Neural Networks Pt. 2: Backpropagation Main Ideas

Summary

In this StatQuest episode, Josh Starmer simplifies the concept of backpropagation in neural networks, focusing on optimizing weights and biases. He explains using the chain rule to calculate derivatives and applying gradient descent to find the optimal parameters. The tutorial illustrates the process with a neural network adjusting to fit a dataset, demonstrating how to minimize the sum of squared residuals by iteratively updating the bias term, b3, until the network's predictions closely match the actual data.

Takeaways

• 🧠 Backpropagation is a method used to optimize the weights and biases in neural networks, despite being a complex process with many details.
• 📈 The script assumes prior knowledge of neural networks, the chain rule, and gradient descent, and provides links for further study.
• 🔍 It focuses on the main ideas of backpropagation, which includes using the chain rule to calculate derivatives and applying these to gradient descent for parameter optimization.
• 📚 The explanation begins by naming each weight and bias in the neural network to clarify which parameters are being discussed.
• 🔄 Backpropagation conceptually starts from the last parameter and works backward to estimate all other parameters, but the script simplifies this by focusing on estimating just the last bias, b3.
• 📉 The process involves adjusting the neural network's output to minimize the sum of squared residuals, which quantify the difference between observed and predicted values.
• 📊 Summation notation is used to simplify the expression for the sum of squared residuals, making it easier to handle mathematically.
• 🔢 The chain rule is essential for finding the derivative of the sum of squared residuals with respect to the unknown parameter, in this case, b3.
• 📈 Gradient descent is used to iteratively adjust the value of b3 to minimize the sum of squared residuals, moving towards the optimal value.
• 📌 The script demonstrates the calculation of the derivative and the application of gradient descent with a learning rate to update the parameter value.
• 🎯 The optimal value for b3 is found when the step size in gradient descent is close to zero, indicating minimal change and convergence on the best fit.

Q & A

What is the main topic of the StatQuest video?

The main topic of the video is Neural Networks, specifically focusing on Part 2: Backpropagation Main Ideas.

What are the prerequisites for understanding the video on backpropagation?

The prerequisites include familiarity with neural networks, the chain rule, and gradient descent.

How does a neural network fit a curve to a dataset?

A neural network fits a curve to a dataset by adjusting weights and biases on the connections to flip and stretch activation functions into new shapes, which are then added together to form a squiggle that fits the data.

What is the purpose of backpropagation in neural networks?

Backpropagation is used to optimize the weights and biases in neural networks to improve the fit of the model to the data.

Why is the chain rule used in backpropagation?

The chain rule is used to calculate the derivatives of the sum of squared residuals with respect to the parameters of the neural network, which is necessary for gradient descent optimization.

What is gradient descent and how does it relate to backpropagation?

Gradient descent is an optimization algorithm used to minimize the cost function in backpropagation by iteratively moving in the direction of the steepest descent as defined by the derivative of the cost function with respect to the parameters.

How does the video simplify the explanation of backpropagation?

The video simplifies backpropagation by initially focusing on estimating the last bias, b3, and then gradually introducing the concepts of the chain rule and gradient descent.

What is the initial value assigned to the bias b3 in the video?

The initial value assigned to the bias b3 is 0, as bias terms are frequently initialized to 0.

How is the sum of squared residuals calculated?

The sum of squared residuals is calculated by taking the difference between observed and predicted values (residuals), squaring each residual, and then summing all the squared residuals together.

What is the role of the learning rate in gradient descent?

The learning rate in gradient descent determines the step size at each iteration, affecting how quickly the algorithm converges to the optimal value of the parameters.

How does the video demonstrate the optimization of the bias b3?

The video demonstrates the optimization of b3 by using gradient descent, starting with an initial value, calculating the derivative of the sum of squared residuals with respect to b3, and iteratively updating the value of b3 until the step size is close to zero.

