The Gradient of Mean Squared Error — Topic 78 of Machine Learning Foundations

Jon Krohn

1 Dec 202124:22

EducationalLearning

32 Likes 10 Comments

TLDRThis video script offers an in-depth look at the process of manually deriving the gradient of mean squared error (MSE) in the context of a regression model. The presenter begins by confirming the partial derivatives of MSE, comparing manual calculations with those obtained through automatic differentiation. The video then delves into a step-by-step explanation of how to calculate the gradient of cost with respect to model parameters, such as the slope (m) and y-intercept (b), using the chain rule. The presenter also discusses the implications of these derivatives, demonstrating how they indicate the direction and magnitude of changes needed to minimize cost. The script concludes with a practical demonstration of gradient descent, showing how the model parameters are adjusted over multiple training rounds to reduce cost and improve the fit of the regression line to the data. The visual representation of this process helps solidify the understanding of gradient descent and its role in machine learning.

Takeaways

📚 The video demonstrates deriving the gradient of mean squared error (MSE) by hand, confirming manual calculations with automatic differentiation.
🔍 The process involves calculating partial derivatives of the cost function with respect to the model parameters, m (slope) and b (y-intercept).
🤖 Automatic differentiation (autodiff) is used for convenience, but the video emphasizes understanding the underlying calculus involved in gradient calculation.
📉 Mean squared error is calculated by averaging the quadratic cost over all instances, which is the sum of the squared differences between predicted and actual values.
🧮 The partial derivative of MSE with respect to the predicted value y hat is found to be 1, simplifying the calculation process.
🔗 The chain rule is applied to find the partial derivatives of the cost function with respect to the model parameters, using previously found partial derivatives.
📈 The gradient descent algorithm is visualized, showing how the model parameters m and b are adjusted to minimize the cost function over multiple training rounds.
⚙️ A learning rate is introduced during the gradient descent process to determine the step size for adjusting the model parameters.
📊 The video includes a visualization tool to plot the regression line, cost, and gradients, aiding in understanding how the model converges to the optimal parameters.
🔢 The final parameter values obtained from the gradient descent optimization match those from previous methods, reinforcing the validity of the manual derivations.
🔬 The video concludes with an emphasis on the importance of understanding partial derivatives and the gradient for machine learning models, setting the stage for further optimization techniques in subsequent tutorials.

Q & A

What is the main focus of the video?
-The video focuses on deriving the gradient of mean squared error manually, comparing it with automatic differentiation results, and visualizing gradient descent in action over multiple training rounds.
What is the formula for calculating mean squared error?
-Mean squared error is calculated by taking the sum of the squared differences between the predicted y values (y hat) and the true y values, dividing by the number of instances (n), and then averaging the quadratic cost.
How does the video approach the calculation of the gradient of cost with respect to model parameters?
-The video first calculates the partial derivatives of the mean squared error with respect to the predicted y (y hat), and then uses the chain rule to find the partial derivatives with respect to the model parameters m (slope) and b (y-intercept).
What is the role of automatic differentiation (autodiff) in this context?
-Automatic differentiation is used to calculate the gradient of the cost function with respect to the model parameters. The video then compares these automatic calculations with manual derivations to confirm their accuracy.
How does the video demonstrate the gradient descent process?
-The video demonstrates gradient descent by adjusting the model parameters (m and b) based on the calculated gradients, using a learning rate, and iterating this process over multiple epochs to minimize the cost function.
What is the significance of visualizing the gradient and cost during the training process?
-Visualizing the gradient and cost helps to understand the optimization process, showing how the model parameters are adjusted to minimize the cost function and how the steepness of the gradient changes as the parameters approach their optimal values.
What is the purpose of the learning rate in gradient descent?
-The learning rate determines the size of the adjustments made to the model parameters during each iteration of gradient descent. It is a crucial hyperparameter that affects the convergence of the model to the optimal parameters.
How does the video ensure that the manual calculations of the gradient match the automatic differentiation results?
-The video performs the manual derivations of the partial derivatives and then implements these derivations in code. It then executes the code and compares the output with the results from automatic differentiation to ensure they match.
What is the significance of the partial derivative of the cost with respect to the model parameters?
-The partial derivatives of the cost with respect to the model parameters indicate the direction and magnitude of the steepest increase in the cost function. In the context of gradient descent, these derivatives guide the optimization process by pointing in the direction of the steepest decrease in the cost.
How does the video handle the complexity of dealing with multiple instances in the dataset?
-The video uses the concept of indexing with 'i' to handle individual instances within the dataset. This allows for the calculation of partial derivatives and the application of the chain rule while maintaining clarity on whether calculations pertain to individual instances or the average across all instances.
What is the final step in the machine learning process demonstrated in the video?
-The final step demonstrated is performing gradient descent to optimize the model parameters. This is done by adjusting the parameters based on the calculated gradients and the learning rate, with the goal of minimizing the cost function.

