title

Lecture 4 | Introduction to Neural Networks

description

In Lecture 4 we progress from linear classifiers to fully-connected neural networks. We introduce the backpropagation algorithm for computing gradients and briefly discuss connections between artificial neural networks and biological neural networks.
Keywords: Neural networks, computational graphs, backpropagation, activation functions, biological neurons
Slides: http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf
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Convolutional Neural Networks for Visual Recognition
Instructors:
Fei-Fei Li: http://vision.stanford.edu/feifeili/
Justin Johnson: http://cs.stanford.edu/people/jcjohns/
Serena Yeung: http://ai.stanford.edu/~syyeung/
Computer Vision has become ubiquitous in our society, with applications in search, image understanding, apps, mapping, medicine, drones, and self-driving cars. Core to many of these applications are visual recognition tasks such as image classification, localization and detection. Recent developments in neural network (aka “deep learning”) approaches have greatly advanced the performance of these state-of-the-art visual recognition systems. This lecture collection is a deep dive into details of the deep learning architectures with a focus on learning end-to-end models for these tasks, particularly image classification. From this lecture collection, students will learn to implement, train and debug their own neural networks and gain a detailed understanding of cutting-edge research in computer vision.
Website:
http://cs231n.stanford.edu/
For additional learning opportunities please visit:
http://online.stanford.edu/

detail

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in this class.", 'start': 32.598, 'duration': 3.143}, {'end': 38.544, 'text': "Before we begin, let's see.", 'start': 36.942, 'duration': 1.602}, {'end': 43.265, 'text': 'So a few administrative details.', 'start': 41.724, 'duration': 1.541}, {'end': 47.088, 'text': 'So assignment one is due Thursday, April 20th.', 'start': 43.846, 'duration': 3.242}, {'end': 50.19, 'text': 'So reminder, we shifted the date back by a little bit.', 'start': 47.188, 'duration': 3.002}, {'end': 53.673, 'text': "And it's gonna be due 11.59 p.m. on Canvas.", 'start': 51.191, 'duration': 2.482}, {'end': 58.316, 'text': 'So you should start thinking about your projects.', 'start': 56.395, 'duration': 1.921}, {'end': 61.899, 'text': 'There are TA specialties listed on the Piazza website.', 'start': 58.937, 'duration': 2.962}, {'end': 70.566, 'text': "So if you have questions about a specific project topic you're thinking about, you can go and try and find the TAs that might be most relevant.", 'start': 61.959, 'duration': 8.607}, {'end': 80.196, 'text': 'And then also for Google Cloud, so all students are going to get $100 in credits to use for Google Cloud for their assignments and project.', 'start': 72.207, 'duration': 7.989}, {'end': 82.959, 'text': 'So you should be receiving an email for that this week.', 'start': 80.876, 'duration': 2.083}], 'summary': 'Assignment one due april 20th, $100 google cloud credits for students.', 'duration': 50.361, 'max_score': 32.598, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k32598.jpg'}], 'start': 4.921, 'title': 'Back propagation and neural networks at stanford', 'summary': 'Covers the announcement of assignment due date, ta specialties for project topics, and the allocation of $100 in credits for google cloud to students for their assignments and projects at stanford university.', 'chapters': [{'end': 82.959, 'start': 4.921, 'title': 'Back propagation and neural networks at stanford', 'summary': 'Covers the announcement of assignment due date, ta specialties for project topics, and the allocation of $100 in credits for google cloud to students for their assignments and projects at stanford university.', 'duration': 78.038, 'highlights': ['Students at Stanford University are allocated $100 in credits to use for Google Cloud for their assignments and projects, with emails expected to be received this week.', 'Assignment one is due on Thursday, April 20th at 11:59 p.m., with the date being shifted back.', 'TA specialties for project topics are listed on the Piazza website for students to seek relevant assistance.']}], 'duration': 78.038, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k4921.jpg', 'highlights': ['Students at Stanford University are allocated $100 in credits to use for Google Cloud for their assignments and projects, with emails expected to be received this week.', 'Assignment one is due on Thursday, April 20th at 11:59 p.m., with the date being shifted back.', 'TA specialties for project topics are listed on the Piazza website for students to seek relevant assistance.']