# Fundamentals¶ This introductory module is focused on introducing several core technologies used for testing and debugging in future modules, and also includes some basic mathematical foundations. In this module, you will start to build up some of the infrastructure for MiniTorch.

All starter code is available in https://github.com/minitorch/Module-0 . Before starting this assignment, make sure to set up your workspace following Setup, and read Contributing to understand how the code should be organized.

Each module has a set of Guides to help with the tasks. We recommend working through the assignment and utilizing the Guides suggested for each task.

This task is designed to help you get comfortable with style checking and testing. We ask you to implement a series of basic mathematical functions. These functions are simple, but they form the basis of MiniTorch. Make sure that you understand each of them as some terminologies might be new.

Todo

Complete the following functions in minitorch/operators.py and pass tests marked as task0_1.

minitorch.operators.mul(x, y)

$$f(x, y) = x * y$$

minitorch.operators.id(x)

$$f(x) = x$$

minitorch.operators.eq(x, y)

$$f(x) =$$ 1.0 if x is equal to y else 0.0

minitorch.operators.neg(x)

$$f(x) = -x$$

minitorch.operators.add(x, y)

$$f(x, y) = x + y$$

minitorch.operators.max(x, y)

$$f(x) =$$ x if x is greater than y else y

minitorch.operators.lt(x, y)

$$f(x) =$$ 1.0 if x is less than y else 0.0

minitorch.operators.sigmoid(x)

$$f(x) = \frac{1.0}{(1.0 + e^{-x})}$$

Calculate as

$$f(x) = \frac{1.0}{(1.0 + e^{-x})}$$ if x >=0 else $$\frac{e^x}{(1.0 + e^{x})}$$

for stability.

Parameters

x (float) -- input

Returns

sigmoid value

Return type

float

minitorch.operators.relu(x)

$$f(x) =$$ x if x is greater than 0, else 0

Parameters

x (float) -- input

Returns

relu value

Return type

float

minitorch.operators.inv(x)

$$f(x) = 1/x$$

minitorch.operators.inv_back(x, d)

If $$f(x) = 1/x$$ compute $$d \times f'(x)$$

minitorch.operators.relu_back(x, d)

If $$f = relu$$ compute $$d \times f'(x)$$

minitorch.operators.log_back(x, d)

If $$f = log$$ as above, compute $$d \times f'(x)$$

### Task 0.2: Testing and Debugging¶

Note

This task requires familiarity with testing and property tests. Make sure to first read the guide on Property Testing, and consult the Hypothesis documentation.

We ask you to implement property tests for your operators from Task 0.1. These tests should ensure that your functions not only work but also obey high-level mathematical properties for any input. Note that you need to change arguments for those test functions.

Todo

Complete the test functions in tests/test_operators.py marked as task0_2.

Note

This task requires familiarity with basic functional programming concepts and notation. This subject is outside the scope of guides provided here, but this tutorial is a good starting place.

To practice the use of higher-order functions in Python, implement three basic functional concepts. Use them in combination with operators described in Task 0.1 to build up more complex mathematical operations that work on lists instead of single values.

Todo

Complete the following functions in minitorch/operators.py and pass tests marked as tasks0_3.

minitorch.operators.map(fn)

Higher-order map. Parameters

fn (one-arg function) -- Function from one value to one value.

Returns

A function that takes a list, applies fn to each element, and returns a new list

Return type

function

minitorch.operators.negList(ls)

Use map() and neg() to negate each element in ls

minitorch.operators.zipWith(fn)

Higher-order zipwith (or map2). Parameters

fn (two-arg function) -- combine two values

Returns

takes two equally sized lists ls1 and ls2, produce a new list by applying fn(x, y) on each pair of elements.

Return type

function

minitorch.operators.addLists(ls1, ls2)

Add the elements of ls1 and ls2 using zipWith() and add()

minitorch.operators.reduce(fn, start)

Higher-order reduce. Parameters
• fn (two-arg function) -- combine two values

• start (float) -- start value $$x_0$$

Returns

function that takes a list ls of elements $$x_1 \ldots x_n$$ and computes the reduction $$fn(x_3, fn(x_2, fn(x_1, x_0)))$$

Return type

function

minitorch.operators.sum(ls)

Sum up a list using reduce() and add().

minitorch.operators.prod(ls)

Product of a list using reduce() and mul().

Note

This task requires familiarity with neural network Modules. Please read Modules to get started. If you want more context for how modules are used, you may find it helpful to skip ahead and read the torch module tutorial.

This task is to implement the core structure of the minitorch.Module class. We ask you to implement a tree data structure that stores named minitorch.Parameter on each node. Such a data structure makes it easy for users to create trees that can be walked to find all of the parameters of interest.

To experiment with the system use the Module Sandbox:

>>> streamlit run app.py -- 0


Todo

Complete the functions in minitorch/module.py and pass tests marked as tasks0_4.

minitorch.Module.train(self)

Set the mode of this module and all descendent modules to train.

minitorch.Module.eval(self)

Set the mode of this module and all descendent modules to eval.

minitorch.Module.named_parameters(self)

Collect all the parameters of this module and its descendents.

Returns

Contains the name and Parameter of each ancestor parameter.

Return type

list of pairs

minitorch.Module.parameters(self)

Enumerate over all the parameters of this module and its descendents.

Note

This task requires familiarity with visualization tools described in Visualization.

For the first few assignments, we use a set of datasets implemented in minitorch/datasets.py, which are 2D point classification datasets. (See TensorFlow Playground for similar examples.) Each of these dataset can be added to the visualization.

To experiment with the system use:

>>> streamlit run app.py -- 0


Read through the code in project/run_torch.py to get a sneak peek of an implementation of a model for these datasets using Torch.

You can also provide a model that attempts to perform the classification by manipulating the parameters. Todo

Start a streamlit server and print an image of the dataset. Hand-create classifiers that split the linear dataset into the correct colors.