A simple Python program for an ANN to cover the MNIST dataset – XI – confusion matrix

Welcome back to my readers who followed me through the (painful?) process of writing a Python class to simulate a “Multilayer Perceptron” [MLP]. The pain in my case resulted from the fact that I am still a beginner in Machine Learning [ML] and Python. Nevertheless, I hope that we have meanwhile acquired some basic understanding of how a MLP works and “learns”. During the course of the last articles we had a close look at such nice things as “forward propagation”, “gradient descent”, “mini-batches” and “error backward propagation”. For the latter I gave you a mathematical description to grasp the background of the matrix operations involved.

Where do we stand after 10 articles and a PDF on the math?

A simple program for an ANN to cover the Mnist dataset – X – mini-batch-shuffling and some more tests
A simple program for an ANN to cover the Mnist dataset – IX – First Tests
A simple program for an ANN to cover the Mnist dataset – VIII – coding Error Backward Propagation
A simple program for an ANN to cover the Mnist dataset – VII – EBP related topics and obstacles
A simple program for an ANN to cover the Mnist dataset – VI – the math behind the „error back-propagation“
A simple program for an ANN to cover the Mnist dataset – V – coding the loss function
A simple program for an ANN to cover the Mnist dataset – IV – the concept of a cost or loss function
A simple program for an ANN to cover the Mnist dataset – III – forward propagation
A simple program for an ANN to cover the Mnist dataset – II – initial random weight values
A simple program for an ANN to cover the Mnist dataset – I – a starting point

We have a working code

  • with some parameters to control layers and node numbers, learning and momentum rates and regularization,
  • with many dummy parts for other output and activation functions than the sigmoid function we used so far,
  • with prepared code fragments for applying MSE instead of “Log Loss” as a cost function,
  • and with dummy parts for handling different input datasets than the MNIST example.

The code is not yet optimized; it includes e.g. many statements for tests which we should eliminate or comment out. A completely open conceptual aspect is the optimization of the adaption of the learning rate; it is very primitive so far. We also need an export/import functionality to be able to perform training with a series of limited epoch numbers per run.
We also should save the weights and accuracy data after a fixed epoch interval to be able to analyze a bit more after training. Another idea – though probably costly – is to even perform intermediate runs on the test data set an get some information on the development of the averaged error on the test data set.

Despite all these deficits, which we need to cover in some more articles, we are already able to perform an insightful task – namely to find out with which numbers and corresponding images of the MNIST data set our MLP has problems with. This leads us to the topics of a confusion matrix and other measures for the accuracy of our algorithm.

However, before we look at these topics, we first create some useful code, which we can save inside cells of the Jupyter notebook we maintain for testing our class “MyANN”.

Some functions to evaluate the prediction capability of our ANN after training

For further analysis we shall apply the following functions later on:

# ------ predict results for all test data 
# *************************
def predict_all_test_data(): 
    size_set = ANN._X_test.shape[0]

    li_Z_in_layer_test  = [None] * ANN._n_total_layers
    li_Z_in_layer_test[0] = ANN._X_test
    
    # Transpose input data matrix  
    ay_Z_in_0T       = li_Z_in_layer_test[0].T
    li_Z_in_layer_test[0] = ay_Z_in_0T
    li_A_out_layer_test  = [None] * ANN._n_total_layers

    # prediction by forward propagation of the whole test set 
    ANN._fw_propagation(li_Z_in = li_Z_in_layer_test, li_A_out = li_A_out_layer_test, b_print = False) 
    ay_predictions_test = np.argmax(li_A_out_layer_test[ANN._n_total_layers-1], axis=0)
    
    # accuracy 
    ay_errors_test = ANN._y_test - ay_predictions_test 
    acc = (np.sum(ay_errors_test == 0)) / size_set
    print ("total acc for test data = ", acc)

def predict_all_train_data(): 
    size_set = ANN._X_train.shape[0]

    li_Z_in_layer_test  = [None] * ANN._n_total_layers
    li_Z_in_layer_test[0] = ANN._X_train
    # Transpose 
    ay_Z_in_0T       = li_Z_in_layer_test[0].T
    li_Z_in_layer_test[0] = ay_Z_in_0T
    li_A_out_layer_test  = [None] * ANN._n_total_layers

    ANN._fw_propagation(li_Z_in = li_Z_in_layer_test, li_A_out = li_A_out_layer_test, b_print = False) 
    Result = np.argmax(li_A_out_layer_test[ANN._n_total_layers-1], axis=0)
    Error = ANN._y_train - Result 
    acc = (np.sum(Error == 0)) / size_set
    print ("total acc for train data = ", acc)    
    

# Plot confusion matrix 
# orginally from Runqi Yang; 
# see https://gist.github.com/hitvoice/36cf44689065ca9b927431546381a3f7
def cm_analysis(y_true, y_pred, filename, labels, ymap=None, figsize=(10,10)):
    """
    Generate matrix plot of confusion matrix with pretty annotations.
    The plot image is saved to disk.
    args: 
      y_true:    true label of the data, with shape (nsamples,)
      y_pred:    prediction of the data, with shape (nsamples,)
      filename:  filename of figure file to save
      labels:    string array, name the order of class labels in the confusion matrix.
                 use `clf.classes_` if using scikit-learn models.
                 with shape (nclass,).
      ymap:      dict: any -> string, length == nclass.
                 if not None, map the labels & ys to more understandable strings.
                 Caution: original y_true, y_pred and labels must align.
      figsize:   the size of the figure plotted.
    """
    if ymap is not None:
        y_pred = [ymap[yi] for yi in y_pred]
        y_true = [ymap[yi] for yi in y_true]
        labels = [ymap[yi] for yi in labels]
    cm = confusion_matrix(y_true, y_pred, labels=labels)
    cm_sum = np.sum(cm, axis=1, keepdims=True)
    cm_perc = cm / cm_sum.astype(float)
 * 100
    annot = np.empty_like(cm).astype(str)
    nrows, ncols = cm.shape
    for i in range(nrows):
        for j in range(ncols):
            c = cm[i, j]
            p = cm_perc[i, j]
            if i == j:
                s = cm_sum[i]
                annot[i, j] = '%.1f%%\n%d/%d' % (p, c, s)
            elif c == 0:
                annot[i, j] = ''
            else:
                annot[i, j] = '%.1f%%\n%d' % (p, c)
    cm = pd.DataFrame(cm, index=labels, columns=labels)
    cm.index.name = 'Actual'
    cm.columns.name = 'Predicted'
    fig, ax = plt.subplots(figsize=figsize)
    ax=sns.heatmap(cm, annot=annot, fmt='')
    #plt.savefig(filename)

    
#
# Plotting 
# **********
def plot_ANN_results(): 
    num_epochs  = ANN._n_epochs
    num_batches = ANN._n_batches
    num_tot = num_epochs * num_batches

    cshape = ANN._ay_costs.shape
    print("n_epochs = ", num_epochs, " n_batches = ", num_batches, "  cshape = ", cshape )
    tshape = ANN._ay_theta.shape
    print("n_epochs = ", num_epochs, " n_batches = ", num_batches, "  tshape = ", tshape )


    #sizing
    fig_size = plt.rcParams["figure.figsize"]
    fig_size[0] = 12
    fig_size[1] = 5

    # Two figures 
    # -----------
    fig1 = plt.figure(1)
    fig2 = plt.figure(2)

    # first figure with two plot-areas with axes 
    # --------------------------------------------
    ax1_1 = fig1.add_subplot(121)
    ax1_2 = fig1.add_subplot(122)

    ax1_1.plot(range(len(ANN._ay_costs)), ANN._ay_costs)
    ax1_1.set_xlim (0, num_tot+5)
    ax1_1.set_ylim (0, 1500)
    ax1_1.set_xlabel("epochs * batches (" + str(num_epochs) + " * " + str(num_batches) + " )")
    ax1_1.set_ylabel("costs")

    ax1_2.plot(range(len(ANN._ay_theta)), ANN._ay_theta)
    ax1_2.set_xlim (0, num_tot+5)
    ax1_2.set_ylim (0, 0.15)
    ax1_2.set_xlabel("epochs * batches (" + str(num_epochs) + " * " + str(num_batches) + " )")
    ax1_2.set_ylabel("averaged error")


 
The first function “predict_all_test_data()” allows us to create an array with the predicted values for all test data. This is based on a forward propagation of the full set of test data; so we handle some relatively big matrices here. The second function delivers prediction values for all training data; the operations of propagation algorithm involve even bigger matrices here. You will nevertheless experience that the calculations are performed very quickly. Prediction is much faster than training!