}, {'end': 484.295, 'segs': [{'end': 174.364, 'src': 'embed', 'start': 135.482, 'weight': 1, 'content': [{'end': 137.082, 'text': 'And we have a preference for simpler models.', 'start': 135.482, 'duration': 1.6}, {'end': 139.078, 'text': 'for better generalization.', 'start': 137.897, 'duration': 1.181}, {'end': 144.743, 'text': 'And so now we want to find the parameters w that correspond to our lowest loss.', 'start': 139.799, 'duration': 4.944}, {'end': 146.264, 'text': 'We want to minimize the loss function.', 'start': 144.783, 'duration': 1.481}, {'end': 150.408, 'text': 'And so to do that, we want to find the gradient of l with respect to w.', 'start': 146.745, 'duration': 3.663}, {'end': 160.773, 'text': 'So, last lecture we talked about how we can do this using optimization, where we want to iteratively take steps in the direction of steepest descent,', 'start': 152.887, 'duration': 7.886}, {'end': 166.178, 'text': 'which is the negative of the gradient, in order to walk down this lost landscape and get to the point of lowest loss.', 'start': 160.773, 'duration': 5.405}, {'end': 174.364, 'text': 'And we saw how this gradient descent can basically take this trajectory, looking like this image on the right,', 'start': 166.818, 'duration': 7.546}], 'summary': 'Prefer simpler models for better generalization; minimize loss function using gradient descent.', 'duration': 38.882, 'max_score': 135.482, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k135482.jpg'}, {'end': 264.634, 'src': 'embed', 'start': 236.763, 'weight': 0, 'content': [{'end': 245.509, 'text': 'what a computational graph is is that we can use this kind of graph in order to represent any function where the nodes of the graph are steps of computation that we go through.', 'start': 236.763, 'duration': 8.746}, {'end': 254.371, 'text': "So, for example, in this example of a linear classifier that we've talked about, the inputs here are x and w,", 'start': 246.209, 'duration': 8.162}, {'end': 264.634, 'text': 'and then this multiplication node represents the matrix multiplier, the multiplication of the parameters w with our data x that we have,', 'start': 254.371, 'duration': 10.263}], 'summary': 'A computational graph represents functions with nodes as computation steps, exemplified with a linear classifier using x and w.', 'duration': 27.871, 'max_score': 236.763, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k236763.jpg'}, {'end': 351.147, 'src': 'embed', 'start': 321.968, 'weight': 2, 'content': [{'end': 327.152, 'text': 'And the input has to go through many layers of transformations in order to get all the way down to the loss function.', 'start': 321.968, 'duration': 5.184}, {'end': 337.157, 'text': 'And this can get even crazier with things like a neural Turing machine, which is another kind of deep learning model.', 'start': 330.552, 'duration': 6.605}, {'end': 345.003, 'text': 'And in this case, you can see that the computational graph for this is really insane, and especially we end up unrolling this over time.', 'start': 337.477, 'duration': 7.526}, {'end': 351.147, 'text': "It's basically completely impractical if you want to compute the gradients for any of these intermediate variables.", 'start': 345.523, 'duration': 5.624}], 'summary': 'Deep learning models undergo complex transformations, impractical for computing gradients.', 'duration': 29.179, 'max_score': 321.968, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k321968.jpg'}, {'end': 495.722, 'src': 'embed', 'start': 462.964, 'weight': 4, 'content': [{'end': 466.907, 'text': "So what back wrap is, it's a recursive application of the chain rule.", 'start': 462.964, 'duration': 3.943}, {'end': 470.891, 'text': "So we're gonna start at the back, the very end of the computational graph,", 'start': 466.947, 'duration': 3.944}, {'end': 474.754, 'text': "and then we're going to work our way backwards and compute all the gradients along the way.", 'start': 470.891, 'duration': 3.863}, {'end': 484.295, 'text': 'So here if we start at the very end, we want to compute the gradient of the output with respect to the last variable, which is just f.', 'start': 475.97, 'duration': 8.325}, {'end': 486.597, 'text': "And so this gradient is just one, it's trivial.", 'start': 484.295, 'duration': 2.302}, {'end': 495.722, 'text': 'So now moving backwards, we want the gradient with respect to z.', 'start': 489.659, 'duration': 6.063}], 'summary': 'Back wrap recursively applies chain rule to compute gradients, starting at the end of the computational graph.', 'duration': 32.758, 'max_score': 462.964, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k462964.jpg'}], 'start': 83.059, 'title': 'Analytic gradient computation, computational graphs, and backpropagation', 'summary': 'Covers the process of defining a classifier, computing gradients, using computational graphs to represent functions in deep learning, and explaining backpropagation using examples, including linear classifiers, convolutional neural networks, and