Outlines

00:00

🧠 Introduction to Backpropagation in Neural Networks

This paragraph introduces the concept of backpropagation in the context of neural networks. It emphasizes that backpropagation is a method for optimizing weights and biases within a neural network. The paragraph explains that the neural network starts with basic activation functions, which are then adjusted through weights and biases to fit a given dataset. The focus is on the main ideas behind backpropagation: using the chain rule to calculate derivatives and applying these derivatives in gradient descent to optimize the network's parameters. The example of optimizing the last bias, b3, is used to illustrate the process, starting with the assumption that all other parameters are already optimized.

05:04

📈 Sum of Squared Residuals and Gradient Descent

The second paragraph delves into the process of quantifying the fit of the neural network to the data using the sum of squared residuals. It explains how residuals are calculated as the difference between observed and predicted values and then squared and summed to provide a measure of the network's performance. The paragraph demonstrates how adjusting the bias term b3 affects the sum of squared residuals, with the goal of minimizing this value to improve the fit. The use of gradient descent is introduced as a method to efficiently find the optimal value for b3 by calculating the derivative of the sum of squared residuals with respect to b3 and iteratively updating the bias term.

10:09

🔍 Applying the Chain Rule for Derivatives in Backpropagation

This paragraph explains the application of the chain rule to calculate the derivative of the sum of squared residuals with respect to the bias term b3. It details the process of breaking down the derivative into two parts: the derivative of the sum of squared residuals with respect to the predicted values and the derivative of the predicted values with respect to b3. The paragraph simplifies the explanation by using summation notation and demonstrates the calculation steps, leading to the final derivative expression that can be used in gradient descent to find the optimal value for b3.

15:10

🚀 Optimizing b3 and Anticipating Future Topics

The final paragraph concludes the explanation of optimizing the bias term b3 using the derivative obtained from the previous discussion and gradient descent. It illustrates the iterative process of updating b3 to minimize the sum of squared residuals, eventually arriving at the optimal value. The paragraph also teases upcoming topics, which will cover optimizing all parameters in a neural network and introduce more advanced concepts and notation. Additionally, it includes a call to action for viewers to support the channel through various means and ends with a signature sign-off.

Keywords

💡Backpropagation

Backpropagation is a method used to calculate the gradient of the loss function with respect to the weights in a neural network. It is central to the training process of neural networks, allowing for the optimization of these weights. In the video, backpropagation is introduced as a means to understand and calculate the derivatives needed for gradient descent, which is used to optimize the neural network's parameters such as weights and biases.

💡Neural Networks

Neural networks are a set of algorithms designed to recognize patterns. They are inspired by the human brain's neural networks and are composed of interconnected nodes or 'neurons'. In the script, a neural network is described as fitting a 'green squiggle' to a dataset, demonstrating how it adjusts its weights and biases to minimize the error between its predictions and the actual data.

💡Chain Rule

The chain rule is a fundamental principle in calculus that allows the computation of the derivative of a composite function. In the context of the video, the chain rule is used to calculate the derivative of the loss function with respect to the network's parameters, which is essential for backpropagation.

💡Gradient Descent

Gradient descent is an optimization algorithm used to minimize a function by iteratively moving in the direction of the steepest descent as defined by the negative of the gradient. In the video, gradient descent is applied to adjust the neural network's parameters, such as the bias term b3, to minimize the sum of squared residuals.

💡Weights

In neural networks, weights are the numerical values that are assigned to the connections between the nodes. They are adjusted during the training process to minimize the error in the network's predictions. The script discusses how backpropagation optimizes these weights, starting with an example where all weights except for the last bias term are assumed to be optimal.

💡Biases

Biases are parameters in a neural network that are added to the weighted sum of inputs before it is passed through an activation function. They are used to provide an adjustable shift in the activation function. The script focuses on optimizing the bias term b3, demonstrating how it affects the position of the 'green squiggle' and ultimately the fit of the neural network to the data.

💡Activation Functions

Activation functions are mathematical equations that determine the output of a neural network node given a set of inputs. They introduce non-linearity into the network, allowing it to model complex patterns. The script mentions the 'soft plus activation function' as part of the process of creating the 'blue curve' in the neural network.