Outlines

00:00

🧮 Deriving the Gradient of Mean Squared Error

This paragraph introduces the video's aim to manually derive the gradient of mean squared error (MSE), compare manual calculations with automatic differentiation (autodiff), and visualize gradient descent. It begins with a review of the previous video, which involved loading data and using a regression model to estimate outputs (y hats). The focus then shifts to calculating MSE and using autodiff to find the gradient of the cost function concerning model parameters m (slope) and b (y-intercept). The speaker also mentions a prior manual calculation of partial derivatives for quadratic cost, indicating a similar approach will be taken for MSE.

05:01

🔗 Chain Rule Application for Partial Derivatives

The second paragraph delves into the mathematical process of calculating partial derivatives using the chain rule. It starts by establishing the partial derivatives of the cost function with respect to the predicted output y hat. The speaker simplifies the expression for MSE with respect to y hat by introducing a new variable u, representing the difference between y hat and the true y value. The partial derivatives of u with respect to y hat and the cost with respect to u are calculated. The paragraph concludes with the expressions needed for calculating the gradient of MSE concerning the model parameters m and b, using the previously found partial derivatives.

10:02

📉 Gradient Descent in Action

The third paragraph discusses the application of the derived partial derivatives in the context of gradient descent. It begins by outlining the process of calculating the partial derivatives of the cost function with respect to the model parameters m and b. The speaker uses the chain rule to combine the previously found partial derivatives to obtain the final expressions needed for gradient descent. The paragraph also includes a brief mention of a prior video that covered the derivation of partial derivatives for quadratic cost. The speaker then transitions to executing the code to confirm the manual derivations match the results from autodiff, providing a practical demonstration of the theoretical concepts.

15:02

📈 Visualizing Gradient Descent and Model Parameters

In this paragraph, the focus is on visualizing the process of gradient descent and the evolution of model parameters over time. The speaker introduces a function to plot the regression line, cost, and gradients, providing a visual representation of the training process. The initial setup includes randomly initialized values for the slope (m) and y-intercept (b) of the regression line. The partial derivatives are calculated and plotted, showing the gradient of the cost function concerning m and b. The speaker then outlines the steps for performing gradient descent, emphasizing the positive correlation between the increase in m or b and the cost, and how adjustments to these parameters will reduce the cost.

20:05

🔍 Optimizing Model Parameters through Gradient Descent

The final paragraph describes the iterative process of gradient descent optimization. It begins with the application of the gradient descent optimizer on the model parameters m and b, using a learning rate to determine the amount of adjustment. The speaker explains that after each training round (epoch), the gradients are reset to zero to prevent accumulation and memory issues. The process involves a forward pass to calculate the model's predictions, followed by the calculation of the cost using MSE, and then the performance of gradient descent to update the parameters. The speaker visualizes the training process over multiple epochs, showing how the cost decreases and the model parameters adjust to better fit the data. The paragraph concludes with the final optimized parameter values for m and b, and an invitation to follow the speaker on various social media platforms for further updates.

Mindmap

Keywords

💡Gradient Descent

Gradient Descent is an optimization algorithm used to minimize a function by iteratively moving towards the steepest descent of the gradient. In the context of the video, it is used to minimize the cost function of a regression model by updating the parameters 'm' (slope) and 'b' (y-intercept). The video demonstrates how the algorithm adjusts these parameters to fit the data points better with each iteration.

💡Mean Squared Error (MSE)

Mean Squared Error is a measure of the average squared differences between the estimated values and the actual value. It is used as a cost function in the video to quantify the difference between the model's predictions (y hat) and the true values (y). The goal is to minimize this cost function, which represents the error of the model.

💡Partial Derivatives

Partial derivatives are derivatives of a function with respect to a single variable while keeping the other variables constant. In the video, partial derivatives of the cost function with respect to the model parameters 'm' and 'b' are calculated. These derivatives are essential for determining the direction and magnitude of the parameter updates in the Gradient Descent algorithm.

