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Tutorial: Algorithmic Differentiation

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Liang Wang edited this page Aug 28, 2017 · 35 revisions

Algorithmic differentiation (AD) is also known as automatic differentiation. It is a powerful tool in many fields, especially useful for fast prototyping in machine learning research. Comparing to numerical differentiation which can only provides approximate results, AD can calculates the exact derivative of a given function.

Owl provides both numerical differentiation (in Algodiff.Numerical module) and algorithmic differentiation (in Algodiff.Generic module). In this tutorial, I will only go through AD module since Numerical module is trivial to use.

Algodiff.Generic is a functor which is able to support both float32 and float64 precision AD. However, you do not need to deal with Algodiff.Generic.Make directly since there are already two ready-made modules.

  • Algodiff.S supports float32 precision;
  • Algodiff.D supports float64 precision;

Algodiff has implemented both forward and backward mode of AD. The complete list of APIs can be found in owl_algodiff_generic.mli. The core APIs are listed below.

  • val diff : (t -> t) -> t -> t : calculate derivative for f : scalar -> scalar
  • val grad : (t -> t) -> t -> t : calculate gradient for f : vector -> scalar
  • val jacobian : (t -> t) -> t -> t : calculate jacobian for f : vector -> vector
  • val hessian : (t -> t) -> t -> t : calculate hessian for f : scalar -> scalar
  • val laplacian : (t -> t) -> t -> t : calculate laplacian for f : scalar -> scalar

Besides, there are also more helper functions such as jacobianv for calculating jacobian vector product; diff' for calculating both f x and diff f x, and etc.

Example 1: Higher-Order Derivatives

The following code first defines a function f0, then calculates from the first to the fourth derivative by calling Algodiff.AD.diff function.

open Algodiff.D;;

let map f x = Vec.map (fun a -> a |> pack_flt |> f |> unpack_flt) x;;

(* calculate derivatives of f0 *)
let f0 x = Maths.(tanh x);;
let f1 = diff f0;;
let f2 = diff f1;;
let f3 = diff f2;;
let f4 = diff f3;;

let x = Vec.linspace (-4.) 4. 200;;
let y0 = map f0 x;;
let y1 = map f1 x;;
let y2 = map f2 x;;
let y3 = map f3 x;;
let y4 = map f4 x;;

(* plot the values of all functions *)
let h = Plot.create "plot_021.png" in
Plot.set_foreground_color h 0 0 0;
Plot.set_background_color h 255 255 255;
Plot.plot ~h x y0;
Plot.plot ~h x y1;
Plot.plot ~h x y2;
Plot.plot ~h x y3;
Plot.plot ~h x y4;
Plot.output h;;

Start your utop, then load and open Owl library. Copy and past the code above, the generated figure will look like this.

If you replace f0 in the previous example with the following definition, then you will have another good-looking figure :)

let f0 x = Maths.(
  let y = exp (neg x) in
  (F 1. - y) / (F 1. + y)
);;

As you see, you can just keep calling diff to get higher and higher-order derivatives. E.g., let g = f |> diff |> diff |> diff |> diff will give you the fourth derivative of f.

Example 2: Gradient Descent Algorithm

Gradient Descent (GD) is a popular numerical method for calculating the optimal value for a given function. Often you need to hand craft the derivative of your function f before plugging into gradient descendent algorithm. With Algodiff, derivation can be done easily. The following several lines of code define the skeleton of GD.

open Algodiff.D

let rec desc ?(eta=F 0.01) ?(eps=1e-6) f x =
  let g = (diff f) x in
  if (unpack_flt g) < eps then x
  else desc ~eta ~eps f Maths.(x - eta * g);;

Now let's define a function we want to optimise, then plug it into desc function.

let f x = Maths.(sin x + cos x);;

let x_min = desc f (F 0.1);;

Because we started searching from 0., the desc function successfully found the local minimum at -2.35619175250552448. You can visually verify that by plotting it out.

let g x = sin x +. cos x in
Plot.plot_fun g (-5.) 5.;;

