Automatic heterogeneous back-propagation that can be used either implicitly (in the style of the ad library) or using explicit graphs built in monadic style. Implements reverse-mode automatic differentiation. Differs from ad by offering full heterogeneity – each intermediate step and the resulting value can have different types. Mostly intended for usage with tensor manipulation libraries to implement automatic back-propagation for gradient descent and other optimization techniques.
At the moment this project is in pre-alpha (v0.0.1.0), and is published/put up on Hackage as a call for comments and thoughts. It has 100% documentation coverage at the moment. Performance was not yet a priority before this, but will be from now on. (Previously, highest priority was API/usability). See the todos section for more information on what’s missing, and how one would be able to contribute!
MNIST Digit Classifier Example
The literate haskell file is a standalone haskell file that you
can compile (preferably with
-O2) on its own with stack or some other
dependency manager. It can also be compiled with the build script in the
project directory (if stack is installed, and appropriate dependencies are
$ ./Build.hs exe
The quick example below describes the running of a neural network with one
hidden layer to calculate its squared error with respect to target
which is parameterized by two weight matrices and two bias vectors.
Vector/matrix types are from the hmatrix package.
logistic :: Floating a => a -> a logistic x = 1 / (1 + exp (-x)) matVec :: (KnownNat m, KnownNat n) => Op '[ L m n, R n ] (R m) neuralNetImplicit :: (KnownNat m, KnownNat n, KnownNat o) => R m -> BPOpI s '[ L n m, R n, L o n, R o ] (R o) neuralNetImplicit inp = \(w1 :< b1 :< w2 :< b2 :< Ø) -> let z = logistic (liftB2 matVec w1 x + b1) in logistic (liftB2 matVec w2 z + b2) where x = constRef inp neuralNetExplicit :: (KnownNat m, KnownNat n, KnownNat o) => R m -> BPOp s '[ L n m, R n, L o n, R o ] (R o) neuralNetExplicit inp = withInps $ \(w1 :< b1 :< w2 :< b2 :< Ø) -> do y1 <- matVec ~$ (w1 :< x1 :< Ø) let x2 = logistic (y1 + b1) y2 <- matVec ~$ (w2 :< x2 :< Ø) return $ logistic (y2 + b2) where x1 = constVar inp
neuralNetImplicit can be “run” with the input
vectors and parameters (a
L n m,
L o n, and
R o) and calculate the
output of the neural net.
runNet :: (KnownNat m, KnownNat n, KnownNat o) => R m -> Tuple '[ L n m, R n, L o n, R o ] -> R o runNet inp = evalBPOp (neuralNetExplicit inp)
But, in defining
neuralNet, we also generated a graph that backprop can
use to do back-propagation, too!
dot :: KnownNat n => Op '[ R n , R n ] Double netGrad :: forall m n o. (KnownNat m, KnownNat n, KnownNat o) => R m -> R o -> Tuple '[ L n m, R n, L o n, R o ] -> Tuple '[ L n m, R n, L o n, R o ] netGrad inp targ params = gradBPOp opError params where -- calculate squared error, in *explicit* style opError :: BPOp s '[ L n m, R n, L o n, R o ] Double opError = do res <- neuralNetExplicit inp err <- bindRef (res - t) dot ~$ (err :< err :< Ø) where t = constRef targ
The result is the gradient of the input tuple’s components, with respect
Double result of
opError (the squared error). We can then use
this gradient to do gradient descent.
The current version isn’t optimized, but 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:
Calculating the gradient using backprop and calculating it by hand (by manual symbolic differentiation) are within an order of magnitude of each-other, time-wise. Using the backprop library takes about 6.5x as long in this case.
However, a full update step (calculate the gradient and update the neural net) adds a lot of constant costs, so for a full training step, the backprop library takes only 2.7x as long as manual symbolic differentation.
This means using this library only slows down your program by a factor of about 2.5x, compared to using only hmatrix.
It’s still definitely not ideal that more than half of the computation time is overhead from the library, but this is just where we stand at the moment. Optimization is just now starting!
Note that at the moment, simply running the network is only slightly slower when using backprop.
Profiling, to gauge where the overhead comes from (compared to “manual” back-propagation) and how to bring it down.
Some simple performance and API tweaks that are probably possible now and would clearly benefit: (if you want to contribute)
a. ~~Providing optimized
BValby supplying known gradients directly instead of relying on ad.~~ (Now finished, since b3898ae)
b. Switch from `ST s` to `IO`, and use `unsafePerformIO` to automatically bind `BVal`s (like *ad* does) when using `liftB`. This might remove some overhead during graph building, and, from an API standpoint, remove the need for explicit binding. c. Switch from `STRef`s/`IORef`s to `Array`. (This one I'm unclear if it would help any)
- Benchmark against competing back-propagation libraries like ad, and auto-differentiating tensor libraries like grenade
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 offer pattern matching on sum types/with different constructors for implicit-graph backprop? It’s possible for explicit-graph versions already, with
choicesVar, but not yet with the implicit-graph interface. Could be similar to an “Applicative vs. Monad” issue where you can only have pre-determined fixed computation paths when using
Applicative, but I’m not sure. Still, it would be nice, because if this was possible, we could possibly do away with explicit-graph mode completely.
b. Though we already have safe sum type support with explicit-graph mode, we can’t support GADTs yet safely. It’d be nice to see if this is possible, because a lot of dependently typed neural network stuff is made much simpler with GADTs.
As of v0.0.3.0, we have a way of dealing with GADTs in explicit-graph mode (using
withGADT) that is unsafe, and requires some ugly manual plumbing by the user that could potentially be confusing. But it would still be nice to have a way that is safe and doesn’t require the manual plumbing and isn’t as easy to mess up.
Removed samples as registered executables in the cabal file, to reduce dependences to a bare minimum. For convenience, build script now also compiles the samples into the local directory if stack is installed.
Added experimental (unsafe) combinators for working with GADTs with existential types,
withGADT, to Numeric.Backprop module.
Fixed broken links in Changelog.
Added optimized numeric
Ops, and re-write
Floatinginstances in terms of them.
Removed all traces of
Unityfrom the library, eliminating a whole swath of “explicit-Summer”/“explicit-Unity” versions of functions. As a consequence, the library now only works with
Numinstances. The API, however, is now much more simple.
Benchmark suite added for MNIST example.
- Initial pre-release, as a request for comments. API is in a usable form and everything is fully documented, but there are definitely some things left to be done. (See README.md)