The third function “cm_analysis()” is not from me, but taken from Github Gist; see below. The fourth function “plot_ANN_results()” creates plots of the evolution of the cost function and the averaged error after training. We come back to these functions below.

To be able to use these functions we need to perform some more imports first. The full list of statements which we should place in the first Jupyter cell of our test notebook now reads:

import numpy as np
import numpy.random as npr
import math 
import sys
import pandas as pd
from sklearn.datasets import fetch_openml
from sklearn.metrics import confusion_matrix
from scipy.special import expit  
import seaborn as sns
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
import matplotlib.patches as mpat 
import time 
import imp
from mycode import myann

Note the new lines for the import of the “pandas” and “seaborn” libraries. Please inform yourself about the purpose of each library on the Internet.

Limited Accuracy

In the last article we performed some tests which showed a thorough robustness of our MLP regarding the MNIST datatset. There was some slight overfitting, but
playing around with hyper-parameters showed no extraordinary jump in “accuracy“, which we defined to be the percentage of correctly predicted records in the test dataset.

In general we can say that an accuracy level of 95% is what we could achieve within the range of parameters we played around with. Regression regularization (Lambda2 > 0) had some positive impact. A structural change to a MLP with just one layer did NOT give us a real breakthrough regarding CPU-time consumption, but when going down to 50 or 30 nodes in the intermediate layer we saw at least some reduction by up to 25%. But then our accuracy started to become worse.

Whilst we did our tests we measured the ANN’s “accuracy” by comparing the number of records for which our ANN did a correct prediction with the total number of records in the test data set. This is a global measure of accuracy; it averages over all 10 digits, i.e. all 10 classification categories. However, if we want to look a bit deeper into the prediction errors our MLP obviously produces it is, however, useful to introduce some more quantities and other measures of accuracy, which can be applied on the level of each output category.

Measures of accuracy, related quantities and classification errors for a specific category

The following quantities and basic concepts are often used in the context of ML algorithms for classification tasks. Predictions of our ANN will not be error free and thus we get an accuracy less than 100%. There are different reasons for this – and they couple different output categories. In the case of MNIST the output categories correspond to the digits 0 to 9. Let us take a specific output category, namely the digit “5”. Then there are two basic types of errors:

  • The network may have predicted a “3” for a MNIST image record, which actually represents a “5” (according to the “y_train”-value for this record). This error case is called a “False Negative“.
  • The network may have predicted a “5” for a MNIST image record, which actually represents a “3” according to its “y_train”-value. This error case is called a “False Positive“.

Both cases mark some difference between an actual and predicted number value for a MNIST test record. Technically, “actual” refers to the number value given by the related record in our array “ANN._y_test”. “Predicted” refers to the related record in an array “ay_prediction_test”, which our function “predict_all_test_data()” returns (see the code above).

Regarding our example digit “5” we obviously can distinguish between the following quantities:

  • AN : The total number of all records in the test data set which actually correspond to our digit “5”.
  • TP: The number of “True Positives”, i.e. the number of those cases correctly detected as “5”s.
  • FP: The number of “False Positives”, i.e. the number of those cases where our ANN falsely predicts a “5”.
  • FN: The number of “False Negatives”, i.e. the number of those cases where our ANN falsely predicts another digit than “5”, but where it actually should predict a “5”.

Then we can calculate the following ratios which all somehow measure “accuracy” for a specific output category:

  • Precision:
    TP / (TP + FP)
  • Recall:
    TP / ( TP + FN))
  • Accuracy:
    TP / AN
  • F1:
    TP / ( TP + 0.5*(FN + TP) )

A careful reader will (rightly) guess that the quantity “recall” corresponds to what we would naively define as “accuracy” – namely the ratio TP/AN.
From its definition it is clear that the quantity “F1” gives us a weighted average between the measures “precision” and “recall”.

How can we get these numbers for all 10 categories from our MLP after training ?

Confusion matrix

When we want to analyze our basic error types per category we need to look at the discrepancy between predicted and actual data. This suggests a presentation in form of a matrix with all for all possible category values both in x- and y-direction. The cells of such a matrix – e.g. a cell for an actual “5” and a predicted “3” – could e.g. be filled with the corresponding FN-number.

We will later on develop our own code to solve the task of creating and displaying such a matrix. But there is a nice guy called Runqi Yang who shared some code for precisely this purpose on GitHub Gist; see https://gist.github.com/hitvoice/36c…
We can use his suggested code as it is in our context. We have already presented it above in form of the function “cm_analysis()“, which uses the pandas and seaborn libraries.

After a training run with the following parameters

try: 
    ANN = myann.MyANN(my_data_set="mnist_keras", n_hidden_layers = 2, 
                 ay_nodes_layers = [0, 70, 30, 0], 
                 n_nodes_layer_out = 10,  
                 my_loss_function = "LogLoss",
                 n_size_mini_batch = 500,
                 n_epochs = 1800, 
                 n_max_batches = 2000,  # small values only for test runs
                 lambda2_reg = 0.2, 
                 lambda1_reg = 0.0,      
                 vect_mode = 'cols', 
                 learn_rate = 0.0001,
                 decrease_const = 0.000001,
                 mom_rate   = 0.00005,  
                 shuffle_batches = True,
                 print_period = 50,         
                 figs_x1=12.0, figs_x2=8.0, 
                 legend_loc='upper right',
                 b_print_test_data = True
                 )
except SystemExit:
    print("stopped")

we get

and

and eventually

When I studied the last plot for a while I found it really instructive. Each of its cell outside the diagonal obviously contains the number of “False Negative” records for these two specific category values – but with respect to actual value.

What more do we learn from the matrix? Well, the numbers in the cells on the diagonal, in a row and in a
column are related to our quantities TP, FN and FP:

  • Cells on the diagonal: For the diagonal we should find many correct “True Positive” values compared to the actual correct MNIST digits. (At least if all numbers are reasonably distributed across the MNIST dataset). We see that this indeed is the case. The ration of “True Positives” and the “Actual Positives” is given as a percentage and with the related numbers inside the respective cells on the diagonal.
  • Cells of a row: The values in the cells of a row (without the cell on the diagonal) of the displayed matrix give us the numbers/ratios for “False Negatives” – with respect to the actual value. If you sum up the individual FN-numbers you get the total number of “False negatives”, which of course is the difference between the total number AN and the number TP for the actual category.
  • Cells of a column: The column cells contain the numbers/ratios for “False Positives” – with respect to the predicted value. If you sum up the individual FN-numbers you get the total number of “False Positives” with respect to the predicted column value.

So, be a bit careful: A FN value with respect to an actual row value is a FP value with respect to the predicted column value – if the cell is one outside the diagonal!

All ratios are calculated with respect to the total actual numbers of data records for a specific category, i.e. a digit.

Looking closely we detect that our code obviously has some problems with distinguishing pictures of “5”s with pictures of “3”s, “6”s and “8”s. The same is true for “8”s and “3”s or “2s”. Also the distinction between “9”s, “3”s and “4”s seems to be difficult sometimes.

Does the confusion matrix change due to random initial weight values and mini-batch-shuffling?

We have seen already that statistical variations have no big impact on the eventual accuracy when training converges to points in the parameter-space close to the point for the minimum of the overall cost-function. Statistical effects between to training runs stem in our case from statistically chosen initial values of the weights and the changes to our mini-batch composition between epochs. But as long as our training converges (and ends up in a global minimum) we should not see any big impact on the confusion matrix. And indeed a second run leads to:

The values are pretty close to those of the first run.

