title

Backpropagation Details Pt. 1: Optimizing 3 parameters simultaneously.

description

The main ideas behind Backpropagation are super simple, but there are tons of details when it comes time to implementing it. This video shows how to optimize three parameters in a Neural Network simultaneously and introduces some Fancy Notation.
NOTE: This StatQuest assumes that you already know the main ideas behind Backpropagation: https://youtu.be/IN2XmBhILt4
...and that also means you should be familiar with...
Neural Networks: https://youtu.be/CqOfi41LfDw
The Chain Rule: https://youtu.be/wl1myxrtQHQ
Gradient Descent: https://youtu.be/sDv4f4s2SB8
LAST NOTE: When I was researching this 'Quest, I found this page by Sebastian Raschka to be helpful: https://sebastianraschka.com/faq/docs/backprop-arbitrary.html
For a complete index of all the StatQuest videos, check out:
https://statquest.org/video-index/
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0:00 Awesome song and introduction
3:01 Derivatives do not change when we optimize multiple parameters
6:28 Fancy Notation
10:51 Derivatives with respect to two different weights
15:02 Gradient Descent for three parameters
17:19 Fancy Gradient Descent Animation
#StatQuest #NeuralNetworks #Backpropagation

detail

{'title': 'Backpropagation Details Pt. 1: Optimizing 3 parameters simultaneously.', 'heatmap': [{'end': 182.178, 'start': 165.49, 'weight': 0.701}, {'end': 1003.999, 'start': 942.81, 'weight': 0.846}], 'summary': 'Tutorial series covers backpropagation, neural network optimization, summation notation, activation functions, derivatives, and gradient descent, with detailed explanations and demonstrations of optimizing parameters, calculating squared residuals, and utilizing the chain rule in neural networks.', 'chapters': [{'end': 399.392, 'segs': [{'end': 91.123, 'src': 'embed', 'start': 64.351, 'weight': 3, 'content': [{'end': 71.157, 'text': 'Then we demonstrated the main ideas behind back propagation by optimizing B sub 3.', 'start': 64.351, 'duration': 6.806}, {'end': 79.684, 'text': 'We first used the chain rule to calculate the derivative of the sum of the squared residuals with respect to the unknown parameter, which,', 'start': 71.157, 'duration': 8.527}, {'end': 83.22, 'text': 'in this case, was B sub 3..', 'start': 79.684, 'duration': 3.536}, {'end': 91.123, 'text': 'Then we initialized the unknown parameter with a number, and in this case we set B sub 3 equal to 0,', 'start': 83.22, 'duration': 7.903}], 'summary': 'Demonstrated back propagation by optimizing b3 using chain rule and setting b3=0.', 'duration': 26.772, 'max_score': 64.351, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii864351.jpg'}, {'end': 150.176, 'src': 'embed', 'start': 118.429, 'weight': 0, 'content': [{'end': 129.257, 'text': 'Note, the goal of this quest is to learn how the chain rule and gradient ascent applies to multiple parameters and to introduce some fancy notation.', 'start': 118.429, 'duration': 10.828}, {'end': 131.568, 'text': 'In the next part,', 'start': 130.426, 'duration': 1.142}, {'end': 139.411, 'text': "we'll go completely bonkers with the chain rule and learn how to optimize all seven parameters in this neural network simultaneously.", 'start': 131.568, 'duration': 7.843}, {'end': 150.176, 'text': "Bam! So let's go back to not knowing the optimal values for W sub 3, W sub 4, and B sub 3.", 'start': 140.491, 'duration': 9.685}], 'summary': 'Learn chain rule and gradient ascent for multiple parameters, optimize all 7 parameters in neural network simultaneously.', 'duration': 31.747, 'max_score': 118.429, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8118429.jpg'}, {'end': 192.107, 'src': 'heatmap', 'start': 165.49, 'weight': 0.701, 'content': [{'end': 171.936, 'text': 'And in this example, that means we randomly select two values from a standard normal distribution.', 'start': 165.49, 'duration': 6.446}, {'end': 182.178, 'text': 'Then we initialize the last bias, b sub 3, to 0, because bias terms frequently start at 0.', 'start': 173.291, 'duration': 8.887}, {'end': 192.107, 'text': 'Now, if we run dosages from 0 to 1 through the connection to the top node in the hidden layer, then, just like before,', 'start': 182.178, 'duration': 9.929}], 'summary': 'Initializing bias to 0 for standard