A simple CNN for the MNIST datasets – II – building the CNN with Keras and a first test

I continue with my series on first exploratory steps with CNNs. After all the discussion of CNN basics in the last article,

A simple CNN for the MNIST datasets – I,

we are well prepared to build a very simple CNN with Keras. By simple I mean simple enough to handle the MNIST digit images. The Keras API for creating CNN models, layers and activation functions is very convenient; a simple CNN does not require much code. So, the Jupyter environment is sufficient for our first experiment.

An interesting topic is the use of a GPU. After a somewhat frustrating experience with a MLP on the GPU of a NV 960 GTX in comparison to a i7 6700K CPU I am eager to see whether we get a reasonable GPU acceleration for a CNN. So, we should prepare our code to use the GPU. This requires a bit of preparation.

We should also ask a subtle question: What do we expect from a CNN in comparison to a MLP regarding the MNIST data? A MLP with 2 hidden layers (with 70 and 30 nodes) provided over 99.5% accuracy on the training data and almost 98% accuracy on a test dataset after some tweaking. Even with basic settings for our MLP we arrived at a value over 97.7% after 35 epochs - below 8 secs. Well, a CNN is probably better in feature recognition than a cluster detection algorithm. But we are talking about the last 2 % of remaining accuracy. I admit that I did not know what to expect ...

A MLP as an important part of a CNN

At the end of the last article I had discussed a very simple layer structure of convolutional and pooling layers:

  • Layer 0: Input layer (tensor of original image data, 3 layers for color channels or one layer for a gray channel)
  • Layer 1: Conv layer (small 3x3 kernel, stride 1, 32 filters => 32 maps (26x26), overlapping filter areas)
  • Layer 2: Pooling layer (2x2 max pooling => 32 (13x13) maps,
    a map node covers 4x4 non overlapping areas per node on the original image)
  • Layer 3: Conv layer (3x3 kernel, stride 1, 64 filters => 64 maps (11x11),
    a map node covers 8x8 overlapping areas on the original image (total effective stride 2))
  • Layer 4: Pooling layer (2x2 max pooling => 64 maps (5x5),
    a map node covers 10x10 areas per node on the original image (total effective stride 5), some border info lost)
  • Layer 5: Conv layer (3x3 kernel, stride 1, 64 filters => 64 maps (3x3),
    a map node covers 18x18 areas per node (effective stride 5), some border info lost )

This is the CNN structure we are going to use in the near future. (Actually, I followed a suggestion of Francois Chollet; see the literature list in the last article). Let us assume that we somehow have established all these convolution and pooling layers for a CNN. Each layer producse some "feature"-related output, structured in form of a tensors. This led to an open question at the end of the last article:

Where and by what do we get a classification of the resulting data with respect to the 10 digit categories of the MNIST images?

Applying filters and extracting "feature hierarchies" of an image alone does not help without a "learned" judgement about these data. But the answer is very simple:

Use a MLP after the last Conv layer and feed it with data from this Conv layer!

When we think in terms of nodes and artificial neurons, we could say: We just have to connect the "nodes" of the feature maps of layer 5 in our special CNN with the nodes of an input layer of a MLP. As a MLP has a flat input layer we need to prepare 9x64 = 576 receiving "nodes" there. We would use weights with a value of "1.0" along these special connections.

Mathematically, this approach can be expressed in terms of a "flattening" operation on the tensor data produced by the the last Conv data. In Numpy terms: We need to reshape the multidimensional tensor containing the values across the stack of maps at the last Conv2D layer into a long 1D array (= a vector).

From a more general perspective we could say: Feeding the output of the Conv part of our CNN into a MLP for classification is quite similar to what we did when we pre-processed the MNIST data by an unsupervised cluster detection algorithm; also there we used the resulting data as input to an MLP. There is one big difference, however:

The optimization of the network's weights during training requires a BW propagation of error terms (more precisely: derivatives of the CNN's loss function) across the MLP AND the convolutional and pooling layers. Error BW propagation should not be stopped at the MLP's input layer: It has to move from the output layer of the MLP back to the MLP's input layer and from there to the convolutional and pooling layers. Remember that suitable filter kernels have to be found during (supervised) training.

If you read my PDF on the error back propagation for a MLP
PDF on the math behind Error Back_Propagation
and think a bit about its basic recipes and results you quickly see that the "input layer" of the MLP is no barrier to error back propagation: The "deltas" discussed in the PDF can be back-propagated right down to the MLP's input layer. Then we just apply the chain rule again. The partial derivatives at the nodes of the input layer with respect to their input values are just "1", because the activation function there is the identity function. The "weights" between the last Conv layer and the input layer of the MLP are no free parameters - we do not need to care about them. And then everything goes its normal way - we apply chain rule after chain rule for all nodes of the maps to determine the gradients of the CNN's loss function with respect to the weights there. But you need not think about the details - Keras and TF2 will take proper care about everything ...

But, you should always keep the following in mind: Whatever we discuss in terms of layers and nodes - in a CNN these are only fictitious interpretations of a series of mathematical operations on tensor data. Not less, not more ..,. Nodes and layers are just very helpful (!) illustrations of non-cyclic graphs of mathematical operations. KI on the level of my present discussion (MLPs, CNNs) "just" corresponds to algorithms which emerge out of a specific deterministic approach to solve an optimization problem.

Using Tensorflow 2 and Keras

Let us now turn to coding. To be able to use a Nvidia GPU we need a Cuda/Cudnn installation and a working Tensorflow backend for Keras. I have already described the installation of CUDA 10.2 and CUDNN on an Opensuse Linux system in some detail in the article Getting a Keras based MLP to run with Cuda 10.2, Cudnn 7.6 and TensorFlow 2.0 on an Opensuse Leap 15.1 system. You can follow the hints there. In case you run into trouble on your Linux distribution try everything with Cuda 10.1.

Some more hints: TF2 in version 2.2 can be installed by the Pip-mechanism in your virtual Python environment ("pip install --upgrade tensorflow"). TF2 contains already a special Keras version - which is the one we are going to use in our upcoming experiment. So, there is no need to install Keras separately with "pip". Note also that, in contrast to TF1, it is NOT necessary to install a separate package "tensorflow-gpu". If all these things are new to you: You find some articles on creating an adequate ML test and development environment based on Python/PyDev/Jupyter somewhere else in this blog.

Imports and settings for CPUs/GPUs

We shall use a Jupyter notebook to perform the basic experiments; but I recommend strongly to consolidate your code in Python files of an Eclipse/PyDev environment afterwards. Before you start your virtual Python environment from a Linux shell you should set the following environment variables:

$>export OPENBLAS_NUM_THREADS=4 # or whatever is reasonable for your CPU (but do not use all CPU cores and/or hyper threads                            
$>export OMP_NUM_THREADS=4                                
$>export TF_XLA_FLAGS=--tf_xla_cpu_global_jit
$>export XLA_FLAGS=--xla_gpu_cuda_data_dir=/usr/local/cuda
$>source bin/activate                                     
(ml_1) $> jupyter notebook

Required Imports

The following commands in a first Jupyter cell perform the required library imports:

import numpy as np
import scipy
import time 
import sys 
import os

import tensorflow as tf
from tensorflow import keras as K
from tensorflow.python.keras import backend as B 
from keras import models
from keras import layers
from keras.utils import to_categorical
from keras.datasets import mnist
from tensorflow.python.client import device_lib

from sklearn.preprocessing import StandardScaler

Do not ignore the statement "from tensorflow.python.keras import backend as B"; we need it later.

The "StandardScaler" of Scikit-Learn will help us to "standardize" the MNIST input data. This is a step which you should know already from MLPs ... You can, of course, also experiment with different normalization procedures. But in my opinion using the "StandardScaler" is just convenient. ( I assume that you already have installed scikit-learn in your virtual Python environment).

Settings for CPUs/GPUs

With TF2 the switching between CPU and GPU is a bit of a mess. Not all new parameters and their settings work as expected. As I have explained in the article on the Cuda installation named above, I, therefore, prefer to an old school, but reliable TF1 approach and use the compatibility interface:

#gpu = False 
gpu = True
if gpu: 
    GPU = True;  CPU = False; num_GPU = 1; num_CPU = 1
else: 
    GPU = False; CPU = True;  num_CPU = 1; num_GPU = 0

config = tf.compat.v1.ConfigProto(intra_op_parallelism_threads=6,
                        inter_op_parallelism_threads=1, 
                        allow_soft_placement=True,
                        device_count = {'CPU' : num_CPU,
                                        'GPU' : num_GPU}, 
                        log_device_placement=True

                       )
config.gpu_options.per_process_gpu_memory_fraction=0.35
config.gpu_options.force_gpu_compatible = True
B.set_session(tf.compat.v1.Session(config=config))

We are brave and try our first runs directly on a GPU. The statement "log_device_placement" will help us to get information about which device - CPU or GP - is actually used.

Loading and preparing MNIST data

We prepare a function which loads and prepares the MNIST data for us. The statements reflect more or less what we did with the MNIST dat when we used them for MLPs.

