Perceptron learning algorithm not converging to 0

Question

Here is my perceptron implementation in ANSI C:

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

float randomFloat()
{
    srand(time(NULL));
    float r = (float)rand() / (float)RAND_MAX;
    return r;
}

int calculateOutput(float weights[], float x, float y)
{
    float sum = x * weights[0] + y * weights[1];
    return (sum >= 0) ? 1 : -1;
}

int main(int argc, char *argv[])
{
    // X, Y coordinates of the training set.
    float x[208], y[208];

    // Training set outputs.
    int outputs[208];

    int i = 0; // iterator

    FILE *fp;

    if ((fp = fopen("test1.txt", "r")) == NULL)
    {
        printf("Cannot open file.\n");
    }
    else
    {
        while (fscanf(fp, "%f %f %d", &x[i], &y[i], &outputs[i]) != EOF)
        {
            if (outputs[i] == 0)
            {
                outputs[i] = -1;
            }
            printf("%f   %f   %d\n", x[i], y[i], outputs[i]);
            i++;
        }
    }

    system("PAUSE");

    int patternCount = sizeof(x) / sizeof(int);

    float weights[2];
    weights[0] = randomFloat();
    weights[1] = randomFloat();

    float learningRate = 0.1;

    int iteration = 0;
    float globalError;

    do {
        globalError = 0;
        int p = 0; // iterator
        for (p = 0; p < patternCount; p++)
        {
            // Calculate output.
            int output = calculateOutput(weights, x[p], y[p]);

            // Calculate error.
            float localError = outputs[p] - output;

            if (localError != 0)
            {
                // Update weights.
                for (i = 0; i < 2; i++)
                {
                    float add = learningRate * localError;
                    if (i == 0)
                    {
                        add *= x[p];
                    }
                    else if (i == 1)
                    {
                        add *= y[p];
                    }
                    weights[i] +=  add;
                }
            }

            // Convert error to absolute value.
            globalError += fabs(localError);

            printf("Iteration %d Error %.2f %.2f\n", iteration, globalError, localError);

            iteration++;
        }

        system("PAUSE");

    } while (globalError != 0);

    system("PAUSE");
    return 0;
}

The training set I'm using: Data Set

I have removed all irrelevant code. Basically what it does now it reads test1.txt file and loads values from it to three arrays: x, y, outputs.

Then there is a perceptron learning algorithm which, for some reason, is not converging to 0 (globalError should converge to 0) and therefore I get an infinite do while loop.

When I use a smaller training set (like 5 points), it works pretty well. Any ideas where could be the problem?

I wrote this algorithm very similar to this C# Perceptron algorithm:

---

EDIT:

Here is an example with a smaller training set:

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

float randomFloat()
{
    float r = (float)rand() / (float)RAND_MAX;
    return r;
}

int calculateOutput(float weights[], float x, float y)
{
    float sum = x * weights[0] + y * weights[1];
    return (sum >= 0) ? 1 : -1;
}

int main(int argc, char *argv[])
{
    srand(time(NULL));

    // X coordinates of the training set.
    float x[] = { -3.2, 1.1, 2.7, -1 };

    // Y coordinates of the training set.
    float y[] = { 1.5, 3.3, 5.12, 2.1 };

    // The training set outputs.
    int outputs[] = { 1, -1, -1, 1 };

    int i = 0; // iterator

    FILE *fp;

    system("PAUSE");

    int patternCount = sizeof(x) / sizeof(int);

    float weights[2];
    weights[0] = randomFloat();
    weights[1] = randomFloat();

    float learningRate = 0.1;

    int iteration = 0;
    float globalError;

    do {
        globalError = 0;
        int p = 0; // iterator
        for (p = 0; p < patternCount; p++)
        {
            // Calculate output.
            int output = calculateOutput(weights, x[p], y[p]);

            // Calculate error.
            float localError = outputs[p] - output;

            if (localError != 0)
            {
                // Update weights.
                for (i = 0; i < 2; i++)
                {
                    float add = learningRate * localError;
                    if (i == 0)
                    {
                        add *= x[p];
                    }
                    else if (i == 1)
                    {
                        add *= y[p];
                    }
                    weights[i] +=  add;
                }
            }

            // Convert error to absolute value.
            globalError += fabs(localError);

            printf("Iteration %d Error %.2f\n", iteration, globalError);          
        }

        iteration++;

    } while (globalError != 0);

    // Display network generalisation.
    printf("X       Y     Output\n");
    float j, k;
    for (j = -1; j <= 1; j += .5)
    {
        for (j = -1; j <= 1; j += .5)
        {
            // Calculate output.
            int output = calculateOutput(weights, j, k);
            printf("%.2f  %.2f  %s\n", j, k, (output == 1) ? "Blue" : "Red");
        }
    }

    // Display modified weights.
    printf("Modified weights: %.2f %.2f\n", weights[0], weights[1]);

    system("PAUSE");
    return 0;
}

Solution 1

In your current code, the perceptron successfully learns the direction of the decision boundary BUT is unable to translate it.

    y                              y
    ^                              ^
    |  - + \\  +                   |  - \\ +   +
    | -    +\\ +   +               | -   \\  + +   +
    | - -    \\ +                  | - -  \\    +
    | -  -  + \\  +                | -  -  \\ +   +
    ---------------------> x       --------------------> x
        stuck like this            need to get like this

