The main difference in a multi-layer perceptron is that each neuron is only activated based on the results of an activation function. This makes use of the kernel trick that maps a non-linear problem into a higher dimensional problem which is linearly separable (see also support vector machines). The mathematics behind this is better explained here [PDF].
A typical activation function is tanh, shown below between -10 and 10 and generated using Wolfram Alpha
So how do we model this network in a functional programming language like Clojure? A typical solution using an imperative/OO language (see here for a good Python implementation) would use arrays representing the weights and modify things in situ, but that's not very functional.
We start by defining some constants (including the activation-function and its derivation).
bp-nnrepresents the network itself.
(def activation-function (fn [x] (Math/tanh x)))
(def activation-function-derivation (fn [y] (- 1.0 (* y y))))
(def num-hidden 2)
(def learning-rate 0.5)
(def momentum 0.1)
(defstruct bp-nn :weight-input :weight-output :change-input :change-output)
Next we create some simple initialization functions to create an initial neural network, together with some helper functions for iterating over matrices (which we'll model as lists of lists). Usually you'd use random weights to initialize things, but allowing fixed values makes testing possible.
"Create a matrix (list of lists)"
(repeat width (repeat height 0)))
"Apply a function to every element in a matrix"
(map (fn [x] (map func x)) m))
"Return a real number within the given range"
(+ (rand (- h l)) l))
([input hidden output]
(create-network input hidden output true))
([input hidden output use-random-weights]
"Create a network with the given number of input, hidden and output nodes"
(let [i (inc input)
w-func (if use-random-weights (fn [_] (rand-range -0.2 0.2)) (fn [_] 0.2))
o-func (if use-random-weights (fn [_] (rand-range -2.0 2.0)) (fn [_] 2.0))]
(matrix-map (make-matrix i hidden) w-func)
(matrix-map (make-matrix hidden output) o-func)
(make-matrix i hidden)
(make-matrix hidden output)))))
The first thing we should do is run a pattern through the network and see what comes out the other end. We're not just interested in the output result, we want to know what happened at the hidden layer, so we return a vector of ao (activation output) and ah (activation hidden).
compis functional composition.
((comp x y) 5)is the equivalent of
(x (y 5))so in the example below we add the numbers together and then apply the activation function. The nested
mapcalls allow us to iterate over the elements in a matrix.
"Calculate the hidden activations"
(apply map (comp activation-function +) (map (fn [col p] (map (partial * p) col)) w i)))
"Run the network with the given pattern and return the output and the hidden values"
(assert (= (count pattern) (dec (count (network :weight-input)))))
(let [p (cons 1 pattern)] ;; ensure bias term added
(let [wi (network :weight-input)
wo (network :weight-output)
ah (apply-activation-function wi p)
ao (apply-activation-function wo ah)]
In order to perform backwards-propagation we need a couple of helper functions that work on matrices and vectors to calculate changes in the output that will be used to update the weights.
These helper functions are pretty sick - (no-one wants to read
(map (partial reduce +) ...). A better design would probably be to introduce a proper matrix abstraction. There's the beginnings of one here but this is a bit too "Java" syntax heavy for more liking.
[wo ah od]
"Calculate the error terms for the hidden"
(let [errors (map (partial reduce +) (map (fn [x] (map * x od)) wo))] ;; Sick.
(map (fn [h e] (* e (activation-function-derivation h))) ah errors)))
[w deltas co ah]
(let [x (map
(fn [wcol ccol h]
(map (fn [wrow crow od]
(let [change (* od h)]
[(+ wrow (* learning-rate change) (* momentum crow)) change]))
wcol ccol deltas))
w co ah)]
[(matrix-map x first) (matrix-map x second)]))
I did warn you...
The next thing we need to implement is the back-propagation algorithm itself. This takes in more parameters than it needs to so that it can be tested standalone (it could be implemented as a local function using a closure to capture some of them). It returns an updated version of the neural network.
[target p results network]
"Back propagate the results to adjust the rates"
(assert (= (count target) (count (first (network :weight-output)))))
(let [pattern (cons 1 p) ;; ensure bias term added
ao (first results)
ah (second results)
error (map - target ao)
wi (network :weight-input)
wo (network :weight-output)
ci (network :change-input)
co (network :change-output)
output-deltas (map (fn [o e] (* e (activation-function-derivation o))) ao error)
hidden-deltas (calculate-hidden-deltas wo ah output-deltas)
updated-output-weights (update-weights wo output-deltas co ah)
updated-input-weights (update-weights wi hidden-deltas ci pattern)]
All that remains is to train the network. We need a set of samples with know results, together with a number of iterations to try. I've split these into
run-patternswhich runs through the patterns once, and
train-network>which creates the initial network and runs it through the patterns the specified number of times.
[network samples expecteds]
(if (empty? samples)
(let [expected (first expecteds)
sample (first samples)
[ah ao] (run-network sample network)
updated-network (back-propagate expected sample [ah ao] network)]
(recur updated-network (rest samples) (rest expecteds)))))
([samples expected iterations]
(train-network (create-network (count (first samples))
num-hidden (count (first expected)))
samples expected iterations))
([network samples expected iterations]
(if (zero? iterations)
(recur (run-patterns network samples expected) samples expected (dec iterations)))))
So how well does it work in practise? Pretty damn good. It correctly converges after a few iterations (this time 100) and consistently gets the XOR test data set correct.
(let [x (apply train-network (conj xor-test-data 100))]
(println (first (run-network [0 0] x)) "-->" 0)
(println (first (run-network [0 1] x)) "-->" 1)
(println (first (run-network [1 0] x)) "-->" 1)
(println (first (run-network [1 1] x)) "-->" 0)))
;;; Example run through the REPL
(0.10717792758953508) --> 0
(0.993502708495955) --> 1
(0.9930515903590437) --> 1
(0.00883530181467182) --> 0
Total code weighs in at ~100 of so lines of purely functional code. This made it a doddle to test. Coding this was an exercise in map masochism. Never have I had to construct so many convoluted
mapexpressions. At least it works in the end. It feels like it would be considerably simpler to implement this using mutability, I might try that for comparison purposes. Any suggestions on improving the code much appreciated! Full code is on GitHub.
(update - read some of Programming Clojure and applied some of the lessons learnt - see here.)
Neural networks are pretty interesting and in the next post I'll look at how to implement basic OCR using them.