  
# load MNIST 
# **********
def load_Mnist():
    mnist = K.datasets.mnist
    (X_train, y_train), (X_test, y_test) = mnist.load_data()

    #print(X_train.shape)
    #print(X_test.shape)

    # preprocess - flatten 
    len_train =  X_train.shape[0]
    len_test  =  X_test.shape[0]
    X_train = X_train.reshape(len_train, 28*28) 
    X_test  = X_test.reshape(len_test, 28*28) 

    #concatenate
    _X = np.concatenate((X_train, X_test), axis=0)
    _y = np.concatenate((y_train, y_test), axis=0)

    _dim_X = _X.shape[0]

    # 32-bit
    _X = _X.astype(np.float32)
    _y = _y.astype(np.int32)

    # normalize  
    scaler = StandardScaler()
    _X = scaler.fit_transform(_X)

    # mixing the training indices - MUST happen BEFORE encoding
    shuffled_index = np.random.permutation(_dim_X)
    _X, _y = _X[shuffled_index], _y[shuffled_index]

    # split again 
    num_test  = 10000
    num_train = _dim_X - num_test
    X_train, X_test, y_train, y_test = _X[:num_train], _X[num_train:], _y[:num_train], _y[num_train:]

    # reshape to Keras tensor requirements 
    train_imgs = X_train.reshape((num_train, 28, 28, 1))
    test_imgs  = X_test.reshape((num_test, 28, 28, 1))
    #print(train_imgs.shape)
    #print(test_imgs.shape)

    # one-hot-encoding
    train_labels = to_categorical(y_train)
    test_labels  = to_categorical(y_test)
    #print(test_labels[4])
    
    return train_imgs, test_imgs, train_labels, test_labels

if gpu:
    with tf.device("/GPU:0"):
        train_imgs, test_imgs, train_labels, test_labels= load_Mnist()
else:
    with tf.device("/CPU:0"):
        train_imgs, test_imgs, train_labels, test_labels= load_Mnist()

 
Some comments:

  • Normalization and shuffling: The "StandardScaler" is used for data normalization. I also shuffled the data to avoid any pre-ordered sequences. We know these steps already from the MLP code we built in another article series.
  • Image data in tensor form: Something, which is different from working with MLPs is that we have to fulfill some requirements regarding the form of input data. From the last article we know already that our data should have a tensor compatible form; Keras expects data from us which have a certain shape. So, no flattening of the data into a vector here as we were used to with MLPs. For images we, instead, need the width, the height of our images in terms of pixels and also the depth (here 1 for gray-scale images). In addition the data samples are to be indexed along the first tensor axis.
    This means that we need to deliver a 4-dimensional array corresponding to a TF tensor of rank 4. Keras/TF2 will do the necessary transformation from a Numpy array to a TF2 tensor automatically for us. The corresponding Numpy shape of the required array is:
    (samples, height, width, depth)
    Some people also use the term "channels" instead of "depth". In the case of MNIST we reshape the input array - "train_imgs" to (num_train, 28, 28, 1), with "num_train" being the number of samples.
  • The use of the function "to_categorical()", more precisely "tf.keras.utils.to_categorical()", corresponds to a one-hot-encoding of the target data. All these concepts are well known from our study of MLPs and MNIST. Keras makes life easy regarding this point ...
  • The statements "with tf.device("/GPU:0"):" and "with tf.device("/CPU:0"):" delegate the execution of (suitable) code on the GPU or the CPU. Note that due to the Python/Jupyter environment some code will of course also be executed on the CPU - even if you delegated execution to the GPU.

If you activate the print statements the resulting output should be:

(60000, 28, 28)
(10000, 28, 28)
(60000, 28, 28, 1)
(10000, 28, 28, 1)
[0. 0. 0. 0. 0. 0. 0. 1. 0. 0.]

The last line proves the one-hot-encoding.

The CNN architecture - and Keras' layer API

Now, we come to a central point: We need to build the 5 central layers of our CNN-architecture. When we build our own MLP code we used a special method to build the different weight arrays, which represented the number of nodes via the array dimensions. A simple method was sufficient as we had no major differences between layers. But with CNNs we have to work with substantially different types of layers. So, how are layers to be handled with Keras?

Well, Keras provides a full layer API with different classes for a variety of layers. You find substantial information on this API and different types of layers at
https://keras.io/api/layers/.

The first section which is interesting for our experiment is https://keras.io/api/ layers/ convolution_layers/ convolution2d/.
You do not need to read much to understand that this is exactly what we need for the "convolutional layers" of our simple CNN model. But how do we instantiate the Conv2D class such that the output works seamlessly together with other layers?

Keras makes our life easy again. All layers are to be used in a purely sequential order. (There are much more complicated layer topologies you can build with Keras! Oh, yes ...). Well, you guess it: Keras offers you a model API; see:
https://keras.io/api/models/.

And there we find a class for a "sequential model" - see https://keras.io/api/ models/sequential/. This class offers us a method "add()" to add layers (and create an instance of the related layer class).

The only missing ingredient is a class for a "pooling" layer. Well, you find it in the layer API documentation, too. The following image depicts the basic structure of our CNN (see the left side of the drawing), as we designed it (see the list above):

Keras code for the Conv and pooling layers

The convolutional part of the CNN can be set up by the following commands:

Convolutional part of the CNN

# Sequential layer model of our CNN
# ***********************************

# Build the Conv part of the CNN
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

# Choose the activation function for the Conv2D layers 
conv_act_func = 1
li_conv_act_funcs = ['sigmoid', 'relu', 'elu', 'tanh']
cact = li_conv_act_funcs[conv_act_func]

# Build the Conv2D layers 
cnn = models.Sequential()
cnn.add(layers.Conv2D(32, (3,3), activation=cact, input_shape=(28,28,1)))
cnn.add(layers.MaxPooling2D((2,2)))
cnn.add(layers.Conv2D(64, (3,3), activation=cact))
cnn.add(layers.MaxPooling2D((2,2)))
cnn.add(layers.Conv2D(64, (3,3), activation=cact))

Easy, isn't it? The nice thing about Keras is that it cares about the required tensor ranks and shapes itself; in a sequential model it evaluates the output of a already defined layer to guess the shape of the tensor data entering the next layer. Thus we have to define an "input_shape" only for data entering the first Conv2D layer!

The first Conv2D layer requires, of course, a shape for the input data. We must also tell the layer interface how many filters and "feature maps" we want to use. In our case we produce 32 maps by first Conv2D layer and 64 by the other two Conv2D layers. The (3x3)-parameter defines the filter area size to be covered by the filter kernel: 3x3 pixels. We define no "stride", so a stride of 1 is automatically used; all 3x3 areas lie close to each other and overlap each other. These parameters result in 32 maps of size 26x26 after the first convolution. The size of the maps of the other layers are given in the layer list at the beginning of this article.

In addition you saw from the code that we chose an activation function via an index of a Python list of reasonable alternatives. You find an explanation of all the different activation functions in the literature. (See also: wikipedia: Activation function). The "sigmoid" function should be well known to you already from my other article series on a MLP.

Now, we have to care about the MLP part of the CNN. The code is simple:

MLP part of the CNN

# Build the MLP part of the CNN
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

# Choose the activation function for the hidden layers of the MLP 
mlp_h_act_func = 0
li_mlp_h_act_funcs = ['relu', 'sigmoid', 'tanh']
mhact = li_mlp_h_act_funcs[mlp_h_act_func]

# Choose the output function for the output layer of the MLP 
mlp_o_act_func = 0
li_mlp_o_act_funcs = ['softmax', 'sigmoid']
moact = li_mlp_o_act_funcs[mlp_o_act_func]

# Build the MLP layers 
cnn.add(layers.Flatten())
cnn.add(layers.Dense(70, activation=mhact))
#cnn.add(layers.Dense(30, activation=mhact))
cnn.add(layers.Dense(10, activation=moact))

This all is very straight forward (with the exception of the last statement). The "Flatten"-layer corresponds to the MLP's inout layer. It just transforms the tensor output of the last Conv2D layer into the flat form usable for the first "Dense" layer of the MLP. The first and only "Dense layer" (MLP hidden layer) builds up connections from the flat MLP "input layer" and associates it with weights. Actually, it prepares a weight-tensor for a tensor-operation on the output data of the feeding layer. Dense means that all "nodes" of the previous layer are connected to the present layer's own "nodes" - meaning: setting the right dimensions of the weight tensor (matrix in our case). As a first trial we work with just one hidden layer. (We shall later see that more layers will not improve accuracy.)

I choose the output function (if you will: the activation function of the output layer) as "softmax". This gives us a probability distribution across the classification categories. Note that this is a different approach compared to what we have done so far with MLPs. I shall comment on the differences in a separate article when I find the time for it. At this point I just want to indicate that softmax combined with the "categorical cross entropy loss" is a generalized version of the combination "sigmoid" with "log loss" as we used it for our MLP.

Parameterization

The above code for creating the CNN would work. However, we want to be able to parameterize our simple CNN. So we include the above statements in a function for which we provide parameters for all layers. A quick solution is to define layer parameters as elements of a Python list - we then get one list per layer. (If you are a friend of clean code design I recommend to choose a more elaborated approach; inject just one parameter object containing all parameters in a structured way. I leave this exercise to you.)

