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

description
For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/3pqkTry This lecture covers supervised learning and linear regression. Andrew Ng Adjunct Professor of Computer Science https://www.andrewng.org/ To follow along with the course schedule and syllabus, visit: http://cs229.stanford.edu/syllabus-autumn2018.html #andrewng #machinelearning Chapters: 00:00 Intro 00:45 Motivate Linear Regression 03:01 Supervised Learning 04:44 Designing a Learning Algorithm 08:27 Parameters of the learning algorithm 14:44 Linear Regression Algorithm 18:06 Gradient Descent 33:01 Gradient Descent Algorithm 42:34 Batch Gradient Descent 44:56 Stochastic Gradient Descent

detail
{'title': 'Stanford CS229: Machine Learning - Linear Regression and Gradient Descent | Lecture 2 (Autumn 2018)', 'heatmap': [{'end': 1600.869, 'start': 1548.779, 'weight': 0.722}, {'end': 2067.521, 'start': 1972.194, 'weight': 0.763}, {'end': 4042.085, 'start': 3986.721, 'weight': 0.755}, {'end': 4651.731, 'start': 4595.275, 'weight': 1}], 'summary': 'This lecture on linear regression and gradient descent covers the basics of linear regression, supervised learning, and the implementation of gradient descent algorithm to minimize the cost function j of theta. it discusses the impact of learning rate on convergence, batch and stochastic gradient descent, matrix derivatives, and normal equations in machine learning.', 'chapters': [{'end': 367.377, 'segs': [{'end': 34.168, 'src': 'embed', 'start': 4.582, 'weight': 0, 'content': [{'end': 5.77, 'text': 'Good morning and welcome back.', 'start': 4.582, 'duration': 1.188}, {'end': 15.716, 'text': "So what we'll see today in class is, um, the first in-depth discussion of a learning algorithm, linear regression.", 'start': 7.751, 'duration': 7.965}, {'end': 20.459, 'text': "And in particular, over the next what hour and a bit you'll see.", 'start': 15.936, 'duration': 4.523}, {'end': 25.763, 'text': 'uh, linear regression, um batch and stochastic gradient descent as an algorithm for fitting linear regression models.', 'start': 20.459, 'duration': 5.304}, {'end': 34.168, 'text': 'And then, uh, the normal equations, um, uh, as a way of- as a very efficient way to let you fit linear models.', 'start': 26.403, 'duration': 7.765}], 'summary': 'Introduction to linear regression and its algorithms in a class session.', 'duration': 29.586, 'max_score': 4.582, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U84582.jpg'}, {'end': 91.228, 'src': 'embed', 'start': 65.024, 'weight': 2, 'content': [{'end': 74.356, 'text': "And the term supervised learning meant that you were given x's, which was a picture of what's in front of the car,", 'start': 65.024, 'duration': 9.332}, {'end': 79.402, 'text': 'and the algorithm had to map that to an output y, which was the steering direction.', 'start': 74.356, 'duration': 5.046}, {'end': 82.146, 'text': 'And that was a regression problem.', 'start': 80.203, 'duration': 1.943}, {'end': 91.228, 'text': 'because the output y that you want is continuous value, right? As opposed to classification problem where y is discrete.', 'start': 84.704, 'duration': 6.524}], 'summary': 'Supervised learning maps x to y for steering direction in regression problem.', 'duration': 26.204, 'max_score': 65.024, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U865024.jpg'}, {'end': 215.325, 'src': 'embed', 'start': 183.889, 'weight': 3, 'content': [{'end': 191.932, 'text': 'um, the process of supervised learning is that you have a training set, uh, such as the dataset that I drew on the left,', 'start': 183.889, 'duration': 8.043}, {'end': 203.982, 'text': 'and you feed this to a learning algorithm, And the job of the learning algorithm is to output a function.', 'start': 191.932, 'duration': 12.05}, {'end': 206.563, 'text': 'uh, to make predictions about housing prices.', 'start': 203.982, 'duration': 2.581}, {'end': 207.723, 'text': 'And by convention.', 'start': 206.703, 'duration': 1.02}, {'end': 215.325, 'text': "um, I'm gonna call this function that it outputs a hypothesis, right?", 'start': 207.723, 'duration': 7.602}], 'summary': 'Supervised learning involves a training set to output a hypothesis for housing price predictions.', 'duration': 31.436, 'max_score': 183.889, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8183889.jpg'}, {'end': 295.263, 'src': 'embed', 'start': 236.043, 'weight': 1, 'content': [{'end': 243.712, 'text': 'The job of the hypothesis is to take as input any size of a house and try to tell you what it thinks should be the price of that house.', 'start': 236.043, 'duration': 7.669}, {'end': 251.861, 'text': 'Now, when designing a learning algorithm, um, and, and you know even though, uh, linear regression right,', 'start': 245.194, 'duration': 6.667}, {'end': 255.845, 'text': 'you may have seen it in a linear algebra class before, in some of the class before um,', 'start': 251.861, 'duration': 3.984}, {'end': 261.329, 'text': 'the way you go about structuring a machine learning algorithm is important and design choices of.', 'start': 255.845, 'duration': 5.484}, {'end': 265.273, 'text': 'you know what is the workflow, what is the dataset, what is the hypothesis, how to represent the hypothesis.', 'start': 261.329, 'duration': 3.944}, {'end': 271.739, 'text': 'These are the key decisions you have to make in, pretty much every supervised learning, every machine learning algorithms design.', 'start': 265.593, 'duration': 6.146}, {'end': 277.084, 'text': "So, uh, as we go through linear regression, I'll try to describe the concepts clearly as well,", 'start': 271.839, 'duration': 5.245}, {'end': 283.13, 'text': "because they'll lay the foundation for the rest of the algorithms, sometimes much more complicated algorithms, you see later this quarter.", 'start': 277.084, 'duration': 6.046}, {'end': 294.563, 'text': "So, when designing a learning algorithm, the first thing we'll need to ask is um how- how do you represent the hypothesis?", 'start': 284.251, 'duration': 10.312}, {'end': 295.263, 'text': 'H, right?', 'start': 294.563, 'duration': 0.7}], 'summary': 'Learning algorithm designs rely on key decisions like workflow, dataset, and hypothesis representation, crucial in supervised and machine learning algorithms.', 'duration': 59.22, 'max_score': 236.043, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8236043.jpg'}], 'start': 4.582, 'title': 'Linear regression in supervised learning', 'summary': "Covers the basics of linear regression as a learning algorithm, its application in supervised learning for regression problems, and the use of a house price dataset to demonstrate the algorithm's usage, laying the foundation for future topics in the course. it also explains the process of supervised learning, the job of the learning algorithm to output a hypothesis, the key decisions in machine learning algorithm design, and the representation of the hypothesis in linear regression.", 'chapters': [{'end': 150.977, 'start': 4.582, 'title': 'Linear regression learning algorithm', 'summary': "Covers the basics of linear regression as a learning algorithm, its application in supervised learning for regression problems, and the use of a house price dataset to demonstrate the algorithm's usage, laying the foundation for future topics in the course.", 'duration': 146.395, 'highlights': ['The chapter introduces linear regression as a learning algorithm for supervised learning regression problems, such as predicting house prices, providing a foundational understanding for future course material.', 'The use of a dataset from Portland, Oregon to predict house prices based on the size of the house, with an example of a 2,104 square feet house priced at $400,000, exemplifies the practical application of linear regression in real-world scenarios.', 'Explanation of supervised learning regression problems, such as predicting the steering direction for a self-driving car, and the differentiation from classification problems, provides context for the application of linear regression in real-world scenarios.']}, {'end': 367.377, 'start': 164.52, 'title': 'Linear regression in supervised learning', 'summary': 'Explains the process of supervised learning, the job of the learning algorithm to output a hypothesis, the key decisions in machine learning algorithm design, and the representation of the hypothesis in linear regression.', 'duration': 202.857, 'highlights': ['The job of the learning algorithm in supervised learning is to output a function, such as a hypothesis to make predictions about housing prices.', 'The key decisions in machine learning algorithm design include structuring the workflow, dataset, and representation of the hypothesis.', 'In linear regression, the hypothesis is represented as a linear function that inputs the size of a house and outputs a number, and in more general cases, with multiple input features such as the number of bedrooms.', 'The lecture lays the foundation for understanding more complicated algorithms in the future and emphasizes the importance of clear concept description in algorithm design.']}], 'duration': 362.795, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U84582.jpg', 'highlights': ['The chapter introduces linear regression as a learning algorithm for supervised learning regression problems, providing a foundational understanding for future course material.', 'The use of a dataset from Portland, Oregon to predict house prices based on the size of the house exemplifies the practical application of linear regression in real-world scenarios.', 'Explanation of supervised learning regression problems, such as predicting the steering direction for a self-driving car, provides context for the application of linear regression in real-world scenarios.', 'The job of the learning algorithm in supervised learning is to output a function, such as a hypothesis to make predictions about housing prices.', 'The key decisions in machine learning algorithm design include structuring the workflow, dataset, and representation of the hypothesis.', 'In linear regression, the hypothesis is represented as a linear function that inputs the size of a house and outputs a number, and in more general cases, with multiple input features such as the number of bedrooms.', 'The lecture lays the foundation for understanding more complicated algorithms in the future and emphasizes the importance of clear concept description in algorithm design.']