Precision, Recall values per digit category and our own confusion matrix

Ok, we now can look at the nice confusion matrix plot and sum up all the values in a row of the confusion matrix to get the total FN-number for the related actual digit value. Or sum up the entries in a column to get the total FP-number. But we want to calculate these values from the ANN’s prediction results without looking at a plot and summation handwork. In addition we want to get the data of the confusion matrix in our own Numpy matrix array independently of foreign code. The following box displays the code for two functions, which are well suited for this task:

# A class to print in color and bold 
class color:
   PURPLE = '\033[95m'
   CYAN = '\033[96m'
   DARKCYAN = '\033[36m'
   BLUE = '\033[94m'
   GREEN = '\033[92m'
   YELLOW = '\033[93m'
   RED = '\033[91m'
   BOLD = '\033[1m'
   UNDERLINE = '\033[4m'
   END = '\033[
0m'

def acc_values(ay_pred_test, ay_y_test):
    ay_x = ay_pred_test
    ay_y = ay_y_test
    # ----- 
    #- dictionary for all false positives for all 10 digits
    fp = {}
    fpnum = {}
    irg = range(10)
    for i in irg:
        key = str(i)
        xfpi = np.where(ay_x==i)[0]
        fpi = np.zeros((10000, 3), np.int64)

        n = 0
        for j in xfpi: 
            if ay_y[j] != i: 
                row = np.array([j, ay_x[j], ay_y[j]])
                fpi[n] = row
                n+=1

        fpi_real   = fpi[0:n]
        fp[key]    = fpi_real
        fpnum[key] = fp[key].shape[0] 

    #- dictionary for all false negatives for all 10 digits
    fn = {}
    fnnum = {}
    irg = range(10)
    for i in irg:
        key = str(i)
        yfni = np.where(ay_y==i)[0]
        fni = np.zeros((10000, 3), np.int64)

        n = 0
        for j in yfni: 
            if ay_x[j] != i: 
                row = np.array([j, ay_x[j], ay_y[j]])
                fni[n] = row
                n+=1

        fni_real = fni[0:n]
        fn[key] = fni_real
        fnnum[key] = fn[key].shape[0] 

    #- dictionary for all true positives for all 10 digits
    tp = {}
    tpnum = {}
    actnum = {}
    irg = range(10)
    for i in irg:
        key = str(i)
        ytpi = np.where(ay_y==i)[0]
        actnum[key] = ytpi.shape[0]
        tpi = np.zeros((10000, 3), np.int64)

        n = 0
        for j in ytpi: 
            if ay_x[j] == i: 
                row = np.array([j, ay_x[j], ay_y[j]])
                tpi[n] = row
                n+=1

        tpi_real = tpi[0:n]
        tp[key] = tpi_real
        tpnum[key] = tp[key].shape[0] 
 
    #- We create an array for the precision values of all 10 digits 
    ay_prec_rec_f1 = np.zeros((10, 9), np.int64)
    print(color.BOLD + "Precision, Recall, F1, Accuracy, TP, FP, FN, AN" + color.END +"\n")
    print(color.BOLD + "i  ", "prec  ", "recall  ", "acc    ", "F1       ", "TP    ", 
          "FP    ", "FN    ", "AN" + color.END) 
    for i in irg:
        key = str(i)
        tpn = tpnum[key]
        fpn = fpnum[key]
        fnn = fnnum[key]
        an  = actnum[key]
        precision = tpn / (tpn + fpn) 
        prec = format(precision, '7.3f')
        recall = tpn / (tpn + fnn) 
        rec = format(recall, '7.3f')
        accuracy = tpn / an
        acc = format(accuracy, '7.3f')
        f1 = tpn / ( tpn + 0.5 * (fnn+fpn) )
        F1 = format(f1, '7.3f')
        TP = format(tpn, '6.0f')
        FP = format(fpn, '6.0f')
        FN = format(fnn, '6.0f')
        AN = format(an,  '6.0f')

        row = np.array([i, precision, recall, accuracy, f1, tpn, fpn, fnn, an])
        ay_prec_rec_f1[i] = row 
        print (i, prec, rec, acc, F1, TP, FP, FN, AN)
        
    return tp, tpnum, fp, fpnum, fn, fnnum, ay_prec_rec_f1 

def create_cf(ay_fn, ay_tpnum):
    ''' fn: array with false negatives row = np.array([j, x[j], y[j]])
    '''
    cf = np.zeros((10, 10), np.int64)
    rgi = range(10)
    rgj = range(10)
    for i in rgi:
        key = str(i)
        fn_i = ay_fn[key][ay_fn[key][:,2] == i]
        for j in rgj:
            if j!= i: 
                fn_ij = fn_i[fn_i[:,1] == j]
                #print(i, j, fn_ij)
                num_fn_ij = fn_ij.shape[0]
                cf[i,j] = num_fn_ij
            if j==i:
                cf[i,j] = ay_tpnum[key]

    cols=["0", "1", "2", "3", "4", "5", "6", "7", "8", "9"]
    df = pd.DataFrame(cf, columns=cols, index=cols)
    # print( "\n", df, "\n")
    # df.style
    
    return cf, df
    
 

 

The first function takes a array with prediction values (later on provided externally
by our “ay_predictions_test”) and compares its values with those of an y_test array which contains the actual values (later provided externally by our “ANN._y_test”). Then it uses array-slicing to create new arrays with information on all error records, related indices and the confused category values. Eventually, the function determines the numbers for AN, TP, FP and FN (per digit category) and prints the gathered information. It also returns arrays with information on records which are “True Positives”, “False Positives”, “False Negatives” and the various numbers.

The second function uses array-slicing of the array which contains all information on the “False Negatives” to reproduce the confusion matrix. It involves Pandas to produce a styled output for the matrix.

Now you can run the above code and the following one in Jupyter cells – of course, only after you have completed a training and a prediction run:

For my last run I got the following data:

We again see that especially “5”s and “9”s have a problem with FNs. When you compare the values of the last printed matrix with those in the plot of the confusion matrix above, you will see that our code produces the right FN/FP/TP-values. We have succeeded in producing our own confusion matrix – and we have all values directly available in our own Numpy arrays.

Some images of “4”-digits with errors

We can use the arrays which we created with functions above to get a look at the images. We use the function “plot_digits()” of Aurelien Geron at handson-ml2 chapter 03 on classification to plot several images in a series of rows and columns. The code is pretty easy to understand; at its center we find the matplotlib-function “imshow()”, which we have already used in other ML articles.

We again perform some array-slicing of the arrays our function “acc_values()” (see above) produces to identify the indices of images in the “X_test”-dataset we want to look at. We collect the first 50 examples of “true positive” images of the “4”-digit, then we take the “false positives” of the 4-digit and eventually the “fales negative” cases. We then plot the images in this order:

def plot_digits(instances, images_per_row=10, **options):
    size = 28
    images_per_row = min(len(instances), images_per_row)
    images = [instance.reshape(size,size) for instance in instances]
    n_rows = (len(instances) - 1) // images_per_row + 1
    row_images = []
    n_empty = n_rows * images_per_row - len(instances)
    images.append(np.zeros((size, size * n_empty)))
    for row in range(n_rows):
        rimages = images[row * images_per_row : (row + 1) * images_per_row]
        row_images.append(np.concatenate(rimages, axis=1))
    image = np.concatenate(row_images, axis=0)
    plt.imshow(image, cmap = mpl.cm.binary, **options)
    plt.axis("off")

ay_tp, ay_tpnum, ay_fp, ay_fpnum, ay_fn, ay_
fnnum, ay_prec_rec_f1 = \
    acc_values(ay_pred_test = ay_predictions_test, ay_y_test = ANN._y_test)

idx_act = str(4)

# fetching the true positives 
num_tp = ay_tpnum[idx_act]
idx_tp = ay_tp[idx_act][:,[0]]
idx_tp = idx_tp[:,0]
X_test_tp = ANN._X_test[idx_tp]

# fetching the false positives 
num_fp = ay_fpnum[idx_act]
idx_fp = ay_fp[idx_act][:,[0]]
idx_fp = idx_fp[:,0]
X_test_fp = ANN._X_test[idx_fp]

# fetching the false negatives 
num_fn = ay_fnnum[idx_act]
idx_fn = ay_fn[idx_act][:,[0]]
idx_fn = idx_fn[:,0]
X_test_fn = ANN._X_test[idx_fn]

# plotting 
# +++++++++++
plt.figure(figsize=(12,12))

# plotting the true positives
# --------------------------
plt.subplot(321)
plot_digits(X_test_tp[0:25], images_per_row=5 )
plt.subplot(322)
plot_digits(X_test_tp[25:50], images_per_row=5 )

# plotting the false positives
# --------------------------
plt.subplot(323)
plot_digits(X_test_fp[0:25], images_per_row=5 )
plt.subplot(324)
plot_digits(X_test_fp[25:], images_per_row=5 )

# plotting the false negatives
# ------------------------------
plt.subplot(325)
plot_digits(X_test_fn[0:25], images_per_row=5 )
plt.subplot(326)
plot_digits(X_test_fn[25:], images_per_row=5 )

 

The first row of the plot shows the (first) 50 “True Positives” for the “4”-digit images in the MNIST test data set. The second row shows the “False Positives”, the third row the “False Negatives”.

Very often you can guess why our MLP makes a mistake. However, in some cases we just have to acknowledge that the human brain is a much better pattern recognition machine than a stupid MLP 🙂 .