We now combine the statements for layer construction in a function:

  
# Sequential layer model of our CNN
# ***********************************

# just for illustration - th ereal parameters are fed later 
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
li_conv_1   = [32, (3,3), 0] 
li_conv_2   = [64, (3,3), 0] 
li_conv_3   = [64, (3,3), 0] 
li_Conv     = [li_conv_1, li_conv_2, li_conv_3]
li_pool_1   = [(2,2)]
li_pool_2   = [(2,2)]
li_Pool     = [li_pool_1, li_pool_2]
li_dense_1  = [70, 0]
li_dense_2  = [10, 0]
li_MLP      = [li_dense_1, li_dense_2]
input_shape = (28,28,1)

# important !!
# ~~~~~~~~~~~
cnn = None

def build_cnn_simple(li_Conv, li_Pool, li_MLP, input_shape ):

    # activation functions to be used in Conv-layers 
    li_conv_act_funcs = ['relu', 'sigmoid', 'elu', 'tanh']
    # activation functions to be used in MLP hidden layers  
    li_mlp_h_act_funcs = ['relu', 'sigmoid', 'tanh']
    # activation functions to be used in MLP output layers  
    li_mlp_o_act_funcs = ['softmax', 'sigmoid']

    
    # Build the Conv part of the CNN
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    num_conv_layers = len(li_Conv)
    num_pool_layers = len(li_Pool)
    if num_pool_layers != num_conv_layers - 1: 
        print("\nNumber of pool layers does not fit to number of Conv-layers")
        sys.exit()
    rg_il = range(num_conv_layers)

    # Define a sequential model 
    cnn = models.Sequential()

    for il in rg_il:
        # add the convolutional layer 
        num_filters = li_Conv[il][0]
        t_fkern_size = li_Conv[il][1]
        cact        = li_conv_act_funcs[li_Conv[il][2]]
        if il==0:
            cnn.add(layers.Conv2D(num_filters, t_fkern_size, activation=cact, input_shape=input_shape))
        else:
            cnn.add(layers.Conv2D(num_filters, t_fkern_size, activation=cact))
        
        # add the pooling layer 
        if il < num_pool_layers:
            t_pkern_size = li_Pool[il][0]
            cnn.add(layers.MaxPooling2D(t_pkern_size))
            

    # Build the MLP part of the CNN
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    num_mlp_layers = len(li_MLP)
    rg_im = range(num_mlp_layers)

    cnn.add(layers.Flatten())

    for im in rg_im:
        # add the dense layer 
        n_nodes = li_MLP[im][0]
        if im < num_mlp_layers - 1:  
            m_act   =  li_mlp_h_act_funcs[li_MLP[im][1]]
        else: 
            m_act   =  li_mlp_o_act_funcs[li_MLP[im][1]]
        cnn.add(layers.Dense(n_nodes, activation=m_act))

    return cnn 

 

We return the model "cnn" to be able to use it afterwards.

How many parameters does our CNN have?

The layers contribute with the following numbers of weight parameters:

  • Layer 1: (32 x (3x3)) + 32 = 320
  • Layer 3: 32 x 64 x (3x3) + 64 = 18496
  • Layer 5: 64 x 64 x (3x3) + 64 = 36928
  • MLP : (576 + 1) x 70 + (70 + 1) x 10 = 41100

Making a total of 96844 weight parameters. Our standard MLP discussed in another article series had (784+1) x 70 + (70 + 1) x 30 + (30 +1 ) x 10 = 57390 weights. So, our CNN is bigger and the CPU time to follow and calculate all the partial derivatives will be significantly higher. So, we should definitely expect some better classification data, shouldn't we?

Compilation

Now comes a thing which is necessary for models: We have not yet defined the loss function and the optimizer or a learning rate. For the latter Keras can choose a proper value itself - as soon as it knows the loss function. But we should give it a reasonable loss function and a suitable optimizer for gradient descent. This is the main purpose of the "compile()"-function.

cnn.compile(optimizer='rmsprop', loss='categorical_crossentropy', metrics=['accuracy'])

Although TF2 can already analyze the graph of tensor operations for partial derivatives, it cannot guess the beginning of the chain rule sequence.

As we have multiple categories "categorial_crossentropy" is a good choice for the loss function. We should also define which optimized gradient descent method is used; we choose "rmsprop" - as this method works well in most cases. A nice introduction is given here: towardsdatascience: understanding-rmsprop-faster-neural-network-learning-62e116fcf29a. But see the books mentioned in the last article on "rmsprop", too.

Regarding the use of different metrics for different tasks see machinelearningmastery.com / custom-metrics-deep-learning-keras-python/. In case of a classification problem, we are interested in the categorical "accuracy". A metric can be monitored during training and will be recorded (besides aother data). We can use it for plotting information on the training process (a topic of the next article).

Training

Training is done by a function model.fit() - here: cnn.fit(). This function accepts a variety of parameters explained here: https://keras.io/ api/ models/ model_training_apis/ #fit-method.

We now can combine compilation and training in one function:

# Training 
def train( cnn, build=False, train_imgs, train_labels, reset, epochs, batch_size, optimizer, loss, metrics,
           li_Conv, li_Poo, li_MLP, input_shape ):
    if build:
        cnn = build_cnn_simple( li_Conv, li_Pool, li_MLP, input_shape)
        cnn.compile(optimizer=optimizer, loss=loss, metrics=metrics)        
        cnn.save_weights('cnn_weights.h5') # save the initial weights 
    # reset weights
    if reset and not build:
        cnn.load_weights('cnn_weights.h5')
    start_t = time.perf_counter()
    cnn.fit(train_imgs, train_labels, epochs=epochs, batch_size=batch_size, verbose=1, shuffle=True) 
    end_t = time.perf_counter()
    fit_t = end_t - start_t
    return cnn, fit_t  # we return cnn to be able to use it by other functions

Note that we save the initial weights to be able to load them again for a new training - otherwise Keras saves the weights as other model data after training and continues with the last weights found. The latter can be reasonable if you want to continue training in defined steps. However, in our forthcoming tests we repeat the training from scratch.

Keras offers a "save"-model and methods to transfer data of a CNN model to files (in two specific formats). For saving weights the given lines are sufficient. Hint: When I specify no path to the file "cnn_weights.h5" the data are - at least in my virtual Python environment - saved in the directory where the notebook is located.

First test

In a further Jupyter cell we place the following code for a test run:

  
# Perform a training run 
# ********************
build = False     
if cnn == None:
    build = True
batch_size=64
epochs=5
reset = True # we want training to start again with the initial weights

optimizer='rmsprop' 
loss='categorical_crossentropy'
metrics=['accuracy']

li_conv_1   = [32, (3,3), 0] 
li_conv_2   = [64, (3,3), 0] 
li_conv_3   = [64, (3,3), 0] 
li_Conv     = [li_conv_1, li_conv_2, li_conv_3]
li_pool_1   = [(2,2)]
li_pool_2   = [(2,2)]
li_Pool     = [li_pool_1, li_pool_2]
li_dense_1  = [70, 0]
li_dense_2  = [10, 0]
li_MLP      = [li_dense_1, li_dense_2]
input_shape = (28,28,1)

try: 
    if gpu:
        with tf.device("/GPU:0"):
            cnn, fit_time = train( cnn, build, train_imgs, train_labels, 
                                   reset, epochs, batch_size, optimizer, loss, metrics, 
                                   li_Conv, li_Pool, li_MLP, input_shape)
        print('Time_GPU: ', fit_time)  
    else:
        with tf.device("/CPU:0"):
            cnn, fit_time = train( cnn, build, train_imgs, train_labels, 
                                   reset, epochs, batch_size, optimizer, loss, metrics, 
                                   li_Conv, li_Pool, li_MLP, input_shape)
        print('Time_CPU: ', fit_time)  
except SystemExit:
    print("stopped due to exception")

You recognize the parameterization of our train()-function. What results do we get ?

Epoch 1/5
60000/60000 [==============================] - 4s 69us/step - loss: 0.1551 - accuracy: 0.9520
Epoch 2/5
60000/60000 [==============================] - 4s 69us/step - loss: 0.0438 - accuracy: 0.9868
Epoch 3/5
60000/60000 [==============================] - 4s 68us/step - loss: 0.0305 - accuracy: 0.9907
Epoch 4/5
60000/60000 [==============================] - 4s 69us/step - loss: 0.0227 - accuracy: 0.9931
Epoch 5/5
60000/60000 [==============================] - 4s 69us/step - loss: 0.0184 - accuracy: 0.9948
Time_GPU:  20.610678611003095

 

And a successive check on the test data gives us:

We can ask Keras also for a description of the model:

Accuracy at the 99% level

We got an accuracy on the test data of 99%! With 5 epochs in 20 seconds - on my old GPU.
This leaves us a very good impression - on first sight ...

Conclusion

We saw today that it is easy to set up a CNN. We used a simple MLP to solve the problem of classification; the data to its input layer are provided by the output of the last convolutional layer. The tensor there has just to be "flattened".

