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 as instead of as 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 Traversals and Prisms:

-- | 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:

  1. “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.
  2. 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

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

2. Write tests!

3. 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 :)

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

5. Some open questions:

   a. Is it possible to support constructors with existential types?