Summary of "Stanford CS229: Machine Learning - Linear Regression and Gradient Descent | Lecture 2 (Autumn 2018)"

Summary of Lecture 2: Linear Regression and Gradient Descent

Main Ideas and Concepts:

Introduction to Linear Regression:
Linear Regression is presented as one of the simplest supervised learning algorithms. It is used to predict continuous output values (e.g., housing prices) based on input features (e.g., size of the house).
Supervised Learning Framework:
In supervised learning, a training dataset is used where each example consists of input features (X) and the corresponding output (Y). The learning algorithm's goal is to output a hypothesis (function) that can predict the output for new input data.
Hypothesis Representation:
The hypothesis in Linear Regression is represented as a linear function of input features. For multiple features, the hypothesis can be expressed as:

h(X) = θ0 + θ1 X1 + θ2 X2 + ... + θn Xn

Using vector notation, this can be compactly written as:

h(X) = ∑j=0n θj Xj
Cost Function:
The Cost Function J(θ) is defined as the mean squared error between the predicted values and the actual values:

J(θ) = (1/2m) ∑i=1m (hθ(x(i)) - y(i))2

The goal is to minimize this Cost Function to find the optimal parameters θ.
Gradient Descent Algorithm:
Gradient Descent is an iterative optimization algorithm used to minimize the Cost Function. The update rule for Gradient Descent is:

θj := θj - α (∂J(θ)/∂θj)

Here, α is the learning rate, which determines the size of the steps taken towards the minimum.
Batch vs. Stochastic Gradient Descent:
- Batch Gradient Descent: Uses the entire dataset to compute the gradient and update parameters, which can be slow for large datasets.
- Stochastic Gradient Descent (SGD): Updates parameters using one training example at a time, leading to faster convergence but with more noise in the updates.
Normal Equation:
For Linear Regression, there exists a direct method to compute the optimal parameters without iteration, known as the Normal Equation:

θ = (XT X)-1 XT y

This method is efficient for smaller datasets but can be problematic if XT X is non-invertible.
Learning Rate Selection:
The learning rate is typically chosen empirically, with common starting values around 0.01. Adjustments may be made based on performance.

Methodology/Instructions:

To implement Linear Regression using Gradient Descent:
- Initialize parameters θ (e.g., to zeros).
- Compute the Cost Function J(θ).
- Update parameters using the Gradient Descent update rule until convergence.
- Optionally, monitor the Cost Function to determine when to stop.
To implement the Normal Equation:
- Construct the design matrix X from the training data.
- Compute the optimal parameters using the formula θ = (XT X)-1 XT y.

Speakers/Sources Featured:

The lecture is presented by an instructor from Stanford University as part of the CS229 Machine Learning course. Specific names are not mentioned in the subtitles.