The level of accuracy reached is impressing. Well, its also a bit frustrating when we think about the efforts we put into our MLP, but we also get a sense for the power and capabilities of CNNs.

In the next article we will care a bit about plotting. We at least want to see the same data which we plotted at the end of our MLP tests.

A simple CNN for the MNIST datasets – I – CNN basics

In a previous article series
A simple Python program for an ANN to cover the MNIST dataset – I – a starting point
we have played with a Python/Numpy code, which created a configurable and trainable "Multilayer Perceptron" [MLP] for us. See also
MLP, Numpy, TF2 – performance issues – Step I – float32, reduction of back propagation
for ongoing code and performance optimization.

A MLP program is useful to study multiple topics in Machine Learning [ML] on a basic level. However, MLPs with dense layers are certainly not at the forefront of ML technology - though they still are fundamental bricks in other more complicated architectures of "Artifical Neural Networks" [ANNs]. During my MLP experiments I became sufficiently acquainted with Python, Jupyter and matplotlib to make some curious first steps into another field of Machine Learning [ML] now: "Convolutional Neural Networks" [CNNs].

CNNs on my level as an interested IT-affine person are most of all fun. Nevertheless, I quickly found out that a somewhat systematic approach is helpful - especially if you later on want to use the Tensorflow's API and not only Keras. When I now write about some experiments I did and do I summarize my own biased insights and sometimes surprises. Probably there are other hobbyists as me out there who also fight with elementary points in the literature and practical experiments. Books alone are not enough ... I hope to deliver some practical hints for this audience. The present articles are, however, NOT intended for ML and CNN experts. Experts will almost certainly not find anything new here.

Although I address CNN-beginners I assume that people who stumble across this article and want to follow me through some experiments have read something about CNNs already. You should know fundamentals about filters, strides and the basic principles of convolution. I shall comment on all these points but I shall not repeat the very basics. I recommend to read relevant chapters in one of the books I recommend at the end of this article. You should in addition have some knowledge regarding the basic structure and functionality of a MLP as well as "gradient descent" as an optimization technique.

The objective of this introductory mini-series is to build a first simple CNN, to apply it to the MNIST dataset and to visualize some of the elementary "features" the CNN detects in the images of handwritten digits. We shall use Keras (with the Tensorflow 2.2 backend and CUDA 10.2) for this purpose. And, of course, a bit of matplotlib and Python/Numpy, too. We are working with MNIST images in the first place - although CNNs can be used to analyze other types of input data. After we have covered the simple standard MNIST image set, we shall also work a bit with the so called "MNIST fashion" set.

But in this article I start with some introductory words on the structure of CNNs and the task of its layers. We shall use the information later on as a reference. In the second article we shall set up and test a simple version of a CNN. Further articles will then concentrate on visualizing what a trained CNN reacts to and how it modifies and analyzes the input data on its layers.

Why CNNs?

When we studied an MLP in combination with the basic MNIST dataset of handwritten digits we found that we got an improvement in accuracy (for the same setup of dense layers) when we pre-processed the data to find "clusters" in the image data before training. Such a process corresponds to detecting parts of an MNIST image with certain gray-white pixel constellations. We used Scikit-Learn's "MiniBatchKMeans" for this purpose.

We saw that the identification of 40 to 70 cluster areas in the images helped the MLP algorithm to analyze the MNIST data faster and better than before. Obviously, training the MLP with respect to combinations of characteristic sub-structures of the different images helped us to classify them as representations of digits. This leads directly to the following question:

What if we could combine the detection of sub-structures in an image with the training process of an ANN?

CNNs are the answer! They are designed to detect elementary structures or features in image data (and other data) systematically. In addition they are enabled to learn something about characteristic compositions of such elementary features during training. I.e., they detect more abstract and composite features specific for the appearance of certain objects within an image. We speak of a "feature hierarchy", which a CNN can somehow grasp and use - e.g. for classification tasks.

While a MLP must learn about pixel constellations and their relations on the whole image area, CNNs are much more flexible and even reusable. They identify and remember elementary sub-structures independent of the exact position of such features within an image. They furthermore learn "abstract concepts" about depicted objects via identifying characteristic and complex composite features on a higher level.

This simplified description of the astonishing capabilities of a CNN indicates that its training and learning is basically a two-fold process:

  • Detecting elementary structures in an image (or other structured data sets) by filtering and extracting patterns within relatively small image areas. We shall call these areas "filter areas".
  • Constructing abstract characteristic features out of the elementary filtered structural elements. This corresponds to building a "hierarchy" of significant features for the classification of images or of distinguished objects or of the positions of such objects within an image.

Now, if you think about the MNIST digit data we understand intuitively that written digits represent some abstract concepts like certain combinations of straight vertical and horizontal line elements, bows and line crossings. The recognition of certain feature combinations of such elementary structures would of course be helpful to recognize and classify written digits better - especially when the recognition of the combination of such features is independent of their exact position on an image.

Filters, kernels and feature maps

An important concept behind CNNs is the systematic application of (various) filters (described and defined by so called "kernels").

A "filter" defines a kind of masking pixel area of limited small size (e.g. 3x3 pixels). A filter combines weighted output values at neighboring nodes of a input layer in a specific defined way. It processes the offered information in a defined area always in the same fixed way - independent of where the filter area is exactly placed on the (bigger) image (or a processed version of it). We call a processed version of an image a "map".

A specific type of CNN layer, called a "Convolution Layer" [Conv layer], and a related operational algorithm let a series of such small masking areas cover the complete surface of an image (or a map). The first Conv layer of a CNN filters the information of the original image information via a multitude of such masking areas. The masks can be arranged overlapping, i.e. they can be shifted against each other by some distance along their axes. Think of the masking filter areas as a bunch of overlapping tiles covering the image. The shift is called stride.

The "filter" mechanism (better: the mathematical recipe) of a specific filter remains the same for all of its small masking areas covering the image. A specific filter emphasizes certain parts of the original information and suppresses other parts in a defined way. If you combine the information of all masks you get a new (filtered) representation of the image - we speak of a "feature map" - sometimes with a somewhat smaller size than the original image (or map) the filter is applied to. The blending of the original data with a filtering mask create a "feature map", i.e. a filtered view onto the input data. The blending process is called "convolution" (due to the related mathematical operations).

The picture below sketches the basic principle of a 3x3-filter which is applied with a constant stride of 2 along each axis of the image:

Convolution is not so complicated as it sounds. It means: You multiply the original data values in the covered small area by factors defined in the filter's kernel and add the resulting values up to get a a distinct value at a defined position inside the map. In the given example with a stride of 2 we get a resulting feature map of 4x4 out of a original 9x9 (image or map).

Note that a filter need not be defined as a square. It can have a rectangular (n x m) shape with (n, m) being integers. (In principle we could also think of other tile forms as e.g. hexagons - as long as they can seamlessly cover a defined plane. Interesting, although I have not seen a hexagon based CNN in the literature, yet).

A filter's kernel defines factors used in the convolution operation - one for each of the (n x m) defined points in the filter area.
Note also that filters may have a "depth" property when they shall be applied to three-dimensional data sets; we may need a depth when we cover colored images (which require 3 input layers). But let us keep to flat filters in this introductory discussion ...

Now we come to a central question: Does a CNN Conv layer use just one filter? The answer is: No!

A Conv layer of a CNN you allows for the construction of multiple different filters. Thus we have to deal with a whole bunch of filters per each convolutional layer. E.g. 32 filters for the first convolutional layer and 64 for the second and 128 for the third. The outcome of the respective filter operations is the creation is of equally many so called "feature maps" (one for each filter) per convolutional layer. With 32 different filters on a Conv layer we would thus build 32 maps at this layer.

This means: A Conv layer has a multitude of sub-layers called "feature maps" which result of the application of different filters on previous image or map data.

You may have guessed already that the next step of abstraction is:
You can apply filters also to "feature maps" of previous filters, i.e. you can chain convolutions. Thus, feature maps are either connected to the image (1st Conv layer) or to the feature maps of a previous layer.

By using a sequence of multiple Conv layers you cover growing areas of the original image. Everything clear? Probably not ...

Filters and their related weights are the end products of the training and optimization of a CNN!

When I first was confronted with the concept of filters, I got confused because many authors only describe the basic technical details of the "convolution" mechanism. They explain with many words how a filter and its kernel work when the filtering area is "moved" across the surface of an image. They give you pretty concrete filter examples; very popular are straight lines and crosses indicated by "ones" as factors in the filter's kernel and zeros otherwise. And then you get an additional lecture on strides and padding. You have certainly read various related passages in books about ML and/or CNNs. A pretty good example for this "explanation" is the (otherwise interesting and helpful!) book of Deru and Ndiaye (see the bottom of this article). I refer to the introductory chapter 3.5.1 on CNN architectures.

Well, the technical procedure is pretty easy to understand from drawings as given above - the real question that nags in your brain is:

"Where the hell do all the different filter definitions come from?"

What many authors forget is a central introductory sentence for beginners:

A filter is not given a priori. Filters (and their kernels) are systematically constructed and build up during the training of a CNN; filters are the end products of a learning and optimization process every CNN must absolve.