Example 3: Newton's Algorithm

Newton's method is a root-finding algorithm by successively searching for better approximation of the root. The Newton's method converges faster than gradient descent. The following implementation calculates the exact hessian of f which in practice is very expensive operation.

open Algodiff.D

let rec newton ?(eta=F 0.01) ?(eps=1e-6) f x =
  let g, h = (gradhessian f) x in
  if (Maths.l2norm g |> unpack_flt) < eps then x
  else newton ~eta ~eps f Maths.(x - eta * g *@ (inv h));;

Now we can apply newton to find the extreme value of Maths.(cos x |> sum).

let _ =
  let f x = Maths.(cos x |> sum) in
  let y = newton f (Mat.uniform 1 2) in
  Mat.print y;;

Example 4: Backpropagation in Neural Network

Now let's talk about the hyped neural network. Backpropagation is the core of all neural networks, actually it is just a special case of reverse mode AD. Therefore, we can write up the backpropagation algorithm from scratch easily with the help of Algodiff module.

let backprop nn eta x y =
  let t = tag () in
  Array.iter (fun l ->
    l.w <- make_reverse l.w t;
    l.b <- make_reverse l.b t;
  ) nn.layers;
  let loss = Maths.(cross_entropy y (run_network x nn) / (F (Mat.row_num y |> float_of_int))) in
  reverse_prop (F 1.) loss;
  Array.iter (fun l ->
    l.w <- Maths.((primal l.w) - (eta * (adjval l.w))) |> primal;
    l.b <- Maths.((primal l.b) - (eta * (adjval l.b))) |> primal;
  ) nn.layers;
  loss |> unpack_flt

Yes, we just used only 13 lines of code to implement the backpropagation. Actually, with some extra coding, we can make a smart application to recognise handwritten digits. E.g., running the application will give you the following prediction on handwritten digit 6. The code has been included in Owl's example and you can find the complete example in test_mnist.ml

Example 5: Computation Graph of Simple Functions

Backward mode generates and maintains a computation graph in order to back propagate the error. The computation graph is very helpful in both debugging and understanding the characteristic of your numerical functions. Owl provides two functions to facilitate you in generating computation graphs.

val to_trace: t list -> string  (* print out the trace in human-readable format *)

val to_dot : tlist -> string    (* print out the computation graph in dot format *)

to_trace is useful when the graph is small and you can print it out on the terminal then observe it directly. to_dot is more useful when the graph grows bigger since you can use specialised visualisation tools to generate professional figures, such as Graphviz.

In the following, I will showcase several computation graphs. However, I will skip the details of how to generate these graphs since you can find out in the Owl's examples.

Let's start with a simple function as below.

let f x y = Maths.((x * sin (x + x) + ( F 1. * sqrt x) / F 7.) * (relu y) |> sum)

The generated computation graph looks like this.

Example 6: Computation Graph of VGG-like Neural Network

Let's define a VGG-like neural network as below.

open Neural.S
open Neural.S.Graph

let make_network input_shape =
  input input_shape
  |> normalisation ~decay:0.9
  |> conv2d [|3;3;3;32|] [|1;1|] ~act_typ:Activation.Relu
  |> conv2d [|3;3;32;32|] [|1;1|] ~act_typ:Activation.Relu ~padding:VALID
  |> max_pool2d [|2;2|] [|2;2|] ~padding:VALID
  |> dropout 0.1
  |> conv2d [|3;3;32;64|] [|1;1|] ~act_typ:Activation.Relu
  |> conv2d [|3;3;64;64|] [|1;1|] ~act_typ:Activation.Relu ~padding:VALID
  |> max_pool2d [|2;2|] [|2;2|] ~padding:VALID
  |> dropout 0.1
  |> fully_connected 512 ~act_typ:Activation.Relu
  |> linear 10 ~act_typ:Activation.Softmax
  |> get_network

The computation graph for this neural network become a bit more complicated now.

Enjoy Owl! Happy hacking!

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