```
In [1]:
```

```
from IPython.display import Image
```

# CNTK 103: Part C - Multi Layer Perceptron with MNIST¶

We assume that you have successfully completed CNTK 103 Part A.

In this tutorial, we train a multi-layer perceptron on MNIST data. This notebook provides the recipe using Python APIs. If you are looking for this example in BrainScript, please look here

## Introduction¶

**Problem** As in CNTK 103B, we will continue to work on the same
problem of recognizing digits in MNIST data. The MNIST data comprises
hand-written digits with little background noise.

```
In [2]:
```

```
# Figure 1
Image(url= "http://3.bp.blogspot.com/_UpN7DfJA0j4/TJtUBWPk0SI/AAAAAAAAABY/oWPMtmqJn3k/s1600/mnist_originals.png", width=200, height=200)
```

```
Out[2]:
```

**Goal**: Our goal is to train a classifier that will identify the
digits in the MNIST dataset. Additionally, we aspire to achieve lower
error rate with Multi-layer perceptron compared to Multi-class logistic
regression.

**Approach**: The same 5 stages we have used in the previous tutorial
are applicable: Data reading, Data preprocessing, Creating a model,
Learning the model parameters and Evaluating (a.k.a. testing/prediction)
the model. - Data reading: We will use the CNTK Text reader - Data
preprocessing: Covered in part A (suggested extension section).

There is a high overlap with CNTK 102. Though this tutorial we adapt the same model to work on MNIST data with 10 classes instead of the 2 classes we used in CNTK 102.

```
In [3]:
```

```
from __future__ import print_function # Use a function definition from future version (say 3.x from 2.7 interpreter)
import matplotlib.image as mpimg
import matplotlib.pyplot as plt
import numpy as np
import sys
import os
import cntk as C
import cntk.tests.test_utils
cntk.tests.test_utils.set_device_from_pytest_env() # (only needed for our build system)
C.cntk_py.set_fixed_random_seed(1) # fix a random seed for CNTK components
%matplotlib inline
```

## Data reading¶

In this section, we will read the data generated in CNTK 103 Part A.

```
In [5]:
```

```
# Define the data dimensions
input_dim = 784
num_output_classes = 10
```

In this tutorial we are using the MNIST data you have downloaded using
CNTK_103A_MNIST_DataLoader notebook. The dataset has 60,000 training
images and 10,000 test images with each image being 28 x 28 pixels. Thus
the number of features is equal to 784 (= 28 x 28 pixels), 1 per pixel.
The variable `num_output_classes`

is set to 10 corresponding to the
number of digits (0-9) in the dataset.

The data is in the following format:

```
|labels 0 0 0 0 0 0 0 1 0 0 |features 0 0 0 0 ...
(784 integers each representing a pixel)
```

In this tutorial we are going to use the image pixels corresponding the
integer stream named “features”. We define a `create_reader`

function
to read the training and test data using the CTF
deserializer.
The labels are 1-hot
encoded. Refer to CNTK 103A
tutorial for data format visualizations.

```
In [6]:
```

```
# Read a CTF formatted text (as mentioned above) using the CTF deserializer from a file
def create_reader(path, is_training, input_dim, num_label_classes):
return C.io.MinibatchSource(C.io.CTFDeserializer(path, C.io.StreamDefs(
labels = C.io.StreamDef(field='labels', shape=num_label_classes, is_sparse=False),
features = C.io.StreamDef(field='features', shape=input_dim, is_sparse=False)
)), randomize = is_training, max_sweeps = C.io.INFINITELY_REPEAT if is_training else 1)
```

```
In [7]:
```

```
# Ensure the training and test data is generated and available for this tutorial.
# We search in two locations in the toolkit for the cached MNIST data set.
data_found = False
for data_dir in [os.path.join("..", "Examples", "Image", "DataSets", "MNIST"),
os.path.join("data", "MNIST")]:
train_file = os.path.join(data_dir, "Train-28x28_cntk_text.txt")
test_file = os.path.join(data_dir, "Test-28x28_cntk_text.txt")
if os.path.isfile(train_file) and os.path.isfile(test_file):
data_found = True
break
if not data_found:
raise ValueError("Please generate the data by completing CNTK 103 Part A")
print("Data directory is {0}".format(data_dir))
```

```
Data directory is ..\Examples\Image\DataSets\MNIST
```

## Model Creation¶

Our multi-layer perceptron will be relatively simple with 2 hidden
layers (`num_hidden_layers`

). The number of nodes in the hidden layer
being a parameter specified by `hidden_layers_dim`

. The figure below
illustrates the entire model we will use in this tutorial in the context
of MNIST data.

If you are not familiar with the terms *hidden layer* and *number of
hidden layers*, please refer back to CNTK 102 tutorial.