This means: For a given problem or dataset you do not know in advance what the "filters" (and their defining kernels) will look like after training (aside of their pixel dimensions already fixed by the CNN's layer definitions). The "factors" of a filter used in the convolution operation are actually weights, whose final values are the outcome of a learning process. Just as in MLPs ...

Noting is really "moved" ...

Another critical point is the somewhat misleading analogy of "moving" a filter across an image's or map's pixel surface. Nothing is ever actually "moved" in a CNN's algorithm. All masks are already in place when the convolution operations are performed:

Every element of a specific e.g. 3x3 kernel corresponds to "factors" for the convolution operation. What are these factors? Again: They are nothing else but weights - in exactly the same sense as we used them in MLPs. A filter kernel represents a set of weight-values to be multiplied with original output values at the "nodes" in other layers or maps feeding input to the nodes of the present map.

Things become much clearer if you imagine a feature map as a bunch of arranged "nodes". Each node of a map is connected to (n x m) nodes of a previous set of nodes on a map or layer delivering input to the Conv layer's maps.

Let us look at an example. The following drawing shows the connections from "nodes" of a feature map "m" of a Conv layer L_(N+1) to nodes of two different maps "1" and "2" of Conv layer L_N. The stride for the kernels is assumed to be just 1.

In the example the related weights are described by two different (3x3) kernels. Note, however, that each node of a specific map uses the same weights for connections to another specific map or sub-layer of the previous (input) layer. This explains the total number of weights between two sequential Conv layers - one with 32 maps and the next with 64 maps - as (64 x 32 x 9) + 64 = 18496. The 64 extra weights account for bias values per map on layer L_(N+1). (As all nodes of a map use fixed bunches of weights, we only need exactly one bias value per map).

Note also that a stride is defined for the whole layer and not per map. Thus we enforce the same size of all maps in a layer. The convolutions between a distinct map and all maps of the previous layer L_N can be thought as operations performed on a column of stacked filter areas at the same position - one above the other across all maps of L_N. See the illustration below:

The weights of a specific kernel work together as an ensemble: They condense the original 3x3 pixel information in the filtered area of the connected input layer or a map to a value at one node of the filter specific feature map. Please note that there is a bias weight in addition for every map; however, at all masking areas of a specific filter the very same 9 weights are applied. See the next drawing for an illustration of the weight application in our example for fictitious node and kernel values.

A CNN learns the appropriate weights (= the filter definitions) for a given bunch of images via training and is guided by the optimization of a loss function. You know these concepts already from MLPs ...

The difference is that the ANN now learns about appropriate "weight ensembles" - eventually (!) working together as a defined convolutional filter between different maps of neighboring Conv (and/or sampling ) Layers. (For sampling see a separate paragraph below.)

The next picture illustrates the column like convolution of information across the identically positioned filter areas across multiple maps of a previous convolution layer:

The fact that the weight ensemble of a specific filter between maps is always the same, explains, by the way, the relatively (!) small number of weight parameters in deep CNNS.

Intermediate summary: The weights, which represent the factors used by a specific filter operation called convolution, are defined during a training process. The filter, its kernel and the respective weight values are the outcome of a mathematical optimization process - mostly guided by gradient descent.

Activation functions

As in MLPs each Conv layer has an associated "activation function" which is applied at each node of all maps after the resulting values of the convolution have been calculated as the nodes input. The output then feeds the connections to the next layer. In CNNs for image handling often "Relu" or "Selu" are used as activation functions - and not "sigmoid" which we applied in our personal MLP code.

Tensors

The above drawings indicate already that we need to arrange the data (of an image) and also the resulting map data in an organized way to be able to apply the required convolutional multiplications and summations the right way.

An colored image is basically a regular 3 dimensional structure with a width "w" (number of pixels along the x-axis), a height "h" (number of pixels along the y-axis) and a (color) depth "d" (d=3 for RGB colors).
If you represent the color value at each pixel and RGB-layer by a float you get a bunch of w x h x d float values which we can organize and index in a 3 dimensional Numpy array. Mathematically such well organized arrays with a defined number of axes (rank), a set of numbers describing the dimension along each axis (shape), a data-type, possible operations (and invariance aspects) define an abstract object called a "tensor". Colored image data can be arranged in 3-dimensional tensors; gray colored images in a pseudo 3D-tensor which has a shape of (n, m, 1). (Keras and Tensorflow want to get imagedata in form of 2D tensors).

Now the important point is: The output data of Conv-layers and their feature maps also represent tensors. A bunch of 32 maps with a defined width and height defines data of a 3D-tensor.

You can imagine each value of such a tensor as the input or output given at a specific node in a layer with a 3-dimensional sub-structure. (In other even more complex data structures than images we would other multi-dimensional data structures.) The weights of a filter kernel describe the connections of the nodes of a feature map on a layer L_N to a specific map of a previous layer. Weights, actually, also define elements of a tensor.

The forward- and backward-propagation operations performed throughout such a complex net during training thus correspond to certain tensor-operations - i.e. generalized versions of the np.dot()-product we got to know in MLPs.
You understood already that e.g strides are important. But you do not need to care about details - Keras and Tensorflow will do the job for you! If you want to read a bit look a the documentation of the TF function "tf.nn.conv2d()".

When we later on train with mini-batches of input data (i.e. batches of images) we get yet another dimension of our tensors. This batch dimension can - quite similar to MLPs - be used to optimize the tensor operations in a vectorized way. See my series on MLPs.

Chained convolutions cover growing areas of the original image

Two sections above I characterized the training of a CNN as a two-fold procedure. From the first drawing it is relatively easy to understand how we get to grasp tiny sub-structures of an image: Just use filters with small kernel sizes!

Fine, but there is probably a second question already arising in your mind:

By what mechanism does a CNN find or recognize a hierarchy of features?

One part of the answer is: Chain convolutions!

Let us assume a first convolutional layer with filters having a stride of 1 and a (3x3) kernel. We get maps with a shape of (26, 26) on this layer. The next Conv layer shall use a (4x4) kernel and also a stride of 1; then we get maps with a shape of (23, 23). A node on the second layer covers (6x6)-arrays on the original image. Two neighboring nodes a total area of (7x7). The individual (6x6)-areas of course overlap.

With a stride of 2 on each Conv-layer the corresponding areas on the original image are (7x7) and (11x11).

So a stack of consecutive (sequential) Conv-layers covers growing areas on the original image. This supports the detection of a feature hierarchy.

However: Small strides require a relatively big number of sequential Conv-layers (for 3x3 kernels and stride 2) at least 13 layers to eventually cover the full image area.

Even if we would not enlarge the number of maps beyond 128 with growing layer number, we would get

(32 x 9 + 32) + (64 x 32 +64) + (128 x 64 + 128) + 10 x (128 x 128 + 128) = 320 + 18496 + 73856 + 10*147584 = 1.568 million weight parameters

to take care of!

This number has to be multiplied by the number of images in a mini-batch - e.g. 500. And - as we know from MLPs we have to keep all intermediate output results in RAM to accelerate the BW propagation for the determination of gradients. Too many data and parameters for the analysis of small 28x28 images!

Big strides, however, would affect the spatial resolution of the first layers in a CNN. What is the way out?

Sub-sampling is necessary!

The famous VGG16 CNN uses pairs and triples of convolution chains in its architecture. How does such a network get control over the number of weight parameters and the RAM requirement for all the output data at all the layers?

To get information in the sense of a feature hierarchy the CNN clearly should not look at details and related small sub-fields of the image, only. It must cover step-wise growing (!) areas of the original image, too. How do we combine these seemingly contradictory objectives in one training algorithm which does not lead to an exploding number of parameters, RAM and CPU time? Well, guys, this is the point where we should pay due respect to all the creative inventors of CNNs:

The answer is: We must accumulate or sample information across larger image or map areas. This is the (underestimated?) task of pooling- or sampling-layers.

For me it was just another confusing point in the beginning - until one grasps the real magic behind it. At first sight a layer like a typical "maxpooling" layer seems to reduce information, only; see the next picture:

The drawing explains that we "sample" the information over multiple pixels e.g. by

  • either calculating an average over pixels (or map node values)
  • or by just picking the maximum value of pixels or map node values (thereby stressing the most important information)

in a certain defined sub-area of an image or map.

The shift or stride used as a default in a pooling layer is exactly the side length of the pooling area. We thus cover the image by adjacent, non-overlapping tiles! This leads to a substantial decrease of the dimensions of the resulting map! With a (2x2) pooling size by a an effective factor of 2. (You can change the default pooling stride - but think about the consequences!)

Of course, averaging or picking a max value corresponds to information reduction.

However: What the CNN really also will do in a subsequent Conv layer is to invest in further weights for the combination of information (features) in and of substantially larger areas of the original image! Pooling followed by an additional convolution obviously supports hierarchy building of information on different scales of image areas!

After we first have concentrated on small scale features (like with a magnifying glass) we now - in a figurative sense - make a step backwards and look at larger scales of the image again.