Conclusion

With the help of a “confusion matrix” it is easy to find out for which MNIST digit-images our algorithm has major problems. A confusion matrix gives us the necessary numbers of those digits (and their images) for which the MLP wrongly predicts “False Positives” or “False Negatives”.

We have also seen that there are three quantities – precision, recall, F1 – which are useful to describe the accuracy of a classification algorithm per classification category.

We have written some code to collect all necessary information about “confused” images into our own Numpy arrays after training. Slicing of Numpy arrays proved to be useful, and matplotlib helped us to visualize examples of the wrongly classified MNIST digit-images.

In the next article
A simple program for an ANN to cover the Mnist dataset – XII – accuracy evolution, learning rate, normalization
we shall extract some more information on the evolution of accuracy during training. We shall also make use of a “clustering” technique to reduce the number of input nodes.

Links

The python code of Runqi Yang (“hitvoice”) at gist.github.com for creating a plot of a confusion-matrix
Information on the function confusion_matrix() provided by sklearn.metrics
Information on the heatmap-functionality provided by “seaborn”
A python seaborn tutorial

 

A simple Python program for an ANN to cover the MNIST dataset – X – mini-batch-shuffling and some more tests

I continue my series on a Python code for a simple multi-layer perceptron [MLP]. During the course of the previous articles we have built a Python class “ANN” with methods to import the MNIST data set and handle forward propagation as well as error backward propagation [EBP]. We also had a deeper look at the mathematics of gradient descent and EBP for MLP training.

A simple program for an ANN to cover the Mnist dataset – IX – First Tests
A simple program for an ANN to cover the Mnist dataset – VIII – coding Error Backward Propagation
A simple program for an ANN to cover the Mnist dataset – VII – EBP related topics and obstacles
A simple program for an ANN to cover the Mnist dataset – VI – the math behind the „error back-propagation“
A simple program for an ANN to cover the Mnist dataset – V – coding the loss function
A simple program for an ANN to cover the Mnist dataset – IV – the concept of a cost or loss function
A simple program for an ANN to cover the Mnist dataset – III – forward propagation
A simple program for an ANN to cover the Mnist dataset – II – initial random weight values
A simple program for an ANN to cover the Mnist dataset – I – a starting point

The code modifications in the last article enabled us to perform a first test on the MNIST dataset. This test gave us some confidence in our training algorithm: It seemed to converge and produce weights which did a relatively good job on analyzing the MNIST images.

We saw a slight tendency of overfitting. But an accuracy level of 96.5% on the test dataset showed that the MLP had indeed “learned” something during training. We needed around 1000 epochs to come to this point.

However, there are a lot of parameters controlling our grid structure and the learning behavior. Such parameters are often called “hyper-parameters“. To get a better understanding of our MLP we must start playing around with such parameters. In this article we shall concentrate on the parameter for (regression) regularization (called Lambda2 in the parameter interface of our class ANN) and then start varying the node numbers on the layers.

But before we start new test runs we add a statistical element to the training – namely the variation of the composition of our mini-batches (see the last article).

General hint: In all of the test runs below we used 4 CPU cores with libOpenBlas on a Linux system with an I7 6700K CPU.

Shuffling the contents of the mini-batches

Let us add some more parameters to the interface of class “ANN”:

shuffle_batches = True
print_period = 20

The first parameter
shall control whether we vary the composition of the mini-batches with each epoch. The second parameter controls for which period of the epochs we print out some intermediate data (costs, averaged error of last mini-batch).

    def __init__(self, 
                 my_data_set = "mnist", 
                 n_hidden_layers = 1, 
                 ay_nodes_layers = [0, 100, 0], # array which should have as much elements as n_hidden + 2
                 n_nodes_layer_out = 10,  # expected number of nodes in output layer 
                                                  
                 my_activation_function = "sigmoid", 
                 my_out_function        = "sigmoid",   
                 my_loss_function       = "LogLoss",   
                 
                 n_size_mini_batch = 50,  # number of data elements in a mini-batch 
                 
                 n_epochs      = 1,
                 n_max_batches = -1,  # number of mini-batches to use during epochs - > 0 only for testing 
                                      # a negative value uses all mini-batches 
                 
                 lambda2_reg = 0.1,     # factor for quadratic regularization term 
                 lambda1_reg = 0.0,     # factor for linear regularization term 
                 
                 vect_mode = 'cols', 
                 
                 learn_rate = 0.001,        # the learning rate (often called epsilon in textbooks) 
                 decrease_const = 0.00001,  # a factor for decreasing the learning rate with epochs
                 mom_rate   = 0.0005,       # a factor for momentum learning
                 
                 shuffle_batches = True,    # True: we mix the data for mini-batches in the X-train set at the start of each epoch
                 
                 print_period = 20,         # number of epochs for which to print the costs and the averaged error
                 
                 figs_x1=12.0, figs_x2=8.0, 
                 legend_loc='upper right',
                 
                 b_print_test_data = True
                 
                 ):
        '''
        Initialization of MyANN
        Input: 
            data_set: type of dataset; so far only the "mnist", "mnist_784" datsets are known 
                      We use this information to prepare the input data and learn about the feature dimension. 
                      This info is used in preparing the size of the input layer.     
            n_hidden_layers = number of hidden layers => between input layer 0 and output layer n 
            
            ay_nodes_layers = [0, 100, 0 ] : We set the number of nodes in input layer_0 and the output_layer to zero 
                              Will be set to real number afterwards by infos from the input dataset. 
                              All other numbers are used for the node numbers of the hidden layers.
            n_nodes_out_layer = expected number of nodes in the output layer (is checked); 
                                this number corresponds to the number of categories NC = number of labels to be distinguished
            
            my_activation_function : name of the activation function to use 
            my_out_function : name of the "activation" function of the last layer which produces the output values 
            my_loss_function : name of the "cost" or "loss" function used for optimization 
            
            n_size_mini_batch : Number of elements/samples in a mini-batch of training data 
                                The number of mini-batches will be calculated from this
            
            n_epochs : number of epochs to calculate during training
            n_max_batches : > 0: maximum of mini-batches to use during training 
                      
      < 0: use all mini-batches  
            
            lambda_reg2:    The factor for the quadartic regularization term 
            lambda_reg1:    The factor for the linear regularization term 
            
            vect_mode: Are 1-dim data arrays (vctors) ordered by columns or rows ?
            
            learn rate :     Learning rate - definies by how much we correct weights in the indicated direction of the gradient on the cost hyperplane.
            decrease_const:  Controls a systematic decrease of the learning rate with epoch number 
            mom_const:       Momentum rate. Controls a mixture of the last with the present weight corrections (momentum learning)
            
            shuffle_batches: True => vary composition of mini-batches with each epoch
            
            print_period:    number of periods between printing out some intermediate data 
                             on costs and the averaged error of the last mini-batch   
                       
            
            figs_x1=12.0, figs_x2=8.0 : Standard sizing of plots , 
            legend_loc='upper right': Position of legends in the plots
            
            b_print_test_data: Boolean variable to control the print out of some tests data 
             
         '''
        
        # Array (Python list) of known input data sets 
        self._input_data_sets = ["mnist", "mnist_784", "mnist_keras"]  
        self._my_data_set = my_data_set
        
        # X, y, X_train, y_train, X_test, y_test  
            # will be set by analyze_input_data 
            # X: Input array (2D) - at present status of MNIST image data, only.    
            # y: result (=classification data) [digits represent categories in the case of Mnist]
        self._X       = None 
        self._X_train = None 
        self._X_test  = None   
        self._y       = None 
        self._y_train = None 
        self._y_test  = None
        
        # relevant dimensions 
        # from input data information;  will be set in handle_input_data()
        self._dim_sets     = 0  
        self._dim_features = 0  
        self._n_labels     = 0   # number of unique labels - will be extracted from y-data 
        
        # Img sizes 
        self._dim_img      = 0 # should be sqrt(dim_features) - we assume square like images  
        self._img_h        = 0 
        self._img_w        = 0 
        
        # Layers
        # ------
        # number of hidden layers 
        self._n_hidden_layers = n_hidden_layers
        # Number of total layers 
        self._n_total_layers = 2 + self._n_hidden_layers  
        # Nodes for hidden layers 
        self._ay_nodes_layers = np.array(ay_nodes_layers)
        # Number of nodes in output layer - will be checked against information from target arrays
        self._n_nodes_layer_out = n_nodes_layer_out
        