💡Sum of Squared Residuals

The sum of squared residuals is a measure of the discrepancy between the observed values and the values predicted by a model. It is used as a loss function in regression problems. In the video, the sum of squared residuals is calculated to quantify how well the neural network's 'green squiggle' fits the data, and its minimization is the goal of the optimization process.

💡Residuals

Residuals are the differences between the actual observed values and the values predicted by a model. They provide a measure of the error of the model's predictions. In the script, residuals are calculated and squared to find their contribution to the sum of squared residuals, which is then used to guide the optimization of the neural network.

💡Learning Rate

The learning rate is a hyperparameter in machine learning that controls the step size at each iteration while moving toward a minimum of a loss function. It is crucial in gradient descent as it determines how quickly or slowly the algorithm converges to the optimal solution. In the script, a learning rate of 0.1 is used to calculate the step size for updating the bias term b3.

💡Optimization

Optimization in the context of neural networks refers to the process of adjusting the network's parameters to minimize the loss function. It is the goal of training a neural network. The script demonstrates the optimization of the bias term b3 through the use of gradient descent, which iteratively updates the parameter to reduce the sum of squared residuals.

Highlights

Backpropagation is a method to optimize weights and biases in neural networks.

The process assumes familiarity with neural networks, the chain rule, and gradient descent.

A neural network fits a curve to a dataset by adjusting activation functions with weights and biases.

Backpropagation starts with the last parameter and works backwards to estimate other parameters.

The main ideas of backpropagation involve using the chain rule and gradient descent.

The chain rule is used to calculate derivatives for optimizing parameters.

Gradient descent is used to find the optimal value for parameters like the bias term b3.

Each weight and bias in the network is given a specific name for clarity in backpropagation.

The process of backpropagation is broken down into bite-sized pieces for simplicity.

The sum of squared residuals is used to quantify the fit of the model to the data.

Residuals are the differences between observed and predicted values.

Gradient descent optimizes parameters by iteratively adjusting them to minimize residuals.

The derivative of the sum of squared residuals with respect to a parameter is key for gradient descent.

The chain rule links the derivative of residuals to the predicted values and the parameters.

The derivative of predicted values with respect to a bias term is straightforward to calculate.

Gradient descent updates the parameter value based on the learning rate and the slope of residuals.

The optimal value for a parameter is found when the step size in gradient descent is close to zero.

Backpropagation will be further explained in subsequent videos to optimize all neural network parameters.

StatQuest study guides are available for offline review of statistics and machine learning concepts.

Support for StatQuest can come in various forms such as Patreon contributions or purchasing merchandise.

Transcripts

Backpropagation is a really big word, but it's not a really big deal. StatQuest!

Hello!

I'm Josh Starmer and welcome to StatQuest!

Today we're going to talk about Neural Networks, Part 2: Backpropagation Main Ideas.

Note: this StatQuest assumes that you are already familiar with neural networks, the

chain rule, and gradient descent. If not, check out the quests. The links are in the description below.

In the StatQuest on Neural Networks Part 1, Inside the Black Box, we started with

a simple dataset that showed whether or not different drug dosages were effective against a virus.

The low and high dosages were not effective, but the medium dosage was effective.

Then we talked about how a neural network like this one fits a green squiggle to this data set.

Remember, the neural network starts with identical activation functions, but, using

different weights and biases on the connections, it flips and stretches the activation

functions into new shapes, which are then added together to get a squiggle that is shifted to fit the data.

However, we did not talk about how to estimate the weights and biases.

So let's talk about how backpropagation optimizes the weights and biases in this, and other, neural networks.

Note: backpropagation is relatively simple, but there are a ton of details, so I split it up into bite-sized pieces.

In this part we talk about the main ideas of backpropagation.

One: using the chain rule to calculate derivatives, and Two: plugging the derivatives

into gradient descent to optimize parameters. In the next part we'll talk about how

the chain rule and gradient descent apply to multiple parameters simultaneously, and introduce some fancy notation.

Then we will go completely bonkers with the chain rule and show how to optimize all

seven parameters simultaneously in this neural network.

Bam!

First, so we can be clear about which specific weights we are talking about, let's

give each one a name: we have w1, w2, w3, and w4.

