A simple Python program for an ANN to cover the MNIST dataset – VIII – coding Error Backward Propagation

I continue with my series on a Python program for coding small “Multilayer Perceptrons” [MLPs].

After all the theoretical considerations of the last two articles we now start coding again. Our objective is to extend our methods for training the MLP on the MNIST dataset by methods which perform the “error back propagation” and the correction of the weights. The mathematical prescriptions were derived in the following PDF:
My PDF on “The math behind EBP”

When you study the new code fragments below remember a few things:
We are prepared to use mini-batches. Therefore, the cost functions will be calculated over the data records of each batch and all the matrix operations for back propagation will cover all batch-records in parallel. Training means to loop over epochs and mini-batches – in pseudo-code:

• Loop over epochs
2. check for convergence criteria,
3. Shuffle all data records in the test data set and build new mini-batches
• Loop over mini-batches
1. Perform forward propagation for all records of the mini-batch
2. calculate and save the total cost value for each mini-batch
3. calculate and save an averaged error on the output layer for each mini-batch
4. perform error backward propagation on all records of the mini-batch to get the gradient of the cost function with respect to all weights
5. adjust all weights on all layers

As discussed in the last article: The cost hyperplane changes a bit with each mini-batch. If there is a good mixture of records in a batch then the form of its specific cost hyperplane will (hopefully) resemble the form of an overall cost hyperplane, but it will not be the same. By the second step in the outer loop we want to avoid that the same data records always get an influence on the gradients at the same position in the correction procedure. Both statistical elements help a bit to overcome dominant records and a non-equal distribution of test records. If we had only pictures for number 3 at the end of our MNIST data set we
may start learning “3” representations very well, but not other numbers. Statistical variation also helps to avoid side minima on the overall cost hyperplane for all data records of the test set.

We shall implement the second step and third step in the epoch loop in the next article – when we are sure that the training algorithm works as expected. So, at the moment we will stop our training only after a given number of epochs.

More input parameters

In the first articles we had build an __init__()-method to parameterize a training run. We have to include three more parameters to control the backward propagation.

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

The first parameter controls by how much we change weights with the help of gradient values. See formula (93) in the PDF of article VI (you find the Link to the latest version in the last section of this article). The second parameter will give us an option to decrease the learning rate with the number of training epochs. Note that a constant decrease rate only makes sense, if we can be relatively sure that we do not end up in a side minimum of the cost function.

The third parameter is interesting: It will allow us to mix the presently calculated weight correction with the correction term from the last step. So to say: We extend the “momentum” of the last correction into the next correction. This helps us not to follow indicated direction changes on the cost hyperplanes too fast.

Some hygienic measures regarding variables

In the already written parts of the code we have used a prefix “ay_” for all variables which represent some vector or array like structure – including Python lists and Numpy arrays. For back propagation coding it will be more important to distinguish between lists and arrays. So, I changed the variable prefix for some important Python lists from “ay_” to “li_”. (I shall do it for all lists used in a further version). In addition I have changed the prefix for Python ranges to “rg_”. These changes will affect the contents and the interface of some methods. You will notice when we come to these methods.

The changed __input__()-method

Our modified __init__() function now looks like this:

```    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

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)

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 othr 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

# 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")
sys.exit()

```

The extended method _set_ANN_structure()

I do not change method “_handle_input_data()”. However, I extend method “def _set_ANN_structure()” by a statement to initialize a list with momentum matrices for all layers.

```    '''-- Method to set ANN structure --'''
def _set_ANN_structure(self):
# check consistency of the node-number list with the number of hidden layers (n_hidden)
self._check_layer_and_node_numbers()
# set node numbers for the input layer and the output layer
self._set_nodes_for_input_output_layers()
self._show_node_numbers()

# create the weight matrix between input and first hidden layer
self._create_WM_Input()
# create weight matrices between the
hidden layers and between tha last hidden and the output layer
self._create_WM_Hidden()

# initialize momentum differences
self._create_momentum_matrices()
#print("\nLength of li_mom = ", str(len(self._li_mom)))

# check and set activation functions
self._check_and_set_activation_and_out_functions()
self._check_and_set_loss_function()

return None

```

The following box shows the changed functions _create_WM_Input(), _create_WM_Hidden() and the new function _create_momentum_matrices():

```    '''-- Method to create the weight matrix between L0/L1 --'''
def _create_WM_Input(self):
'''
Method to create the input layer
The dimension will be taken from the structure of the input data
We need to fill self._w[0] with a matrix for conections of all nodes in L0 with all nodes in L1
We fill the matrix with random numbers between [-1, 1]
'''
# the num_nodes of layer 0 should already include the bias node
num_nodes_layer_0 = self._ay_nodes_layers[0]
num_nodes_with_bias_layer_0 = num_nodes_layer_0 + 1
num_nodes_layer_1 = self._ay_nodes_layers[1]

# fill the matrix with random values
#rand_low  = -1.0
#rand_high = 1.0
rand_low  = -0.5
rand_high = 0.5
rand_size = num_nodes_layer_1 * (num_nodes_with_bias_layer_0)

randomizer = 1 # method np.random.uniform

w0 = self._create_vector_with_random_values(rand_low, rand_high, rand_size, randomizer)
w0 = w0.reshape(num_nodes_layer_1, num_nodes_with_bias_layer_0)

# put the weight matrix into array of matrices
self._li_w.append(w0)
print("\nShape of weight matrix between layers 0 and 1 " + str(self._li_w[0].shape))

#
'''-- Method to create the weight-matrices for hidden layers--'''
def _create_WM_Hidden(self):
'''
Method to create the weights of the hidden layers, i.e. between [L1, L2] and so on ... [L_n, L_out]
We fill the matrix with random numbers between [-1, 1]
'''

# The "+1" is required due to range properties !
rg_hidden_layers = range(1, self._n_hidden_layers + 1, 1)

# for random operation
rand_low  = -1.0
rand_high = 1.0

for i in rg_hidden_layers:
print ("Creating weight matrix for layer " + str(i) + " to layer " + str(i+1) )

num_nodes_layer = self._ay_nodes_layers[i]
num_nodes_with_bias_layer = num_nodes_layer + 1

# the number of the next layer is taken without the bias node!
num_nodes_layer_next = self._ay_nodes_layers[i+1]

# assign random values
rand_size = num_nodes_layer_next * num_nodes_with_bias_layer

randomizer = 1 # np.random.uniform

w_i_next = self._create_vector_with_random_values(rand_low, rand_high, rand_size, randomizer)
w_i_next = w_i_next.reshape(num_nodes_layer_next, num_nodes_with_bias_layer)

# put the weight matrix into our array of matrices
self._li_w.append(w_i_next)
print("Shape of weight matrix between layers " + str(i) + " and " + str(i+1) + " = " + str(self._li_w[i].shape))
#
'''-- Method to create and initialize matrices for momentum learning (differences) '''
def _create_momentum_matrices(self):
rg_layers = range(0, self._n_total_layers - 1)
for i in rg_layers:
r
self._li_mom[i] = np.zeros(self._li_w[i].shape)
#print("shape of li_mom[" + str(i) + "] = ", self._li_mom[i].shape)
```