The trick is to evaluate large scale information by sampling layers in addition to the small scale information information already extracted by the previous convolutions. Yes, we drop resolution information - but by introducing a suitable mix of convolutions and sampling layers we also force the network systematically to concentrate on combined large scale features, which in the end are really important for the image classification as a whole!

As sampling counterbalances an explosion of parameters we can invest into a growing number of feature maps with growing scales of covered image areas. I.e. we add more and new filters reacting to combinations of larger scale information.

Look at the second to last illustration: Assume that the 32 maps on layer L_N depicted there are the result of a sampling operation. The next convolution gathers new knowledge about more, namely 64 different combinations of filtered structures over a whole vertical stack of small filter areas located at the same position on the 32 maps of layer N. The new information is in the course of training conserved into 64 weight ensembles for 64 maps on layer N+1.

Resulting options for architectures

We can think of multiple ways of combining Conv layers and pooling layers. A simple recipe for small images could be

  • Layer 0: Input layer (tensor of original image data, 3 color layers or one gray layer)
  • >Layer 1: Conv layer (small 3x3 kernel, stride 1, 32 filters, 32 maps (26x26), analyzes 3x3 overlapping areas)
  • Layer 2: Pooling layer (2x2 max pooling => 32 (13x13) maps,
    a node covers 4x4 non overlapping areas per node on the original image)
  • Layer 3: Conv layer (3x3 kernel, stride 1, 64 filters, 64 maps (11x11),
    a node covers 8x8 overlapping areas on the original image (total effective stride 2))
  • Layer 4: Pooling layer (2x2 max pooling => 64 maps (5x5),
    a node covers 10x10 areas per node on the original image (total effective stride 5), some border info lost)
  • Layer 5: Conv layer (3x3 kernel, stride 1, 64 filters, 64 maps (3x3),
    a node covers 18x18 per node (effective stride 5), some border info lost )

The following picture illustrates the resulting successive combinations of nodes along one axis of a 28x28 image.

Note that I only indicated the connections to border nodes of the Conv filter areas.

The kernel size decides on the smallest structures we look at - especially via the first convolution. The sampling decides on the sequence of steadily growing areas which we then analyze for specific combinations of smaller structures.

Again: It is most of all the (down-) sampling which allows for an effective hierarchical information building over growing larger image areas! Actually we do not really drop information by sampling - instead we give the network a chance to collect and code new information on a higher, more abstract level (via a whole bunch of numerous new weights).

The big advantages of the sampling layers get obvious:

  • They reduce the numbers of required weights
  • They reduce the amount of required memory - not only for weights but also for the output data, which must be saved for every layer, map and node.
  • They reduce the CPU load for FW and BW propagation
  • They also limit the risk of overfitting as some detail information is dropped.

Of course there are many other sequences of layers one could think about. E.g., we could combine 2 to 3 Conv layers before we apply a pooling layer. Such a layer sequence is characteristic of the VGG nets.

Further aspects

Just as MLPs a CNN represents an acyclic graph, where the maps contain increasingly fewer nodes but where the number of maps per layer increases on average.

Questions and objectives for this article series

An interesting question, which seldom is answered in introductory books, is whether two totally independent training runs for a given CNN-architecture applied on the same input data will produce the same filters in the same order. We shall investigate this point in the forthcoming articles.

Another interesting point is: What does a CNN see at which convolution layer? What do the "features" (= basic structural elements) in an image which triggers a specific filter, look like?

If we could look into the output at some maps we could possibly see what filters do with the original image. And if we found a way to construct a structured image which triggers a specific filter then we could better understand what patterns the CNN reacts to. Examples for these different types of visualizations with respect to convolution in a CNN are objectives of this article series.

Conclusion

Today we covered a lot of "theory" on some aspects of CNNs. But we have a sufficiently solid basis regarding the structure and architecture now.

CNNs obviously have a much more complex structure than MLPs: They are deep in the sense of many sequential layers. And each convolutional layer has a complex structure in form of many parallel sub-layers (feature maps) itself. Feature maps are associated with filters, whose parameters (weights) get learned during the training. A map results from covering the original image or a map of a previous layer with small (overlapping) tiles of small filtering areas.

A mix of convolution and pooling layers allows for a look at detail features of the image in small areas in lower layers, whilst later layers can focus on feature combinations of larger image areas. The involved filters thus allow for the "awareness" of a hierarchy of features with translational invariance.

Pooling layers are important because they help to control the amount of weight parameters - and they enhance the effectiveness of detecting the most important feature correlations on larger image scales.

All nice and convincing - but the attentive reader will ask: Where and how do we do the classification?
Try to answer this question yourself first.

In the next article we shall build a concrete CNN and apply it to the MNIST dataset of images of handwritten digits. And whilst we do it I deliver the answer to the question posed above. Stay tuned ...

Literature

"Advanced Machine Learning with Python", John Hearty, 2016, Packt Publishing - See chapter 4.

"Deep Learning mit Python und Keras", Francois Chollet, 2018, mitp Verlag - See chapter 5.

"Hands-On Machine learning with SciKit-Learn, Keras & Tensorflow", 2nd edition, Aurelien Geron, 2019, O'Reilly - See chapter 14.

"Deep Learning mit Tensorflow, keras und Tensorflow.js", Matthieu Deru, Alassane Ndiaye, 2019, Rheinwerk Verlag, Bonn - see chapter 3

MLP, Numpy, TF2 – performance issues – Step III – a correction to BW propagation

In the last articles of this series

MLP, Numpy, TF2 – performance issues – Step II – bias neurons, F- or C- contiguous arrays and performance
MLP, Numpy, TF2 – performance issues – Step I – float32, reduction of back propagation

we looked at the FW-propagation of the MLP code which I discussed in another article series. We found that the treatment of bias neurons in the input layer was technically inefficient due to a collision of C- and F-contiguous arrays. By circumventing the problem we could accelerate the FW-propagation of big batches (as the complete training or test data set) by more than a factor of 2.

In this article I want to turn to the BW propagation and do some analysis regarding CPU consumption there. We will find a simple (and stupid) calculation step there which we shall replace. This will give us another 15% to 22% performance improvement in comparison to what we have reached in the last article for MNIST data:

  • 9.6 secs for 35 epochs and a batch-size of 500
  • and 8.7 secs for a batch-size of 20000.

Present CPU time relation between the FW- and the BW-propagation

The central training of mini-batches is performed by the method "_handle_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):
        ''' .... '''
        # 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
        
        # Major steps for the mini-batch during one epoch iteration 
        # **********************************************************
        
        #ts=time.perf_counter()
        # 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
        # ******
        # ts=time.perf_counter()
        # slicing 
        li_Z_in_layer[0] = self._X_train[ay_idx_batch] # numpy arrays can be indexed by an array of integers
        
        # transposition 
        #~~~~~~~~~~~~~~
        li_Z_in_layer[0] = li_Z_in_layer[0].T
        #te=time.perf_counter(); t_batch = te - ts;
        #print("\nti - transposed inputbatch =", t_batch)
        
        # Step 2: Call forward propagation method for the present mini-batch of training records
        # *******
        #tsa = time.perf_counter() 
        self._fw_propagation(li_Z_in = li_Z_in_layer, li_A_out = li_A_out_layer) 
        #tea = time.perf_counter(); ta = tea - tsa;  print("ta - FW-propagation", "%10.8f"%ta)
        
        # Step 3: Cost calculation for the mini-batch 
        # ********
        #tsb = time.perf_counter() 
        ay_y_enc = self._ay_onehot[:, ay_idx_batch]
        ay_ANN_out = li_A_out_layer[self._n_total_layers-1]
        total_costs_batch, rel_reg_contrib = 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
        self._ay_reg_cost_contrib[num_epoch, num_batch] = rel_reg_contrib
        #teb = time.perf_counter(); tb = teb - tsb; print("tb - cost calculation", "%10.8f"%tb)
        
        
        # Step 4: Avg-error for later plotting 
        # ********
        #tsc = time.perf_counter() 
        # 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
        ay_theta_avg = np.average(np.abs(ay_theta_out)) 
        self._ay_theta[num_epoch, num_batch] = ay_theta_avg 
        #tec = time.perf_counter(); tc = tec - tsc; print("tc - error", "%10.8f"%tc)
        
        
        # Step 5: Perform gradient calculation via back propagation of errors
        # ******* 
        #tsd = time.perf_counter() 
        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, 
                              li_grad = li_grad_layer, 
                              b_print = b_print,
                              b_internal_timing = False 
                              ) 
        #ted = time.perf_counter(); td = ted - tsd; print("td - BW propagation", "%10.8f"%td)
        
        # Step 7: Adjustment of weights  
        # *******        
        #tsf = time.perf_counter() 
        rg_layer=range(0, self._n_total_layers -1)
        for N in rg_layer:
            delta_w_N = self._learn_rate * li_grad_layer[N]
            self._li_w[N] -= ( delta_w_N + (self._mom_rate * self._li_mom[N]) )
            
            # save momentum
            self._li_mom[N] = delta_w_N
        #tef = time.perf_counter(); tf = tef - tsf; print("tf - weight correction", "%10.8f"%tf)
        
        return None

 

I took some time measurements there:

ti - transposed inputbatch = 0.0001785
ta - FW-propagation 0.00080975
tb - cost calculation 0.00030705
tc - error 0.00016182
td - BW propagation 0.00112558
tf - weight correction 0.00020079

ti - transposed inputbatch = 0.00018144
ta - FW-propagation 0.00082022
tb - cost calculation 0.00031284
tc - error 0.00016652
td - BW propagation 0.00106464
tf - weight correction 0.00019576

You see that the FW-propagation is a bit faster than the BW-propagation. This is a bit strange as the FW-propagation is dominated meanwhile by a really expensive operation which we cannot accelerate (without choosing a new activation function): The calculation of the sigmoid value for the inputs at layer L1.

