benchmarks use microlens and microlens-th instead of lens.


May 28, 2018

NOTE Major breaking changes to Explicit modules, and some re-shuffling of typeclass constraints on various non-explicit functions that should only affect polymorphic usage.

  • Huge improvements in performance! Around 20-40% 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 dropping zeroFuncs to the appropriate functions.
  • Added warnings to Explicit modules that the API is “semi-stable”.
  • overVar and %~~, for modifying fields. Essentially a wrapper over a viewVar and setVar.
  • Argument order in the backpropWith family of functions changed again; breaking change for those using any backpropWith 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 “un-breaking” 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, namely one = gradBP id.
  • Fixity declarations for ^^?, ^^?!, and <$>.
  • Added fmap . const and <$ to Prelude modules.
  • Backprop instances for Expr from simple-reflect
  • Added zeroVecNum and oneVecNum to Numeric.Backprop.Class, which is potentially more efficient than zeroVec and oneVec if the items are instances of Num and the vectors are larger. Also added NumVec newtype wrapper giving Backprop instances to vectors using zeroVecNum and oneVecNum instead of zeroVec and oneVec.
  • Build.hs build script now also builds profiling results


May 25, 2018

  • Argument order in backpropWith family of functions switched around to allow for final gradient to be given after-the-fact. Breaking change for anyone using any backpropWith 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 a ZeroFunc for the container, and so all versions of collectVar and functions that use it (fmap, liftA2, liftA3, traverse, mapAccumL, mapAccumR) no longer require Backprop or Num 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 of splitBV and joinBV.
  • foldr and foldl' added to Prelude modules, for convenience.
  • round and fromIntegral' (“unround”) added to Prelude modules.


May 12, 2018

  • evalBP0 added, for convenience for no-argument values that need to be evaluated without backpropagation.
  • splitBV and joinBV for “higher-kinded data” style BVar manipulation, via the BVGroup helper typeclass.
  • toList, mapAccumL, and mapAccumR for Prelude.Backprop modules
  • Backprop instance for BVar
  • COMPLETE pragmas for T2 and T3
  • Un-exported gzero, gadd, and gone from Numeric.Backprop.Class
  • Many, many more instances of Backprop
  • Backprop instance for Proxy made non-strict for add
  • Swapped type variable order for a few library functions, which might potentially be breaking changes.


  • 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)


May 8, 2018

  • Added ABP newtype wrapper to Numeric.Backprop.Class (re-exported from Numeric.Backprop and Numeric.Backprop.Explicit) to give free Backprop instances for Applicative actions.
  • Added NumBP newtype wrapper to Numeric.Backprop.Class (re-exported in the same places as ABP) to give free Backprop instances for Num instances.
  • Added ^^?! (unsafe access) to Numeric.Backprop and Numeric.Backprop.Num.
  • Backprop instance for Natural from Numeric.Natural. Should actually be safe, unlike its Num instance!
  • zfFunctor and ofFunctor for instances of Functor for Numeric.Backprop.Explicit.
  • realToFrac and fromIntegral to Prelude modules
  • T2 and T3 patterns for Numeric.Backprop, for conveniently constructing and deconstructing tuples.


May 1, 2018

  • Added Backprop class in Numeric.Backprop.Class, which is a typeclass specifically for “backpropagatable” values. This will replace Num.
  • API of Numeric.Backprop completely re-written to require values be instances of Backprop instead of Num. This closes some outstanding issues with the reliance of Num, and allows backpropagation to work with non-Num instances like variable-length 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 of Backprop 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 one-ing works on a per-value basis. Useful for those who wish to avoid writing orphan instances of Backprop for types with no Num 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 for constVar 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.


Apr 26, 2018

  • Added coerceVar to Numeric.Backprop
  • Added Random instaces for all tuple types. Same as for Binary, 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.


Apr 9, 2018

  • Fixed NFData instance for T; before, was shallow.
  • Added Typeable instances for all tuple types, and for BVar.
  • Added Eq, Ord, Show, etc. instances for T.
  • 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.


Mar 30, 2018

  • T added to Numeric.Backprop.Tuple: basically an HList with a Num instance.
  • Eq and Ord instances for BVar. Is this sound?


  • Refactored Monoid instances in Numeric.Backprop.Tuple


Mar 25, 2018

  • isoVar, isoVar2, isoVar3, and isoVarN: convenient aliases for applying isomorphisms to BVars. Helpful for use with constructors and deconstructors.
  • opIso2 and opIso3 added to Numeric.Backprop.Op, for convenience.
  • T0 (Unit with numeric instances) added to Numeric.Backprop.Tuple.


  • 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 from Num as well.


Feb 12, 2018

  • 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 backwards-compatibility.
  • opIsoN added to export list of Numeric.Backprop
  • noGrad and noGrad1 added to Numeric.Backprop.Op, for functions with no defined gradient.


  • Completely decoupled the internal implementation from Num, which showed some performance benefits.


Feb 7, 2018

  • 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 ghc-prim instead of Some I from type-combinators


Feb 6, 2018

  • Added canonical strict tuple types with Num instances, in the module Numeric.Backprop.Tuple. This is meant to be a band-aid for the problem of orphan instances and potential mismatched tuple types.
  • Fixed bug in collectVar that occurs if container sizes change


  • 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.


Feb 5, 2018

  • First non-alpha 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 BVars is now lens-, prism-, and traversal- based, instead of iso- and generics-based.
    • Op is no longer monadic
    • Mono modules are removed.
    • Implicit modules are removed, since they are the default
    • Iso module is removed, since Isos no longer play major role in the implementation of the library.
  • Removed dependency on ad and ad-based ops, which had been pulling in the vast majority of dependencies.
  • Moved from .cabal file to hpack system.



  • 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.



  • Added optimized numeric Ops, and re-write Num/Fractional/Floating instances in terms of them.

  • Removed all traces of Summer/Unity from the library, eliminating a whole swath of “explicit-Summer”/“explicit-Unity” versions of functions. As a consequence, the library now only works with Num instances. The API, however, is now much more simple.

  • Benchmark suite added for MNIST example.



  • Initial pre-release, 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