# backprop

**Introductory blog post**

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. Mostly intended for usage
with gradient descent and other numeric optimization techniques.

Currently up on hackage (with 100% documentation coverage), but more
up-to-date documentation is currently rendered on github pages!

## 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 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*:

```
data Network i h o = Net { _weight1 :: L h i
, _bias1 :: R h
, _weight2 :: L o h
, _bias2 :: R o
}
makeLenses ''Network
```

Normally, we might write code to "run" a neural network on an input like this:

```
neuralNet
:: R i
-> Network i h o
-> R h
neuralNet x n = z
where
y = logistic $ (n ^. weight1) #> x + (n ^. bias1)
z = logistic $ (n ^. weight2) #> y + (n ^. bias2)
logistic :: Floating a => a -> a
logistic x = 1 / (1 + exp (-x))
```

(`R i`

is an i-length vector, `L h i`

is an h-by-i matrix, etc., `#>`

is
matrix-vector multiplication, and `^.`

is access to a field via lens.)

When given an input vector and the network, we compute the result of the neural
network ran on the input vector.

We can write it, instead, using *backprop*:

```
neuralNet
:: Reifies s W
=> BVar s (R i)
-> BVar s (Network i h o)
-> BVar s (R o)
neuralNet x n = z
where
y = logistic $ (n ^^. weight1) #> x + (n ^^. bias1)
z = logistic $ (n ^^. weight2) #> y + (n ^^. bias2)
logistic :: Floating a => a -> a
logistic x = 1 / (1 + exp (-x))
```

(`#>!`

is a backprop-aware version of `#>`

, and `^^.`

is access to a field via
lens in a `BVar`

)

And that's it! `neuralNet`

is now backpropagatable!

We can "run" it using `evalBP`

:

```
evalBP (neuralNet (constVar x)) :: Network i h o -> R o
```

And we can find the gradient using `gradBP`

:

```
gradBP (neuralNet (constVar x)) :: Network i h o -> Network i h o
```

If we write a function to compute errors:

```
netError
:: Reifies s W
=> BVar s (R i)
-> BVar s (R o)
-> BVar s (Network i h o)
-> BVar s Double
netError x targ n = norm_2 (neuralNet x - t)
```

(`norm_2`

is a backprop-aware euclidean norm)

Now, we can perform gradient descent!

```
gradDescent
:: R i
-> R o
-> Network i h o
-> Network i h o
gradDescent x targ n0 = n0 - 0.1 * gradient
where
gradient = gradBP (netError (constVar x) (constVar targ)) 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 my blog post and the MNIST
tutorial (also rendered as a pdf)

## Lens Access

A lot of the friction of dealing with `BVar s a`

s instead of `a`

s directly is
alleviated with the lens interface.

With a lens, you can "view" and "set" items inside a `BVar`

, as if they were
the actual values:

```
(^.) :: a -> Lens' a b -> b
(^^.) :: BVar s a -> Lens' a b -> BVar s b
(.~) :: Lens' a b -> b -> a -> a
(.~~) :: Lens' a b -> BVar s b -> BVar s a -> BVar s a
```

And you can also extract multiple potential targets, as well, using
`Traversal`

s and `Prism`

s:

```
-- | Actually takes a Traversal, to be more general.
-- Can be used to implement "pattern matching" on BVars
(^?) :: a -> Prism' a b -> Maybe ( b)
(^^?) :: BVar s a -> Prism' a b -> Maybe (BVar s b)
(^..) :: a -> Traversal' a b -> [ b]
(^^..) :: BVar s a -> Traversal' a b -> [BVar s b]
```

Note that the library itself has no *lens* dependency, using *microlens*
instead.

## 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 potentially ditching `Num`

for another typeclass that only has `+`

,
`0`

, and `1`

. Currently, `Num`

is required for all backpropagated types,
but only `+`

, `fromInteger 0`

, and `fromInteger 1`

are ever used.

The main upside to using `Num`

is that it integrates well with the rest of
the Haskell ecosystem, and many things already have useful `Num`

instances.

There are two downsides -- one minor and one major.

It requires more work to make a type backpropagatable. Instead of
writing only `+`

, `0`

and `1`

, users must also define `*`

, `-`

or
`negate`

, `abs`

, `signum`

, and all of `fromInteger`

. However, I don't
see this being a big issue in practice, since most values that will be
used with *backprop* would presumably also benefit from having a full
`Num`

instance even without the need to backprop.

Automatically generated prisms (used with `^^?`

) work with tuples, and
so cannot work out-of-the-box without a `Num`

instance for tuples. In
addition, it's often useful to have anonymous products and tuples in
general.

This is bandaided-over by having *backprop* provide canonical
tuple-with-`Num`

types for different libraries to use, but it's not a
perfect solution.

This can be resolved by using the orphan instances in the
*NumInstances* package. Still, there might be some headache for
application developers if different libraries using *backprop*
accidentally pull in their orphan instances from different places.

Alternatively, one day we can get `Num`

instances for tuples into
*base*!

The extra complexity that would come from adding a custom typeclass just
for `+`

/ `0`

/ `1`

, though, I feel, might not be worth the benefit. The
entire numeric Haskell ecosystem, at the time, revolves around `Num`

.

However, it is worth noting that it wouldn't be too hard to add "Additive
Typeclass" instances for any custom types -- one would just need to define
`(<+>) = (+)`

, `zero = fromInteger 0`

, and `one = fromInteger 1`

(a
three-liner), so it might not be too bad.

But really, a lot of this would all resolve itself if we got `Num`

instances for tuples in base :)

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?