💡Automatic Differentiation

Automatic Differentiation, often abbreviated as autodiff, is a set of techniques to compute derivatives of numerical functions. In the video, it is used to calculate the gradient of the cost function with respect to the model parameters. The script compares the results from autodiff with manual calculations to confirm their correctness.

💡Model Parameters

Model parameters are the variables that define the model's behavior. In the context of the video, 'm' (slope) and 'b' (y-intercept) are the parameters of the linear regression model. The video focuses on finding the optimal values for these parameters to minimize the cost function.

💡Chain Rule

The Chain Rule is a fundamental theorem in calculus used to compute the derivative of a composite function. In the video, the Chain Rule is applied to calculate the partial derivatives of the cost function with respect to the model parameters 'm' and 'b', which are nested inside the predicted values 'y hat'.

💡Cost Function

A Cost Function, also known as a loss function or objective function, measures how well a model's predictions match the actual data. In the video, the Mean Squared Error is used as the cost function. The process of minimizing this function through Gradient Descent is central to training the regression model.

💡Regression Model

A Regression Model is a statistical model used to predict a continuous dependent variable from one or more independent variables. In the video, a simple linear regression model is used, represented by the equation 'y hat = m * x + b', where 'm' is the slope and 'b' is the y-intercept.

💡Learning Rate

The Learning Rate is a hyperparameter used in Gradient Descent to control the step size at each iteration while moving toward a minimum of the cost function. A properly chosen learning rate helps in efficiently converging to the optimal solution, as discussed in the video.

💡Epoch

An Epoch refers to one complete pass through the entire training dataset. In the context of the video, each round of training where the Gradient Descent algorithm is applied to all training examples is considered one epoch. The process is repeated for multiple epochs until the model parameters converge to their optimal values.

💡Stochastic Gradient Descent (SGD)

Stochastic Gradient Descent is a variant of the Gradient Descent algorithm where the gradient is calculated using only one training example (or a small batch) at a time, rather than the entire dataset. Although the video primarily discusses the standard Gradient Descent, it mentions the use of an optimizer which could be an implementation of SGD.

Highlights

The video demonstrates the manual derivation of the gradient of mean squared error.

Comparison of manually computed partial derivatives with those calculated using automatic differentiation.

Visualization of gradient descent's impact over multiple training rounds.

Explanation of how to calculate the partial derivatives of mean squared error with respect to the predicted value, y hat.

Derivation of the partial derivative of cost with respect to the model parameters m (slope) and b (y-intercept).

Use of the chain rule to find the gradient of mean squared error concerning model parameters.

Confirmation of the manual derivation's accuracy by comparing it with autodiff results in a code demonstration.

Introduction of a function to visualize labeled regression plots, including model parameters, cost, and optionally the gradient.

Illustration of how the gradient's direction influences the adjustment of model parameters during gradient descent.

Discussion on the positive correlation between an increase in the model parameter and the cost, and its implication on parameter tuning.

Application of a learning rate in the stochastic gradient descent optimizer for model training.

Reiteration of the machine learning process involving forward pass, cost calculation, gradient descent, and parameter update.

Execution of multiple epochs of training to observe the descent of cost and the convergence of model parameters.

Analysis of the relationship between cost and model parameters, highlighting when to increase or decrease parameters for cost reduction.

Achievement of a low, near-zero cost value indicating a well-fitted model to the data points.

Final model parameters obtained after extensive training, showcasing the effectiveness of the gradient descent method.

Emphasis on understanding the underlying calculus of partial derivatives for a deeper comprehension of gradient descent.

Invitation to follow the creator on social media and subscribe for the next video in the series for further insights into optimization.

Transcripts

Browse More Related Video

Gradient Descent (Hands-on with PyTorch) — Topic 77 of Machine Learning Foundations

Machine Learning from First Principles, with PyTorch AutoDiff — Topic 66 of ML Foundations

Backpropagation — Topic 79 of Machine Learning Foundations

Gradient Descent, Step-by-Step

The Chain Rule for Derivatives — Topic 59 of Machine Learning Foundations

Multiple Linear Regression: Three independent variables case

Related Tags

Machine Learning Gradient Descent Mean Squared Error Autodiff Derivation Model Optimization Partial Derivatives Regression Analysis Data Fitting Learning Rate Cost Minimization