        # Weights 
        # --------
        # empty List for all weight-matrices for all layer-connections
        # Numbering : 
        # w[0] contains the weight matrix which connects layer 0 (input layer ) to hidden layer 1 
        # w[1] contains the weight matrix which connects layer 1 (input layer ) to (hidden?) layer 2 
        self._li_w = []
        
        # Arrays for encoded output labels - will be set in _encode_all_mnist_labels()
        # -------------------------------
        self._ay_onehot = None
        self._ay_oneval = None
        
        # Known Randomizer methods ( 0: np.random.randint, 1: np.random.uniform )  
        # ------------------
        self.__ay_known_randomizers = [0, 1]

        # Types of activation functions and output functions 
        # ------------------
        self.__ay_activation_functions = ["sigmoid"] # later also relu 
        self.__ay_output_functions     = ["sigmoid"] # later also 
softmax 
        
        # Types of cost functions 
        # ------------------
        self.__ay_loss_functions = ["LogLoss", "MSE" ] # later also other types of cost/loss functions  


        # the following dictionaries will be used for indirect function calls 
        self.__d_activation_funcs = {
            'sigmoid': self._sigmoid, 
            'relu':    self._relu
            }
        self.__d_output_funcs = { 
            'sigmoid': self._sigmoid, 
            'softmax': self._softmax
            }  
        self.__d_loss_funcs = { 
            'LogLoss': self._loss_LogLoss, 
            'MSE': self._loss_MSE
            }  
        # Derivative functions 
        self.__d_D_activation_funcs = {
            'sigmoid': self._D_sigmoid, 
            'relu':    self._D_relu
            }
        self.__d_D_output_funcs = { 
            'sigmoid': self._D_sigmoid, 
            'softmax': self._D_softmax
            }  
        self.__d_D_loss_funcs = { 
            'LogLoss': self._D_loss_LogLoss, 
            'MSE': self._D_loss_MSE
            }  
        
        
        # The following variables will later be set by _check_and set_activation_and_out_functions()            
        
        self._my_act_func  = my_activation_function
        self._my_out_func  = my_out_function
        self._my_loss_func = my_loss_function
        self._act_func = None    
        self._out_func = None    
        self._loss_func = None    
        
        # number of data samples in a mini-batch 
        self._n_size_mini_batch = n_size_mini_batch
        self._n_mini_batches = None  # will be determined by _get_number_of_mini_batches()

        # maximum number of epochs - we set this number to an assumed maximum 
        # - as we shall build a backup and reload functionality for training, this should not be a major problem 
        self._n_epochs = n_epochs
        
        # maximum number of batches to handle ( if < 0 => all!) 
        self._n_max_batches = n_max_batches
        # actual number of batches 
        self._n_batches = None

        # regularization parameters
        self._lambda2_reg = lambda2_reg
        self._lambda1_reg = lambda1_reg
        
        # parameter for momentum learning 
        self._learn_rate = learn_rate
        self._decrease_const = decrease_const
        self._mom_rate   = mom_rate
        self._li_mom = [None] *  self._n_total_layers
        
        # shuffle data in X_train? 
        self._shuffle_batches = shuffle_batches
        
        # epoch period for printing 
        self._print_period = print_period
        
        # book-keeping for epochs and mini-batches 
        # -------------------------------
        # range for epochs - will be set by _prepare-epochs_and_batches() 
        self._rg_idx_epochs = None
        # range for mini-batches 
        self._rg_idx_batches = None
        # dimension of the numpy arrays for book-keeping - will be set in _prepare_epochs_and_batches() 
        self._shape_epochs_batches = None    # (n_epochs, n_batches, 1) 

        # list for error values at outermost layer for minibatches and epochs during training
        # we use a numpy array here because we can redimension it
        self._ay_theta = None
        # list for cost values of mini-batches during training 
        # The list will later be split into sections for epochs 
        self._ay_costs = None
        
        
        # Data elements for back propagation
        # ----------------------------------
        
        # 2-dim array of partial derivatives of the elements of an additive cost function 
        # The derivative is taken with respect to the output results a_j = ay_ANN_out[j]
        # The array dimensions account for nodes and sampls of a 
mini_batch. The array will be set in function 
        # self._initiate_bw_propagation()
        self._ay_delta_out_batch = None
        

        # parameter to allow printing of some test data 
        self._b_print_test_data = b_print_test_data

        # Plot handling 
        # --------------
        # Alternatives to resize plots 
        # 1: just resize figure  2: resize plus create subplots() [figure + axes] 
        self._plot_resize_alternative = 1 
        # Plot-sizing
        self._figs_x1 = figs_x1
        self._figs_x2 = figs_x2
        self._fig = None
        self._ax  = None 
        # alternative 2 does resizing and (!) subplots() 
        self.initiate_and_resize_plot(self._plot_resize_alternative)        
        
        
        # ***********
        # operations 
        # ***********
        
        # check and handle input data 
        self._handle_input_data()
        # set the ANN structure 
        self._set_ANN_structure()
        
        # Prepare epoch and batch-handling - sets ranges, limits num of mini-batches and initializes book-keeping arrays
        self._rg_idx_epochs, self._rg_idx_batches = self._prepare_epochs_and_batches()
        
        # perform training 
        start_c = time.perf_counter()
        self._fit(b_print=True, b_measure_batch_time=False)
        end_c = time.perf_counter()
        print('\n\n ------') 
        print('Total training Time_CPU: ', end_c - start_c) 
        print("\nStopping program regularily")

 
Both parameters affect our method “_fit()” in the following way :

    ''' -- Method to set the number of batches based on given batch size -- '''
    def _fit(self, b_print = False, b_measure_batch_time = False):
        '''
        Parameters: 
            b_print:                 Do we print intermediate results of the training at all? 
            b_print_period:          For which period of epochs do we print? 
            b_measure_batch_time:    Measure CPU-Time for a batch
        '''
        rg_idx_epochs  = self._rg_idx_epochs 
        rg_idx_batches = self._rg_idx_batches
        if (b_print):    
            print("\nnumber of epochs = " + str(len(rg_idx_epochs)))
            print("max number of batches = " + str(len(rg_idx_batches)))
       
        # loop over epochs
        for idxe in rg_idx_epochs:
            if (b_print and (idxe % self._print_period == 0) ):
                print("\n ---------")
                print("\nStarting epoch " + str(idxe+1))
                
            # sinmple adaption of the learning rate 
            self._learn_rate /= (1.0 + self._decrease_const * idxe)
            
            # shuffle indices for a variation of the mini-batches with each epoch
            if self._shuffle_batches:
                shuffled_index = np.random.permutation(self._dim_sets)
                self._X_train, self._y_train, self._ay_onehot = self._X_train[shuffled_index], self._y_train[shuffled_index], self._ay_onehot[:, shuffled_index]
                
            
            # loop over mini-batches
            for idxb in rg_idx_batches:
                if b_measure_batch_time: 
                    start_0 = time.perf_counter()
                # deal with a mini-batch
                self._handle_mini_batch(num_batch = idxb, num_epoch=idxe, b_print_y_vals = False, b_print = False)
                if b_measure_batch_time: 
                    end_0 = time.perf_counter()
                    print('Time_CPU for batch ' + str(idxb+1), end_0 - start_0) 
                
            if (b_print and (idxe % self._print_period == 0) ):
                print("\ntotal costs of mini_batch = ", self._ay_costs[idxe, idxb])
      
          print("avg total error of mini_batch = ", self._ay_theta[idxe, idxb])
                
        return None

 

Results for shuffling the contents of the mini-batches

With shuffling we expect a slightly broader variation of the costs and the averaged error. But the accuracy should no change too much in the end. We start a new test run with the following parameters:

     ay_nodes_layers = [0, 70, 30, 0], 
     n_nodes_layer_out = 10,  
     my_loss_function = "LogLoss",
     n_size_mini_batch = 500,
     n_epochs = 1500, 
     n_max_batches = 2000,  # small values only for test runs
     lambda2_reg = 0.1, 
     lambda1_reg = 0.0,      
     vect_mode = 'cols', 
     learn_rate = 0.0001,
     decrease_const = 0.000001,
     mom_rate   = 0.00005,  
     shuffle_batches = True,
     print_period = 20,         
...