And let's name each bias: b1, b2, and b3.

Note: conceptually, backpropagation starts with the last parameter and works its way

backwards to estimate all of the other parameters.

However, we can discuss all of the main ideas behind a backpropagation by just estimating the last bias, b3.

So, in order to start from the back, let's assume that we already have optimal values

for all of the parameters except for the last bias term, b3.

Note: throughout this, and the next StatQuests, I'll make the parameter values that

have already been optimized green, and unoptimized parameters will be red.

Also, note: to keep the math simple, let's assume dosages go from 0, for low, to 1, for high.

Now, if we run dosages from 0 to 1 through the connection to the top node in the hidden

layer, then we get the x-axis coordinates for the activation function, that are all

inside this red box and when we plug the x-axis coordinates into the activation function

which, in this example, is the soft plus activation function, we get the corresponding

y-axis coordinates, and this blue curve.

Then we multiply the y-axis coordinates on the blue curve by negative 1.22 and we get the final blue curve.

Bam!

Now, if we run dosages from zero to one through the connection to the bottom node

in the hidden layer, then we get x-axis coordinates inside this red box.

Now we plug those x-axis coordinates into the activation function to get the corresponding

y-axis coordinates for this orange curve.

Now we multiply the y-axis coordinates on the orange curve by negative 2.3 and we end up with this final orange curve.

Bam!

Now we add the blue and orange curves together to get this green squiggle.

Now we are ready to add the final bias, b3, to the green squiggle.

Because we don't yet know the optimal value for b3, we have to give it an initial

value, and because bias terms are frequently initialized to 0, we will set b3 equal to 0.

Now, adding zero to all of the y-axis coordinates on the green squiggle leaves it right where it is.

However, that means the green squiggle is pretty far from the data that we observed.

We can quantify how good the green squiggle fits the data by calculating the sum of the squared residuals.

A residual is the difference between the observed and predicted values.

For example, this residual is the observed value, zero, minus the predicted value

from the green squiggle, negative 2.6. This residual is the observed value, one,

minus the predicted value from the green squiggle, negative 1.61.

Lastly, this residual is the observed value, 0, minus the predicted value from the green squiggle, negative 2.61.

Now we square each residual and add them all together to get 20.4 for the sum of the squared residuals.

So when b3 equals 0, the sum of the squared residuals equals 20.4. And that corresponds

to this location on this graph that has the sum of the squared residuals on the y -axis and the bias, b3, on the x-axis.

Now, if we increase b3 to 1, then we would add one to the y-axis coordinates on the

green squiggle and shift the green squiggle up one.

And we end up with shorter residuals.

When we do the math, the sum of the squared residuals equals 7.8, and that corresponds to this point on our graph.

If we increase b3 to 2, then the sum of the squared residuals equals 1.11. And if

we increase b3 to 3, then the sum of the squared residuals equals 0.46. And if we

had time to plug in tons of values for b3, we would get this pink curve, and we could

find the lowest point, which corresponds to the value for b3 that results in the

lowest sum of the squared residuals, here.

However, instead of plugging in tons of values to find the lowest point in the pink

curve, we use gradient descent to find it relatively quickly.

And that means we need to find the derivative of the sum of the squared residuals with respect to b3.

Now, remember the sum of the squared residuals equals the first residual squared,

plus all of the other squared residuals.

Now, because this equation takes up a lot of space, we can make it smaller by using summation notation.

The greek symbol sigma tells us to sum things together, and 'i' is an index for the

observed and predicted values that starts at one.

And the index goes from one to the number of values, 'n', which in this case is set to 3.

So, when 'i' equals one, we're talking about the first residual.

When 'i' equals two, we're talking about the second residual.

And when 'i' equals three, we are talking about the third residual.

Now let's talk a little bit more about the predicted values.

Each predicted value comes from the green squiggle, and the green squiggle comes from the last part of the neural network.

In other words, the green squiggle is the sum of the blue and orange curves, plus b3.

Now remember, we want to use gradient descent to optimize b3, and that means we need

to take the derivative of the sum of the squared residuals with respect to b3.