The modified functions _fit() and _handle_mini_batch()

The _fit()-function is modified to include a systematic decrease of the learning rate.

```    ''' -- Method to set the number of batches based on given batch size -- '''
def _fit(self, b_print = False, b_measure_batch_time = False):

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):
print("\n ---------")
print("\nStarting epoch " + str(idxe+1))

self._learn_rate /= (1.0 + self._decrease_const * idxe)

# loop over mini-batches
for idxb in rg_idx_batches:
if (b_print):
print("\n ---------")
print("\nDealing with mini-batch " + str(idxb+1))
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 idxb == 100:
#    sys.exit()

return None
```

Note that the number of epochs is determined by an external parameter as an upper limit of the range “rg_idx_epochs”.

Method “_handle_mini_batch()” requires several changes: First we define lists which are required to save matrix data of the backward propagation. And, of course, we call a method to perform the BW propagation (see step 6 in the code). Some statements print shapes, if required. At step 7 of the code we correct the weights by using the learning rate and the calculated gradient of the loss function.

Note, that we mix the correction evaluated at the last batch-record with the correction evaluated for the present record! This corresponds to a simple form of momentum learning. We then have to save the present correction values, of course. Note that the list for momentum correction “li_mom” is, therefore, not deleted at the end of a mini-batch treatment !

In addition to saving the total costs per mini-batch we now also save a mean error at the output level. The average is done by the help of Numpy’s function numpy.average() for matrices. Remember, we build the average over errors at all output nodes and all records of the mini-batch.

```
''' -- Method to deal with a batch -- '''
def _handle_mini_batch(self, num_batch = 0, num_epoch = 0, b_print_y_vals = False, b_print = False, b_keep_bw_matrices = True):
'''
For each batch we keep the input data array Z and the output data A (output of activation function!)
for all layers in Python lists
We can use this as input variables in function calls - mutable variables are handled by reference values !
We receive the A and Z data from propagation functions and proceed them to cost and gradient calculation functions

As an initial step we define the Python lists li_Z_
in_layer and li_A_out_layer
and fill in the first input elements for layer L0

Forward propagation:
--------------------
li_Z_in_layer : List of layer-related 2-dim matrices for input values z at each node (rows) and all batch-samples (cols).
li_A_out_layer: List of layer-related 2-dim matrices for output alues z at each node (rows) and all batch-samples (cols).
The output is created by Phi(z), where Phi represents an activation or output function

Note that the matrices in ay_A_out will be permanently extended by a row (over all samples)
to account for a bias node of each inner layer. This happens during FW propagation.

Note that the matrices ay_Z_in will be temporarily extended by a row (over all samples)
to account for a bias node of each inner layer. This happens during BW propagation.

Backward propagation:
--------------------
li_delta_out:  Startup matrix for _out_delta-values at the outermost layer
li_grad_layer: List of layer-related matrices with gradient values for the correction of the weights

Depending on parameter "b_keep_bw_matrices" we keep
- a list of layer-related matrices D with values for the derivatives of the act./output functions
- a list of layer-related matrices for the back propagated delta-values
in lists during back propagation. This can support error analysis.

All matrices in the lists are 2 dimensional with dimensions for nodes (rows) and training samples (cols)
All these lists be deleted at the end of the function to accelerate garbadge handling

Input parameters:
----------------
num_epoch:     Number of present epoch
num_batch:    Number of present mini-batch
'''
# Layer-related lists to be filled with 2-dim Numpy matrices during FW propagation
# ********************************************************************************
li_Z_in_layer  = [None] * self._n_total_layers # List of matrices with z-input values for each layer; filled during FW-propagation
li_A_out_layer = li_Z_in_layer.copy()          # List of matrices with results of activation/output-functions for each layer; filled during FW-propagation
li_delta_out   = li_Z_in_layer.copy()          # Matrix with out_delta-values at the outermost layer
li_delta_layer = li_Z_in_layer.copy()          # List of the matrices for the BW propagated delta values
li_D_layer     = li_Z_in_layer.copy()          # List of the derivative matrices D containing partial derivatives of the activation/ouput functions
li_grad_layer  = li_Z_in_layer.copy()          # List of the matrices with gradient values for weight corrections

if b_print:
len_lists = len(li_A_out_layer)
print("\nnum_epoch = ", num_epoch, "  num_batch = ", num_batch )
print("\nhandle_mini_batch(): length of lists = ", len_lists)
self._info_point_print("handle_mini_batch: point 1")

# Print some infos
# ****************
if b_print:
self._print_batch_infos()
self._info_point_print("handle_mini_batch: point 2")

# Major steps for the mini-batch during one epoch iteration
# **********************************************************

# Step 0: List of indices for data records in the present mini-batch
# ******
ay_idx_batch = self._ay_mini_batches[num_batch]

# Step 1: Special preparation of the Z-input to the MLP's input Layer L0
# ******
# Layer L0: Fill in the input vector for the ANN's input layer L0
li_
Z_in_layer[0] = self._X_train[ay_idx_batch] # numpy arrays can be indexed by an array of integers
if b_print:
print("\nPropagation : Shape of X_in = li_Z_in_layer = ", li_Z_in_layer[0].shape)
#print("\nidx, expected y_value of Layer L0-input :")
#for idx in self._ay_mini_batches[num_batch]:
#    print(str(idx) + ', ' + str(self._y_train[idx]) )
self._info_point_print("handle_mini_batch: point 3")

# Step 2: Layer L0: We need to transpose the data of the input layer
# *******
ay_Z_in_0T       = li_Z_in_layer[0].T
li_Z_in_layer[0] = ay_Z_in_0T
if b_print:
print("\nPropagation : Shape of transposed X_in = li_Z_in_layer = ", li_Z_in_layer[0].shape)
self._info_point_print("handle_mini_batch: point 4")

# Step 3: Call forward propagation method for the present mini-batch of training records
# *******
# this function will fill the ay_Z_in- and ay_A_out-lists with matrices per layer
self._fw_propagation(li_Z_in = li_Z_in_layer, li_A_out = li_A_out_layer, b_print = b_print)

if b_print:
ilayer = range(0, self._n_total_layers)
print("\n ---- ")
print("\nAfter propagation through all " + str(self._n_total_layers) + " layers: ")
for il in ilayer:
print("Shape of Z_in of layer L" + str(il) + " = " + str(li_Z_in_layer[il].shape))
print("Shape of A_out of layer L" + str(il) + " = " + str(li_A_out_layer[il].shape))
if il < self._n_total_layers-1:
print("Shape of W of layer L" + str(il) + " = " + str(self._li_w[il].shape))
print("Shape of Mom of layer L" + str(il) + " = " + str(self._li_mom[il].shape))
self._info_point_print("handle_mini_batch: point 5")

# Step 4: Cost calculation for the mini-batch
# ********
ay_y_enc = self._ay_onehot[:, ay_idx_batch]
ay_ANN_out = li_A_out_layer[self._n_total_layers-1]
# print("Shape of ay_ANN_out = " + str(ay_ANN_out.shape))

total_costs_batch = self._calculate_loss_for_batch(ay_y_enc, ay_ANN_out, b_print = False)
# we add the present cost value to the numpy array
self._ay_costs[num_epoch, num_batch] = total_costs_batch
if b_print:
print("\n total costs of mini_batch = ", self._ay_costs[num_epoch, num_batch])
self._info_point_print("handle_mini_batch: point 6")
print("\n total costs of mini_batch = ", self._ay_costs[num_epoch, num_batch])

# Step 5: Avg-error for later plotting
# ********
# mean "error" values - averaged over all nodes at outermost layer and all data sets of a mini-batch
ay_theta_out = ay_y_enc - ay_ANN_out
if (b_print):
print("Shape of ay_theta_out = " + str(ay_theta_out.shape))
ay_theta_avg = np.average(np.abs(ay_theta_out))
self._ay_theta[num_epoch, num_batch] = ay_theta_avg

if b_print:
print("\navg total error of mini_batch = ", self._ay_theta[num_epoch, num_batch])
self._info_point_print("handle_mini_batch: point 7")
print("avg total error of mini_batch = ", self._ay_theta[num_epoch, num_batch])

# Step 6: Perform gradient calculation via back propagation of errors
# *******
self._bw_propagation( ay_y_enc = ay_y_enc,
li_Z_in = li_Z_in_layer,
li_A_out = li_A_out_layer,
li_delta_out = li_delta_out,
li_delta = li_delta_
layer,
li_D = li_D_layer,
b_print = b_print,
b_internal_timing = False
)

# Step 7: Adjustment of weights
# *******
rg_layer=range(0, self._n_total_layers -1)
for N in rg_layer:
self._li_w[N] -= ( delta_w_N + (self._mom_rate * self._li_mom[N]) )
# save momentum
self._li_mom[N] = delta_w_N

# try to accelerate garbage handling
# **************
if len(li_Z_in_layer) > 0:
del li_Z_in_layer
if len(li_A_out_layer) > 0:
del li_A_out_layer
if len(li_delta_out) > 0:
del li_delta_out
if len(li_delta_layer) > 0:
del li_delta_layer
if len(li_D_layer) > 0:
del li_D_layer

return None
```