}, {'end': 1031.015, 'segs': [{'end': 450.855, 'src': 'embed', 'start': 401.591, 'weight': 0, 'content': [{'end': 409.118, 'text': 'where x1 is the size of the house and x2 is- is the number of bedrooms okay?', 'start': 401.591, 'duration': 7.527}, {'end': 412.061, 'text': 'Um, so, in order to?', 'start': 410.019, 'duration': 2.042}, {'end': 431.657, 'text': 'So, in order to simplify the notation, um, in order to make that notation a little bit more compact, um,', 'start': 418.433, 'duration': 13.224}, {'end': 450.855, 'text': "I'm also gonna introduce this other notation where, um, we want to write the hypothesis as sum from J equals 0 to 2 of Theta J XJ.", 'start': 431.657, 'duration': 19.198}], 'summary': 'Introducing compact notation for hypothesis with variables x1 and x2.', 'duration': 49.264, 'max_score': 401.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8401591.jpg'}, {'end': 574.634, 'src': 'embed', 'start': 514.533, 'weight': 1, 'content': [{'end': 518.496, 'text': 'And the job of the learning algorithm is to choose parameters.', 'start': 514.533, 'duration': 3.963}, {'end': 523.84, 'text': 'Theta that allows you to make good predictions about you know prices of houses right?', 'start': 518.496, 'duration': 5.344}, {'end': 533.418, 'text': "Um, and just to lay out some more uh notation that we're gonna use throughout this quarter, I'm gonna use a standard that,", 'start': 524.561, 'duration': 8.857}, {'end': 543.721, 'text': 'uh M will define as the number of training examples.', 'start': 533.418, 'duration': 10.303}, {'end': 556.264, 'text': 'So M is going to be the number of rows right in the table above. um, where you know each house you have in your training set is one training example.', 'start': 544.021, 'duration': 12.243}, {'end': 561.812, 'text': "Um, you've already seen me use x to denote the inputs.", 'start': 557.811, 'duration': 4.001}, {'end': 568.293, 'text': 'Um, and often the inputs are called features.', 'start': 564.532, 'duration': 3.761}, {'end': 574.634, 'text': "Um, you know, I think I don't know as-, as- as a- as a emerging discipline grows up right.", 'start': 569.033, 'duration': 5.601}], 'summary': 'Learning algorithm chooses parameters theta for predicting house prices, with m as the number of training examples.', 'duration': 60.101, 'max_score': 514.533, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8514533.jpg'}, {'end': 653.909, 'src': 'embed', 'start': 619.401, 'weight': 3, 'content': [{'end': 637.675, 'text': "Um. and uh, I'm going to use this notation um x superscript i, comma y superscript i in parentheses to denote the i-th training example.", 'start': 619.401, 'duration': 18.274}, {'end': 642.259, 'text': "okay?. So the superscript parentheses i, that's not exponentiation.", 'start': 637.675, 'duration': 4.584}, {'end': 648.304, 'text': 'Uh, I think that as you build as it is, this is, um, this notation x i, comma y i.', 'start': 642.279, 'duration': 6.025}, {'end': 653.909, 'text': 'this is just a way of, uh, writing an index into the table of training examples above, okay?', 'start': 648.304, 'duration': 5.605}], 'summary': 'The notation x^i, y^i represents the i-th training example in the table.', 'duration': 34.508, 'max_score': 619.401, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8619401.jpg'}, {'end': 744.44, 'src': 'embed', 'start': 709.17, 'weight': 4, 'content': [{'end': 717.832, 'text': 'So in this example, uh, n is equal to 2, right? Uh, because we have two features, which is, um, the size of the house and the number of bedrooms.', 'start': 709.17, 'duration': 8.662}, {'end': 729.276, 'text': 'So two features, which is why you can take this, right, and write this, um, as a sum from j equals 0 to n.', 'start': 717.852, 'duration': 11.424}, {'end': 744.44, 'text': 'Um, and so here x and Theta are n plus 1-dimensional, because we added the extra um x0 and Theta 0, okay?', 'start': 733.496, 'duration': 10.944}], 'summary': 'Example with 2 features: house size and bedrooms', 'duration': 35.27, 'max_score': 709.17, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8709170.jpg'}, {'end': 934.049, 'src': 'embed', 'start': 899.155, 'weight': 5, 'content': [{'end': 914.461, 'text': 'Minimize the squared difference between what the hypothesis outputs H, subscript, Theta of x minus y squared, right?', 'start': 899.155, 'duration': 15.306}, {'end': 921.124, 'text': "So let's say we want to minimize the squared difference between the prediction, which is h of x and y, which is the correct price.", 'start': 914.521, 'duration': 6.603}, {'end': 929.627, 'text': 'Um, and so what we want to do is choose values of Theta that minimizes that.', 'start': 921.904, 'duration': 7.723}, {'end': 934.049, 'text': 'Um, to fill this out, you know, you have M training examples.', 'start': 929.647, 'duration': 4.402}], 'summary': 'Minimize squared difference between h(theta) and y, with m training examples.', 'duration': 34.894, 'max_score': 899.155, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8899155.jpg'}, {'end': 1011.948, 'src': 'embed', 'start': 979.303, 'weight': 6, 'content': [{'end': 989.055, 'text': "And so, in linear regression, I'm gonna define the cost function J of Theta to be equal to that.", 'start': 979.303, 'duration': 9.752}, {'end': 997.644, 'text': "and uh, we'll find parameters Theta that minimizes the cost function J of Theta.", 'start': 989.055, 'duration': 8.589}, {'end': 1004.665, 'text': "Okay, Um and questions I've often gotten is you know why squared error??", 'start': 998.442, 'duration': 6.223}, {'end': 1005.965, 'text': 'Why not absolute error?', 'start': 1004.705, 'duration': 1.26}, {'end': 1007.746, 'text': 'or this error to the power of 4??', 'start': 1005.965, 'duration': 1.781}, {'end': 1011.948, 'text': "Uh, we'll talk more about that when we talk about um.", 'start': 1007.746, 'duration': 4.202}], 'summary': 'In linear regression, the cost function j of theta is defined to find parameters that minimize the squared error, addressing the question of why squared error is used instead of other types of error.', 'duration': 32.645, 'max_score': 979.303, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8979303.jpg'}], 'start': 368.057, 'title': 'Linear regression and supervised learning basics', 'summary': 'Covers the basics of linear regression with two input features, explaining notations for hypothesis, parameters, and training examples. it also introduces supervised learning, discussing training examples, input features, target variable, parameters selection for a hypothesis, and the significance of squared error in linear regression.', 'chapters': [{'end': 585.316, 'start': 368.057, 'title': 'Linear regression basics', 'summary': 'Introduces the concept of linear regression with two input features and explains the notations for hypothesis, parameters, and training examples.', 'duration': 217.259, 'highlights': ['The hypothesis is expressed as h of x equals Theta 0 plus Theta 1, x1 plus Theta 2 x2, where x1 is the size of the house and x2 is the number of bedrooms.', 'The number of training examples (M) is defined as the number of rows in the training set.', 'The parameters (Theta) of the learning algorithm are responsible for making good predictions about house prices.']}, {'end': 1031.015, 'start': 585.316, 'title': 'Supervised learning basics', 'summary': 'Introduces the concept of supervised learning, explaining the notation for training examples, the role of input features, the target variable, and the parameters selection for a hypothesis. it also discusses the cost function and highlights the significance of squared error in linear regression.', 'duration': 445.699, 'highlights': ['The chapter defines the notation for training examples as X being the input features and Y being the output, denoting each training example as X^i,Y^i, and explains the role of X^i,Y^i in representing the i-th training example.', "It introduces the use of notation 'n' to denote the number of features for the supervised learning problem, with an example of n=2 for the size of the house and the number of bedrooms, and explains the dimensional aspects of X and Theta based on the number of features.", 'The text discusses the significance of choosing parameters Theta for the hypothesis, emphasizing the objective to make H of X close to Y for the training examples by minimizing the squared difference between H subscript Theta of X and Y, and defining the cost function J of Theta as the squared difference over the training examples.', 'It mentions the convention of adding a one-half constant in the cost function to simplify the math for later minimization, and explains the significance of using squared error in linear regression, addressing potential questions about alternative error measures in the context of generalized linear models.']