neural turing machines.', 'chapters': [{'end': 236.763, 'start': 83.059, 'title': 'Analytic gradient computation', 'summary': 'Discusses the process of defining a classifier using a function parametrized by weights, introducing a loss function, and computing the gradient of the loss function to minimize the loss in order to find the parameters corresponding to the lowest loss.', 'duration': 153.704, 'highlights': ['The chapter discusses the process of defining a classifier using a function parametrized by weights, introducing a loss function, and computing the gradient of the loss function to minimize the loss in order to find the parameters corresponding to the lowest loss.', 'The function f takes data x as input and outputs a vector of scores for each class.', 'The total loss term L is a combination of the data term and a regularization term expressing model simplicity.', 'The process involves finding the gradient of the loss function with respect to the parameters w to minimize the loss.', 'Different methods for computing the gradient are discussed, including numerical and analytic approaches.']}, {'end': 345.003, 'start': 236.763, 'title': 'Computational graphs in deep learning', 'summary': 'Explains how computational graphs are used to represent functions in deep learning, using examples of linear classifiers and complex models like convolutional neural networks and neural turing machines.', 'duration': 108.24, 'highlights': ['The computational graph is used to represent any function, with nodes representing computation steps, such as matrix multiplication for linear classifiers and layers of transformations for complex models.', 'Backpropagation is a technique used with computational graphs to compute gradients with respect to every variable in the graph, making it useful for working with complex functions like convolutional neural networks and neural Turing machines.', 'Convolutional neural networks involve multiple layers of transformations from input image to loss function, demonstrating the complexity of functions that can be represented using computational graphs.']}, {'end': 484.295, 'start': 345.523, 'title': 'Backpropagation in computational graphs', 'summary': 'Explains backpropagation in computational graphs using a simple example of f(x, y, z) = x + y * z, demonstrating the recursive application of the chain rule to compute gradients, and it showcases the intermediate variable naming and gradient computation with respect to x, y, and z.', 'duration': 138.772, 'highlights': ['The chapter explains the recursive application of the chain rule to compute gradients in a computational graph.', 'It showcases the intermediate variable naming and gradient computation with respect to x, y, and z.', 'The example function f(x, y, z) = x + y * z is used to demonstrate the concept of backpropagation in computational graphs.']}], 'duration': 401.236, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k83059.jpg', 'highlights': ['The computational graph represents any function, aiding in backpropagation (5)', 'Defining a classifier involves minimizing loss by computing gradients (4)', 'Convolutional neural networks demonstrate the complexity of computational graphs (3)', 'The process includes finding the gradient of the loss function to minimize loss (3)', 'Backpropagation involves the recursive application of the chain rule (2)']}, {'end': 856.281, 'segs': [{'end': 575.625, 'src': 'heatmap', 'start': 489.659, 'weight': 2, 'content': [{'end': 495.722, 'text': 'So now moving backwards, we want the gradient with respect to z.', 'start': 489.659, 'duration': 6.063}, {'end': 499.625, 'text': 'And we know that df over dz is equal to q.', 'start': 495.722, 'duration': 3.903}, {'end': 507.604, 'text': 'So the value of q is just three, and so we have here, df over dz equals three.', 'start': 499.625, 'duration': 7.979}, {'end': 512.246, 'text': 'And so next, if we want to do df over dq, what is the value of that??', 'start': 508.464, 'duration': 3.782}, {'end': 517.467, 'text': 'What is df over dq?', 'start': 516.307, 'duration': 1.16}, {'end': 527.911, 'text': 'So we have here df over dq is equal to z and the value of z is negative four.', 'start': 517.508, 'duration': 10.403}, {'end': 531.953, 'text': 'So here we have df over dq is equal to negative four.', 'start': 529.512, 'duration': 2.441}, {'end': 541.986, 'text': 'Okay, so now continuing to move backwards through the graph, we want to find df over dy.', 'start': 537.102, 'duration': 4.884}, {'end': 548.59, 'text': 'But here in this case, the gradient with respect to y, y is not connected directly to f.', 'start': 543.427, 'duration': 5.163}, {'end': 552.453, 'text': "It's connected through an intermediate node of z.", 'start': 548.59, 'duration': 3.863}, {'end': 563.541, 'text': "And so the way we're going to do this is we can leverage the chain rule, which says that df over dy can be written