**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 differential 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

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:

For computing the gradient, there is about a 2.5ms overhead (or about 3.5x)
compared to computing the gradients by hand. Some more profiling and
investigation can be done, since there are two main sources of potential
slow-downs:

- "Inefficient" gradient computations, because of automated
differentiation not being as efficient as what you might get from doing
things by hand and simplifying. This sort of cost is probably not
avoidable.
- Overhead incurred by the book-keeping and actual automatic
differentiating system, which involves keeping track of a dependency
graph and propagating gradients backwards in memory. This sort of
overhead is what we would be aiming to reduce.

It is unclear which one dominates the current slowdown.

However, it may be worth noting that this isn't necessarily a significant
bottleneck. *Updating* the networks using *hmatrix* actually dominates the
runtime of the training. Manual gradient descent takes 3.2ms, so the extra
overhead is about 60%-70%.

Running the network (and the backprop-aware functions) incurs virtually
zero overhead (about 4%), meaning that library authors could actually
export backprop-aware functions by default and not lose any performance.

## 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.