If we look at the intermediate printout for the last mini-batch of some epochs and compare it to the results given in the last article, we see a stronger variation in the costs and averaged error. The reason is that the composition of last mini-batch of an epoch changes with every epoch.

number of epochs = 1500
max number of batches = 120
---------
Starting epoch 1
total costs of mini_batch =  1757.7650929607967
avg total error of mini_batch =  0.17086198431410954
---------
Starting epoch 61
total costs of mini_batch =  511.7001121819204
avg total error of mini_batch =  0.030287362041332373
---------
Starting epoch 121
total costs of mini_batch =  435.2513093033654
avg total error of mini_batch =  0.023445601362614754
----------
Starting epoch 181
total costs of mini_batch =  361.8665831722295
avg total error of mini_batch =  0.018540003201911136
---------
Starting epoch 241
total costs of mini_batch =  293.31230634431023
avg total error of mini_batch =  0.0138237366634751
---------
Starting epoch 301
total costs of mini_batch =  332.70394217467936
avg total error of mini_batch =  0.017697548541363246
---------
Starting epoch 361
total costs of mini_batch =  249.26400606039937
avg total error of mini_batch =  0.011765164578232358
---------
Starting epoch 421
total costs of mini_batch =  240.0503762160913
avg total error of mini_batch =  0.011650843329895542
---------
Starting epoch 481
total costs of mini_batch =  222.89422430417295
avg total error of mini_batch =  0.011503859412784031
---------
Starting epoch 541
total costs of mini_batch =  200.1195962051405
avg total error of mini_batch =  0.009962020519104173
---------
tarting epoch 601
total costs of mini_batch =  206.74753168607685
avg total error of mini_batch =  0.01067995191155135
---------
Starting epoch 661
total costs of mini_batch =  171.14090717705736
avg total error of mini_batch =  0.0077091934178393105
---------
Starting epoch 721
total costs of mini_batch =  158.44967190977957
avg total error of mini_batch =  0.0070760922760890735
---------
Starting epoch 781
total costs of mini_batch =  165.4047453537401
avg total error of mini_batch =  0.008622788115637027
---------
Starting epoch 841
total costs of mini_batch =  140.52762105883642
avg total error of mini_batch =  0.0067360505574077766
---------
Starting epoch 901
total costs of mini_batch =  163.9117184790982
avg total error of mini_batch =  0.007431666926365192
---------
Starting epoch 961
total costs of mini_batch =  126.05539161877512
avg total error of mini_batch =  0.005982378079899406
---------
Starting epoch 1021
total costs of mini_batch =  114.89943308334199
avg total error of mini_batch =  0.005122976288751798
---------
Starting epoch 1081
total costs of mini_batch =  117.22051220670932
avg total error of mini_batch 
=  0.005185936692097749
---------
Starting epoch 1141
total costs of mini_batch =  140.88969853048422
avg total error of mini_batch =  0.007665464508660714
---------
Starting epoch 1201
total costs of mini_batch =  113.27223303239667
avg total error of mini_batch =  0.0059791015452599705
---------
Starting epoch 1261
total costs of mini_batch =  105.55343407063131
avg total error of mini_batch =  0.005000503315756879
---------
Starting epoch 1321
total costs of mini_batch =  130.48116668827038
avg total error of mini_batch =  0.006287118265324945
---------
Starting epoch 1381
total costs of mini_batch =  109.04042315247389
avg total error of mini_batch =  0.005874339148860562
---------
Starting epoch 1441
total costs of mini_batch =  121.01379412127089
avg total error of mini_batch =  0.0065105907117289944
---------
Starting epoch 1461
total costs of mini_batch =  103.08774822996196
avg total error of mini_batch =  0.005299079778792264
---------
Starting epoch 1481
total costs of mini_batch =  106.21334182056928
avg total error of mini_batch =  0.005343967730134955
-------
Total training Time_CPU:  1963.8792177759988

 
Note that the averaged error values result from averaging of the absolute values of the errors of all records in a batch! The small numbers are not due to a cancelling of positive by negative deviations. A contribution to the error at an output node is given by the absloute value of the difference between the predicted real output value and the encoded target output value. We then first calculate an average over all output nodes (=10) per record and then average these values over all records of a batch. Such an “averaged error” gives us a first indication of the accuracy level reached.

Note that this averaged error is not becoming a constant. The last values in the above list indicate that we do not get much better with the error than 0.0055 on the training data. Our approached minimum points on the various cost hyperplanes of the mini-batches obviously hop around the global minimum on the hyperplane of the total costs. One of the reasons is the varying composition of the mini-batches; another reason is that the cost hyperplanes of the various mini-batches themselves are different from the hyperplane of the total costs of all records of the test data set. We see the effects of a mixture of “batch gradient descent” and “stochastic gradient descent” here; i.e., we do not get rid of stochastic elements even if we are close to a global minimum.

Still we observe an overall convergent behavior at around 1050 epochs. There our curves get relatively flat.

Accuracy values are:

total accuracy for training data = 0.9914
total accuracy for test data        = 0.9611

So, this is pretty much the same as in our original run in the last article without data shuffling.

Dropping regularization

In the above run we had used a quadratic from of the regularization (often called Ridge regularization). In the next test run we shall drop regularization completely (Lambda2 = 0, Lambda1 = 0) and find out whether this hampers the generalization of our MLP and the resulting accuracy with respect to the test data set.

Resulting data for the last epochs of the test run are

Starting epoch 1001
total costs of mini_batch =  67.98542512352101
avg total error of mini_batch =  0.007449654093429594
---------
nStarting epoch 1051
total costs of mini_batch =  56.69195783294443
avg total error of mini_batch =  0.0063384571747725415
---------
Starting epoch 1101
total costs of mini_batch =  51.81035466850738
avg total error of mini_batch =  0.005939699354987233
---------
Starting epoch 1151
total costs of mini_batch =  52.23157716632318
avg total error of mini_batch =  0.006373981433882217
---------
Starting epoch 1201
total costs of mini_batch =  48.40298652277855
avg total error of mini_batch =  0.005653856253701317
---------
Starting epoch 1251
total costs of mini_batch =  45.00623540189525
avg total error of mini_batch =  0.005245339176038497
---------
Starting epoch 1301
total costs of mini_batch =  36.88409532579881
avg total error of mini_batch =  0.004600719544961844
---------
Starting epoch 1351
total costs of mini_batch =  36.53543045554845
avg total error of mini_batch =  0.003993852242709943
---------
Starting epoch 1401
total costs of mini_batch =  38.80422469954769
avg total error of mini_batch =  0.00464620714991714
---------
Starting epoch 1451
total costs of mini_batch =  42.39371261881638
avg total error of mini_batch =  0.005294796697150631
------
Total training Time_CPU:  2118.4527089519997

 
Note, by the way, that the absolute values of the costs depend on the regularization parameter; therefore we see somewhat lower values in the end than before. But the absolute cost values are not so important regarding the general convergence and the accuracy of the network reached.

We omit the plots and just give the accuracy values:

total accuracy for training data = 0.9874
total accuracy for test data        = 0.9514

We get a slight drop in accuracy for the test data set – small (1%), but notable. It is interesting the even the accuracy on the training data became influenced.

Why might it be interesting to check the impact of the regularization?

We learn from literature that regularization helps with overfitting. Another aspect discussed e.g. by Jörg Frochte in his book “Maschinelles Lernen” is, whether we have enough training data to determine the vast amount of weights in complicated networks. He suggests on page 190 of his book to consider the number of weights in an MLP and compare it with the number of available data points.

He suggests that one may run into trouble if the difference between the number of weights (number of degrees of freedom) and the number of data records (number of independent information data) becomes too big. However, his test example was a rather limited one and for regression not classification. He also notes that if the data are well distributed may not be as big as assumed. If one thinks about it, one may also come to the question whether the real amount of data provided by the records is not by a factor of 10 larger – as we use 10 output values per record ….

Anyway, I think it is worthwhile to have a look at regularization.

Enlarging the regularization factor

We double the value of Lambda2: Lambda2 = 0.2.

Starting epoch 1251
total costs of mini_batch =  128.00827405482318
avg total error of mini_batch =  0.007276206815511017
---------
Starting epoch 1301
total costs of mini_batch =  107.62983581797556
avg total error of mini_batch =  0.005535653858885446
---------
Starting epoch 1351
total costs of mini_batch =  107.83630092292944
avg total error of mini_batch =  0.
005446805325519184
---------
Starting epoch 1401
total costs of mini_batch =  119.7648277329852
avg total error of mini_batch =  0.00729466852297802
---------
Starting epoch 1451
total costs of mini_batch =  106.74254206278933
avg total error of mini_batch =  0.005343124456075227

 
We get a slight improvement of the accuracy compared to our first run with data shuffling:

total accuracy for training data = 0.9950
total accuracy for test data        = 0.964

So, regularization does have its advantages. I recommend to investigate the impact of this parameter closely, if you need to get the “last percentages” in generalization and accuracy for a given MLP-model.