Each Dense layer (as illustrated below) shows the input dimensions, output dimensions and activation function that layer uses. Specifically, the layer below shows: input dimension = 784 (1 dimension for each input pixel), output dimension = 400 (number of hidden nodes, a parameter specified by the user) and activation function being relu.

In this model we have 2 dense layer called the hidden layers each with
an activation function of `relu`

and one output layer with no
activation.

The output dimension (a.k.a. number of hidden nodes) in the 2 hidden layer is set to 400 and 200 in the illustration above. In the code below we keep both layers to have the same number of hidden nodes (set to 400). The number of hidden layers is 2. Fill in the following values: - num_hidden_layers - hidden_layers_dim

The final output layer emits a vector of 10 values. Since we will be using softmax to normalize the output of the model we do not use an activation function in this layer. The softmax operation comes bundled with the loss function we will be using later in this tutorial.

```
In [8]:
```

```
num_hidden_layers = 2
hidden_layers_dim = 400
```

Network input and output: - **input** variable (a key CNTK concept): >An
**input** variable is a container in which we fill different
observations in this case image pixels during model learning
(a.k.a.training) and model evaluation (a.k.a. testing). Thus, the shape
of the `input`

must match the shape of the data that will be provided.
For example, when data are images each of height 10 pixels and width 5
pixels, the input feature dimension will be 50 (representing the total
number of image pixels). More on data and their dimensions to appear in
separate tutorials.

**Question** What is the input dimension of your chosen model? This is
fundamental to our understanding of variables in a network or model
representation in CNTK.

```
In [9]:
```

```
input = C.input_variable(input_dim)
label = C.input_variable(num_output_classes)
```

## Multi-layer Perceptron setup¶

The cell below is a direct translation of the illustration of the model shown above.

```
In [10]:
```

```
def create_model(features):
with C.layers.default_options(init = C.layers.glorot_uniform(), activation = C.ops.relu):
h = features
for _ in range(num_hidden_layers):
h = C.layers.Dense(hidden_layers_dim)(h)
r = C.layers.Dense(num_output_classes, activation = None)(h)
return r
z = create_model(input)
```

`z`

will be used to represent the output of a network.

We introduced sigmoid function in CNTK 102, in this tutorial you should try different activation functions in the hidden layer. You may choose to do this right away and take a peek into the performance later in the tutorial or run the preset tutorial and then choose to perform the suggested activity.

** Suggested Activity ** - Record the training error you get with
`sigmoid`

as the activation function - Now change to `relu`

as the
activation function and see if you can improve your training error

*Quiz*: Name some of the different supported activation functions. Which
activation function gives the least training error?

```
In [11]:
```

```
# Scale the input to 0-1 range by dividing each pixel by 255.
z = create_model(input/255.0)
```

### Learning model parameters¶

Same as the previous tutorial, we use the `softmax`

function to map
the accumulated evidences or activations to a probability distribution
over the classes (Details of the softmax
function).

## Training¶

Similar to CNTK 102, we minimize the cross-entropy between the label and predicted probability by the network. If this terminology sounds strange to you, please refer to the tutorial CNTK 102 for a refresher.

```
In [12]:
```

```
loss = C.cross_entropy_with_softmax(z, label)
```

### Evaluation¶

In order to evaluate the classification, one can compare the output of
the network which for each observation emits a vector of evidences (can
be converted into probabilities using `softmax`

functions) with
dimension equal to number of classes.

```
In [13]:
```

```
label_error = C.classification_error(z, label)
```

### Configure training¶

The trainer strives to reduce the `loss`

function by different
optimization approaches, Stochastic Gradient
Descent
(`sgd`

) being a basic one. Typically, one would start with random
initialization of the model parameters. The `sgd`

optimizer would
calculate the `loss`

or error between the predicted label against the
corresponding ground-truth label and using
gradient-decent
generate a new set model parameters in a single iteration.

The aforementioned model parameter update using a single observation at
a time is attractive since it does not require the entire data set (all
observation) to be loaded in memory and also requires gradient
computation over fewer datapoints, thus allowing for training on large
data sets. However, the updates generated using a single observation
sample at a time can vary wildly between iterations. An intermediate
ground is to load a small set of observations and use an average of the
`loss`

or error from that set to update the model parameters. This
subset is called a *minibatch*.

With minibatches we often sample observation from the larger training
dataset. We repeat the process of model parameters update using
different combination of training samples and over a period of time
minimize the `loss`

(and the error). When the incremental error rates
are no longer changing significantly or after a preset number of maximum
minibatches to train, we claim that our model is trained.