normal distribution values.', 'duration': 26.617, 'max_score': 165.49, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8165490.jpg'}, {'end': 326.931, 'src': 'embed', 'start': 273.291, 'weight': 1, 'content': [{'end': 281.373, 'text': 'Now, just like before, we can quantify how well the green squiggle fits the data by calculating the sum of the squared residuals.', 'start': 273.291, 'duration': 8.082}, {'end': 287.717, 'text': 'And we get the sum of the squared residuals equals 1.4.', 'start': 282.694, 'duration': 5.023}, {'end': 299.525, 'text': 'Now, even though we have not yet optimized W sub 3 and W sub 4, we can still plot the sum of the squared residuals with respect to B sub 3.', 'start': 287.717, 'duration': 11.808}, {'end': 305.249, 'text': 'And just like before, if we change B sub 3, then we will change the sum of the squared residuals.', 'start': 299.525, 'duration': 5.724}, {'end': 309.229, 'text': 'And that means, just like before,', 'start': 306.846, 'duration': 2.383}, {'end': 322.628, 'text': 'we can optimize B3 by finding the derivative of the sum of the squared residuals with respect to B3 and plugging the derivative into the gradient descent algorithm to find the optimal value for B3..', 'start': 309.229, 'duration': 13.399}, {'end': 326.931, 'text': 'And just like before,', 'start': 325.03, 'duration': 1.901}], 'summary': 'The sum of squared residuals is 1.4, with optimization of b3 using gradient descent.', 'duration': 53.64, 'max_score': 273.291, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8273291.jpg'}, {'end': 399.392, 'src': 'embed', 'start': 373.732, 'weight': 2, 'content': [{'end': 379.194, 'text': 'The point of this is that, even though we are now optimizing more than one parameter,', 'start': 373.732, 'duration': 5.462}, {'end': 385.836, 'text': 'the derivatives that we have already calculated with respect to the sum of the squared residuals do not change.', 'start': 379.194, 'duration': 6.642}, {'end': 399.392, 'text': "Bam! Now let's talk about how to calculate the derivatives of the sum of the squared residuals with respect to the weights w sub 3 and w sub 4.", 'start': 386.876, 'duration': 12.516}], 'summary': 'Optimizing multiple parameters does not change derivatives calculated for sum of squared residuals.', 'duration': 25.66, 'max_score': 373.732, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8373732.jpg'}], 'start': 0.129, 'title': 'Backpropagation and neural network optimization', 'summary': 'Delves into backpropagation, covering optimizing parameters, gradient ascent, and multiple parameter optimization in a neural network. it also explains neural network optimization, including calculating the sum of squared residuals as 1.4, iterative parameter optimization, and the use of the chain rule.', 'chapters': [{'end': 182.178, 'start': 0.129, 'title': 'Backpropagation details, part 1', 'summary': 'Discusses the main ideas behind backpropagation, including optimizing parameters using the chain rule and gradient ascent, with a focus on optimizing the last bias term, b3, and introducing notation for multiple parameter optimization in a neural network.', 'duration': 182.049, 'highlights': ['The chapter demonstrates the main ideas behind backpropagation by optimizing the last bias term, B3, using the chain rule and gradient descent.', 'It introduces the goal of learning how the chain rule and gradient ascent applies to multiple parameters and fancy notation in neural networks.', 'The next part will focus on optimizing all seven parameters in the neural network simultaneously.']}, {'end': 399.392, 'start': 182.178, 'title': 'Neural network optimization', 'summary': 'Explains the process of optimizing a neural network through the example of calculating the sum of squared residuals as 1.4, and the iterative process of optimizing parameters such as b3, with the derivative of the sum of squared residuals with respect to b3 being calculated using the chain rule, and the fact that the derivatives calculated with respect to the sum of squared residuals do not change when optimizing more than one parameter.', 'duration': 217.214, 'highlights': ['The sum of the squared residuals equals 1.4, indicating the fitting of the green squiggle to the data.', 'The iterative process of optimizing B3 involves finding the