Forward Propagation

The method for forward propagation remains unchanged in its structure. We only changed the prefix for the Python lists.

```    ''' -- Method to handle FW propagation for a mini-batch --'''
def _fw_propagation(self, li_Z_in, li_A_out, b_print= False):

b_internal_timing = False

# index range of layers
#    Note that we count from 0 (0=>L0) to E L(=>E) /
#    Careful: during BW-propgation we may need a correct indexing of lists filled during FW-propagation
ilayer = range(0, self._n_total_layers-1)

# propagation loop
# ***************
for il in ilayer:
if b_internal_timing: start_0 = time.perf_counter()

if b_print:
print("\nStarting propagation between L" + str(il) + " and L" + str(il+1))
print("Shape of Z_in of layer L" + str(il) + " (without bias) = " + str(li_Z_in[il].shape))

# Step 1: Take input of last layer and apply activation function
# ******
if il == 0:
A_out_il = li_Z_in[il] # L0: activation function is identity
else:
A_out_il = self._act_func( li_Z_in[il] ) # use real activation function

# Step 2: Add bias node
# ******
# save in array
li_A_out[il] = A_out_il
if b_print:
print("Shape of A_out of layer L" + str(il) + " (with bias) = " + str(li_A_out[il].shape))

# Step 3: Propagate by matrix operation
# ******
Z_in_ilp1 = np.dot(self._li_w[il], A_out_il)
li_Z_in[il+1] = Z_in_ilp1

if b_internal_timing:
end_0 = time.perf_counter()
print('Time_CPU for layer propagation L' + str(il) + ' to L' + str(il+1), end_0 - start_0)

# treatment of the last layer
# ***************************
il = il + 1
if b_print:
print("\nShape of Z_in of layer L" + str(il) + " = " + str(li_Z_in[il].shape))
A_out_il = self._out_func( li_Z_in[il] ) # use the output function
li_A_out[il] = A_out_il
if b_print:
print("Shape of A_out of last layer L" + str(il) + " = " + str(li_A_out[il].shape))

return None

```

We shall later learn that the treatment of bias neurons can be done more efficiently. The present way of coding it reduces performance – especially at the input layer. See the article series starting with
MLP, Numpy, TF2 – performance issues – Step I – float32, reduction of back propagation
for more information. At the present stage of our discussion we are, however, more interested in getting a working code first – and not so much in performance optimization.

Methods for Error Backward Propagation

In contrast to the recipe given in my PDF on the EBP-math we cannot calculate the matrices with the derivatives of the activation functions “ay_D” in advance for all layers. The reason was discussed in the last article VII: Some matrices have to be intermediately adjusted for a bias-neuron, which is ignored in the analysis of the PDF.