One of the key parameter for
optimization
is called the `learning_rate`

. For now, we can think of it as a
scaling factor that modulates how much we change the parameters in any
iteration. We will be covering more details in later tutorial. With this
information, we are ready to create our trainer.

```
In [14]:
```

```
# Instantiate the trainer object to drive the model training
learning_rate = 0.2
lr_schedule = C.learning_parameter_schedule(learning_rate)
learner = C.sgd(z.parameters, lr_schedule)
trainer = C.Trainer(z, (loss, label_error), [learner])
```

First let us create some helper functions that will be needed to visualize different functions associated with training.

```
In [15]:
```

```
# Define a utility function to compute the moving average sum.
# A more efficient implementation is possible with np.cumsum() function
def moving_average(a, w=5):
if len(a) < w:
return a[:] # Need to send a copy of the array
return [val if idx < w else sum(a[(idx-w):idx])/w for idx, val in enumerate(a)]
# Defines a utility that prints the training progress
def print_training_progress(trainer, mb, frequency, verbose=1):
training_loss = "NA"
eval_error = "NA"
if mb%frequency == 0:
training_loss = trainer.previous_minibatch_loss_average
eval_error = trainer.previous_minibatch_evaluation_average
if verbose:
print ("Minibatch: {0}, Loss: {1:.4f}, Error: {2:.2f}%".format(mb, training_loss, eval_error*100))
return mb, training_loss, eval_error
```

### Run the trainer¶

We are now ready to train our fully connected neural net. We want to decide what data we need to feed into the training engine.

In this example, each iteration of the optimizer will work on
`minibatch_size`

sized samples. We would like to train on all 60000
observations. Additionally we will make multiple passes through the data
specified by the variable `num_sweeps_to_train_with`

. With these
parameters we can proceed with training our simple multi-layer
perceptron network.

```
In [16]:
```

```
# Initialize the parameters for the trainer
minibatch_size = 64
num_samples_per_sweep = 60000
num_sweeps_to_train_with = 10
num_minibatches_to_train = (num_samples_per_sweep * num_sweeps_to_train_with) / minibatch_size
```

```
In [17]:
```

```
# Create the reader to training data set
reader_train = create_reader(train_file, True, input_dim, num_output_classes)
# Map the data streams to the input and labels.
input_map = {
label : reader_train.streams.labels,
input : reader_train.streams.features
}
# Run the trainer on and perform model training
training_progress_output_freq = 500
plotdata = {"batchsize":[], "loss":[], "error":[]}
for i in range(0, int(num_minibatches_to_train)):
# Read a mini batch from the training data file
data = reader_train.next_minibatch(minibatch_size, input_map = input_map)
trainer.train_minibatch(data)
batchsize, loss, error = print_training_progress(trainer, i, training_progress_output_freq, verbose=1)
if not (loss == "NA" or error =="NA"):
plotdata["batchsize"].append(batchsize)
plotdata["loss"].append(loss)
plotdata["error"].append(error)
```

```
Minibatch: 0, Loss: 2.3106, Error: 81.25%
Minibatch: 500, Loss: 0.2747, Error: 7.81%
Minibatch: 1000, Loss: 0.0964, Error: 1.56%
Minibatch: 1500, Loss: 0.1252, Error: 4.69%
Minibatch: 2000, Loss: 0.0086, Error: 0.00%
Minibatch: 2500, Loss: 0.0387, Error: 1.56%
Minibatch: 3000, Loss: 0.0206, Error: 0.00%
Minibatch: 3500, Loss: 0.0486, Error: 3.12%
Minibatch: 4000, Loss: 0.0178, Error: 0.00%
Minibatch: 4500, Loss: 0.0107, Error: 0.00%
Minibatch: 5000, Loss: 0.0077, Error: 0.00%
Minibatch: 5500, Loss: 0.0042, Error: 0.00%
Minibatch: 6000, Loss: 0.0045, Error: 0.00%
Minibatch: 6500, Loss: 0.0292, Error: 0.00%
Minibatch: 7000, Loss: 0.0190, Error: 1.56%
Minibatch: 7500, Loss: 0.0060, Error: 0.00%
Minibatch: 8000, Loss: 0.0031, Error: 0.00%
Minibatch: 8500, Loss: 0.0019, Error: 0.00%
Minibatch: 9000, Loss: 0.0006, Error: 0.00%
```

Let us plot the errors over the different training minibatches. Note that as we iterate the training loss decreases though we do see some intermediate bumps.