derivative of the sum of the squared residuals with respect to B3 and plugging the derivative into the gradient descent algorithm to find the optimal value for B3.', 'The derivative of the sum of the squared residuals with respect to B3 is calculated using the chain rule, linking it to the predicted values and the derivative of the predicted values with respect to B3.', 'The derivatives calculated with respect to the sum of squared residuals do not change when optimizing more than one parameter, maintaining consistency in the optimization process.', 'The process of optimizing the weights W sub 3 and W sub 4 is explained, although detailed calculations are not provided.']}], 'duration': 399.263, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8129.jpg', 'highlights': ['The next part will focus on optimizing all seven parameters in the neural network simultaneously.', 'The iterative process of optimizing B3 involves finding the derivative of the sum of the squared residuals with respect to B3 and plugging the derivative into the gradient descent algorithm to find the optimal value for B3.', 'The derivatives calculated with respect to the sum of squared residuals do not change when optimizing more than one parameter, maintaining consistency in the optimization process.', 'The chapter demonstrates the main ideas behind backpropagation by optimizing the last bias term, B3, using the chain rule and gradient descent.', 'The sum of the squared residuals equals 1.4, indicating the fitting of the green squiggle to the data.']}, {'end': 728.248, 'segs': [{'end': 432.467, 'src': 'embed', 'start': 399.392, 'weight': 0, 'content': [{'end': 405.673, 'text': 'Unfortunately, before we can do that, we have to introduce some fancy notation.', 'start': 399.392, 'duration': 6.281}, {'end': 414.355, 'text': "First, let's remember that the i in this summation notation is an index for the data in the data set.", 'start': 406.993, 'duration': 7.362}, {'end': 422.276, 'text': 'For example, when i equals 1, we are talking about observed sub 1, which is 0.', 'start': 415.275, 'duration': 7.001}, {'end': 428.086, 'text': 'And we are talking about predicted sub 1, which is 0.72.', 'start': 422.276, 'duration': 5.81}, {'end': 432.467, 'text': 'However, we can also talk about dosage sub i.', 'start': 428.086, 'duration': 4.381}], 'summary': 'Introduction of fancy notation for data set indexing and examples of observed and predicted values.', 'duration': 33.075, 'max_score': 399.392, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8399392.jpg'}, {'end': 499.822, 'src': 'embed', 'start': 466.572, 'weight': 1, 'content': [{'end': 477.621, 'text': 'And it adds bias sub 1, which is negative 1.43, to get an x-axis coordinate for the activation function in the top node in the hidden layer.', 'start': 466.572, 'duration': 11.049}, {'end': 494.24, 'text': 'Meanwhile, the other connection multiplies input sub i by weight w sub 2, which is negative 3.53, and adds bias b sub 2, which is 0.57,', 'start': 479.355, 'duration': 14.885}, {'end': 499.822, 'text': 'to get an x-axis coordinate for the activation function in the bottom node, in the hidden layer.', 'start': 494.24, 'duration': 5.582}], 'summary': 'The activation function in the hidden layer uses different weights and biases to calculate x-axis coordinates for top and bottom nodes.', 'duration': 33.25, 'max_score': 466.572, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8466572.jpg'}, {'end': 663.066, 'src': 'embed', 'start': 631.894, 'weight': 2, 'content': [{'end': 639.737, 'text': 'Likewise, in order to get the y-axis coordinates for the activation function in the bottom node, we plug x, sub two,', 'start': 631.894, 'duration': 7.843}, {'end': 642.118, 'text': 'comma i into the activation function.', 'start': 639.737, 'duration': 2.381}, {'end': 647.22, 'text': 'And that gives us y sub two comma i.', 'start': 643.479, 'duration': 3.741}, {'end': 653.443, 'text': 'Bam!. Now that we understand the fancy notation,', 'start': 647.22, 'duration': 6.223}, {'end': 663.066, 'text': 'we can talk about how to calculate the derivatives of the sum of the squared residuals with respect to the weights w sub three and w sub four.', 'start': 654.517, 'duration': 8.549}], 'summary': 'Explaining activation function and calculating