**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
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 -> Network
```

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 fully-connected
feed-forward ANN.
- Automatic differentiation using
*backprop* and the lens-based accessor
interface
- Automatic differentiation using
*backprop* and the "higher-kinded
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* 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.

## 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 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.

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
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.

## Todo

Benchmark against competing back-propagation libraries like *ad*, and
auto-differentiating 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,
third-order differentiation, as well? This might be useful for certain
ODE solvers which rely on second order gradients and hessians.