Enlarging node numbers

We have 60000 data records in the training set. In our example case we needed to fix around 784*70 + 70*30 + 30*10 = 57280 weight values. This is pretty close to the total amount of training data (60000). What happens if we extend the number of weights beyond the number of training records?
E.g. 784*100 + 100*50 + 50*10 = 83900. Do we get some trouble?

The results are:

          
Starting epoch 1151
total costs of mini_batch =  109.77341617599176
avg total error of mini_batch =  0.005494982077591186
---------
Starting epoch 1201
total costs of mini_batch =  113.5293680548904
avg total error of mini_batch =  0.005352117137100675
---------
Starting epoch 1251
total costs of mini_batch =  116.26371170820423
avg total error of mini_batch =  0.0072335516486698
---------
Starting epoch 1301
total costs of mini_batch =  99.7268420386945
avg total error of mini_batch =  0.004850817052601995
---------
Starting epoch 1351
total costs of mini_batch =  101.16579732551999
avg total error of mini_batch =  0.004831600835072556
---------
Starting epoch 1401
total costs of mini_batch =  98.45208584213253
avg total error of mini_batch =  0.004796133492821962
---------
Starting epoch 1451
total costs of mini_batch =  99.279344780807
avg total error of mini_batch =  0.005289728162205425
------
Total training Time_CPU:  2159.5880855739997

 

Ooops, there appears a glitch in the data around epoch 1250. Such things happen! So, we should have a look at the graphs before we decide to take the weights of a special epoch for our MLP model!

But in the end, i.e. with the weights at epoch 1500 the accuracy values are:

total accuracy for training data = 0.9962
total accuracy for test data        = 0.9632

So, we were NOT punished by extending our network, but we gained nothing worth the effort.

Now, let us go up with node numbers much more: 300 and 100 => 784*300 + 300*100 + 100*10 = 266200; ie. substantially more individual weights than training samples! First with Lambda2 = 0.2:

Starting epoch 1201
total costs of mini_batch =  104.4420759423322
avg total error of mini_batch =  0.0037985801450468246
---------
Starting epoch 1251
total costs of mini_batch =  102.80878647657674
avg total error of mini_batch =  0.003926855904089417
---------
Starting epoch 1301
total costs of mini_batch =  100.01189950545002
avg total error of mini_batch =  0.0037743225368465773
---------
Starting epoch 1351
total 
costs of mini_batch =  97.34294880936079
avg total error of mini_batch =  0.0035513092392408865
---------
Starting epoch 1401
total costs of mini_batch =  93.15432903284587
avg total error of mini_batch =  0.0032916082743134206
---------
Starting epoch 1451
total costs of mini_batch =  89.79127326241868
avg total error of mini_batch =  0.0033628384147655283
------
Total training Time_CPU:  4254.479082876998

 

total accuracy for training data = 0.9987
total accuracy for test data        = 0.9630

So , much CPU-time for no gain!

Now, what happens if we set Lambda2 = 0? We get:

total accuracy for training data = 0.9955
total accuracy for test data        = 0.9491

This is a small change around 1.4%! I would say:

In the special case of MNIST, a MLP-network with 2 hidden layers and a small learning rate we see neither a major problem regarding the amount of available data, nor a dramatic impact of the regularization. Regularization brings around a 1% gain in accuracy.

Reducing the node number on the hidden layers

Another experiment could be to reduce the number of nodes on the hidden layers. Let us go down to 40 nodes on L1 and 20 on L2, then to 30 and 20. Lambda2 is set to 0.2 in both runs. acc1 in the following listing means the accuracy for the training data, acc2 for the test data.

The results are:

40 nodes on L1, 20 nodes on L2, 1500 epochs => 1600 sec, acc1 = 0.9898, acc2 = 0.9578
30 nodes on L1, 20 nodes on L2, 1800 epochs => 1864 sec, acc1 = 0.9861, acc2 = 0.9513

We loose around 1% in accuracy!

The plot for 30, 20 nodes on layers L1, L2 shows that we got a convergence only beyond 1500 epochs.

Working with just one hidden layer

To get some more indications regarding efficiency let us now turn to networks with just one layer.
We investigate three situations with the following node numbers on the hidden layer: 100, 50, 30

The plot for 100 nodes on the hidden layer; we get convergence at around 1050 epochs.

Interestingly the CPU time for 100 nodes is with 1850 secs not smaller than for the network with 70 and 30 nodes on the hidden layers. As the dominant matrices are the ones connecting layer L0 and layer L1 this is quite understandable. (Note the CPU time also depends on the consumption of other jobs on the system.

The plots for 50 and 30 nodes on the hidden layer; we get convergence at around 1450 epochs. The
CPU time for 1500 epochs goes down to 1500 sec and XXX sec, respectively.

Plot for 50 nodes on the hidden layer:

Plot for 30 nodes:

We get the following accuracy values:

100 nodes, 1950 sec (1500 epochs), acc1 = 0.9947, acc2 = 0.9619,
50 nodes, 1600 sec (1500 epochs), acc1 = 0.9880, acc2 = 0.9566,
30 nodes, 1450 sec (1500 epochs), acc1 = 0.9780, acc2 = 0.9436

Well, now we see a drop in the accuracy by around 2% compared to our best cases. You have to decide yourself whether the gain in CPU time is worth it.

Note, by the way, that the accuracy value for 50 nodes is pretty close to the value S. Rashka got in his book “Python Machine Learning”. If you compare such values with your own runs be aware of the rather small learning rate (0.0001) and momentum rates (0.00005) I used. You can probably become faster with smaller learning rates. But then you may need another type of adaption for the learning rate compared to our simple linear one.

Conclusion

We saw that our original guess of a network with 2 hidden layers with 70 and 30 nodes was not a bad one. A network with just one layer with just 50 nodes or below does not give us the same accuracy. However, we neither saw a major improvement if we went to grids with 300 nodes on layer L1 and 100 nodes on layer L2. Despite some discrepancy between the number of weights in comparison to the number of test records we saw no significant loss in accuracy either – with or without regularization.
We also learned that we should use regularization (here of the quadratic type) to get the last 1% to 2% bit of accuracy on the test data in our special MNIST case.

In the next article

A simple program for an ANN to cover the Mnist dataset – XI – confusion matrix

we shall have a closer look at those MNIST number images where our MLP got problems.

 

A simple Python program for an ANN to cover the MNIST dataset – IX – First Tests

This brief article continues my series on a Python program for simple MLPs.

A simple program for an ANN to cover the Mnist dataset – VIII – coding Error Backward Propagation
A simple program for an ANN to cover the Mnist dataset – VII – EBP related topics and obstacles
A simple program for an ANN to cover the Mnist dataset – VI – the math behind the „error back-propagation“
A simple program for an ANN to cover the Mnist dataset – V – coding the loss function
A simple program for an ANN to cover the Mnist dataset – IV – the concept of a cost or loss function
A simple program for an ANN to cover the Mnist dataset – III – forward propagation
A simple program for an ANN to cover the Mnist dataset – II – initial random weight values
A simple program for an ANN to cover the Mnist dataset – I – a starting point

With the code for “Error Backward Propagation” we have come so far that we can perform first tests. As planned from the beginning we take the MNIST dataset as a test example. In a first approach we do not rebuild the mini-batches with each epoch. Neither do we vary the MLP setup.

What we are interested in is the question whether our artificial neural network converges with respect to final weight values during training. I.e. we want to see whether the training algorithm finds a reasonable global minimum on the cost hyperplane over the parameter space of all weights.

Test results

We use the following parameters

    ANN = myann.MyANN(my_data_set="mnist_keras", n_hidden_layers = 2, 
                 ay_nodes_layers = [0, 70, 30, 0], 
                 n_nodes_layer_out = 10,  

                 #my_loss_function = "MSE",
                 my_loss_function = "LogLoss",
                 n_size_mini_batch = 500,
     
                 n_epochs = 1500, 
                 n_max_batches = 2000,
                     
                 lambda2_reg = 0.1, 
                 lambda1_reg = 0.0,      

                 vect_mode = 'cols', 
                      
                 learn_rate = 0.0001,
                 decrease_const = 0.000001,
                 mom_rate   = 0.00005,  

                 figs_x1=12.0, figs_x2=8.0, 
                 legend_loc='upper right',
                 b_print_test_data = True
                 )

We use two hidden layers with 70 and 30 nodes. Note that we pick a rather small learning rate, which gets diminished even further. The number of epochs (1500) is quite high; with 4 CPU threads on an i7-6700K processor training takes around 40 minutes if started from a Jupyter notebook. (Our program is not yet optimized.)