derivatives for weights w3 and w4', 'duration': 31.172, 'max_score': 631.894, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8631894.jpg'}], 'start': 399.392, 'title': 'Summation notation and neural network activation', 'summary': 'Introduces summation notation to represent data and discusses the use of index i to denote different elements in the dataset, with specific examples and corresponding weights provided. it also explores the calculation of x-axis and y-axis coordinates for the activation functions in a neural network, along with the derivatives for the sum of the squared residuals with respect to certain weights.', 'chapters': [{'end': 466.572, 'start': 399.392, 'title': 'Introduction to summation notation', 'summary': 'Introduces summation notation to represent data and discusses the use of index i to denote different elements in the dataset, with specific examples and corresponding weights provided.', 'duration': 67.18, 'highlights': ['The i in summation notation acts as an index for the data set, with examples such as observed sub 1 and predicted sub 1 provided.', 'Dosage sub i is discussed with specific examples, including dosage sub 1, dosage sub 2, and dosage sub 3, which are 0, 0.5, and 1 respectively.', 'The connection between input sub i and weight w sub 1, which is 3.34, is explained.']}, {'end': 728.248, 'start': 466.572, 'title': 'Neural network activation and derivatives', 'summary': 'Discusses the calculation of x-axis and y-axis coordinates for the activation functions in a neural network, using specific examples and the calculation of derivatives for the sum of the squared residuals with respect to certain weights.', 'duration': 261.676, 'highlights': ['The x-axis coordinates for the activation functions in the top and bottom nodes are calculated by adding bias and multiplying input by weight, resulting in x sub 1 comma i values and x sub 2 comma i values, such as x sub is equal to 1.91 and x sub is equal to negative 2.96.', 'Understanding the fancy notation for y-axis coordinates, y sub 1 comma i and y sub 2 comma i, and their relation to the activation functions in the top and bottom nodes, which are used to calculate the final blue and orange curve by multiplying with weights w sub 3 and w sub 4.', 'Discussing the calculation of derivatives for the sum of the squared residuals with respect to weights w sub three and w sub four, involving the multiplication of y-axis coordinates by the respective weights for the predicted values.']}], 'duration': 328.856, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8399392.jpg', 'highlights': ['The i in summation notation acts as an index for the data set, with examples provided.', 'The x-axis coordinates for the activation functions are calculated by adding bias and multiplying input by weight.', 'Discussing the calculation of derivatives for the sum of the squared residuals with respect to weights.']}, {'end': 1110.889, 'segs': [{'end': 825.392, 'src': 'embed', 'start': 729.158, 'weight': 0, 'content': [{'end': 736.324, 'text': 'then the sum of the squared residuals are linked to W sub 3 and W sub 4 by the predicted values.', 'start': 729.158, 'duration': 7.166}, {'end': 750.735, 'text': 'That means we can use the chain rule to determine the derivative of the sum of the squared residuals with respect to W sub 3 and with respect to W sub 4..', 'start': 737.745, 'duration': 12.99}, {'end': 769.27, 'text': 'The chain rule says that the derivative of the sum of the squared residuals with respect to W sub 3 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 W sub 3..', 'start': 750.735, 'duration': 18.535}, {'end': 779.336, 'text': 'Likewise, the derivative with respect to W sub 4 is the derivative of the sum of the squared residuals with respect to the predicted values times.', 'start': 769.27, 'duration': 10.066}, {'end': 784.249, 'text': 'the derivative of the predicted values with respect to W sub 4..', 'start': 779.336, 'duration': 4.913}, {'end': 786.19, 'text': 'Double bam? Not yet.', 'start': 784.249, 'duration': 1.941}, {'end': 788.971, 'text': 'Note in both cases,', 'start': 787.03, 'duration': 1.941}, {'end': 799.554, 'text': 'the derivative of the sum of the squared residuals with respect to the predicted values is the exact same as the derivative used for b, sub 3..', 