}], 'duration': 662.958, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U8368057.jpg', 'highlights': ['The hypothesis is expressed as h of x equals Theta 0 plus Theta 1, x1 plus Theta 2 x2, where x1 is the size of the house and x2 is the number of bedrooms.', 'The number of training examples (M) is defined as the number of rows in the training set.', 'The parameters (Theta) of the learning algorithm are responsible for making good predictions about house prices.', 'The chapter defines the notation for training examples as X being the input features and Y being the output, denoting each training example as X^i,Y^i, and explains the role of X^i,Y^i in representing the i-th training example.', "It introduces the use of notation 'n' to denote the number of features for the supervised learning problem, with an example of n=2 for the size of the house and the number of bedrooms, and explains the dimensional aspects of X and Theta based on the number of features.", 'The text discusses the significance of choosing parameters Theta for the hypothesis, emphasizing the objective to make H of X close to Y for the training examples by minimizing the squared difference between H subscript Theta of X and Y, and defining the cost function J of Theta as the squared difference over the training examples.', 'It mentions the convention of adding a one-half constant in the cost function to simplify the math for later minimization, and explains the significance of using squared error in linear regression, addressing potential questions about alternative error measures in the context of generalized linear models.']}, {'end': 1600.869, 'segs': [{'end': 1138.733, 'src': 'embed', 'start': 1080.536, 'weight': 0, 'content': [{'end': 1084.718, 'text': 'J of Theta, that minimizes the cost function J of Theta.', 'start': 1080.536, 'duration': 4.182}, {'end': 1087.78, 'text': "Um, we're going to use an algorithm called gradient descent.", 'start': 1085.298, 'duration': 2.482}, {'end': 1095.343, 'text': 'And, um, sorry.', 'start': 1093.582, 'duration': 1.761}, {'end': 1099.766, 'text': "You know, this is my first class teaching in this classroom, so I'm trying to figure out logistics like this.", 'start': 1095.683, 'duration': 4.083}, {'end': 1100.386, 'text': 'All right.', 'start': 1100.146, 'duration': 0.24}, {'end': 1101.667, 'text': "Let's get rid of the chair.", 'start': 1100.846, 'duration': 0.821}, {'end': 1103.168, 'text': 'Okay, cool.', 'start': 1102.687, 'duration': 0.481}, {'end': 1107.47, 'text': 'Um, all right.', 'start': 1104.428, 'duration': 3.042}, {'end': 1118.199, 'text': 'And so with uh, gradient descent, we are going to start with some value of Theta.', 'start': 1107.63, 'duration': 10.569}, {'end': 1120.561, 'text': 'um, and it could be.', 'start': 1118.199, 'duration': 2.362}, {'end': 1124.163, 'text': 'you know, Theta equals the vector of all zeros would be a reasonable default.', 'start': 1120.561, 'duration': 3.602}, {'end': 1126.505, 'text': "We could initialize it randomly, kind of doesn't really matter.", 'start': 1124.183, 'duration': 2.322}, {'end': 1135.051, 'text': "But, uh, Theta is this three-dimensional vector, and I'm writing zero with an arrow on top to denote the vector of all zeros.", 'start': 1126.985, 'duration': 8.066}, {'end': 1138.733, 'text': "So zero, with an arrow on top there's a vector, there's a zero, zero, zero everywhere, right?", 'start': 1135.091, 'duration': 3.642}], 'summary': 'Teaching about gradient descent and initializing theta with zeros or randomly.', 'duration': 58.197, 'max_score': 1080.536, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81080536.jpg'}, {'end': 1442.821, 'src': 'embed', 'start': 1410.632, 'weight': 1, 'content': [{'end': 1411.892, 'text': "There's a function of parameters theta.", 'start': 1410.632, 'duration': 1.26}, {'end': 1416.413, 'text': "And the only thing you're gonna do is tweak or modify the parameters theta.", 'start': 1412.072, 'duration': 4.341}, {'end': 1429.268, 'text': "one step of gradient descent, um, can be implemented as follows, which is Theta j gets updated as Theta j minus, I'll just write this out.", 'start': 1417.754, 'duration': 11.514}, {'end': 1437.039, 'text': 'Um, so bit more notation.', 'start': 1435.078, 'duration': 1.961}, {'end': 1442.821, 'text': "I'm gonna use colon equals, I'm gonna use this notation to denote assignment.", 'start': 1437.579, 'duration': 5.242}], 'summary': 'Implement one step of gradient descent to tweak parameters theta.', 'duration': 32.189, 'max_score': 1410.632, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81410632.jpg'}, {'end': 1600.869, 'src': 'heatmap', 'start': 1548.779, 'weight': 0.722, 'content': [{'end': 1554.181, 'text': "if you've taken a calculus class a while back, you may remember that the derivative of a function is, you know,", 'start': 1548.779, 'duration': 5.402}, {'end': 1556.242, 'text': 'defines the direction of steepest descent.', 'start': 1554.181, 'duration': 2.061}, {'end': 1561.964, 'text': 'So it defines the direction that allows you to go downhill as steeply as possible, uh, on the- on the hill like that.', 'start': 1556.322, 'duration': 5.642}, {'end': 1568.446, 'text': 'Question? How do you determine the learning rate? Oh, how do you determine the learning rate? Uh, let me get back to that.', 'start': 1562.024, 'duration': 6.422}, {'end': 1569.106, 'text': "It's a good question.", 'start': 1568.466, 'duration': 0.64}, {'end': 1573.268, 'text': "Uh, for now, um, uh, you know, there's a theory and there's a practice.", 'start': 1569.266, 'duration': 4.002}, {'end': 1579.296, 'text': 'Uh, in practice, you set to 0.01.', 'start': 1573.548, 'duration': 5.748}, {'end': 1580.597, 'text': 'let me say a bit more about that later.', 'start': 1579.296, 'duration': 1.301}, {'end': 1589.943, 'text': 'Um, if if you actually, if, if you scale all the features between 0 and 1, you know minus 1 and plus 1 or something like that, then then yeah,', 'start': 1583.499, 'duration': 6.444}, {'end': 1591.363, 'text': 'then then try.', 'start': 1589.943, 'duration': 1.42}, {'end': 1594.085, 'text': 'you could try a few values and see what lets you minimize the function best.', 'start': 1591.363, 'duration': 2.722}, {'end': 1600.869, 'text': 'But if the features are scaled to plus minus 1, I usually start with 0.01 and then, and then try increasing and decreasing it.', 'start': 1594.125, 'duration': 6.744}], 'summary': 'Derivative defines direction of steepest descent, practice sets learning rate to 0.01.', 'duration': 52.09, 'max_score': 1548.779, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81548779.jpg'}, {'end': 1589.943, 'src': 'embed', 'start': 1562.024, 'weight': 3, 'content': [{'end': 1568.446, 'text': 'Question? How do you determine the learning rate? Oh, how do you determine the learning rate? Uh, let me get back to that.', 'start': 1562.024, 'duration': 6.422}, {'end': 1569.106, 'text': "It's a good question.", 'start': 1568.466, 'duration': 0.64}, {'end': 1573.268, 'text': "Uh, for now, um, uh, you know, there's a theory and there's a practice.", 'start': 1569.266, 'duration': 4.002}, {'end': 1579.296, 'text': 'Uh, in practice, you set to 0.01.', 'start': 1573.548, 'duration': 5.748}, {'end': 1580.597, 'text': 'let me say a bit more about that later.', 'start': 1579.296, 'duration': 1.301}, {'end': 1589.943, 'text': 'Um, if if you actually, if, if you scale all the features between 0 and 1, you know minus 1 and plus 1 or something like that, then then yeah,', 'start': 1583.499, 'duration': 6.444}], 'summary': 'Learning rate is set to 0.01 in practice and features are scaled between 0 and 1.', 'duration': 27.919, 'max_score': 1562.024, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81562024.jpg'}], 'start': 1031.015, 'title': 'Gradient descent algorithm and visualization', 'summary': 'Covers the implementation of gradient descent algorithm to minimize the cost function j of theta, with mention of y squared error and error to the power of 4, and the initialization of the theta vector. it also explains the visualization of the algorithm for minimizing a function j of theta, involving updating parameters theta to reduce the value of j of theta using partial derivatives, with the learning rate typically set to 0.01 in practice.', 'chapters': [{'end': 1138.733, 'start': 1031.015, 'title': 'Gradient descent algorithm', 'summary': 'Covers the implementation of the gradient descent algorithm to minimize the cost function j of theta, with a mention of y squared error and error to the power of 4, and the initialization of the theta vector.', 'duration': 107.718, 'highlights': ['The chapter introduces the implementation of the gradient descent algorithm to minimize the cost function J of Theta, including a discussion on Y squared error and error to the power of 4.', 'The instructor mentions the initialization of the Theta vector, suggesting that setting it to the vector of all zeros could be a reasonable default.', "The instructor briefly touches on logistics and classroom setup, indicating it's their first class teaching in this environment."]}, {'end': 1600.869, 'start': 1139.294, 'title': 'Gradient descent visualization', 'summary': 'Explains gradient descent algorithm for minimizing a function j of theta, which involves updating parameters theta to reduce the value of j of theta using partial derivatives, with the learning rate typically set to 0.01 in practice.', 'duration': 461.575, 'highlights': ['Gradient descent algorithm for minimizing a function J of Theta involves updating parameters Theta to reduce the value of J of Theta using partial derivatives', 'Learning rate typically set to 0.01 in practice for gradient descent']}], 'duration': 569.854, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81031015.jpg', 'highlights': ['The chapter introduces the implementation of the gradient descent algorithm to minimize the cost function J of Theta, including a discussion on Y squared error and error to the power of 4.', 'Gradient descent algorithm for minimizing a function J of Theta involves updating parameters Theta to reduce the value of J of Theta using partial derivatives', 'The instructor mentions the initialization of the Theta vector, suggesting that setting it to the vector of all zeros could be a reasonable default.', 'Learning rate typically set to 0.01 in practice for gradient descent', "The instructor briefly touches on logistics and classroom setup, indicating it's their first class teaching in this environment."]