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Deep Learning From Scratch - Theory and Implementation

DanielSabinasz
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Next: Perceptrons

Deep Learning From Scratch: Theory and Implementation

In this tutorial, we develop the mathematical and algorithmic underpinnings of deep neural networks from scratch and implement our own neural network library in Python, mimicing the TensorFlow API. I do not assume that you have any preknowledge about machine learning or neural networks. However, you should have some preknowledge of calculus, linear algebra, fundamental algorithms and probability theory on an undergraduate level. If you get stuck at some point, please leave a comment.

By the end of this text, you will have a deep understanding of the math behind neural networks and how deep learning libraries work under the hood.

I have tried to keep the code as simple and concise as possible, favoring conceptual clarity over efficiency. Since our API mimics the TensorFlow API, you will know how to use TensorFlow once you have finished this text, and you will know how TensorFlow works under the hood conceptually (without all the overhead that comes with an omnipotent, maximally efficient machine learning API).

In order to stay updated when further lessons are added, you can subscribe to my blog at deepideas.net via Facebook, Twitter or Newsletter.

Computational Graphs

We shall start by defining the concept of a computational graph, since neural networks are a special form thereof. A computational graph is a directed graph where the nodes correspond to operations or variables. Variables can feed their value into operations, and operations can feed their output into other operations. This way, every node in the graph defines a function of the variables.

The values that are fed into the nodes and come out of the nodes are called tensors, which is just a fancy word for a multi-dimensional array. Hence, it subsumes scalars, vectors and matrices as well as tensors of a higher rank.

Let's look at an example. The following computational graph computes the sum z of two inputs x and y. Here, x and y are input nodes to z and z is a consumer of x and y. z therefore defines a function z:R2R where z(x,y)=x+y.

Image

The concept of a computational graph becomes more useful once the computations become more complex. For example, the following computational graph defines an affine transformation z(A,x,b)=Ax+b.

Image

Operations

Every operation is characterized by three things:

  • A compute function that computes the operation's output given values for the operation's inputs
  • A list of input_nodes which can be variables or other operations
  • A list of consumers that use the operation's output as their input

Let's put this into code:

Operation
class Operation:
"""Represents a graph node that performs a computation.
An `Operation` is a node in a `Graph` that takes zero or
more objects as input, and produces zero or more objects
as output.
"""
def __init__(self, input_nodes=[]):
"""Construct Operation
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Some elementary operations

Let's implement some elementary operations in order to become familiar with the Operation class (and because we will need them later).

Addition
class add(Operation):
"""Returns x + y element-wise.
"""
def __init__(self, x, y):
"""Construct add
Args:
x: First summand node
y: Second summand node
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Matrix multiplication
class matmul(Operation):
"""Multiplies matrix a by matrix b, producing a * b.
"""
def __init__(self, a, b):
"""Construct matmul
Args:
a: First matrix
b: Second matrix
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

In both of these operations, we assume that the tensors are NumPy arrays, in which the element-wise addition and matrix multiplication (.dot) are already implemented for us.

Placeholders

Not all the nodes in a computational graph are operations. For example, in the affine transformation graph, A, x and b are not operations. Rather, they are inputs to the graph that have to be supplied with a value once we want to compute the output of the graph. To provide such values, we introduce placeholders.

Placeholder
class placeholder:
"""Represents a placeholder node that has to be provided with a value
when computing the output of a computational graph
"""
def __init__(self):
"""Construct placeholder
"""
self.consumers = []
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Variables

In the affine transformation graph, there is a qualitative difference between x on the one hand and A and b on the other hand. While x is an input to the operation, A and b are parameters of the operation, i.e. they are intrinsic to the graph. We will refer to such parameters as Variables.

Variable
class Variable:
"""Represents a variable (i.e. an intrinsic, changeable parameter of a computational graph).
"""
def __init__(self, initial_value=None):
"""Construct Variable
Args:
initial_value: The initial value of this variable
"""
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

The Graph class

Finally, we'll need a class that bundles all the operations, placeholders and variables together. When creating a new graph, we can call its as_default method to set the _default_graph to this graph. This way, we can create operations, placeholders and variables without having to pass in a reference to the graph everytime.

Graph
class Graph:
"""Represents a computational graph
"""
def __init__(self):
"""Construct Graph"""
self.operations = []
self.placeholders = []
self.variables = []
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Example

Let's now use the classes we have built to create a computational graph for the following affine transformation:

z=(1001)x+(11)
Example
# Create a new graph
Graph().as_default()
# Create variables
A = Variable([[1, 0], [0, -1]])
b = Variable([1, 1])
# Create placeholder
x = placeholder()
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Computing the output of an operation

Now that we are confident creating computational graphs, we can start to think about how to compute the output of an operation.

Let's create a Session class that encapsulates an execution of an operation. We would like to be able to create a session instance and call a run method on this instance, passing the operation that we want to compute and a dictionary containing values for the placeholders:

session = Session()
output = session.run(z, {
    x: [1, 2]
})

This should compute the following value:

z=(1001)(12)+(11)=(21)

In order to compute the function represented by an operation, we need to apply the computations in the right order. For example, we cannot compute z before we have computed y as an intermediate result. Therefore, we have to make sure that the operations are carried out in the right order, such that the values of every node that is an input to an operation o has been computed before o is computed. This can be achieved via post-order traversal.

Session
import numpy as np
class Session:
"""Represents a particular execution of a computational graph.
"""
def run(self, operation, feed_dict={}):
"""Computes the output of an operation
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Let's test our class on the example from above:

Example
session = Session()
output = session.run(z, {
x: [1, 2]
})
print(output)
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
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