'start': 788.971, 'duration': 10.583}, {'end': 804.976, 'text': 'Just to remind you, we start by substituting the sum of the squared residuals with its equation.', 'start': 799.554, 'duration': 5.422}, {'end': 819.789, 'text': 'Then we 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 1..', 'start': 806.121, 'duration': 13.668}, {'end': 825.392, 'text': 'Lastly, we simplify by multiplying 2 by negative 1.', 'start': 819.789, 'duration': 5.603}], 'summary': 'Using chain rule to find derivatives of squared residuals with respect to w sub 3 and w sub 4.', 'duration': 96.234, 'max_score': 729.158, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8729158.jpg'}, {'end': 1003.999, 'src': 'heatmap', 'start': 942.81, 'weight': 0.846, 'content': [{'end': 949.392, 'text': 'Then we plug in the observed values and plug in the predicted values from the green squiggle.', 'start': 942.81, 'duration': 6.582}, {'end': 956.675, 'text': 'Remember, we get the predicted values on the green squiggle by running the dosages through the neural network.', 'start': 950.413, 'duration': 6.262}, {'end': 966.559, 'text': 'Now we plug in the y-axis coordinates for the activation function in the top node, y sub 1 comma i.', 'start': 958.236, 'duration': 8.323}, {'end': 971.641, 'text': 'Lastly, we do the math and get 2.58.', 'start': 966.559, 'duration': 5.082}, {'end': 982.147, 'text': 'Likewise, we calculate the derivative of the sum of the squared residuals with respect to w sub 4 and with respect to b sub 3.', 'start': 971.641, 'duration': 10.506}, {'end': 1003.999, 'text': 'Now we use the derivatives to calculate the new values for w sub 3, w sub 4, and b sub 3.', 'start': 982.147, 'duration': 21.852}], 'summary': 'Using neural network, predicted values yield 2.58; derivatives used to update w3, w4, and b3.', 'duration': 61.189, 'max_score': 942.81, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8942810.jpg'}, {'end': 1073.368, 'src': 'embed', 'start': 1040.41, 'weight': 4, 'content': [{'end': 1047.135, 'text': 'Now, watch how the green squiggle fits the data after 175 steps in gradient descent.', 'start': 1040.41, 'duration': 6.725}, {'end': 1055.382, 'text': 'Bam! So, after a bunch of steps, we see how gradient descent optimizes the parameters.', 'start': 1047.856, 'duration': 7.526}, {'end': 1057.723, 'text': 'Triple bam!.', 'start': 1056.243, 'duration': 1.48}, {'end': 1068.026, 'text': "In the next StatQuest we'll go totally bonkers with the chain rule and show how to optimize all of the parameters in a neural network simultaneously.", 'start': 1058.924, 'duration': 9.102}, {'end': 1073.368, 'text': "Now it's time for some shameless self-promotion.", 'start': 1069.487, 'duration': 3.881}], 'summary': 'Gradient descent optimizes parameters after 175 steps, with future focus on optimizing all parameters in a neural network.', 'duration': 32.958, 'max_score': 1040.41, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii81040410.jpg'}], 'start': 729.158, 'title': 'Derivatives and gradient descent in neural networks', 'summary': 'Covers the application of the chain rule in determining derivatives of the sum of squared residuals with respect to w sub 3 and w sub 4, and explains the use of derivatives in gradient descent to optimize w and b, with a demonstration using a fancy animation.', 'chapters': [{'end': 799.554, 'start': 729.158, 'title': 'Chain rule in residuals', 'summary': 'Explains the application of the chain rule in determining the derivatives of the sum of squared residuals with respect to w sub 3 and w sub 4 in the context of predicted values.', 'duration': 70.396, 'highlights': ['The derivative of the sum of the squared residuals with respect to W sub 3 is determined using the chain rule, involving the derivative of the sum of the squared residuals with respect to the predicted values multiplied by the derivative of the predicted values with respect to W sub 3.', 'Similarly, the derivative with respect to W sub 4 involves the derivative of the sum of the squared residuals with respect to the predicted values multiplied by the derivative of the predicted values with respect to W sub 4.', 'The chapter emphasizes that the derivative of the sum of the squared residuals with respect to the predicted values is the same as the derivative used for b, sub 3.']