}, {'end': 2155.434, 'segs': [{'end': 1706.891, 'src': 'embed', 'start': 1678.107, 'weight': 3, 'content': [{'end': 1680.788, 'text': 'So if you have only one training example um.', 'start': 1678.107, 'duration': 2.681}, {'end': 1688.224, 'text': 'And so from calculus, if you take the derivative of a square, you know the 2 comes down, and so that cancels out with the half.', 'start': 1681.781, 'duration': 6.443}, {'end': 1691.105, 'text': 'So 2 times 1 half times.', 'start': 1688.464, 'duration': 2.641}, {'end': 1695.527, 'text': 'um, uh, the thing inside right?', 'start': 1691.105, 'duration': 4.422}, {'end': 1700.25, 'text': 'Uh, and then by the uh chain rule of uh derivatives.', 'start': 1696.308, 'duration': 3.942}, {'end': 1704.792, 'text': "uh, that's times the partial derivative of Theta j of x, Theta of x.", 'start': 1700.25, 'duration': 4.542}, {'end': 1706.891, 'text': 'minus y right?', 'start': 1705.95, 'duration': 0.941}], 'summary': 'Explaining the derivative calculation with one training example.', 'duration': 28.784, 'max_score': 1678.107, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81678107.jpg'}, {'end': 1818.379, 'src': 'embed', 'start': 1791.135, 'weight': 2, 'content': [{'end': 1796.92, 'text': 'And so, um, when you take the partial derivative of this big sum with respect to Theta j uh,', 'start': 1791.135, 'duration': 5.785}, {'end': 1808.476, 'text': 'instead of just j equals 1 with respect to Theta j in general then the only term that even depends on Theta j is the term Theta J, XJ,', 'start': 1796.92, 'duration': 11.556}, {'end': 1818.379, 'text': 'and so the partial derivative of all the other terms end up being 0, and partial derivative of this term with respect to Theta J is equal to XJ,', 'start': 1808.476, 'duration': 9.903}], 'summary': 'Partial derivative of the sum with respect to theta j equals xj', 'duration': 27.244, 'max_score': 1791.135, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81791135.jpg'}, {'end': 1929.678, 'src': 'embed', 'start': 1906.248, 'weight': 4, 'content': [{'end': 1913.791, 'text': 'but uh, this was- I kind of used definition of the cost function J of Theta, defined using just one single training example.', 'start': 1906.248, 'duration': 7.543}, {'end': 1916.092, 'text': 'but you actually have M training examples.', 'start': 1913.791, 'duration': 2.301}, {'end': 1926.116, 'text': 'And so, um the- the- the correct formula for the derivative is actually, if you take this thing and sum it over all M training examples, um,', 'start': 1916.372, 'duration': 9.744}, {'end': 1929.678, 'text': 'the derivative of- you know the- the derivative of the sum is the sum of the derivatives, right?', 'start': 1926.116, 'duration': 3.562}], 'summary': 'Correct formula for derivative involves summing over all m training examples', 'duration': 23.43, 'max_score': 1906.248, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81906248.jpg'}, {'end': 2067.521, 'src': 'heatmap', 'start': 1972.194, 'weight': 0.763, 'content': [{'end': 1979.318, 'text': "when it's defined using um uh, all of the um uh when it's defined using all of the training examples.", 'start': 1972.194, 'duration': 7.124}, {'end': 1994.973, 'text': 'Okay. And so the gradient descent algorithm is to repeat until convergence.', 'start': 1979.338, 'duration': 15.635}, {'end': 2009.999, 'text': 'carry out this update and in each iteration of gradient descent uh, you do this update for j equals uh, 0,, 1, up to n.', 'start': 1994.973, 'duration': 15.026}, {'end': 2011.74, 'text': 'uh, where n is the number of features.', 'start': 2009.999, 'duration': 1.741}, {'end': 2014.021, 'text': 'So n was 2 in our example.', 'start': 2011.78, 'duration': 2.241}, {'end': 2020.563, 'text': 'Um, and if you do this then, uh, uh, you know, actually let me see.', 'start': 2016.002, 'duration': 4.561}, {'end': 2026.104, 'text': "Then what will happen is, um, I'll show you the animation.", 'start': 2021.503, 'duration': 4.601}, {'end': 2028.344, 'text': 'As you fit.', 'start': 2027.484, 'duration': 0.86}, {'end': 2032.105, 'text': 'hopefully you find a pretty good value of the parameters Theta, right?', 'start': 2028.344, 'duration': 3.761}, {'end': 2045.107, 'text': 'So, um, it turns out that when you plot the cost function J of Theta for a linear regression model, um, it turns out that,', 'start': 2033.945, 'duration': 11.162}, {'end': 2050.568, 'text': 'unlike the earlier diagram I had shown, which has local optima,', 'start': 2046.385, 'duration': 4.183}, {'end': 2058.353, 'text': 'it turns out that if J of Theta is defined the way that you know we just defined it for linear regression is the sum of squared terms.', 'start': 2050.568, 'duration': 7.785}, {'end': 2061.436, 'text': 'um, then J of Theta turns out to be a quadratic function, right?', 'start': 2058.353, 'duration': 3.083}, {'end': 2063.177, 'text': "It's the sum of these squares of terms.", 'start': 2061.476, 'duration': 1.701}, {'end': 2067.521, 'text': 'And so J of Theta will always look like- look like a big bowl like this.', 'start': 2063.318, 'duration': 4.203}], 'summary': 'Gradient descent minimizes j(theta) to find optimal parameters for linear regression with n=2 features.', 'duration': 95.327, 'max_score': 1972.194, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81972194.jpg'}, {'end': 2050.568, 'src': 'embed', 'start': 1979.338, 'weight': 0, 'content': [{'end': 1994.973, 'text': 'Okay. And so the gradient descent algorithm is to repeat until convergence.', 'start': 1979.338, 'duration': 15.635}, {'end': 2009.999, 'text': 'carry out this update and in each iteration of gradient descent uh, you do this update for j equals uh, 0,, 1, up to n.', 'start': 1994.973, 'duration': 15.026}, {'end': 2011.74, 'text': 'uh, where n is the number of features.', 'start': 2009.999, 'duration': 1.741}, {'end': 2014.021, 'text': 'So n was 2 in our example.', 'start': 2011.78, 'duration': 2.241}, {'end': 2020.563, 'text': 'Um, and if you do this then, uh, uh, you know, actually let me see.', 'start': 2016.002, 'duration': 4.561}, {'end': 2026.104, 'text': "Then what will happen is, um, I'll show you the animation.", 'start': 2021.503, 'duration': 4.601}, {'end': 2028.344, 'text': 'As you fit.', 'start': 2027.484, 'duration': 0.86}, {'end': 2032.105, 'text': 'hopefully you find a pretty good value of the parameters Theta, right?', 'start': 2028.344, 'duration': 3.761}, {'end': 2045.107, 'text': 'So, um, it turns out that when you plot the cost function J of Theta for a linear regression model, um, it turns out that,', 'start': 2033.945, 'duration': 11.162}, {'end': 2050.568, 'text': 'unlike the earlier diagram I had shown, which has local optima,', 'start': 2046.385, 'duration': 4.183}], 'summary': 'Gradient descent algorithm updates parameters for linear regression to find optimal values.', 'duration': 71.23, 'max_score': 1979.338, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81979338.jpg'}], 'start': 1600.949, 'title': 'Linear regression and gradient descent', 'summary': 'Discusses derivative calculation, partial derivatives, chain rule, and gradient descent algorithm for updating theta. it explains the cost function j of theta for linear regression and its optimization through gradient descent.', 'chapters': [{'end': 1875.536, 'start': 1600.949, 'title': 'Derivative calculation and partial derivatives', 'summary': 'Discusses derivative calculation, partial derivatives, and the chain rule, highlighting the process of taking the partial derivative with respect to theta j and explaining the resulting expression.', 'duration': 274.587, 'highlights': ['The process of taking the partial derivative with respect to Theta j is explained, resulting in the expression H(Theta x) - y times xj.', 'The lecture notes with detailed derivations and definitions are available on the CS229 course website, providing additional resources for understanding the material.', 'The lecture includes the explanation of taking the derivative of a square, where the 2 comes down, leading to the expression 2 times 1 half times the function inside.', 'The chapter emphasizes the simplification of the derivative calculation assuming just one training example, skipping the sum over all training examples for now.', "The chain rule of derivatives is discussed, demonstrating the process of taking the derivative of a square and multiplying it with the derivative of what's inside."]}, {'end': 2026.104, 'start': 1875.536, 'title': 'Gradient descent update', 'summary': 'Explains the process of updating theta using the gradient descent algorithm, demonstrating the correct formula for the partial derivative with respect to the cost function j of theta, and iterating through the update for each feature.', 'duration': 150.568, 'highlights': ['The correct formula for the partial derivative with respect to the cost function J of Theta is the sum of the derivative over all M training examples, with x_i as input features and y_i as the target label.', 'The gradient descent algorithm iterates through the update for each feature, where n is the number of features, and repeats the process until convergence.']}, {'end': 2155.434, 'start': 2027.484, 'title': 'Understanding linear regression cost function', 'summary': 'Explains how the cost function j of theta for a linear regression model forms a quadratic function with no local optima, and using gradient descent, the algorithm takes steps downhill in the direction of steepest descent which is always orthogonal to the contour direction.', 'duration': 127.95, 'highlights': ['The cost function J of Theta for a linear regression model forms a quadratic function with no local optima, and the only local optima is also the global optima.', 'Gradient descent algorithm takes steps downhill in the direction of steepest descent, which is always orthogonal to the contour direction.', 'The contours of the quadratic function will be ellipses or ovals, and the algorithm takes steps downhill in the direction of steepest descent.']