The resulting code of our method for EBP looks like given below:

```
''' -- Method to handle error BW propagation for a mini-batch --'''
def _bw_propagation(self,
ay_y_enc, li_Z_in, li_A_out,
b_print = True, b_internal_timing = False):

# List initialization: All parameter lists or arrays are filled or to be filled by layers
# Note: the lists li_Z_in, li_A_out were already filled by _fw_propagation() for the present batch

# Initiate BW propagation - provide delta-matrices for outermost layer
# ***********************
# Input Z at outermost layer E  (4 layers -> layer 3)
ay_Z_E = li_Z_in[self._n_total_layers-1]
# Output A at outermost layer E (was calculated by output function)
ay_A_E = li_A_out[self._n_total_layers-1]

# Calculate D-matrix (derivative of output function) at outmost the layer - presently only D_sigmoid
ay_D_E = self._calculate_D_E(ay_Z_E=ay_Z_E, b_print=b_print )

# Get the 2 delta matrices for the outermost layer (only layer E has 2 delta-matrices)
ay_delta_E, ay_delta_out_E = self._calculate_delta_E(ay_y_enc=ay_y_enc, ay_A_E=ay_A_E, ay_D_E=ay_D_E, b_print=b_print)

# We check the shapes
shape_theory = (self._n_nodes_layer_out, self._n_size_mini_batch)
if (b_print and ay_delta_E.shape != shape_theory):
print("\nError: Shape of ay_delta_E is wrong:")
print("Shape = ", ay_delta_E.shape, "  ::  should be = ", shape_theory )
if (b_print and ay_D_E.shape != shape_theory):
print("\nError: Shape of ay_D_E is wrong:")
print("Shape = ", ay_D_E.shape, "  ::  should be = ", shape_theory )

# add the matrices to their lists ; li_delta_out gets only one element
idxE = self._n_total_layers - 1
li_delta_out[idxE] = ay_delta_out_E # this happens only once
li_delta[idxE]     = ay_delta_E
li_D[idxE]         = ay_D_E
li_grad[idxE]      = None    # On the outermost layer there is no gradient !

if b_print:
print("bw: Shape delta_E = ", li_delta[idxE].shape)
print("bw: Shape D_E = ", ay_D_E.shape)
self._info_point_print("bw_propagation: point bw_1")

# Loop over all layers in reverse direction
# ******************************************
# index range of target layers N in BW direction (starting with E-1 => 4 layers -> layer 2))
if b_print:
range_N_bw_layer_test = reversed(range(0,
self._n_total_layers-1))   # must be -1 as the last element is not taken
rg_list = list(range_N_bw_layer_test) # Note this exhausts the range-object
print("range_N_bw_layer = ", rg_list)

range_N_bw_layer = reversed(range(0, self._n_total_layers-1))   # must be -1 as the last element is not taken

# loop over layers
for N in range_N_bw_layer:
if b_print:
print("\n N (layer) = " + str(N) +"\n")
# start timer
if b_internal_timing: start_0 = time.perf_counter()

# Back Propagation operations between layers N+1 and N
# *******************************************************
# this method handles the special treatment of bias nodes in Z_in, too
ay_delta_N, ay_D_N, ay_grad_N = self._bw_prop_Np1_to_N( N=N, li_Z_in=li_Z_in, li_A_out=li_A_out, li_delta=li_delta, b_print=False )

if b_internal_timing:
end_0 = time.perf_counter()
print('Time_CPU for BW layer operations ', end_0 - start_0)

# add matrices to their lists
li_delta[N] = ay_delta_N
li_D[N]     = ay_D_N
#sys.exit()

return

```

We first handle the necessary matrix evaluations for the outermost layer. We use two helper functions there to calculate the derivative of the output function with respect to the a-term [ _calculate_D_E() ] and to calculate the values for the “delta“-terms at all nodes and for all records [ _calculate_delta_E() ] according to the prescription in the PDF:

```
''' -- Method to calculate the matrix with the derivative values of the output function at outermost layer '''
def _calculate_D_E(self, ay_Z_E, b_print= True):
'''
This method calculates and returns the D-matrix for the outermost layer
The D matrix contains derivatives of the output function with respect to local input "z_j" at outermost nodes.

Returns
------
ay_D_E:    Matrix with derivative values of the output function
with respect to local z_j valus at the nodes of the outermost layer E
Note: This is a 2-dim matrix over layer nodes and training samples of the mini-batch
'''
if self._my_out_func == 'sigmoid':
ay_D_E = self._D_sigmoid(ay_Z=ay_Z_E)

else:
print("The derivative for output function " + self._my_out_func + " is not known yet!" )
sys.exit()

return ay_D_E

''' -- Method to calculate the delta_E matrix as a starting point of the backward propagation '''
def _calculate_delta_E(self, ay_y_enc, ay_A_E, ay_D_E, b_print= False):
'''
This method calculates and returns the 2 delta-matrices for the outermost layer

Returns
------
delta_E:     delta_matrix of the outermost layer (indicated by E)
delta_out:   delta_out matrix => elements are local derivative values of the cost function
with respect to the output "a_j" at an outermost node
!!! delta_out will only be returned if calculable !!!

Note: these are 2-dim matrices over layer nodes and training samples of the mini-batch
'''

if self._my_loss_func == 'LogLoss':
# Calculate delta_S_E directly to avoid problems with zero denominators
ay_delta_E = ay_A_E - ay_y_enc
# delta_out is fetched but may be None
ay_delta_out, ay_D_
numerator, ay_D_denominator = self._D_loss_LogLoss(ay_y_enc, ay_A_E, b_print = False)

# To be done: Analyze critical values in D_denominator

# Release variables explicitly
del ay_D_numerator
del ay_D_denominator

if self._my_loss_func == 'MSE':
# First calculate delta_out and then the delta_E
delta_out = self._D_loss_MSE(ay_y_enc, ay_A_E, b_print=False)
# calculate delta_E via matrix multiplication
ay_delta_E = delta_out * ay_D_E

return ay_delta_E, ay_delta_out