}, {'end': 1110.889, 'start': 799.554, 'title': 'Derivative and gradient descent in neural networks', 'summary': 'Explains the derivative of the sum of the squared residuals with respect to the predicted values, using the chain rule, and the use of these derivatives in gradient descent to optimize w and b. it also showcases the application of gradient descent through a fancy animation.', 'duration': 311.335, 'highlights': ['The derivative of the sum of the squared residuals with respect to the predicted values is calculated using the chain rule, and the resulting derivatives are used in gradient descent to optimize W and B, repeating the process until predictions no longer improve or reaching a maximum number of steps.', 'The process involves substituting the sum of the squared residuals with its equation, using the chain rule to move the square to the front, and then multiplying by the derivative of the stuff inside the parentheses with respect to the predicted values. This calculation results in a value of 2.58 for W sub 3.', 'The application of gradient descent is demonstrated through a fancy animation, showcasing how the process optimizes the parameters after 175 steps, and the promise of going further in the next StatQuest to optimize all parameters in a neural network simultaneously.']}], 'duration': 381.731, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/iyn2zdALii8/pics/iyn2zdALii8729158.jpg', 'highlights': ['The derivative of the sum of the squared residuals with respect to W sub 3 is determined using the chain rule.', 'The derivative with respect to W sub 4 involves the derivative of the sum of the squared residuals with respect to the predicted values multiplied by the derivative of the predicted values with respect to W sub 4.', 'The derivative of the sum of the squared residuals with respect to the predicted values is calculated using the chain rule, and the resulting derivatives are used in gradient descent to optimize W and B.', 'The process involves substituting the sum of the squared residuals with its equation, using the chain rule to move the square to the front, and then multiplying by the derivative of the stuff inside the parentheses with respect to the predicted values.', 'The application of gradient descent is demonstrated through a fancy animation, showcasing how the process optimizes the parameters after 175 steps.']}], 'highlights': ['The iterative process of optimizing B3 involves finding the derivative of the sum of the squared residuals with respect to B3 and plugging the derivative into the gradient descent algorithm to find the optimal value for B3.', 'The derivatives calculated with respect to the sum of squared residuals do not change when optimizing more than one parameter, maintaining consistency in the optimization process.', 'The chapter demonstrates the main ideas behind backpropagation by optimizing the last bias term, B3, using the chain rule and gradient descent.', 'The sum of the squared residuals equals 1.4, indicating the fitting of the green squiggle to the data.', 'The i in summation notation acts as an index for the data set, with examples provided.', 'The x-axis coordinates for the activation functions are calculated by adding bias and multiplying input by weight.', 'The derivative of the sum of the squared residuals with respect to W sub 3 is determined using the chain rule.', 'The derivative with respect to W sub 4 involves the derivative of the sum of the squared residuals with respect to the predicted values multiplied by the derivative of the predicted values with respect to W sub 4.', 'The derivative of the sum of the squared residuals with respect to the predicted values is calculated using the chain rule, and the resulting derivatives are used in gradient descent to optimize W and B.', 'The process involves substituting the sum of the squared residuals with its equation, using the chain rule to move the square to the front, and then multiplying by the derivative of the stuff inside the parentheses with respect to the predicted values.', 'The application of gradient descent is demonstrated through a fancy animation, showcasing how the process optimizes the parameters after 175 steps.', 'The next part will focus on optimizing all seven parameters in the neural network simultaneously.']}