}], 'duration': 554.485, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U81600949.jpg', 'highlights': ['The cost function J of Theta for linear regression forms a quadratic function with no local optima', 'The gradient descent algorithm iterates through the update for each feature until convergence', 'The process of taking the partial derivative with respect to Theta j is explained', 'The chain rule of derivatives is discussed, demonstrating the process of taking the derivative of a square', 'The correct formula for the partial derivative with respect to the cost function J of Theta is the sum of the derivative over all M training examples']}, {'end': 2441.127, 'segs': [{'end': 2196.64, 'src': 'embed', 'start': 2155.434, 'weight': 1, 'content': [{'end': 2159.198, 'text': "uh, uh, because there's only one global minimum.", 'start': 2155.434, 'duration': 3.764}, {'end': 2163.582, 'text': 'um, this algorithm will eventually converge to the global minimum.', 'start': 2159.198, 'duration': 4.384}, {'end': 2168.286, 'text': 'Okay Um, and so the question just now about the choice of the learning rate Alpha.', 'start': 2163.962, 'duration': 4.324}, {'end': 2174.432, 'text': 'Um, if you set Alpha to be very, very large, to be too large, then it can overshoot right?', 'start': 2168.847, 'duration': 5.585}, {'end': 2178.876, 'text': 'The steps you take can be too large and you can run past the minimum.', 'start': 2174.472, 'duration': 4.404}, {'end': 2183.497, 'text': 'uh, if you set it to be too small, then you need a lot of iterations and the error will be slow.', 'start': 2179.296, 'duration': 4.201}, {'end': 2193.679, 'text': 'And so what happens in practice is, uh, usually you try a few values and- and- and see what value of the learning rate allows you to most efficiently.', 'start': 2184.117, 'duration': 9.562}, {'end': 2196.64, 'text': 'you know, drive down the value of J of Theta, right?', 'start': 2193.679, 'duration': 2.961}], 'summary': 'Converging to global minimum with appropriate learning rate is crucial for efficient error reduction.', 'duration': 41.206, 'max_score': 2155.434, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82155434.jpg'}, {'end': 2260.799, 'src': 'embed', 'start': 2219.18, 'weight': 0, 'content': [{'end': 2226.143, 'text': 'So try 0.01, 0.02, 0.04, 0.08, kind of like a doubling scale or some uh, uh, uh,', 'start': 2219.18, 'duration': 6.963}, {'end': 2231.766, 'text': 'doubling scale or tripling scale and try a few values and see what value allows you to drive down the learning rate fastest.', 'start': 2226.143, 'duration': 5.623}, {'end': 2241.09, 'text': 'Okay Um, let me just- so I just want to visualize this in one other way, um, which is with the data.', 'start': 2232.127, 'duration': 8.963}, {'end': 2243.79, 'text': 'So, uh, this is- this is the actual dataset.', 'start': 2241.43, 'duration': 2.36}, {'end': 2247.011, 'text': 'Uh, there are, um, there are actually 49 points in this dataset.', 'start': 2243.991, 'duration': 3.02}, {'end': 2250.453, 'text': 'So M, the number of training examples is 49.', 'start': 2247.051, 'duration': 3.402}, {'end': 2259.037, 'text': 'And so if you initialize the parameters to 0, that means initializing your hypothesis or initializing a straight line fit to the data,', 'start': 2250.453, 'duration': 8.584}, {'end': 2260.799, 'text': 'to be that horizontal line, right?', 'start': 2259.037, 'duration': 1.762}], 'summary': 'Experiment with learning rates (0.01, 0.02, 0.04, 0.08) on a dataset with 49 points to drive down the learning rate fastest.', 'duration': 41.619, 'max_score': 2219.18, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82219180.jpg'}, {'end': 2314.731, 'src': 'embed', 'start': 2286.017, 'weight': 3, 'content': [{'end': 2290.08, 'text': 'So the parameters went from this value to this value, to this value, to this value, and so on.', 'start': 2286.017, 'duration': 4.063}, {'end': 2296.299, 'text': 'And so the other way of visualizing gradient descent is if gradient descent starts off.', 'start': 2290.996, 'duration': 5.303}, {'end': 2303.964, 'text': 'with this hypothesis, with each iteration of gradient descent, you are trying to find different values of the parameters.', 'start': 2296.299, 'duration': 7.665}, {'end': 2308.027, 'text': 'theta, uh, that allows the straight line to fit the data better.', 'start': 2303.964, 'duration': 4.063}, {'end': 2311.849, 'text': 'So after one iteration of gradient descent, this is a new hypothesis.', 'start': 2308.167, 'duration': 3.682}, {'end': 2314.731, 'text': 'You now have different values of Theta 0 and Theta 1.', 'start': 2311.929, 'duration': 2.802}], 'summary': 'Gradient descent iteratively updates parameters to fit data better.', 'duration': 28.714, 'max_score': 2286.017, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82286017.jpg'}, {'end': 2400.941, 'src': 'embed', 'start': 2370.182, 'weight': 5, 'content': [{'end': 2370.842, 'text': 'Why is the-?', 'start': 2370.182, 'duration': 0.66}, {'end': 2375.867, 'text': 'why are you subtracting alpha times the gradient rather than adding alpha times the gradient??', 'start': 2370.842, 'duration': 5.025}, {'end': 2379.249, 'text': 'Um, let me suggest- actually let me erase the screen.', 'start': 2376.347, 'duration': 2.902}, {'end': 2384.774, 'text': 'Um, so let me suggest you work through one example.', 'start': 2380.23, 'duration': 4.544}, {'end': 2391.4, 'text': "Um, uh, it turns out that if you add a multiple times the gradient, you'll be going uphill rather than going downhill.", 'start': 2384.794, 'duration': 6.606}, {'end': 2398.18, 'text': 'And maybe one way to see that would be if, um, you know, take a quadratic function.', 'start': 2391.56, 'duration': 6.62}, {'end': 2400.941, 'text': 'um, excuse me, right?', 'start': 2398.18, 'duration': 2.761}], 'summary': 'The transcript discusses the concept of subtracting alpha times the gradient to go downhill rather than uphill.', 'duration': 30.759, 'max_score': 2370.182, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82370182.jpg'}], 'start': 2155.434, 'title': 'Learning rate and gradient descent', 'summary': 'Discusses the impact of learning rate on convergence, emphasizing the need for an optimal learning rate for efficient convergence. it also explains gradient descent with 49 data points and the process of changing parameters theta to fit the data better.', 'chapters': [{'end': 2243.79, 'start': 2155.434, 'title': 'Choosing learning rate for convergence', 'summary': 'Discusses the impact of learning rate on convergence to the global minimum in an algorithm, emphasizing the significance of choosing an optimal learning rate for efficient convergence.', 'duration': 88.356, 'highlights': ['Setting the learning rate to a very large value can cause overshooting, leading to steps being too large and running past the minimum.', 'Choosing a very small learning rate can result in slow convergence, requiring a high number of iterations.', 'Practical approach involves trying multiple values of the learning rate, such as 0.01, 0.02, 0.04, and 0.08 on an exponential scale to identify the most efficient value for driving down the learning rate fastest.']}, {'end': 2441.127, 'start': 2243.991, 'title': 'Gradient descent explained', 'summary': 'Explains gradient descent with 49 data points, the process of changing parameters theta to fit the data better, and the reason for subtracting alpha times the gradient instead of adding it.', 'duration': 197.136, 'highlights': ['The chapter explains the process of changing parameters theta to fit the data better, with each iteration of gradient descent trying to find different values of the parameters theta that allows the straight line to fit the data better.', 'The chapter provides an example of subtracting alpha times the gradient instead of adding it by working through a quadratic function and demonstrates the reason for subtracting it multiple times the gradient to go downhill rather than uphill.', 'The chapter starts off with 49 points in the dataset and initializes the parameters to 0, explaining how the hypothesis starts with a horizontal line and then changes through gradient descent iterations.']}], 'duration': 285.693, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82155434.jpg', 'highlights': ['Practical approach involves trying multiple values of the learning rate, such as 0.01, 0.02, 0.04, and 0.08 on an exponential scale to identify the most efficient value for driving down the learning rate fastest.', 'Setting the learning rate to a very large value can cause overshooting, leading to steps being too large and running past the minimum.', 'Choosing a very small learning rate can result in slow convergence, requiring a high number of iterations.', 'The chapter explains the process of changing parameters theta to fit the data better, with each iteration of gradient descent trying to find different values of the parameters theta that allows the straight line to fit the data better.', 'The chapter starts off with 49 points in the dataset and initializes the parameters to 0, explaining how the hypothesis starts with a horizontal line and then changes through gradient descent iterations.', 'The chapter provides an example of subtracting alpha times the gradient instead of adding it by working through a quadratic function and demonstrates the reason for subtracting it multiple times the gradient to go downhill rather than uphill.']