So let us look into the BW-propagation; the code for it is momentarily:

    ''' -- Method to handle error BW propagation for a mini-batch --'''
    def _bw_propagation(self, 
                        ay_y_enc, li_Z_in, li_A_out, 
                        li_delta_out, li_delta, li_D, li_grad, 
                        b_print = True, b_internal_timing = False):
        
        # List initialization: All parameter lists or arrays are filled or to be filled by layer operations 
        # 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
        # *********************** 
        tsa = time.perf_counter() 
        # 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 )
        #ay_D_E = ay_A_E * (1.0 - ay_A_E)

        # 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) 
        
        # 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 ! 
        
        tea = time.perf_counter(); ta = tea - tsa; print("\nta-bp", "%10.8f"%ta)
        
        # Loop over all layers in reverse direction 
        # ******************************************
        # index range of target layers N in BW direction (starting with E-1 => 4 layers -> layer 2))
        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 
        tsb = time.perf_counter() 
        for N in range_N_bw_layer:
            
            # Back Propagation operations between layers N+1 and N 
            # *******************************************************
            # this method handles the special treatment of bias nodes in Z_in, too
            tsib = time.perf_counter() 
            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 )
            teib = time.perf_counter(); tib = teib - tsib; print("N = ", N, " tib-bp", "%10.8f"%tib)
            
            # add matrices to their lists 
            #tsic = time.perf_counter() 
            li_delta[N] = ay_delta_N
            li_D[N]     = ay_D_N
            li_grad[N]= ay_grad_N
            #teic = time.perf_counter(); tic = teic - tsic; print("\nN = ", N, " tic = ", "%10.8f"%tic)
        teb = time.perf_counter(); tb = teb - tsb; print("tb-bp", "%10.8f"%tb)
       
        return

 

Typical timing results are:

ta-bp 0.00007112
N =  2  tib-bp 0.00025399
N =  1  tib-bp 0.00051683
N =  0  tib-bp 0.00035941
tb-bp 0.00126436

ta-bp 0.00007492
N =  2  tib-bp 0.00027644
N =  1  tib-bp 0.00090043
N =  0  tib-bp 0.00036728
tb-bp 0.00168378

We see that the CPU consumption of "_bw_prop_Np1_to_N()" should be analyzed in detail. It is relatively time consuming at every layer, but especially at layer L1. (The list adds are insignificant.)
What does this method presently look like?

    ''' -- 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 between layer N+1 and N 
        Version 1.5 - partially accelerated 

        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 
        '''
        
        # Prepare required quantities - and add bias neuron to ay_Z_in 
        # ****************************
        
        # Weight matrix meddling between layers N and N+1 
        ay_W_N = self._li_w[N]

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

        # fetch output value saved during FW propagation 
        ay_A_N = li_A_out[N]

        # Optimization V1.5 ! 
        if N > 0: 
            
            #ts=time.perf_counter()
            ay_Z_N = li_Z_in[N]
            # !!! Add intermediate row (for bias) to Z_N !!!
            ay_Z_N = self._add_bias_neuron_to_layer(ay_Z_N, 'row')
            #te=time.perf_counter(); t1 = te - ts; print("\nBW t1 = ", t1, " N = ", N) 
        
            # Derivative matrix for the activation function (with extra bias node row)
            # ********************
            #    can only be calculated now as we need the z-values
            #ts=time.perf_counter()
            ay_D_N = self._calculate_D_N(ay_Z_N)
            #te=time.perf_counter(); t2 = te - ts; print("\nBW t2 = ", t2, " N = ", N) 
            
            # Propagate delta
            # **************

            # intermediate delta 
            # ~~~~~~~~~~~~~~~~~~
            #ts=time.perf_counter()
            ay_delta_w_N = ay_W_N.T.dot(ay_delta_Np1)
            #te=time.perf_counter(); t3 = te - ts; print("\nBW t3 = ", t3) 
            
            # final delta 
            # ~~~~~~~~~~~
            #ts=time.perf_counter()
            ay_delta_N = ay_delta_w_N * ay_D_N
            
            # Orig reduce dimension again
            # **************************** 
            ay_delta_N = ay_delta_N[1:, :]
            #te=time.perf_counter(); t4 = te - ts; print("\nBW t4 = ", t4) 
            
        else: 
            ay_delta_N = None
            ay_D_N = None
        
        # Calculate gradient
        # ********************
        #ts=time.perf_counter()
        ay_grad_N = np.dot(ay_delta_Np1, ay_A_N.T)
        #te=time.perf_counter(); t5 = te - ts; print("\nBW t5 = ", t5) 
        
        # regularize gradient (!!!! without adding bias nodes in the L1, L2 sums) 
        #ts=time.perf_counter()
        if self._lambda2_reg > 0: 
            ay_grad_N[:, 1:] += self._li_w[N][:, 1:] * self._lambda2_reg 
        if self._lambda1_reg > 0: 
            ay_grad_N[:, 1:] += np.sign(self._li_w[N][:, 1:]) * self._lambda1_reg 
        #te=time.perf_counter(); t6 = te - ts; print("\nBW t6 = ", t6) 
        
        return ay_delta_N, ay_D_N, ay_grad_N

 
Timing data for a batch-size of 500 are:

N =  2
BW t1 =  0.0001169009999557602  N =  2
BW t2 =  0.00035331499998392246  N =  2
BW t3 =  0.00018078099992635543
BW t4 =  0.00010234199999104021
BW t5 =  9.928200006470433e-05
BW t6 =  2.4267000071631628e-05
N =  2  tib-bp 0.00124414

N =  1
BW t1 =  0.0004323499999827618  N =  1
BW t2 =  0.000781415999881574  N =  1
BW t3 =  4.2077999978573644e-05
BW t4 =  0.00022921000004316738
BW t5 =  9.376399998473062e-05
BW t6 =  0.00012183700005152787
N =  1  tib-bp 0.00216281

N =  0
BW t5 =  0.0004289769999559212
BW t6 =  0.00015404999999191205
N =  0  tib-bp 0.00075249
....
N =  2
BW t1 =  0.00012802800006284087  N =  2
BW t2 =  0.00034988200013685855  N =  2
BW t3 =  0.0001854429999639251
BW t4 =  0.00010359299994888715
BW t5 =  0.00010210400000687514
BW t6 =  2.4010999823076418e-05
N =  2  tib-bp 0.00125854

N =  1
BW t1 =  0.0004407169999467442  N =  1
BW t2 =  0.0007845899999665562  N =  1
BW t3 =  0.00025684100000944454
BW t4 =  0.00012409999999363208
BW t5 =  0.00010345399982725212
BW t6 =  0.00012994100006835652
N =  1  tib-bp 0.00221321

N =  0
BW t5 =  0.00044504700008474174
BW t6 =  0.00016473000005134963
N =  0  tib-bp 0.00071442

....
N =  2
BW t1 =  0.000292730999944979  N =  2
BW t2 =  0.001102525000078458  N =  2
BW t3 =  2.9429999813146424e-05
BW t4 =  8.547999868824263e-06
BW t5 =  3.554099998837046e-05
BW t6 =  2.5041999833774753e-05
N =  2  tib-bp 0.00178565

N =  1
BW t1 =  3.143399999316898e-05  N =  1
BW t2 =  0.0006720640001276479  N =  1
BW t3 =  5.4785999964224175e-05
BW t4 =  9.756200006449944e-05
BW t5 =  0.0001605449999715347
BW t6 =  1.8391000139672542e-05
N =  1  tib-bp 0.00147566

N =  0
BW t5 =  0.0003641810001226986
BW t6 =  6.338999992294703e-05
N =  0  tib-bp 0.00046542

 
It seems that we should care about t1, t2, t3 for hidden layers and maybe about t5 at layers L1/L0.