```

Further required helper methods to calculate the cost functions and related derivatives are :

```    ''' method to calculate the logistic regression loss function '''
def _loss_LogLoss(self, ay_y_enc, ay_ANN_out, b_print = False):
'''
Method which calculates LogReg loss function in a vectorized form on multidimensional Numpy arrays
'''
b_test = False

if b_print:
print("From LogLoss: shape of ay_y_enc =  " + str(ay_y_enc.shape))
print("From LogLoss: shape of ay_ANN_out =  " + str(ay_ANN_out.shape))
print("LogLoss: ay_y_enc = ", ay_y_enc)
print("LogLoss: ANN_out = \n", ay_ANN_out)
print("LogLoss: log(ay_ANN_out) =  \n", np.log(ay_ANN_out) )

# The following means an element-wise (!) operation between matrices of the same shape!
Log1 = -ay_y_enc * (np.log(ay_ANN_out))
# The following means an element-wise (!) operation between matrices of the same shape!
Log2 = (1 - ay_y_enc) * np.log(1 - ay_ANN_out)

# the next operation calculates the sum over all matrix elements
# - thus getting the total costs for all mini-batch elements
cost = np.sum(Log1 - Log2)

#if b_print and b_test:
# Log1_x = -ay_y_enc.dot((np.log(ay_ANN_out)).T)
# print("From LogLoss: L1 =   " + str(L1))
# print("From LogLoss: L1X =  " + str(L1X))

if b_print:
print("From LogLoss: cost =  " + str(cost))

# The total costs is just a number (scalar)
return cost
#
''' method to calculate the derivative of the logistic regression loss function
with respect to the output values '''
def _D_loss_LogLoss(self, ay_y_enc, ay_ANN_out, b_print = False):
'''
This function returns the out_delta_S-matrix which is required to initialize the
BW propagation (EBP)
Note ANN_out is the A_out-list element ( a 2-dim matrix) for the outermost layer
In this case we have to take care of denominators = 0
'''
D_numerator = ay_ANN_out - ay_y_enc
D_denominator = -(ay_ANN_out - 1.0) * ay_ANN_out
n_critical = np.count_nonzero(D_denominator < 1.0e-8)
if n_critical > 0:
delta_s_out = None
else:
delta_s_out = np.divide(D_numerator, D_denominator)
return delta_s_out, D_numerator, D_denominator
#
''' method to calculate the MSE loss function '''
def _loss_MSE(self, ay_y_enc, ay_ANN_out, b_print = False):
'''
Method which calculates LogReg loss function in a vectorized form on multidimensional Numpy arrays
'''
if b_print:
print("From loss_MSE: shape of ay_y_enc =  " + str(ay_y_enc.shape))
print("From loss_MSE: shape of ay_ANN_out =  " + str(ay_ANN_out.shape))
#print("LogReg: ay_y_enc = ", ay_y_enc)
#print("LogReg: ANN_out = \n", ay_
ANN_out)
#print("LogReg: log(ay_ANN_out) =  \n", np.log(ay_ANN_out) )

cost = 0.5 * np.sum( np.square( ay_y_enc - ay_ANN_out ) )

if b_print:
print("From loss_MSE: cost =  " + str(cost))

return cost
#
''' method to calculate the derivative of the MSE loss function
with respect to the output values '''
def _D_loss_MSE(self, ay_y_enc, ay_ANN_out, b_print = False):
'''
This function returns the out_delta_S - matrix which is required to initialize the
BW propagation (EBP)
Note ANN_out is the A_out-list element ( a 2-dim matrix) for the outermost layer
In this case the output is harmless (no critical denominator)
'''
delta_s_out = ay_ANN_out - ay_y_enc
return delta_s_out
```

You see that we are a bit careful to avoid zero denominators for the Logarithmic loss function in all of our helper functions.

The check statements for shapes can be eliminated in a future version when we are sure that everything works correctly. Keeping the layer specific matrices during the handling of a mini-batch will be also good for potentially required error analysis in the beginning. In the end we only may keep the gradient-matrices and the layer specific matrices required to process the local calculations during back propagation.

Then we turn to loop over all other layers down to layer L0. The matrix operation to be done for all these layers are handled in a further method:

```
''' -- Method to calculate the BW-propagated delta-matrix and the gradient matrix to/for layer N '''
def _bw_prop_Np1_to_N(self, N, li_Z_in, li_A_out, li_delta, b_print=False):
'''
BW-error-propagation bewtween layer N+1 and N
Inputs:
li_Z_in:  List of input Z-matrices on all layers - values were calculated during FW-propagation
li_A_out: List of output A-matrices - values were calculated during FW-propagation
li_delta: List of delta-matrices - values for outermost ölayer E to layer N+1 should exist

Returns:
ay_delta_N - delta-matrix of layer N (required in subsequent steps)
ay_D_N     - derivative matrix for the activation function on layer N
ay_grad_N  - matrix with gradient elements of the cost fnction with respect to the weights on layer N
'''

if b_print:
print("Inside _bw_prop_Np1_to_N: N = " + str(N) )

# Prepare required quantities - and add bias neuron to ay_Z_in
# ****************************

# Weight matrix meddling betwen layer N and N+1
ay_W_N = self._li_w[N]
shape_W_N   = ay_W_N.shape # due to bias node first dim is 1 bigger than Z-matrix
if b_print:
print("shape of W_N = ", shape_W_N )

# delta-matrix of layer N+1
ay_delta_Np1 = li_delta[N+1]
shape_delta_Np1 = ay_delta_Np1.shape

# !!! Add intermediate row (for bias) to Z_N !!!
ay_Z_N = li_Z_in[N]
shape_Z_N_orig = ay_Z_N.shape
shape_Z_N = ay_Z_N.shape # dimensions should fit now with W- and A-matrix

# Derivative matrix for the activation function (with extra bias node row)
#    can only be calculated now as we need the z-values
ay_D_N = self._calculate_D_N(ay_Z_N)
shape_D_N = ay_D_N.shape

ay_A_N = li_A_out[N]
shape_A_N = ay_A_N.shape

# print shapes
if b_print:
print("shape of W_N = ", shape_W_N)
print("
shape of delta_(N+1) = ", shape_delta_Np1)
print("shape of Z_N_orig = ", shape_Z_N_orig)
print("shape of Z_N = ", shape_Z_N)
print("shape of D_N = ", shape_D_N)
print("shape of A_N = ", shape_A_N)

# Propagate delta
# **************
if li_delta[N+1] is None:
print("BW-Prop-error:\n No delta-matrix found for layer " + str(N+1) )
sys.exit()

# Check shapes for np.dot()-operation - here for element [0] of both shapes - as we operate with W.T !
if ( shape_W_N[0] != shape_delta_Np1[0]):
print("BW-Prop-error:\n shape of W_N [", shape_W_N, "]) does not fit shape of delta_N+1 [", shape_delta_Np1, "]" )
sys.exit()

# intermediate delta
# ~~~~~~~~~~~~~~~~~~
ay_delta_w_N = ay_W_N.T.dot(ay_delta_Np1)
shape_delta_w_N = ay_delta_w_N.shape

# Check shapes for element wise *-operation !
if ( shape_delta_w_N != shape_D_N ):
print("BW-Prop-error:\n shape of delta_w_N [", shape_delta_w_N, "]) does not fit shape of D_N [", shape_D_N, "]" )
sys.exit()

# final delta
# ~~~~~~~~~~~
ay_delta_N = ay_delta_w_N * ay_D_N
# reduce dimension again
ay_delta_N = ay_delta_N[1:, :]
shape_delta_N = ay_delta_N.shape

# Check dimensions again - ay_delta_N.shape should fit shape_Z_in_orig
if shape_delta_N != shape_Z_N_orig:
print("BW-Prop-error:\n shape of delta_N [", shape_delta_N, "]) does not fit original shape Z_in_N [", shape_Z_N_orig, "]" )
sys.exit()

if N > 0:
shape_W_Nm1 = self._li_w[N-1].shape
if shape_delta_N[0] != shape_W_Nm1[0] :
print("BW-Prop-error:\n shape of delta_N [", shape_delta_N, "]) does not fit shape of W_Nm1 [", shape_W_Nm1, "]" )
sysexit()

# ********************
#     required for all layers down to 0
# check shapes
if shape_delta_Np1[1] != shape_A_N[1]:
print("BW-Prop-error:\n shape of delta_Np1 [", shape_delta_Np1, "]) does not fit shape of A_N [", shape_A_N, "] for matrix multiplication" )
sys.exit()

# regularize gradient (!!!! without adding bias nodes in the L1, L2 sums)
ay_grad_N[:, 1:] += (self._li_w[N][:, 1:] * self._lambda2_reg + np.sign(self._li_w[N][:, 1:]) * self._lambda1_reg)

#
# Check shape
print("BW-Prop-error:\n shape of grad_N [", shape_grad_N, "]) does not fit shape of W_N [", shape_W_N, "]" )
sys.exit()

# print shapes
if b_print:
print("shape of delta_N = ", shape_delta_N)