}, {'end': 3059.234, 'segs': [{'end': 2610.2, 'src': 'embed', 'start': 2581.324, 'weight': 2, 'content': [{'end': 2587.886, 'text': "And so every single step of gradient descent becomes very slow because you're scanning over, you're reading over right,", 'start': 2581.324, 'duration': 6.562}, {'end': 2597.372, 'text': 'like 100 million training examples, uh, uh, and uh, before you can even, you know, make one tiny little step of gradient descent, okay?, Um yeah,', 'start': 2587.886, 'duration': 9.486}, {'end': 2603.135, 'text': "And by the way, I think, I don't know, I- I- I feel like, uh, in today's era of big data, people start to lose intuitions about what's a big data set.", 'start': 2597.392, 'duration': 5.743}, {'end': 2610.2, 'text': "I think even by today's standards, like 100 million examples is still very big, right? I- I- I rarely, only rarely use 100 million examples.", 'start': 2603.156, 'duration': 7.044}], 'summary': "Scanning 100 million training examples slows gradient descent, even in today's big data era.", 'duration': 28.876, 'max_score': 2581.324, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82581324.jpg'}, {'end': 2892.314, 'src': 'embed', 'start': 2858.571, 'weight': 0, 'content': [{'end': 2865.634, 'text': 'Uh, but when you have a very large dataset, um stochastic gradient descent allows your implementation,', 'start': 2858.571, 'duration': 7.063}, {'end': 2868.316, 'text': 'allows your algorithm to make much faster progress.', 'start': 2865.634, 'duration': 2.682}, {'end': 2874.278, 'text': 'Uh, and so, um, uh and and so when you have very large datasets,', 'start': 2868.936, 'duration': 5.342}, {'end': 2890.852, 'text': 'stochastic gradient descent is used much more in practice than batch gradient descent.', 'start': 2874.278, 'duration': 16.574}, {'end': 2892.314, 'text': 'Uh, yeah.', 'start': 2890.852, 'duration': 1.462}], 'summary': 'Stochastic gradient descent is much faster than batch gradient descent for large datasets.', 'duration': 33.743, 'max_score': 2858.571, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82858571.jpg'}, {'end': 3059.234, 'src': 'embed', 'start': 3011.358, 'weight': 3, 'content': [{'end': 3012.818, 'text': 'So it takes smaller and smaller steps.', 'start': 3011.358, 'duration': 1.46}, {'end': 3018.14, 'text': 'So if you do that, then what happens is the size of the oscillations will decrease, Uh,', 'start': 3012.878, 'duration': 5.262}, {'end': 3021.761, 'text': 'and so you end up oscillating or bouncing around the smaller regions.', 'start': 3018.14, 'duration': 3.621}, {'end': 3027.903, 'text': "So wherever you end up may not be the global- global minimum, but at least it'll be- it'll be closer to the global minimum.", 'start': 3021.841, 'duration': 6.062}, {'end': 3030.584, 'text': 'Yeah So decreasing the learning rate is used much more often.', 'start': 3028.343, 'duration': 2.241}, {'end': 3033.205, 'text': 'Cool Question? Yeah.', 'start': 3032.165, 'duration': 1.04}, {'end': 3040.716, 'text': 'Oh, sure.', 'start': 3040.316, 'duration': 0.4}, {'end': 3046.682, 'text': "When do you stop the sun's rain descent? Uh, uh, plot J of Theta, uh, over time.", 'start': 3040.757, 'duration': 5.925}, {'end': 3049.565, 'text': "So J of Theta is a cost function that you're trying to drive down.", 'start': 3046.962, 'duration': 2.603}, {'end': 3052.127, 'text': 'So monitor J of Theta.', 'start': 3050.165, 'duration': 1.962}, {'end': 3057.332, 'text': "as you know, it's going down over time and then if it looks like it stopped going down, then you can say oh,", 'start': 3052.127, 'duration': 5.205}, {'end': 3059.234, 'text': 'it looks like it stopped going down and it stopped raining.', 'start': 3057.332, 'duration': 1.902}], 'summary': 'Decreasing learning rate reduces oscillations, gets closer to global minimum.', 'duration': 47.876, 'max_score': 3011.358, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83011358.jpg'}], 'start': 2441.447, 'title': 'Gradient descent in linear regression and stochastic gradient descent', 'summary': 'Discusses the batch gradient descent algorithm in linear regression, emphasizing its efficiency for large datasets and potential slowdown with millions of examples, while also exploring the drawbacks of batch gradient descent and the benefits of stochastic gradient descent in training algorithms with large datasets.', 'chapters': [{'end': 2613.613, 'start': 2441.447, 'title': 'Gradient descent in linear regression', 'summary': 'Discusses the batch gradient descent algorithm used in linear regression, highlighting its efficiency for large datasets and the potential slowdown due to processing millions of examples.', 'duration': 172.166, 'highlights': ['The batch gradient descent algorithm is widely used in learning and can be employed for various purposes with a focus on processing large datasets, such as those containing millions of examples.', "The term 'batch gradient descent' refers to processing the entire training set as one batch of data, which can result in slow computation when dealing with massive datasets, such as those with millions of examples.", 'In the era of big data, processing a data set with millions of examples using batch gradient descent can lead to slow computation, as each step requires scanning through the entire dataset, potentially resulting in a significant slowdown.']}, {'end': 3059.234, 'start': 2613.613, 'title': 'Gradient descent & stochastic gradient descent', 'summary': 'Explores the drawbacks of batch gradient descent, the benefits of stochastic gradient descent in training algorithms with large datasets, and the common practice of slowly decreasing the learning rate to minimize oscillations and get closer to the global minimum.', 'duration': 445.621, 'highlights': ['Stochastic gradient descent allows your algorithm to make much faster progress with very large datasets, making it more commonly used in practice than batch gradient descent.', 'Slowly decreasing the learning rate is a common practice to minimize oscillations and get closer to the global minimum in stochastic gradient descent.', 'Monitoring the cost function J of Theta over time is used to determine when to stop stochastic gradient descent.']}], 'duration': 617.787, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U82441447.jpg', 'highlights': ['Stochastic gradient descent allows faster progress with large datasets', 'Batch gradient descent is widely used for processing large datasets', 'Processing millions of examples with batch gradient descent can lead to slow computation', 'Slowly decreasing learning rate minimizes oscillations in stochastic gradient descent', 'Monitoring cost function J of Theta over time determines when to stop stochastic gradient descent']}, {'end': 3357.017, 'segs': [{'end': 3117.229, 'src': 'embed', 'start': 3060.1, 'weight': 1, 'content': [{'end': 3065.323, 'text': 'Uh, and then, um, you know, one nice thing about linear regression is, uh, it has no local optimum.', 'start': 3060.1, 'duration': 5.223}, {'end': 3066.784, 'text': 'And so,', 'start': 3065.623, 'duration': 1.161}, {'end': 3076.569, 'text': "um uh it- you run into these convergence debugging types of issues less often when you're training highly non-linear things like neural networks,", 'start': 3066.784, 'duration': 9.785}, {'end': 3078.47, 'text': "which we'll talk about later in CS2299 as well.", 'start': 3076.569, 'duration': 1.901}, {'end': 3080.971, 'text': 'Uh, these issues become more acute.', 'start': 3079.09, 'duration': 1.881}, {'end': 3086.934, 'text': 'Cool Okay, great.', 'start': 3083.532, 'duration': 3.402}, {'end': 3090.624, 'text': 'So, um, oh, yeah.', 'start': 3088.243, 'duration': 2.381}, {'end': 3098.667, 'text': 'Oh, would your learning rate be 1 over n times the learning rate for batch descent? Not really.', 'start': 3095.726, 'duration': 2.941}, {'end': 3099.808, 'text': "It's usually much bigger than that.", 'start': 3098.687, 'duration': 1.121}, {'end': 3101.489, 'text': 'Uh, uh, yeah.', 'start': 3100.288, 'duration': 1.201}, {'end': 3102.069, 'text': 'Uh, yeah.', 'start': 3101.589, 'duration': 0.48}, {'end': 3108.852, 'text': 'Um, because if your learning rate was 1 over n times that of what you use with batch descent, then it end up being as slow as batch descent.', 'start': 3102.209, 'duration': 6.643}, {'end': 3109.932, 'text': "So it's usually much bigger.", 'start': 3108.872, 'duration': 1.06}, {'end': 3117.229, 'text': "Okay So, um, So that's stochastic gradient descent.", 'start': 3112.033, 'duration': 5.196}], 'summary': 'Linear regression has no local optimum, making convergence debugging less common in training non-linear models like neural networks.', 'duration': 57.129, 'max_score': 3060.1, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83060100.jpg'}, {'end': 3187.481, 'src': 'embed', 'start': 3134.915, 'weight': 0, 'content': [{'end': 3136.816, 'text': "because it's one less thing to fiddle with right?", 'start': 3134.915, 'duration': 1.901}, {'end': 3138.696, 'text': "It's just one less thing to have to worry about.", 'start': 3136.836, 'duration': 1.86}, {'end': 3140.397, 'text': 'uh, the parameters oscillating.', 'start': 3138.696, 'duration': 1.701}, {'end': 3149.42, 'text': 'But if your dataset is too large, that batch gradient descent becomes prohibitive- prohibitively slow, then, uh, almost everyone would use, you know,', 'start': 3140.817, 'duration': 8.603}, {'end': 3151.181, 'text': 'stochastic gradient descent instead, right?', 'start': 3149.42, 'duration': 1.761}, {'end': 3168.308, 'text': 'Or- or, however, more like stochastic gradient descent, um, okay?, All right.', 'start': 3151.341, 'duration': 16.967}, {'end': 3175.687, 'text': 'So um gradient descent, both Bastian descent and Sarkozy.', 'start': 3170.661, 'duration': 5.026}, {'end': 3181.714, 'text': 'gradient descent is an iterative algorithm, meaning that you have to take multiple steps to get to.', 'start': 3175.687, 'duration': 6.027}, {'end': 3184.437, 'text': 'you know, get near, hopefully the global optimum.', 'start': 3181.714, 'duration': 2.723}, {'end': 3187.481, 'text': 'It turns out, this is another algorithm.', 