Gradient Descent, Step-by-Step

This paragraph introduces the concept of Gradient Descent as a method for estimating parameters in various data science fields, such as statistics, machine learning, and optimization problems. It explains that Gradient Descent can optimize a wide range of things, from fitting a line in linear regression to optimizing clusters in t-SNE. The paragraph sets the stage for a detailed exploration of the Gradient Descent algorithm, starting with a simple data set of weight and height to illustrate how the algorithm finds optimal values for the intercept and slope of a line. It emphasizes the importance of understanding least squares and linear regression basics before diving into Gradient Descent.

This section delves into the specifics of using Gradient Descent to optimize the intercept of a line. It explains the process of starting with a random initial guess for the intercept and using the sum of squared residuals as a loss function to evaluate the line's fit to the data. The paragraph highlights the concept of residuals, showing how they are calculated and summed to find the optimal intercept. It also introduces the idea of plotting the sum of squared residuals against different intercept values and discusses the efficiency of Gradient Descent over manual optimization methods.

This paragraph focuses on the mathematical aspect of Gradient Descent, particularly the derivation of the sum of squared residuals with respect to the intercept. It explains the process of taking the derivative of the loss function and how it is used to find the optimal value for the intercept. The paragraph introduces the concept of the learning rate and how it determines the step size in the optimization process. It also touches on the idea of taking smaller steps as the algorithm nears the optimal value to fine-tune the solution. The explanation includes a step-by-step walkthrough of the derivative calculation, emphasizing the importance of the chain rule and the role of the slope in guiding the search for the minimum.

This section describes the iterative process of Gradient Descent, starting with the calculation of the derivative at a given intercept and using it to update the intercept value. It explains how the algorithm takes larger steps when far from the optimal value and smaller steps as it approaches the optimum. The paragraph illustrates the process with an example, showing how the intercept is updated through multiple iterations and how the sum of squared residuals decreases with each step. It also introduces the concept of stopping criteria for Gradient Descent, either when the step size is very small or a maximum number of steps have been reached.

This paragraph extends the discussion to include both the intercept and the slope in the optimization process using Gradient Descent. It describes the 3D graph representation of the loss function with respect to the intercept and slope, and the need to take derivatives with respect to both parameters. The section explains the process of calculating new intercept and slope values using the derivatives and learning rate. It also touches on the concept of the gradient, which is a collection of derivatives with respect to each parameter, and how it is used to guide the descent to the lowest point in the loss function.

This final paragraph discusses the sensitivity of Gradient Descent to the learning rate and how it can affect the convergence to the optimal solution. It highlights the importance of choosing an appropriate learning rate and introduces the concept of automatically adjusting the learning rate during the optimization process. The section also explains the iterative process for updating the intercept and slope using the calculated derivatives and learning rate. It concludes with an introduction to Stochastic Gradient Descent as a method to reduce computation time by using a subset of the data in each step, making the process more efficient for large data sets.