```

This function does more or less exactly what we have requested by our theoretical analysis in the last two articles. Note the intermediate handling of bias nodes! Note also that bias nodes are NOT regarded in regularization terms L1 and L2! The function to calculate the derivative of the activation function is:

```
#
''' -- Method to calculate the matrix with the derivative values of the output function at outermost layer '''

def _calculate_D_N(self, ay_Z_N, b_print= False):
'''
This method calculates and returns the D-matrix for the outermost layer
The D matrix contains derivatives of the output function with respect to local input "z_j" at outermost nodes.

Returns
------
ay_D_E:    Matrix with derivative values of the output function
with respect to local z_j valus at the nodes of the outermost layer E
Note: This is a 2-dim matrix over layer nodes and training samples of the mini-batch
'''
if self._my_out_func == 'sigmoid':
ay_D_E = self._D_sigmoid(ay_Z = ay_Z_N)

else:
print("The derivative for output function " + self._my_out_func + " is not known yet!" )
sys.exit()

return ay_D_E

```

The methods to calculate regularization terms for the loss function are:

```
#
''' method do calculate the quadratic regularization term for the loss function '''
def _regularize_by_L2(self, b_print=False):
'''
The L2 regularization term sums up all quadratic weights (without the weight for the bias)
over the input and all hidden layers (but not the output layer)
The weight for the bias is in the first column (index 0) of the weight matrix -
as the bias node's output is in the first row of the output vector of the layer
'''
ilayer = range(0, self._n_total_layers-1) # this excludes the last layer
L2 = 0.0
for idx in ilayer:
L2 += (np.sum( np.square(self._li_w[idx][:, 1:])) )
L2 *= 0.5 * self._lambda2_reg
if b_print:
print("\nL2: total L2 = " + str(L2) )
return L2
#
''' method do calculate the linear regularization term for the loss function '''
def _regularize_by_L1(self, b_print=False):
'''
The L1 regularization term sums up all weights (without the weight for the bias)
over the input and all hidden layers (but not the output layer
The weight for the bias is in the first column (index 0) of the weight matrix -
as the bias node's output is in the first row of the output vector of the layer
'''
ilayer = range(0, self._n_total_layers-1) # this excludes the last layer
L1 = 0.0
for idx in ilayer:
L1 += np.sum(np.abs( self._li_w[idx][:, 1:]))
L1 *= 0.5 * self._lambda1_reg
if b_print:
print("\nL1: total L1 = " + str(L1))
return L1
```

Also the BW-propagation code presented here will later be the target of optimization steps. We shall see that it – despite working correctly – can be criticized regarding efficiency at several points. See again the article series starting with
MLP, Numpy, TF2 – performance issues – Step I – float32, reduction of back propagation.

Conclusion

We have extended our set of methods quite a bit. At the core of the operations we perform matrix operations which are supported by the Openblas library on a Linux system with multiple CPU cores. In the next article

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

we shall test the convergence of our training for the MNIST data set. We shall see that a MLP with two hidden layers with 70 and 30 nodes can give us a convergence
of the averaged relative error down to 0.006 after 1000 epochs on the test data. However, we have to analyze such results for overfitting. Stay tuned …

My PDF on “The math behind EBP”

A simple Python program for an ANN to cover the MNIST dataset – VII – EBP related topics and obstacles

I continue with my series about a Python program to build simple MLPs:

On our tour we have already learned a lot about multiple aspects of MLP usage. I name forward propagation, matrix operations, loss or cost functions. In the last article of this series
A simple program for an ANN to cover the Mnist dataset – VI – the math behind the „error back-propagation“
I tried to explain some of the math which governs “Error Back Propagation” [EBP]. See the PDF attached to the last article.

EBP is an algorithm which applies the “Gradient Descent” method for the optimization of the weights of a Multilayer Perceptron [MLP]. “Gradient Descent” itself is a method where we step-wise follow short tracks perpendicular to contour lines of a hyperplane in a multidimensional parameter space to hopefully approach a global minimum. A step means a change of of parameter values – in our context of weights. In our case the hyperplane is the surface formed by the cost function over the weights. If we have m weights we get a hyperplane in an (m+1) dimensional space.

To apply gradient descent we have to calculate partial derivatives of the cost function with respect to the weights. We have discussed this in detail in the last article. If you read the PDF you certainly have noted: Most of the time we shall execute matrix operations to provide the components of the weight gradient. Of course, we must guarantee that the matrices’ dimensions fit each other such that the required operations – as an element-wise multiplication and the numpy.dot(X,Y)-operation – become executable.

Unfortunately, there are some challenges regarding this point which we have not covered, yet. One objective of this article is to get prepared for these potential problems before we start coding EBP.

