This is part 1 of a series of tutorials, in which 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.

- Part I: Computational Graphs
- Part II: Perceptrons
- Part III: Training criterion
- Part IV: Gradient Descent and Backpropagation
- Part V: Multi-Layer Perceptrons
- Part VI: TensorFlow

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).

The full source code of the API can be found at https://github.com/danielsabinasz/TensorSlow. You also find a Jupyter Notebook there, which is equivalent to this blog post but allows you to fiddle with the code.

# 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 $z : \mathbb{R^2} \rightarrow \mathbb{R}$ where $z(x, y) = x + y$.

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 $z(A, x, b) = Ax + b$.

## 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:

### 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). 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.

#### Addition

#### Matrix multiplication

## 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**.

## 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**.

## 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.

## Example

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

$$

z = \begin{pmatrix}

1 & 0 \\

0 & -1

\end{pmatrix}

\cdot

x

+

\begin{pmatrix}

1 \\

1

\end{pmatrix}

$$

## 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 = \begin{pmatrix}

1 & 0 \\

0 & -1

\end{pmatrix}

\cdot

\begin{pmatrix}

1 \\

2

\end{pmatrix}

+

\begin{pmatrix}

1 \\

1

\end{pmatrix}

=

\begin{pmatrix}

2 \\

-1

\end{pmatrix}

$$

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.

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

Looks good.

If you have any questions, feel free to leave a comment. Otherwise, continue with the next part: II: Perceptrons.

Valentine Michael Smith12th September 2017 at 10:49 amThis is the website I’ve been looking for for months. Really, really nicely done. Thanks!

thecity213th September 2017 at 7:32 pmWhere is `dot()` coming from?

Daniel Sabinasz16th September 2017 at 8:22 pmWe assume that our multi-dimensional data are numpy arrays, which have such a dot function. I’ll try to incorporate that into the text.

Emile Petrone22nd September 2017 at 7:35 pmWhere is operation.output defined?

Daniel Sabinasz24th September 2017 at 3:14 pmIn the Session.run method, we iterate over all nodes in order and set their output (node.output). “operation” is one of these nodes, so after the loop we know that its output has been set and we can return it.

Huang Pu26th September 2017 at 7:32 amExcellent blogs! Can I translate your blog into Chinese? Of course I will keep your original blog URL. 🙂

Daniel Sabinasz26th September 2017 at 1:33 pmI’d very much welcome a translation. However, please do not create a Chinese copy of the website under the same name (like deepideas.cn). If you send me your translations, I can post them as translations to deepideas.net in your name. Or, if you prefer, you could post them on your own personal blog and link back to the original version.

geo200911th November 2017 at 9:25 pmWhen i try to compile the code i am getting error “NameError: name ‘add’ is not defined”

can someone tell me what i am missing

thanks.

geo200911th November 2017 at 9:35 pmsorry, i missed the definition off add in a code. Had matmul though 🙂

So please ignore question.

Victor Abreu29th July 2020 at 1:51 pmDaniel, that is a really good blog post. Really in line with what I was looking for.

I implemented your code, but I got a error. I tried to debug the code, but I did not have much luck:

AttributeError: ‘add’ object has no attribute ‘output’

any ideas of why phyton is returning this error?

Mike29th July 2020 at 1:51 pmHave the following error: ‘Variable’ object has no attribute ‘dot’

for both hand-copied and direct-copied code (into a single file)

Any ideas?

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Iakolev Ivan29th July 2020 at 2:01 pmThank you for this very nice tutorial!

Hızır Can Bayram29th July 2020 at 2:01 pmQuestion 1:

Can we say placeholders contain variables as a set? Or these two are seperated from each other?

While A and b are variables, x is a placeholder. According to the definition, any computational graph has to be consisted of operations or variables. How about placeholders? It seems to me that you also said that placeholders and variables are different:

“In the affine transformation graph, there is a qualitative difference between x on the one hand and A and b on the other hand. “

Placeholders are in this graph too. Should we change the definition of computatinal graph from “… correspon to operations or variables” to “… correspond to operations or inputs”, since you call inputs both variables and placeholders:

“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.”

Actually, there is another statement that you made , which is

“While x is an input to the operation, A and b are parameters of the operation, i.e. they are intrinsic to the graph.”

It seems inputs dont contain both placeholders and variables.

Summary: There is a confusion for me here. I hope you will take care of it by explaning following keywords in more detail once for me: Input, variable, placeholder, parameter.

Question 2:

About tensors… Does a tensor have to provide the following two conditions?

-: values that are fed into the nodes

–: values that are come out of the nodes.

Cant we define A, x, Z tensor, while we can y? Or… We can define all these nodes in the graph as a tensor?

MS29th July 2020 at 2:01 pm# Iterate all nodes to determine their value

for node in nodes_postorder:

if type(node) == placeholder:

# Set the node value to the placeholder value from feed_dict

node.output = feed_dict[node]

elif type(node) == Variable:

# Set the node value to the variable’s value attribute

node.output = node.value

else: # Operation

# Get the input values for this operation from node_values

node.inputs = [input_node.output for input_node in node.input_nodes]

# Compute the output of this operation

node.output = node.compute(*node.inputs)

# Convert lists to numpy arrays

if type(node.output) == list:

node.output = np.array(node.output)

# Return the requested node value

return operation.output

Precisely, what are “.inputs” & “.output” from Python perspective?