I'm training myself over learning neural network. There is a function that I can't make my neural network learn: f(x) = max(x_1, x_2). It seems like a very simple function with 2 inputs and 1 input but yet a 3 layer neural network trained over a thousand sample with 2000 epochs get it completly wrong. I'm using deeplearning4j.
Is there any reason why the max function would be very hard to learn for a neural network or am I just tuning it wrong?
Just wanted to point out: If you use relu instead of tanh than there is actually an exact solution, and I guess if you would shrink down the network to this exact same size (1 hidden layer with 3 nodes), you would always end up with these weights (module permutations of nodes and scaling of weights (first layer scaled by gamma, second by 1/gamma)):
max(a,b) = ((1, 1, -1)) * relu( ((1,-1), (0,1), (0,-1)) * ((a,b)) )
where * is the matrix multiplication.
This equation translates the following human-readable version into NN-language:
max(a,b) = relu(a-b) + b = relu(a-b) + relu(b) - relu(-b)
I have not actually tested it, my point is, that it should theoretically be very easy for networks to learn this function.
EDIT: I just tested this and the result was as I expected it:
[[-1.0714666e+00 -7.9943770e-01 9.0549403e-01]
[ 1.0714666e+00 -7.7552663e-08 2.6146751e-08]]
and
[[ 0.93330014]
[-1.250879 ]
[ 1.1043695 ]]
where the corresponding first and second layer. Transposing the second and multiplying with the first set of weights one ends up with a normalized version which can be compared to my theoretic results very easily:
[[-9.9999988e-01 9.9999988e-01 1.0000000e+00]
[ 9.9999988e-01 9.7009000e-08 2.8875675e-08]]
It isnt that hard, at least, if you restrict x1 and x2 to be within an interval, e.g. between [0,3]. Taken the "RegressionSum" example from the deeplearning4j examples I quickly rewrote it to learn max instead sum and it works quite ok giving me that results:
Max(0.6815540048808918,0.3112081053899819) = 0.64
Max(2.0073597506364407,1.93796211086664) = 2.09
Max(1.1792029272560556,2.5514324329058233) = 2.58
Max(2.489185375059013,0.0818746888836388) = 2.46
Max(2.658169689797984,1.419135581889197) = 2.66
Max(2.855509810112818,2.9661811672685086) = 2.98
Max(2.774757710538552,1.3988513143140069) = 2.79
Max(1.5852295273047565,1.1228662895771744) = 1.56
Max(0.8403435207065576,2.5595015474951195) = 2.60
Max(0.06913178775631723,2.61883825802004) = 2.54
Below is my modified Version of the RegressionSum example, which was originally from Anwar 3/15/16:
public class RegressionMax {
//Random number generator seed, for reproducability
public static final int seed = 12345;
//Number of iterations per minibatch
public static final int iterations = 1;
//Number of epochs (full passes of the data)
public static final int nEpochs = 200;
//Number of data points
public static final int nSamples = 10000;
//Batch size: i.e., each epoch has nSamples/batchSize parameter updates
public static final int batchSize = 100;
//Network learning rate
public static final double learningRate = 0.01;
// The range of the sample data, data in range (0-1 is sensitive for NN, you can try other ranges and see how it effects the results
// also try changing the range along with changing the activation function
public static int MIN_RANGE = 0;
public static int MAX_RANGE = 3;
public static final Random rng = new Random(seed);
public static void main(String[] args){
//Generate the training data
DataSetIterator iterator = getTrainingData(batchSize,rng);
//Create the network
int numInput = 2;
int numOutputs = 1;
int nHidden = 10;
MultiLayerNetwork net = new MultiLayerNetwork(new NeuralNetConfiguration.Builder()
.seed(seed)
.iterations(iterations)
.optimizationAlgo(OptimizationAlgorithm.STOCHASTIC_GRADIENT_DESCENT)
.learningRate(learningRate)
.weightInit(WeightInit.XAVIER)
.updater(Updater.NESTEROVS).momentum(0.9)
.list()
.layer(0, new DenseLayer.Builder().nIn(numInput).nOut(nHidden)
.activation("tanh")
.build())
.layer(1, new OutputLayer.Builder(LossFunctions.LossFunction.MSE)
.activation("identity")
.nIn(nHidden).nOut(numOutputs).build())
.pretrain(false).backprop(true).build()
);
net.init();
net.setListeners(new ScoreIterationListener(1));
//Train the network on the full data set, and evaluate in periodically
for( int i=0; i<nEpochs; i++ ){
iterator.reset();
net.fit(iterator);
}
// Test the max of some numbers (Try different numbers here)
Random rand = new Random();
for (int i= 0; i< 10; i++) {
double d1 = MIN_RANGE + (MAX_RANGE - MIN_RANGE) * rand.nextDouble();
double d2 = MIN_RANGE + (MAX_RANGE - MIN_RANGE) * rand.nextDouble();
INDArray input = Nd4j.create(new double[] { d1, d2 }, new int[] { 1, 2 });
INDArray out = net.output(input, false);
System.out.println("Max(" + d1 + "," + d2 + ") = " + out);
}
}
private static DataSetIterator getTrainingData(int batchSize, Random rand){
double [] max = new double[nSamples];
double [] input1 = new double[nSamples];
double [] input2 = new double[nSamples];
for (int i= 0; i< nSamples; i++) {
input1[i] = MIN_RANGE + (MAX_RANGE - MIN_RANGE) * rand.nextDouble();
input2[i] = MIN_RANGE + (MAX_RANGE - MIN_RANGE) * rand.nextDouble();
max[i] = Math.max(input1[i], input2[i]);
}
INDArray inputNDArray1 = Nd4j.create(input1, new int[]{nSamples,1});
INDArray inputNDArray2 = Nd4j.create(input2, new int[]{nSamples,1});
INDArray inputNDArray = Nd4j.hstack(inputNDArray1,inputNDArray2);
INDArray outPut = Nd4j.create(max, new int[]{nSamples, 1});
DataSet dataSet = new DataSet(inputNDArray, outPut);
List<DataSet> listDs = dataSet.asList();
Collections.shuffle(listDs,rng);
return new ListDataSetIterator(listDs,batchSize);
}
}