Hence, we use smaller minibatches and using `sgd`

enables us to have a
great scalability while being performant for large data sets. There are
advanced variants of the optimizer unique to CNTK that enable harnessing
computational efficiency for real world data sets and will be introduced
in advanced tutorials.

```
In [18]:
```

```
# Compute the moving average loss to smooth out the noise in SGD
plotdata["avgloss"] = moving_average(plotdata["loss"])
plotdata["avgerror"] = moving_average(plotdata["error"])
# Plot the training loss and the training error
import matplotlib.pyplot as plt
plt.figure(1)
plt.subplot(211)
plt.plot(plotdata["batchsize"], plotdata["avgloss"], 'b--')
plt.xlabel('Minibatch number')
plt.ylabel('Loss')
plt.title('Minibatch run vs. Training loss')
plt.show()
plt.subplot(212)
plt.plot(plotdata["batchsize"], plotdata["avgerror"], 'r--')
plt.xlabel('Minibatch number')
plt.ylabel('Label Prediction Error')
plt.title('Minibatch run vs. Label Prediction Error')
plt.show()
```

### Run evaluation / testing¶

Now that we have trained the network, let us evaluate the trained
network on the test data. This is done using `trainer.test_minibatch`

.

```
In [19]:
```

```
# Read the training data
reader_test = create_reader(test_file, False, input_dim, num_output_classes)
test_input_map = {
label : reader_test.streams.labels,
input : reader_test.streams.features,
}
# Test data for trained model
test_minibatch_size = 512
num_samples = 10000
num_minibatches_to_test = num_samples // test_minibatch_size
test_result = 0.0
for i in range(num_minibatches_to_test):
# We are loading test data in batches specified by test_minibatch_size
# Each data point in the minibatch is a MNIST digit image of 784 dimensions
# with one pixel per dimension that we will encode / decode with the
# trained model.
data = reader_test.next_minibatch(test_minibatch_size,
input_map = test_input_map)
eval_error = trainer.test_minibatch(data)
test_result = test_result + eval_error
# Average of evaluation errors of all test minibatches
print("Average test error: {0:.2f}%".format(test_result*100 / num_minibatches_to_test))
```

```
Average test error: 1.74%
```

Note, this error is very comparable to our training error indicating that our model has good “out of sample” error a.k.a. generalization error. This implies that our model can very effectively deal with previously unseen observations (during the training process). This is key to avoid the phenomenon of overfitting.

**Huge** reduction in error compared to multi-class LR (from CNTK 103B).

We have so far been dealing with aggregate measures of error. Let us now
get the probabilities associated with individual data points. For each
observation, the `eval`

function returns the probability distribution
across all the classes. The classifier is trained to recognize digits,
hence has 10 classes. First let us route the network output through a
`softmax`

function. This maps the aggregated activations across the
network to probabilities across the 10 classes.

```
In [20]:
```

```
out = C.softmax(z)
```

Let us a small minibatch sample from the test data.

```
In [21]:
```

```
# Read the data for evaluation
reader_eval = create_reader(test_file, False, input_dim, num_output_classes)
eval_minibatch_size = 25
eval_input_map = {input: reader_eval.streams.features}
data = reader_test.next_minibatch(eval_minibatch_size, input_map = test_input_map)
img_label = data[label].asarray()
img_data = data[input].asarray()
predicted_label_prob = [out.eval(img_data[i]) for i in range(len(img_data))]
```

```
In [22]:
```

```
# Find the index with the maximum value for both predicted as well as the ground truth
pred = [np.argmax(predicted_label_prob[i]) for i in range(len(predicted_label_prob))]
gtlabel = [np.argmax(img_label[i]) for i in range(len(img_label))]
```

```
In [23]:
```

```
print("Label :", gtlabel[:25])
print("Predicted:", pred)
```

```
Label : [4, 5, 6, 7, 8, 9, 7, 4, 6, 1, 4, 0, 9, 9, 3, 7, 8, 4, 7, 5, 8, 5, 3, 2, 2]
Predicted: [4, 6, 6, 7, 8, 9, 7, 4, 6, 1, 4, 0, 9, 9, 3, 7, 8, 0, 7, 5, 8, 5, 3, 2, 2]
```

Let us visualize some of the results

```
In [24]:
```

```
# Plot a random image
sample_number = 5
plt.imshow(img_data[sample_number].reshape(28,28), cmap="gray_r")
plt.axis('off')
img_gt, img_pred = gtlabel[sample_number], pred[sample_number]
print("Image Label: ", img_pred)
```

```
Image Label: 9
```

**Exploration Suggestion** - Try exploring how the classifier behaves
with different parameters, e.g. changing the `minibatch_size`

parameter from 25 to say 64 or 128. What happens to the error rate? How
does the error compare to the logistic regression classifier? - Try
increasing the number of sweeps - Can you change the network to reduce
the training error rate? When do you see *overfitting* happening?

**Code link**

If you want to try running the tutorial from Python command prompt please run the SimpleMNIST.py example.