# A Detailed Look

So, what's really going on?

## The BVar

The entire library revolves around the BVar, a variable holding a "backpropagatable value". As you use a BVar, the backprop library will track how it is used and where you use it. You can use evalBP to simply get the result, but using gradBP will perform backpropagation ("reverse-mode automatic differentiation")

For example, we looked earlier at a function that computes the square root of a quadrupled number:

myFunc :: Double
-> Double
myFunc x = sqrt (x * 4)


As we are using it, its type is "really":

myFunc' :: Reifies s W
=> BVar s Double
-> BVar s Double
myFunc' x = sqrt (x * 4)


myFunc' takes a BVar s Double (a BVar containing a Double) and returns a new one that is the square root of the quadrupled number. You can think of the Reifies s W as being a necessary constraint that allows backpropagation to happen.

BVars have Num, Fractional, and Floating instances, and so can be used with addition, multiplication, square rooting, etc. The "most general" type of myFunc is myFunc :: Floating a => a -> a, and since BVar s Double has a Floating instance, you could even just use it directly as a backpropagatable function.

This means you can basically treat a BVar s Double almost exactly like it was a Double --- you'll practically never tell the difference! BVars also have Ord and Eq instances, so you can compare them and branch on the results, too.

myAbs :: Reifies s W
=> BVar s Double
-> BVar s Double
myAbs x | x < 0     = negate x
| otherwise = x


The goal of the BVar interface is that you should be able to treat a BVar s a (a BVar containing an a) as if it was an a, with no easily noticeable differences.

## Runners

The entire point of the library is to write your computation as a normal function taking a BVar (or many) and returning a single BVar. Just treat BVars as if they actually were the value they are containing, and you can't go wrong.

Once you do this, you can use evalBP to "run" the function itself:

evalBP :: (forall s. Reifies s W => BVar s a -> BVar s b)
-> (a -> b)


This can be read as taking a BVar s a -> BVar s b and returning the a -> b that that function encodes. The RankN type there (the forall s.) is mostly there to prevent leakage of BVars (same as it is used in Control.Monad.ST and runST). It ensures that no BVars "escape" the function somehow.

evalBP is extremely efficient, and usually carries virtually zero overhead over writing your function directly on your values without BVars.

But, the more interesting thing of course is computing the gradient of your function. This is done with gradBP:

gradBP :: (Backprop a, Backprop b)
=> (forall s. Reifies s W => BVar s a -> BVar s b)
-> a
-> a


Which takes a BVar s a -> BVar s b backpropagatable function and an input, and returns the gradient at that input. It gives the direction of greatest positive change (in the output) of your input, and also how much a variation in your input will affect your output.

And that's all there is to it! Instead of a -> b's, write BVar s a -> BVar s b's to compute what you want to know the gradient of. These are normal functions, so you can use all of your favorite higher order functions and combinators (like (.), map, etc.). And once you're done, use gradBP to compute that gradient.

Note that gradBP requires a Backprop constraint on the input and output of your function. Backprop is essentially the typeclass of values that can be "backpropagated". For product types, this instance is automatically derivable. But writing your own custom instances for your own types is also fairly straightforward. More on this later!

The rest of the package really is just ways to manipulate BVar s as as if they were just as, to make everything as smooth as possible. Let's move on to learning about ways to manipulate BVars!