However, for a batch-size of 15000 things look a bit different:

N =  2
BW t1 =  0.0005776280000304723  N =  2
BW t2 =  0.004995969999981753  N =  2
BW t3 =  0.0003165199999557444
BW t4 =  0.0005244750000201748
BW t5 =  0.000518499999998312
BW t6 =  2.2458999978880456e-05
N =  2  tib-bp 0.00736144

N =  1
BW t1 =  0.0010120430000029046  N =  1
BW t2 =  0.010797029000002567  N =  1
BW t3 =  0.0005006920000028003
BW t4 =  0.0008704929999794331
BW t5 =  0.0010805200000163495
BW t6 =  3.0326000000968634e-05
N =  1  tib-bp 0.01463436

N =  0
BW t5 =  0.006987539000022025
BW t6 =  0.00023552499999368592
N =  0  tib-bp 0.00730959


N =  2
BW t1 =  0.0006299790000525718  N =  2
BW t2 =  0.005081416999985322  N =  2
BW t3 =  0.00018547400003399162
BW t4 =  0.0005970070000103078
BW t5 =  0.000564008000026206
BW t6 =  2.3311000006742688e-05
N =  2  tib-bp 0.00737899

N =  1
BW t1 =  0.0009376909999900818  N =  1
BW t2 =  0.010650266999959968  N =  1
BW t3 =  0.0005232729999988806
BW t4 =  0.0009100700000317374
BW t5 =  0.0011237720000281115
BW t6 =  0.00016643800000792908
N =  1  tib-bp 0.01466144

N =  0
BW t5 =  0.006987463000029948
BW t6 =  0.00023978600000873485
N =  0  tib-bp 0.00734308

 
For big batch-sizes "t2" dominates everything. It seems that we have found another code area which causes the trouble with big batch-sizes which we already observed before!

What operations do the different CPU times stand for?

To keep an overview without looking into the code again, I briefly summarize which operations cause which of the measured time differences:

  • "t1" - which contributes for small batch-sizes stands for adding a bias neuron to the input data Z_in at each layer.
  • "t2" - which is by far dominant for big batch sizes stands for calculating the derivative of the output/activation function (in our case of the sigmoid function) at the various layers.
  • "t3" - which contributes at some layers stands for a dot()-matrix multiplication with the transposed weight-matrix,
  • "t4" - covers an element-wise matrix-multiplication,
  • "t5" - contributes at the BW-transition from layer L1 to L0 and covers the matrix multiplication there (including the full output matrix with the bias neurons at L0)

Use the output values calculated at each layer during FW-propagation!

Why does the calculation of the derivative of the sigmoid function take so much time? Answer: Because I coded it stupidly! Just look at it:

    ''' -- 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

    ''' -- method for the derivative of the sigmoid function-- '''
    def _D_sigmoid(self, ay_Z):
        ''' 
        Derivative of sigmoid function with respect to Z-values 
        - works via expit element-wise on matrices
        Input:  Z - Matrix with Input values for the activation function Phi() = sigmoid() 
        Output: D - Matrix with derivative values 
        '''
        S_Z = self._sigmoid(ay_Z)
        return S_Z * (1.0 - S_Z)

 
We first call an intermediate function which then directs us to the right function for a chosen activation function. Well meant: So far, we use only the sigmoid function, but it could e.g. also be the relu() or tanh()-function. So, we did what we did for the sake of generalization. But we did it badly because of two reasons:

  • We did not keep up a function call pattern which we introduced in the FW-propagation.
  • The calculation of the derivative is inefficient.

The first point is a minor one: During FW-propagation we called the right (!) activation function, i.e. the one we choose by input parameters to our ANN-object, by an indirect call. Why not do it the same way here? We would avoid an intermediate function call and keep up a pattern. Actually, we prepared the necessary definitions already in the __init__()-function.

The second point is relevant for performance: The derivative function produces the correct results for a given "ay_Z", but this is totally inefficient in our BW-situation. The code repeats a really expensive operation which we have already performed during FW-propagation: calling sigmoid(ay_Z) to get "A_out"-values per layer then. We even put the A_out-values [=sigmoid(ay_Z_in)] per layer and batch (!) with some foresight into a list in "li_A_out[]" at that point of the code (see the FW-propagation code discussed in the last article).

So, of course, we should use these "A_out"-values now in the BW-steps! No further comment .... you see what we need to do.

Hint: Actually, also other activation functions "act(Z)" like e.g. the "tanh()"-function have derivatives which depend on on "A=act(Z)", only. So, we should provide Z and A via an interface to the derivative function and let the respective functions take what it needs.
But, my insight into my own dumbness gets worse.

Eliminate the bias neuron operation!

Why did we need a bias-neuron operation? Answer: We do not need it! It was only introduced due to insufficient cleverness. In the article

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

I have already indicated that we use the function for adding a row of bias-neurons again only to compensate one deficit: The matrix of the derivative values did not fit the shape of the weight matrix for the required element-wise operations. However, I also said: There probably is an alternative.

Well, let me make a long story short: The steps behind t1 up to t4 to calculate "ay_delta_N" for the present layer L_N (with N>=1) can be compressed into two relatively simple lines:

ay_delta_w_N = ay_W_N.T.dot(ay_delta_Np1)
ay_delta_N = ay_delta_w_N[1:,:] * ay_A_N[1:,:] * (1.0 - ay_A_N[1:,:]); ay_D_N = None;

No bias back and forth corrections! Instead we use simple slicing to compensate for our weight matrices with a shape covering an extra row of bias node output. No Z-based derivative calculation; no sigmoid(Z)-call. The last statement is only required to support the present output interface. Think it through in detail; the shortcut does not cause any harm.

Code change for tests

Before we bring the code into a new consolidated form with re-coded methods let us see what we gain by just changing the code to the two lines given above in terms of CPU time and performance. Our function "_bw_prop_Np1_to_N()" then gets reduced to the following lines:

    ''' -- 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):
        
        # Weight matrix meddling between layers N and N+1 
        ay_W_N = self._li_w[N]
        ay_delta_Np1 = li_delta[N+1]

        # fetch output value saved during FW propagation 
        ay_A_N = li_A_out[N]

        # Optimization from previous version  
        if N > 0: 
            #ts=time.perf_counter()
            ay_Z_N = li_Z_in[N]
            
            # Propagate delta
            # ~~~~~~~~~~~~~~~~~
            ay_delta_w_N = ay_W_N.T.dot(ay_delta_Np1)
            ay_delta_N = ay_delta_w_N[1:,:] * ay_A_N[1:,:] * (1.0 - ay_A_N[1:,:])
            ay_D_N = None; 
            
        else: 
            ay_delta_N = None
            ay_D_N = None
        
        # Calculate gradient
        # ********************
        ay_grad_N = np.dot(ay_delta_Np1, ay_A_N.T)
        
        if self._lambda2_reg > 0: 
            ay_grad_N[:, 1:] += self._li_w[N][:, 1:] * self._lambda2_reg 
        if self._lambda1_reg > 0: 
            ay_grad_N[:, 1:] += np.sign(self._li_w[N][:, 1:]) * self._lambda1_reg 
        
        return ay_delta_N, ay_D_N, ay_grad_N

 

Performance gain

What run times do we get with this setting? We perform our typical test runs over 35 epochs - but this time for two different batch-sizes:

Batch-size = 500

 
------------------
Starting epoch 35

Time_CPU for epoch 35 0.2169024469985743
Total CPU-time:  7.52385053600301

learning rate =  0.0009994051838157095

total costs of training set   =  -1.0
rel. reg. contrib. to total costs =  -1.0

total costs of last mini_batch   =  65.43618
rel. reg. contrib. to batch costs =  0.12302863

mean abs weight at L0 :  -10.0
mean abs weight at L1 :  -10.0
mean abs weight at L2 :  -10.0

avg total error of last mini_batch =  0.00758
presently batch averaged accuracy   =  0.99272

-------------------
Total training Time_CPU:  7.5257336139984545

Not bad! We became faster by around 2 secs compared to the results of the last article! This is close to an improvement of 20%.

But what about big batch sizes? Here is the result for a relatively big batch size:

Batch-size = 20000

------------------
Starting epoch 35

Time_CPU for epoch 35 0.2019189490019926
Total CPU-time:  6.716679593999288

learning rate =  9.994051838157101e-05

total costs of training set   =  -1.0
rel. reg. contrib. to total costs =  -1.0

total costs of last mini_batch   =  13028.141
rel. reg. contrib. to batch costs =  0.00021923862

mean abs weight at L0 :  -10.0
mean abs weight at L1 :  -10.0
mean abs weight at L2 :  -10.0

avg total error of last mini_batch =  0.04389
presently batch averaged accuracy   =  0.95602

-------------------
Total training Time_CPU:  6.716954112998792

Again an acceleration by roughly 2 secs - corresponding to an improvement of 22%!

In both cases I took the best result out of three runs.

Conclusion

Enough for today! We have done a major step with regard to performance optimization also in the BW-propagation. It remains to re-code the derivative calculation in form which uses indirect function calls to remain flexible. I shall give you the code in the next article.

We learned today is that we, of course, should reuse the results of the FW-propagation and that it is indeed a good investment to save the output data per layer in some Python list or other suitable structures during FW-propagation. We also saw again that a sufficiently efficient bias neuron treatment can be achieved by a more efficient solution than provisioned so far.

All in all we have meanwhile gained more than a factor of 6.5 in performance since we started with optimization. Our new standard values are 7.3 secs and 6.8 secs for 35 epochs on MNIST data and batch sizes of 500 and 20000, respectively.

We have reached the order of what Keras and TF2 can deliver on a CPU for big batch sizes. For small batch sizes we are already faster. This indicates that we have done no bad job so far ...

In the next article we shall look a bit at the matrix operations and evaluate further optimization options.