Another point worth discussing is: Is there really just one cost function when we use mini-batches in combination with gradient descent? Regarding the descriptions and the formulas in the PDF of the last article this was and is not fully clear. We only built sums there over cost contributions of all the records in a mini-batch. We did NOT use a loss function which assigned to costs to deviations of the predicted result (after forward propagation) from known values for all training data records.

This triggers the question what our
code in the end really does if and when it works with mini-batches during weight optimization … We start with this point.

In the following I try to keep the writing close to the quantity notations in the PDF. Sorry for a bad display of the δs in HTML.

Gradient descent and mini-batches – one or multiple cost functions?

Regarding the formulas given so far, we obviously handle costs and gradient descent batch-wise. I.e. each mini-batch has its own cost function – with fewer contributions than a cost function for all records would have. Each cost function has (hopefully) a defined position of a global minimum in the weights’ parameter space. Taking this into consideration the whole mini-batch approach is obviously based on some conceptually important assumptions:

• The basic idea is that the positions of the global minima of all the cost-functions for the different batches do not deviate too much from each other in the basic parameter space.
• If we additionally defined a cost function for all test data records (over all batches) then this cost function should display a global minimum positioned in between the ones of the batches’ cost functions.
• This also means that there should be enough records in each batch with a really statistical distribution and no specialties associated with them.
• Contour lines and gradients on the hyperplanes defined by the loss functions will differ from each other. On average over all mini-batches this should not hinder convergence into a common optimum.

To understand the last point let us assume that we have a batch for MNIST dataset where all records of handwritten digits show a tendency to be shifted to the left border of the basic 28×28 pixel frames. Then this batch would probably give us other weights than other batches.

To get a deeper understanding, let us take only two batches. By chance their cost functions may deviate a bit. In the plots below I have just simulated this by two assumed “cost” functions – each forming a hyperplane in 3 dimensions over only two parameter (=weight) dimensions x and y. You see that the “global” minima of the blue and the red curve deviate a bit in their position.

The next graph shows the sum, i.e. the full “cost function”, in green in comparison to the (vertically shifted and scaled) original functions.

Also here you clearly see the differences in the minimas’ positions. What does this mean for gradient descent?

Firstly, the contour lines on the total cost function would deviate from the ones on the cost function hyperplanes of our 2 batches. So would the directions of the different gradients at the point presently reached in the parameter space during optimization! Working with batches therefore means jumping around on the surface of the total cost function a bit erratically and not precisely along the direction of steepest descent there. By the way: This behavior can be quite helpful to overcome local minima.

Secondly, in our simplified example we would in the end not converge completely, but jump or circle around the minimum of the total cost function. Reason: Each batch forces the weight corrections for x,y into different directions namely those of
its own minimum. So, a weight correction induced by one bath would be countered by corrections imposed by the optimization for the other batch. (Regarding MNIST it would e.g. be interesting to run a batch with handwritten digits of Europeans against a batch with digits written by Americans and see how the weights differ after gradient descent has converged for each batch.)

This makes us understand multiple things:

• Mini-batches should be built with a statistical distribution of records and their composition should be changed statistically from epoch to epoch.
• We need a criterion to stop iterating over too many epochs senselessly.
• We should investigate whether the number and thus the size of mini-batches influences the results of EBP.
• At the end of an optimization run we could invest in some more iterations not for the batches, but for the full cost function of all training records and see if we can get a little deeper into the minimum of this total cost function.
• We should analyze our batches – if we keep them up and do not create them statistically anew at the beginning of each epoch – for special data records whose properties are off of the normal – and maybe eliminate those data records.

Repetition: Why Back-propagation of 2 dimensional matrices and not vectors?

The step wise matrix operations of EBP are to be performed according to a scheme with the following structure:

• On a given layer N apply a layer specific matrix “NW.T” (depending on the weights there) by some operational rule on some matrix “(N+1)δS“, which contains some data already calculated for layer (N+1).
• Take the results and modify it properly by multiplying it element-wise with some other matrix ND (containing derivative expressions for the activation function) until you get a new NδS.
• Get partial derivatives of the cost function with respect to the weights on layer (N-1) by a further matrix operation of NδS on a matrix with output values (N-1)A.TS on layer (N-1).
• Proceed to the next layer in backward direction.

The input into this process is a matrix of error-dependent quantities, which are defined at the output layer. These values are then back-propagated in parallel to the inner layers of our MLP.

Now, why do we propagate data matrices and not just data vectors? Why are we allowed to combine so many different multiplications and summations described in the last article when we deal with partial derivatives with respect to variables deep inside the network?

The answer to the first question is numerical efficiency. We operate on all data records of a mini-batch in parallel; see the PDF. The answer to the second question is 2-fold:

• We are allowed to perform so many independent operations because of the linear structure of our cost-functions with respect to contributions coming from the records of a mini-batch and the fact that we just apply linear operations between layers during forward propagation. All contributions – however non-linear each may be in itself – are just summed up. And propagation itself between layers is defined to be linear.
• The only non-linearity occurring – namely in the form of non-linear activation functions – is to be applied just on layers. And there it works only node-wise! We do not
couple values for nodes on one and the same layer.

In this sense MLPs are very simple by definition – although they may look complex! (By the way and if you wonder why MLPs are nevertheless so powerful: One reason has to do with the “Universal Approximation Theorem”; see the literature hint at the end.)

Consequence of the simplicity: We can deal with δ-values (see the PDF) for both all nodes of a layer and all records of a mini-batch in parallel.

Results derived in the last article would change dramatically if we had rules that coupled the Z- or A-values of different nodes! E.g. if the squared value at node 7 in layer X must always be the sum of squared values at nodes 5 an 6. Believe me: There are real networks in this world where such a type of node coupling occurs – not only in physics.

Note: As we have explained in the PDF, the nodes of a layer define one dimension of the NδS“-matrices,
the number of mini-batch records the other. The latter remains constant. So, during the process the δ-matrices change only one of their 2 dimensions.

Some possible pitfalls to tackle before EBP-coding

Now, my friends, we can happily start coding … Nope, there are actually some minor pitfalls, which we have to explain first.

Special cost-, activation- and output-functions

I refer to the PDF mentioned above and its formulas. The example explained there referred to the “Log Loss” function, which we took as an example cost function. In this case the outδS and the 3δS-terms at the nodes of the outermost layer turned out to be quite simple. See formula (21), (22), (26) and (27) in the PDF.

However, there may be other cost functions for which the derivative with respect to the output vector “a” at the outermost nodes is more complicated.