And because the sum of the squared residuals are linked to b3 by the predicted values,

we can use the chain rule to solve for the derivative of the sum of the squared residuals with respect to b3.

The chain rule says that the derivative of the sum of the squared residuals with respect

to b3 is the derivative of the sum of the squared residuals with respect to the predicted

values, times the derivative of the predicted values with respect to b3.

Now, before we calculate the derivative of the sum of the squared residuals with respect

to the predicted values, let's clean up our workspace and move these equations out of the way.

Now we can solve for the derivative of the sum of the squared residuals with respect

to the predicted values by first substituting in the equation, and then use the chain

rule to move the square to the front, and then we multiply that by the derivative

of the stuff inside the parentheses with respect to the predicted values, negative one.

Now we simplify by multiplying two by negative 1, and we have the derivative of the

sum of the squared residuals with respect to the predicted values.

So let's move that up here, and now we are done with the first part.

Now let's solve for the second part: the derivative of the predicted values with respect to b3.

We start by plugging in the equation for the predicted values.

Remember, the blue and orange curves were created before we got to b3.

So the derivative of the blue curve with respect to b3 is 0, because the blue curve is independent of b3.

And the derivative of the orange curve with respect to b3 is also 0.

Lastly, the derivative of b3, with respect to b3, is 1.

Now we just add everything up, and the derivative of the predicted values with respect to b3, is one.

So we multiply the derivative of the sum of the squared residuals with respect to the predicted values by 1.

Note: this times 1 part in the equation doesn't do anything, but I'm leaving it in

to remind us that the derivative of the sum of the squared residuals with respect

to b3 consists of two parts: the derivative of the sum of the squared residuals with

respect to the predicted values, and the derivative of the predicted values with respect to b3.

Bam! And at long last we have the derivative of the sum of the squared residuals with respect to b3.

And that means we can plug this derivative into gradient descent to find the optimal value for b3.

So let's move this equation up and show how we can use this equation with gradient descent.

Note: if you're not familiar with gradient descent, check out the quest

the link is in the description below.

Anyway, first, we expand the summation.

Then, we plug in the observed values and the values predicted by the green squiggle.

Remember, we get the predicted values on the green squiggle by running the dosages through the neural network.

Now, we just do the math and get negative 15.7. And that corresponds to the slope for when b3 equals zero.

Now we plug the slope into the gradient descent equation for step size, and, in this

example, we'll set the learning rate to 0.1. And that means the step size is -1.57.

Now we use the step size to calculate the new value for b3 by plugging in the current

value for b3, zero, and the step size, -1.57. And the new value for b3 is 1.57.

Changing b3 to 1.57 shifts the green squiggle up, and that shrinks the residuals.

Now, plugging in the new predicted values and doing the math gives us -6.26, which

corresponds to the slope when b3 equals 1.57.

Then, we calculate the step size and the new value for b3, which is 2.19.

Changing b3 to 2.19 shifts the green squiggle up further, and that shrinks the residuals even more.

Now we just keep taking steps until the step size is close to zero. And because the

step size is close to 0 when b3 equals 2.61, we decide that 2.61 is the optimal value for b3.

Double bam!

So, the main ideas for backpropagation are that, when a parameter is unknown, like

b3, we use the chain rule to calculate the derivative of the sum of the squared residuals

with respect to the unknown parameter, which in this case was b3.

Then we initialize the unknown parameter with a number, and in this case we set b3

equal to zero, and used gradient descent to optimize the unknown parameter.

Triple bam!

In the next StatQuest we'll show how these ideas can be used to optimize all of the parameters in a neural network.

Now it's time for some shameless self-promotion.

If you want to review statistics and machine learning offline, check out the StatQuest study guides at statquest.org.

There's something for everyone.

Hooray!

We've made it to the end of another exciting StatQuest.

If you like this StatQuest and want to see more, please subscribe. And if you want

to support StatQuest, consider contributing to my patreon campaign, becoming a channel

member, buying one or two of my original songs, or a t-shirt or a hoodie, or just

donate the links are in the description below. Alright, until next time. Quest on!