'start': 3185.138, 'duration': 2.343}], 'summary': 'Gradient descent is a popular iterative algorithm, with stochastic gradient descent being preferred for large datasets due to its speed.', 'duration': 52.566, 'max_score': 3134.915, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83134915.jpg'}, {'end': 3251.842, 'src': 'embed', 'start': 3225.308, 'weight': 4, 'content': [{'end': 3231.091, 'text': 'to just jump in one step to the global optimum uh without needing to use an iterative algorithm.', 'start': 3225.308, 'duration': 5.783}, {'end': 3235.533, 'text': "Right And- and this- this- this one I'm gonna present next is called the normal equation.", 'start': 3231.451, 'duration': 4.082}, {'end': 3237.314, 'text': 'It works only for linear regression.', 'start': 3235.613, 'duration': 1.701}, {'end': 3239.996, 'text': "It doesn't work for any of the other algorithms we'll talk about later this quarter.", 'start': 3237.334, 'duration': 2.662}, {'end': 3251.842, 'text': 'But, um, uh, but let me quickly show you the derivation of that.', 'start': 3240.556, 'duration': 11.286}], 'summary': 'Normal equation finds global optima for linear regression.', 'duration': 26.534, 'max_score': 3225.308, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83225308.jpg'}, {'end': 3321.919, 'src': 'embed', 'start': 3296.809, 'weight': 5, 'content': [{'end': 3303.052, 'text': 'um, what some linear algebra classes do is cover the board with, you know, pages and pages of matrix derivatives.', 'start': 3296.809, 'duration': 6.243}, {'end': 3314.197, 'text': 'Um, what I wanna do is describe to you a matrix derivative notation that allows you to derive the normal equation in roughly four lines of linear algebra.', 'start': 3303.612, 'duration': 10.585}, {'end': 3316.698, 'text': 'uh, rather than sort of pages and pages of linear algebra.', 'start': 3314.197, 'duration': 2.501}, {'end': 3321.919, 'text': "And in the work I've done in machine learning, you know, sometimes notation really matters right?", 'start': 3317.258, 'duration': 4.661}], 'summary': 'Introducing a concise matrix derivative notation to derive the normal equation in roughly four lines of linear algebra, simplifying the process from pages and pages.', 'duration': 25.11, 'max_score': 3296.809, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83296809.jpg'}], 'start': 3060.1, 'title': 'Gradient descent techniques', 'summary': 'Compares stochastic and batch gradient descent methods, emphasizing the advantages of using stochastic gradient descent for large datasets. it also discusses issues related to learning rates and convergence debugging in training highly non-linear models like neural networks. additionally, it explains the iterative algorithm of gradient descent and its use in various algorithms, including the special case of the normal equation for linear regression, which offers a one-step solution to the global optimum.', 'chapters': [{'end': 3151.181, 'start': 3060.1, 'title': 'Gradient descent comparison', 'summary': 'Discusses the advantages of using stochastic gradient descent over batch gradient descent, especially when dealing with large datasets and the potential issues related to learning rates and convergence debugging in training highly non-linear models like neural networks.', 'duration': 91.081, 'highlights': ['Stochastic gradient descent is preferred for large datasets due to the prohibitively slow nature of batch gradient descent, which is more efficient for small datasets.', 'Learning rate for stochastic gradient descent is usually much bigger than 1 over n times the learning rate for batch descent to avoid being as slow as batch descent.', 'Linear regression has no local optimum, leading to less convergence debugging issues when training highly non-linear models like neural networks.']}, {'end': 3357.017, 'start': 3151.341, 'title': 'Gradient descent and normal equation', 'summary': 'Explains the iterative algorithm of gradient descent and its use in various algorithms, along with the special case of the normal equation for linear regression, offering a one-step solution to the global optimum.', 'duration': 205.676, 'highlights': ['Gradient descent is an iterative algorithm used in various algorithms including generalized linear models and neural networks.', 'The normal equation provides a one-step solution to the global optimum for linear regression, unlike iterative algorithms like gradient descent.', 'Matrix derivative notation allows for the derivation of the normal equation in just four lines of linear algebra, simplifying the process compared to traditional methods.']}], 'duration': 296.917, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83060100.jpg', 'highlights': ['Stochastic gradient descent is preferred for large datasets due to its efficiency compared to batch gradient descent.', 'Learning rate for stochastic gradient descent is usually much bigger than 1 over n times the learning rate for batch descent.', 'Linear regression has no local optimum, leading to less convergence debugging issues when training highly non-linear models like neural networks.', 'Gradient descent is an iterative algorithm used in various algorithms including generalized linear models and neural networks.', 'The normal equation provides a one-step solution to the global optimum for linear regression, unlike iterative algorithms like gradient descent.', 'Matrix derivative notation allows for the derivation of the normal equation in just four lines of linear algebra, simplifying the process compared to traditional methods.']}, {'end': 4692.184, 'segs': [{'end': 3412.977, 'src': 'embed', 'start': 3387.302, 'weight': 0, 'content': [{'end': 3394.966, 'text': "Where- remember, Theta is a three-dimensional vector, so it's R3, or actually it's R, n plus 1, right?", 'start': 3387.302, 'duration': 7.664}, {'end': 3399.929, 'text': 'If you have, uh, two features of the hulls, if n equals 2, then Theta is three-dimensional.', 'start': 3395.046, 'duration': 4.883}, {'end': 3401.771, 'text': "it's n plus 1 dimensional, so it's a vector.", 'start': 3399.929, 'duration': 1.842}, {'end': 3408.134, 'text': "And so I'm gonna define the derivative with respect to Theta of J of Theta as follows.", 'start': 3402.351, 'duration': 5.783}, {'end': 3412.977, 'text': 'Um, this is going to be itself, uh, 3 by 1 vector.', 'start': 3408.475, 'duration': 4.502}], 'summary': 'Theta is a 3d vector in r, n+1 dimensional, with its derivative being a 3x1 vector.', 'duration': 25.675, 'max_score': 3387.302, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83387302.jpg'}, {'end': 3896.028, 'src': 'embed', 'start': 3865.12, 'weight': 1, 'content': [{'end': 3867.123, 'text': 'So trace just means sum of diagonal entries.', 'start': 3865.12, 'duration': 2.003}, {'end': 3871.77, 'text': 'And um, some facts about the trace of a matrix.', 'start': 3868.044, 'duration': 3.726}, {'end': 3878.675, 'text': 'you know, trace of A is equal to the trace of A transpose, because if you transpose the matrix right,', 'start': 3871.77, 'duration': 6.905}, {'end': 3884.88, 'text': "you're just flipping it along the- the 45 degree axis and so the- the diagonal entries actually stay the same when you transpose the matrix.", 'start': 3878.675, 'duration': 6.205}, {'end': 3888.183, 'text': 'So the trace of A is equal to the trace of A transpose.', 'start': 3884.9, 'duration': 3.283}, {'end': 3896.028, 'text': 'Um, and then, uh, there- there- there are some other useful properties of, um, the trace operator.', 'start': 3889.043, 'duration': 6.985}], 'summary': 'Trace of a matrix equals trace of its transpose, useful properties of trace operator.', 'duration': 30.908, 'max_score': 3865.12, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83865120.jpg'}, {'end': 4042.085, 'src': 'heatmap', 'start': 3986.721, 'weight': 0.755, 'content': [{'end': 3994.128, 'text': 'And um, another one that is a little bit harder to prove.', 'start': 3986.721, 'duration': 7.407}, {'end': 4007.495, 'text': 'is that the trace, excuse me, derivative of A trans- of A, A transpose C is Okay.', 'start': 3994.128, 'duration': 13.367}, {'end': 4015.881, 'text': 'Yeah. So I think just as- just as for you know ordinary um calculus, we know the derivative of x squared is 2x right?', 'start': 4007.775, 'duration': 8.106}, {'end': 4021.164, 'text': 'And so we all figured out that rule and we just use it too much without- without having to re-derive it every time.', 'start': 4015.921, 'duration': 5.243}, {'end': 4022.986, 'text': 'Uh, this is a little bit like that.', 'start': 4021.785, 'duration': 1.201}, {'end': 4031.271, 'text': "The trace of a squared c is, you know, 2 times cA, right? It's a little bit like that, but- but with- with matrix notation is there.", 'start': 4023.046, 'duration': 8.225}, {'end': 4042.085, 'text': 'So think of this as analogous to, DDA of A squared C equals 2AC, right? But this is like the matrix version of that.', 'start': 4031.331, 'duration': 10.754}], 'summary': 'Derivative of a transpose times a times c is analogous to the derivative of x squared, 2ac.', 'duration': 55.364, 'max_score': 3986.721, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83986721.jpg'}, {'end': 4341.491, 'src': 'embed', 'start': 4312.215, 'weight': 2, 'content': [{'end': 4315.959, 'text': 'And um, if you- you- you remember so Z transpose.', 'start': 4312.215, 'duration': 3.744}, {'end': 4318.701, 'text': 'Z is equal to sum over i.', 'start': 4315.959, 'duration': 2.742}, {'end': 4319.582, 'text': 'Z squared right?', 'start': 4318.701, 'duration': 0.881}, {'end': 4322.525, 'text': 'A vector transpose itself is the sum of squares of elements.', 'start': 4319.822, 'duration': 2.703}, {'end': 4328.711, 'text': 'And so this vector transpose itself is the sum of squares of the elements.', 'start': 4323.105, 'duration': 5.606}, {'end': 4332.406, 'text': 'Right So- so, which is y uh so- so.', 