In addition we may have other output or activation functions than the sigmoid function discussed in the PDF’s example. Further, the output function may differ from the activation function at inner layers. Thus, we find that the partial derivatives of these functions with respect to their variables “z” must be calculated explicitly and as needed for each layer during back propagation; i.e., we have to provide separate and specific functions for the provision of the required derivatives.

At the outermost layer we apply the general formulas (84) to (88) with matrix ED containing derivatives of the output-function Eφ(z) with respect to the input z to find EδS with E marking the outermost layer. Afterwards, however, we apply formula (92) – but this time with D-elements referring to derivatives of the standard activation-function φ used at nodes of inner layers.

The special case of the Log Loss function and other loss functions with critical denominators in their derivative

Formula (21) shows something interesting for the quantity outδS, which is a starting point for backward propagation: a denominator depending on critical factors, which directly involve output “a” at the outer nodes or “a” in a difference term. But in our one-hot-approach “a” may become zero or come close to it – during training by accident or by convergence! This is a dangerous thing; numerically we absolutely want to avoid any division by zero or by small numbers close to the numerical accuracy of a programming language.

What mathematically saves us in the special case of Log Loss are formulas (26) and (27), where due to some “magic” the dangerous denominator is cancelled by a corresponding factor in the numerator when we evaluate EδS.

In the general case, however,
we must investigate what numerical dangers the functional form of the derivative of the loss function may bring with it. In the end there are two things we should do:

• Build a function to directly calculate EδS and put as much mathematical knowledge about the involved functions and operations into it as possible, before employing an explicit calculation of values of the cost function’s derivative.
• Check the involved matrices, whose elements may appear in denominators, for elements which are either zero or close to it in the sense of the achievable accuracy.

For our program this means: Whether we calculate the derivative of a cost function to get values for “outδS” will depend on the mathematical nature of the cost function. In case of Log Loss we shall avoid it. In case of MSE we shall perform the numerical operation.

Handling of bias nodes

A further complication of our aspired coding has its origin in the existence of bias nodes on every inner layer of the MLP. A bias node of a layer adds an additional degree of freedom whilst adjusting the layer’s weights; a bias node has no input, it produces only a constant output – but is connected with weights to all normal nodes of the next layer.

Some readers who are not so familiar with “artificial neural networks” may ask: Why do we need bias nodes at all?

Well, think about a simple matrix operation on a 2 dim-vector; it changes its direction and length. But if we want to approximate a function for regression or a separation hyperplanes for classification by a linear operation then we need another element which corresponds to a constant translation part in a linear transformation: z = w1*x1 + w2*x2 + const.. Take a simple function y=w*x + c. The “c” controls where the line crosses the y axis. We need such a parameter if our line should separate clusters of points separably distributed somewhere in the (x,y)-plane; the w is not sufficient to orientate and position the hyperplane in the (x,y)-plane.

This very basically what bias neurons are good for regarding the basically linear operation between two MLP-layers. They add a constant to an otherwise linear transformation.

Do we need a bias node on all layers? Definitely on the input layer. However, on the hidden layers a trained network could by learning evolve weights in such a way that a bias neuron comes about – with almost zero weights on one side. At least in principle; however, we make it easier for the MLP to converge by providing explicit “bias” neurons.

What did we do to account for bias nodes in our Python code so far? We extended the matrices describing the output arrays ay_A_out of the activation function (for input ay_Z_in) on the input and all hidden layers by elements of an additional row. This was done by the method “add_bias_neuron_to_layer()” – see the codes given in article III.

The important point is that our weight matrices already got a corresponding dimension when we built them; i.e. we defined weights for the bias nodes, too. Of course, during optimization we must calculate partial derivatives of the cost function with respect to these weights.

The problem is:

We need to back propagate a delta-matrix Nδ for layer N via
( (NW.T).dot(Nδ) ). But then we can not apply a simple element-wise matrix multiplication with the (N-1)D(z)-matrix at layer N-1. Reason: The dimensions do not fit, if we calculate the elements of D only for the existing Z-Values at layer N-1.

There are two solutions for coding:

• We can add a row artificially and intermediately to the Z-matrix to calculate the D-matrix, then calculate NδS as
( (NW.T).dot(Nδ) ) * (N_1)D
and eliminate the first artificial row appearing in NδS afterwards.
• The other option is to reduce the weight-matrix (NW) by a row intermediately and restore it again afterwards.

What we do is a matter of efficiency; in our coding we shall follow the first way and test the difference to the second way afterwards.

Check the matrix dimensions

As all steps to back-propagate and to circumvent the pitfalls require a bit of matrix wizardry we should at least check at every step during EBP backward-propagation that the dimensions of the involved matrices fit each other.

Outlook

Guys, after having explained some of the matrix math in the previous article of this series and the problems we have to tackle whilst programming the EBP-algorithm we are eventually well prepared to add EBP-methods to our Python class for MLP simulation. We are going to to this in the next article:
A simple program for an ANN to cover the Mnist dataset – VIII – coding Error Backward Propagation

Literature

“Machine Learning – An Applied Mathematics Introduction”, Paul Wilmott, 2019, Panda Ohana Publishing

A simple Python program for an ANN to cover the MNIST dataset – VI – the math behind the “error back-propagation”

I continue with my article series on how to program a training algorithm for a multi-layer perceptron [MLP]. In the course of my last articles

we have already created code for the “Feed Forward Propagation” algorithm [FFPA] and two different cost functions – “Log Loss” and “MSE”. In both cases we took care of a vectorized handling of multiple data records in mini-batches of training data.

Before we turn to the coding of the so called “error back-propagation” [EBP], I found it usefull to clarify the math behind behind this method for ANN/MLP-training. Understanding the basic principles of the gradient descent method for the optimization of MLP-weights is easy. But comprehending

• why and how gradient descent method leads to the back propagation of error terms
• and how we cover multiple training data records at the same time

is not – at least not in my opinion. So, I have discussed the required analysis and resulting algorithmic steps in detail in a PDF which you find attached to this article. I used a four layer MLP as an example for which I derived the partial derivatives of the “Log Loss” cost function for weights of the hidden layers in detail. I afterwards generalized the formalism. I hope the contents of the PDF will help beginners in the field of ML to understand what kind of matrix operations gradient descent leads to.

PDF on the math behind Error Back_Propagation

In the next article we shall encode the surprisingly compact algorithm for EBP. In the meantime I wish all readers Merry Christmas …

Addendum 01.01.2020 / 23.02.2020 : Corrected a missing “-” for the cost function and resulting terms in the above PDF.