^{(as someone pointed out, here is a more accurate version)}

The problem lies in the fact that your perceptron has no bias term, i.e. a third weight component connected to an input of value 1.

       w0   -----
    x ---->|     |
           |  f  |----> output (+1/-1)
    y ---->|     |
       w1   -----
               ^ w2
    1(bias) ---|

The following is how I corrected the problem:

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>

#define LEARNING_RATE    0.1
#define MAX_ITERATION    100

float randomFloat()
{
    return (float)rand() / (float)RAND_MAX;
}

int calculateOutput(float weights[], float x, float y)
{
    float sum = x * weights[0] + y * weights[1] + weights[2];
    return (sum >= 0) ? 1 : -1;
}

int main(int argc, char *argv[])
{
    srand(time(NULL));

    float x[208], y[208], weights[3], localError, globalError;
    int outputs[208], patternCount, i, p, iteration, output;

    FILE *fp;
    if ((fp = fopen("test1.txt", "r")) == NULL) {
        printf("Cannot open file.\n");
        exit(1);
    }

    i = 0;
    while (fscanf(fp, "%f %f %d", &x[i], &y[i], &outputs[i]) != EOF) {
        if (outputs[i] == 0) {
            outputs[i] = -1;
        }
        i++;
    }
    patternCount = i;

    weights[0] = randomFloat();
    weights[1] = randomFloat();
    weights[2] = randomFloat();

    iteration = 0;
    do {
        iteration++;
        globalError = 0;
        for (p = 0; p < patternCount; p++) {
            output = calculateOutput(weights, x[p], y[p]);

            localError = outputs[p] - output;
            weights[0] += LEARNING_RATE * localError * x[p];
            weights[1] += LEARNING_RATE * localError * y[p];
            weights[2] += LEARNING_RATE * localError;

            globalError += (localError*localError);
        }

        /* Root Mean Squared Error */
        printf("Iteration %d : RMSE = %.4f\n",
            iteration, sqrt(globalError/patternCount));
    } while (globalError > 0 && iteration <= MAX_ITERATION);

    printf("\nDecision boundary (line) equation: %.2f*x + %.2f*y + %.2f = 0\n",
        weights[0], weights[1], weights[2]);

    return 0;
}

... with the following output:

Iteration 1 : RMSE = 0.7206
Iteration 2 : RMSE = 0.5189
Iteration 3 : RMSE = 0.4804
Iteration 4 : RMSE = 0.4804
Iteration 5 : RMSE = 0.3101
Iteration 6 : RMSE = 0.4160
Iteration 7 : RMSE = 0.4599
Iteration 8 : RMSE = 0.3922
Iteration 9 : RMSE = 0.0000

Decision boundary (line) equation: -2.37*x + -2.51*y + -7.55 = 0

And here's a short animation of the code above using MATLAB, showing the decision boundary at each iteration:

Solution 2

It might help if you put the seeding of the random generator at the start of yout main instead of reseeding on every call to randomFloat, i.e.

float randomFloat()
{
    float r = (float)rand() / (float)RAND_MAX;
    return r;
}

// ...

int main(int argc, char *argv[])
{
    srand(time(NULL));

    // X, Y coordinates of the training set.
    float x[208], y[208];

Solution 3

Some small errors I spotted in your source code:

int patternCount = sizeof(x) / sizeof(int);

Better change this to

int patternCount = i;

so you doesn't have to rely on your x array to have the right size.

You increase iterations inside the p loop, whereas the original C# code does this outside the p loop. Better move the printf and the iteration++ outside the p loop before the PAUSE statement - also I'd remove the PAUSE statement or change it to

if ((iteration % 25) == 0) system("PAUSE");

Even doing all those changes, your program still doesn't terminate using your data set, but the output is more consistent, giving an error oscillating somewhere between 56 and 60.

The last thing you could try is to test the original C# program on this dataset, if it also doesn't terminate, there's something wrong with the algorithm (because your dataset looks correct, see my visualization comment).

Comments

[Dijar]

This is a very good advice, although it doesn't help (running it here leads to >= 1 million iterations without end in sight). I think there still is some problem with the algorithm here or with the assumption that it should converge to 0.

[Richard Knop]

I have added an example with smaller training set at the end of my post. You can try to compile that to see how it should work. I have no idea why it fails with larger training sets.

[Richard Knop]

Thanks for help, the problem is that the training set is lineary separable and therefor, the error should converge to 0 and potentially become 0 and the do while loop should end. There must be some mistake in my implementation of the Perceptron algorithm.

[Buksy]

How should I draw separating line? If y = ax + c is equation for separating line. How do I get a and c constants from weights of learned perceptron?

[Amro]

@Buksy: the equation of the line is: w0*x + w1*y + w2 = 0 where w_i are the weights learned (weight components connected to the x/y inputs + the bias; Refer to the diagram at the beginning of the post). Obviously you can reorder the terms to look like y=ax+b

[MathuSum Mut]

Why does it not converge if the statement if (outputs[i] == 0) outputs[i] = -1; is removed?

[Amro]

@MathuSumMut The activation function used calculateOutput returns -1 or +1, which I kept from the original code. The class target in the original dataset file is encoded as 0/1, hence the need to replace 0 with -1.