'start': 4329.485, 'duration': 2.921}, {'end': 4337.909, 'text': 'the cost function J of Theta is computed by taking the sum of squares of all of these elements, of all of these errors.', 'start': 4332.406, 'duration': 5.503}, {'end': 4341.491, 'text': 'And- and the way you do that is to take this vector.', 'start': 4338.249, 'duration': 3.242}], 'summary': 'The cost function j of theta is computed by taking the sum of squares of errors.', 'duration': 29.276, 'max_score': 4312.215, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U84312215.jpg'}, {'end': 4655.357, 'src': 'heatmap', 'start': 4595.275, 'weight': 3, 'content': [{'end': 4608.293, 'text': 'And uh, the optimum value for Theta is Theta equals X transpose X, inverse X transpose Y.', 'start': 4595.275, 'duration': 13.018}, {'end': 4621.165, 'text': 'okay?. Um, and if you implement this, um, then you know you can, in basically one step, get the value of Theta that corresponds to the global minimum.', 'start': 4608.293, 'duration': 12.872}, {'end': 4631.044, 'text': 'Um, and-, and- and again common question I get at this point is well, what if X is non-invertible?', 'start': 4626.042, 'duration': 5.002}, {'end': 4634.145, 'text': 'Uh, what that usually means is you have, uh, redundant features.', 'start': 4631.064, 'duration': 3.081}, {'end': 4635.965, 'text': 'uh, that your features are linearly dependent.', 'start': 4634.145, 'duration': 1.82}, {'end': 4641.247, 'text': "Uh, uh, but if you use something called the pseudo-inverse, you- you kind of get the right answer if that's the case.", 'start': 4636.786, 'duration': 4.461}, {'end': 4647.489, 'text': 'Although I think the even more right answer is if you have linearly dependent features, probably means you have the same feature repeated twice,', 'start': 4641.287, 'duration': 6.202}, {'end': 4651.731, 'text': 'and I would usually go and figure out what features actually repeated leading to this problem.', 'start': 4647.489, 'duration': 4.242}, {'end': 4655.357, 'text': 'Okay All right.', 'start': 4651.751, 'duration': 3.606}], 'summary': 'Optimum value for theta is x transpose x inverse x transpose y, ensuring global minimum.', 'duration': 18.571, 'max_score': 4595.275, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U84595275.jpg'}], 'start': 3357.077, 'title': 'Matrix derivatives in machine learning', 'summary': 'Introduces derivative notation for learning algorithms, discusses matrix function mapping, explains matrix trace properties, and outlines matrix operations and normal equations in machine learning.', 'chapters': [{'end': 3449.607, 'start': 3357.077, 'title': 'Derivative notation for learning algorithms', 'summary': 'Introduces a derivative notation for learning algorithms, defining the derivative of j of theta with respect to theta, as a 3 by 1 vector representing a three-dimensional vector with three components, used for partial derivatives of j with respect to each element.', 'duration': 92.53, 'highlights': ['The derivative of J of Theta with respect to Theta is defined as a 3 by 1 vector, representing a three-dimensional vector with three components, used for partial derivatives of J with respect to each element.', 'The three components of the 3 by 1 vector represent the partial derivative of J with respect to each of the three elements.']}, {'end': 3827.984, 'start': 3449.627, 'title': 'Matrix derivative and function mapping', 'summary': 'Discusses the concept of matrix functions mapping to real numbers and the derivative of a matrix function with respect to the matrix, along with an overview of the derivation of the normal equation for minimizing the cost function j of theta.', 'duration': 378.357, 'highlights': ['Matrix functions mapping to real numbers and the derivative of a matrix function with respect to the matrix are explained, using a specific example of a function that inputs a 2 by 2 matrix and maps the values of the matrix to a real number.', 'The process of deriving the normal equation for minimizing the cost function J of Theta, by taking derivatives with respect to Theta, setting the derivatives equal to 0, and solving for Theta, is outlined, emphasizing the goal of finding the global minimum of the cost function.', 'The consistent application of the matrix derivative definition to a column vector, treating it as an n by 1 matrix, is discussed, highlighting its alignment with the previously described definition for the derivative of J with respect to Theta.', 'The concept of a matrix derivative and its relevance to the derivation of the normal equation for minimizing the cost function J of Theta is emphasized, with a focus on the process of taking derivatives with respect to Theta, setting them equal to 0, and solving for Theta to attain the global minimum of the cost function.']}, {'end': 4114.444, 'start': 3828.725, 'title': 'Matrix trace properties', 'summary': 'Explains the properties of the trace operator, including its definition as the sum of diagonal entries, its equality to the trace of the transposed matrix, and its derivative properties, such as the derivative of f of a equaling b transpose. it also covers the cyclic permutation property and the derivative of a squared c.', 'duration': 285.719, 'highlights': ['The trace of a matrix A is defined as the sum of its diagonal entries, denoted as the trace of A.', 'The trace of A is equal to the trace of A transpose, showcasing the equality of the trace of a matrix and its transposed version.', 'The derivative of f of A, where f of A equals the trace of A times B, is equal to B transpose, providing a significant derivative property of the trace operator.', 'The trace of AB is equal to the trace of BA, demonstrating the commutative property of the trace operator for matrix multiplication.', 'The trace of A times B times C is equal to the trace of C times A times B, showcasing the cyclic permutation property of the trace operator for matrix multiplication.', 'The derivative of A squared C is analogous to the derivative of x squared, but with matrix notation, representing a significant derivative property of the trace operator in matrix notation.']}, {'end': 4692.184, 'start': 4120.185, 'title': 'Matrix operations and normal equations', 'summary': 'Explains matrix operations and the normal equations in machine learning, outlining the process of defining the design matrix x, computing the cost function j of theta, and deriving the normal equations to obtain the optimum value for theta.', 'duration': 571.999, 'highlights': ['The chapter explains the concept of the design matrix X, which involves stacking up M training examples in rows and defining the vector y to be the labels from the training examples, providing a fundamental understanding of matrix operations in machine learning.', 'The cost function J of Theta is computed as 1 half of X Theta minus Y transpose X Theta minus Y, which represents the sum of squares of the errors made by the learning algorithm on the M examples, offering insight into the calculation of the cost function in machine learning.', 'The derivation of the normal equations is outlined, where the optimum value for Theta is obtained as Theta equals X transpose X, inverse X transpose Y, providing a crucial method to find the global minimum in machine learning.', 'The explanation of addressing non-invertible X by using the pseudo-inverse and identifying redundant features in linearly dependent features, highlighting practical solutions to handle such scenarios in machine learning.']}], 'duration': 1335.107, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/4b4MUYve_U8/pics/4b4MUYve_U83357077.jpg', 'highlights': ['The derivative of J of Theta with respect to Theta is defined as a 3 by 1 vector, representing a three-dimensional vector with three components, used for partial derivatives of J with respect to each element.', 'The trace of A is equal to the trace of A transpose, showcasing the equality of the trace of a matrix and its transposed version.', 'The cost function J of Theta is computed as 1 half of X Theta minus Y transpose X Theta minus Y, which represents the sum of squares of the errors made by the learning algorithm on the M examples, offering insight into the calculation of the cost function in machine learning.', 'The explanation of addressing non-invertible X by using the pseudo-inverse and identifying redundant features in linearly dependent features, highlighting practical solutions to handle such scenarios in machine learning.']}], 'highlights': ['The normal equation provides a one-step solution to the global optimum for linear regression, unlike iterative algorithms like gradient descent.', 'The chapter introduces the implementation of the gradient descent algorithm to minimize the cost function J of Theta, including a discussion on Y squared error and error to the power of 4.', 'The cost function J of Theta for linear regression forms a quadratic function with no local optima', 'The process of taking the partial derivative with respect to Theta j is explained', 'The hypothesis is expressed as h of x equals Theta 0 plus Theta 1, x1 plus Theta 2 x2, where x1 is the size of the house and x2 is the number of bedrooms.', 'Stochastic gradient descent allows faster progress with large datasets', 'The derivative of J of Theta with respect to Theta is defined as a 3 by 1 vector, representing a three-dimensional vector with three components, used for partial derivatives of J with respect to each element.', 'The instructor mentions the initialization of the Theta vector, suggesting that setting it to the vector of all zeros could be a reasonable default.', 'The job of the learning algorithm in supervised learning is to output a function, such as a hypothesis to make predictions about housing prices.', 'The key decisions in machine learning algorithm design include structuring the workflow, dataset, and representation of the hypothesis.']}