I supplemented my code with some statements to print out data for the total costs and the averaged error for mini-batches. We get the
following series of data for the last mini-batch within every 50th epoch.

Starting epoch 1
total costs of mini_batch =  1757.1499806500506
avg total error of mini_batch =  0.17150838451718683
---------
Starting epoch 51
total costs of mini_batch =  532.5817607913532
avg total error of mini_batch =  0.034658195573307196
---------
Starting epoch 101
total costs of mini_batch =  436.67115522484687
avg total error of mini_batch =  0.023496458964699055
---------
Starting epoch 151
total costs of mini_batch =  402.381331415108
avg total error of mini_batch =  0.020836159866695597
---------
Starting epoch 201
total costs of mini_batch =  342.6296325512483
avg total error of mini_batch =  0.016565121882126693
---------
Starting epoch 251
total costs of mini_batch =  319.5995117831668
avg total error of mini_batch =  0.01533372596379799
---------
Starting epoch 301
total costs of mini_batch =  288.2201307002896
avg total error of mini_batch =  0.013799141451102469
---------
Starting epoch 351
total costs of mini_batch =  272.40526022720826
avg total error of mini_batch =  0.013499221607285198
---------
Starting epoch 401
total costs of mini_batch =  251.02417188663628
avg total error of mini_batch =  0.012696943309314687
---------
Starting epoch 451
total costs of mini_batch =  231.92274565746214
avg total error of mini_batch =  0.011152542115360705
---------
Starting epoch 501
total costs of mini_batch =  216.34658280101385
avg total error of mini_batch =  0.010692239864121407
---------
Starting epoch 551
total costs of mini_batch =  215.21791509166042
avg total error of mini_batch =  0.010999255316821901
---------
Starting epoch 601
total costs of mini_batch =  207.79645393570436
avg total error of mini_batch =  0.011123079894527222
---------
Starting epoch 651
total costs of mini_batch =  188.33965068903723
avg total error of mini_batch =  0.009868734062493835
---------
Starting epoch 701
total costs of mini_batch =  173.07625091642274
avg total error of mini_batch =  0.008942065167336382
---------
Starting epoch 751
total costs of mini_batch =  174.98264336120369
avg total error of mini_batch =  0.009714870291761567
---------
Starting epoch 801
total costs of mini_batch =  161.10229359519792
avg total error of mini_batch =  0.008844419847237179
---------
Starting epoch 851
total costs of mini_batch =  155.4186141788981
avg total error of mini_batch =  0.008244783820578621
---------
Starting epoch 901
total costs of mini_batch =  158.88876607392308
avg total error of mini_batch =  0.008970678691005138
---------
Starting epoch 951
total costs of mini_batch =  148.61870772570722
avg total error of mini_batch =  0.008124438423034456
---------
Starting epoch 1001
total costs of mini_batch =  152.16976618516264
avg total error of mini_batch =  0.009151413825781066
---------
Starting epoch 1051
total costs of mini_batch =  142.24802525081637
avg total error of mini_batch =  0.008297161160449798
---------
Starting epoch 1101
total costs of mini_batch =  137.3828515603569
avg total error of mini_batch =  0.007659755348989629
---------
Starting epoch 1151
total costs of mini_batch =  129.8472897084494
avg total error of mini_batch =  0.007254892176613871
---------
Starting epoch 1201
total costs of mini_batch =  139.30002497623792
avg total error of mini_batch =  0.007881199505625214
---------
Starting epoch 1251
total costs of mini_batch =  138.0323454321882
avg total error of mini_batch =  0.00807373439996105
---------
Starting epoch 1301
total costs of mini_batch =  117.95701570484076
avg total error of mini_batch =  0.006378071703153664
---------
Starting epoch 1351
total costs of mini_batch =  125.
71869046937177
avg total error of mini_batch =  0.0072716189968114265
---------
Starting epoch 1401
total costs of mini_batch =  117.3485602627176
avg total error of mini_batch =  0.006291182169676069
---------
Starting epoch 1451
total costs of mini_batch =  118.09317470010767
avg total error of mini_batch =  0.0066519021636054195
---------
Starting epoch 1491
total costs of mini_batch =  112.69566736699439
avg total error of mini_batch =  0.006151660466611035

 ------
Total training Time_CPU:  2430.0504785089997

 
We can display the results also graphically; a code fragemnt to do this, may look like follows in a Jupyter cell:

# Plotting 
# **********
num_epochs  = ANN._ay_costs.shape[0]
num_batches = ANN._ay_costs.shape[1]
num_tot = num_epochs * num_batches

ANN._ay_costs = ANN._ay_costs.reshape(num_tot)
ANN._ay_theta = ANN._ay_theta .reshape(num_tot)

#sizing
fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 5

# Two figures 
# -----------
fig1 = plt.figure(1)
fig2 = plt.figure(2)

# first figure with two plot-areas with axes 
# --------------------------------------------
ax1_1 = fig1.add_subplot(121)
ax1_2 = fig1.add_subplot(122)

ax1_1.plot(range(len(ANN._ay_costs)), ANN._ay_costs)
ax1_1.set_xlim (0, num_tot+5)
ax1_1.set_ylim (0, 2000)
ax1_1.set_xlabel("epochs * batches (" + str(num_epochs) + " * " + str(num_batches) + " )")
ax1_1.set_ylabel("costs")

ax1_2.plot(range(len(ANN._ay_theta)), ANN._ay_theta)
ax1_2.set_xlim (0, num_tot+5)
ax1_2.set_ylim (0, 0.2)
ax1_2.set_xlabel("epochs * batches (" + str(num_epochs) + " * " + str(num_batches) + " )")
ax1_2.set_ylabel("averaged error")

 

We get:

This looks quite promising!
The vertical spread of both curves is due to the fact that we plotted cost and error data for each mini-batch. As we know the cost hyperplanes of the batches differ from each other and the hyperplane of the total costs for all training data. So do the cost and error values.

Secondary test: Rate of correctly and wrongly predicted values of the training and the test data sets

With the following code in a Jupyter cell we can check the relative percentage of correctly predicted MNIST numbers for the training data set and the test data set:

# ------ all training data 
# *************************
size_set = ANN._X_train.shape[0]

li_Z_in_layer_test  = [None] * ANN._n_total_layers
li_Z_in_layer_test[0] = ANN._X_train
# Transpose 
ay_Z_in_0T       = li_Z_in_layer_test[0].T
li_Z_in_layer_test[0] = ay_Z_in_0T
li_A_out_layer_test  = [None] * ANN._n_total_layers

ANN._fw_propagation(li_Z_in = li_Z_in_layer_test, li_A_out = li_A_out_layer_test, b_print = True) 
Result = np.argmax(li_A_out_layer_test[3], axis=0)
Error = ANN._y_train - Result 
acc_train = (np.sum(Error == 0)) / size_set
print ("total accuracy for training data = ", acc_train)

# ------ all test data 
# *************************
size_set = ANN._X_test.shape[0]

li_Z_in_layer_test  = [None] * ANN._n_total_layers
li_Z_in_layer_test[0] = ANN._X_test
# Transpose 
ay_Z_in_0T       = li_Z_in_layer_test[0].T
li_Z_in_layer_test[0] = ay_Z_in_0T
li_A_out_layer_test  = [None] * ANN._n_total_layers

ANN._fw_propagation(li_Z_in = li_Z_in_layer_test, li_A_out = li_A_out_layer_test, b_print = True) 
Result = np.argmax(li_A_out_layer_
test[3], axis=0)
Error = ANN._y_test - Result 
acc_test = (np.sum(Error == 0)) / size_set
print ("total accuracy for test data = ", acc_test)

 

“acc” stands for “accuracy”.

We get

total accuracy for training data = 0.9919
total accuracy for test data        = 0.9645

So, there is some overfitting – but not much.

Conclusion

Our training algorithm and the error backward propagation seem to work.

The real question is, whether we produced the accuracy values really efficiently: In our example case we needed to fix around 786*70 + 70*30 + 30*10 = 57420 weight values. This is close to the total amount of training data (60000). A smaller network with just one hidden layer would require much fewer values – and the training would be much faster regarding CPU time.

So, in the next article
A simple program for an ANN to cover the Mnist dataset – X – mini-batch-shuffling and some more tests
we shall extend our tests to different setups of the MLP.