as df over dq times dq over dy.", 'start': 552.453, 'duration': 11.088}, {'end': 575.625, 'text': 'And so the intuition of this is that, in order to find the effect of y on f, this is actually equivalent to if we take the effect of q times, q on f,', 'start': 564.682, 'duration': 10.943}], 'summary': 'Using chain rule to find gradients, df/dy can be expressed as df/dq * dq/dy.', 'duration': 91.33, 'max_score': 489.659, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k489659.jpg'}, {'end': 686.93, 'src': 'embed', 'start': 661.365, 'weight': 0, 'content': [{'end': 666.768, 'text': 'Again, we have negative four times one, and the gradient with respect to x is going to be negative four.', 'start': 661.365, 'duration': 5.403}, {'end': 676.962, 'text': "Okay, so what we're doing in backprop is we basically have all of these nodes in our computational graph,", 'start': 671.058, 'duration': 5.904}, {'end': 679.744, 'text': 'but each node is only aware of its immediate surroundings.', 'start': 676.962, 'duration': 2.782}, {'end': 686.93, 'text': 'So we have at each node we have the local inputs that are connected to this node, the values that are flowing into the node,', 'start': 680.405, 'duration': 6.525}], 'summary': 'Backpropagation computes local gradients for each node in the computational graph.', 'duration': 25.565, 'max_score': 661.365, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k661365.jpg'}, {'end': 746.116, 'src': 'embed', 'start': 710.894, 'weight': 1, 'content': [{'end': 716.419, 'text': 'Each node is going to be something like the addition or the multiplication that we had in that earlier example,', 'start': 710.894, 'duration': 5.525}, {'end': 722.405, 'text': "which is something where we can just write down the gradient and we don't have to go through very complex calculus in order to find this.", 'start': 716.419, 'duration': 5.986}, {'end': 738.529, 'text': 'Yeah, so basically, if we go back, hold on.', 'start': 734.912, 'duration': 3.617}, {'end': 746.116, 'text': 'So if we go back here, we could exactly find all of these using just calculus.', 'start': 741.334, 'duration': 4.782}], 'summary': 'Nodes in the example simplify calculus for finding gradients.', 'duration': 35.222, 'max_score': 710.894, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k710894.jpg'}], 'start': 484.295, 'title': 'Backpropagation in computational graphs', 'summary': 'Explains the process of backpropagation in a computational graph, illustrating the calculation of gradients with specific values and the application of the chain rule, with a specific example demonstrating the computation of gradients for simple operations.', 'chapters': [{'end': 625.76, 'start': 484.295, 'title': 'Backward propagation in graph', 'summary': 'Explains the process of backward propagation in a graph, demonstrating the calculation of gradients with respect to z, q, and y using chain rule, with key points including specific values for q, z, and y, and the application of the chain rule.', 'duration': 141.465, 'highlights': ['The gradient with respect to z is calculated as df over dz equals three, where q is equal to three.', 'The calculation of df over dq is explained as df over dq equals negative four, where z is negative four.', 'The application of the chain rule to find df over dy is demonstrated, with dq over dy equal to one, resulting in the effect of y on f being negative four.']}, {'end': 856.281, 'start': 630.443, 'title': 'Backpropagation and computational graphs', 'summary': 'Explains backpropagation using a computational graph, demonstrating how to compute gradients and values through simple operations and the chain rule, with a specific example showing the calculation of gradients for simple computations.', 'duration': 225.838, 'highlights': ['The backpropagation process involves using a computational graph where each node computes local inputs, outputs, and gradients, simplifying the computation of gradients for complex expressions using simple operations and the chain rule.', 'The example illustrates the computation of gradients for simple operations, such as additions and multiplications, by applying the chain rule, thereby avoiding the need to derive the entire complex expression for the gradient computation.', 'The specific example in the transcript demonstrates the calculation of gradients for simple computations, showcasing the simplicity and effectiveness of using computational nodes and the chain rule for gradient computation.']}], 'duration': 371.986, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k484295.jpg', 'highlights': ['The backpropagation process involves using a computational graph where each node computes local inputs, outputs, and gradients, simplifying the computation of gradients for complex expressions using simple operations and the chain rule.', 'The specific example in the transcript demonstrates the calculation of gradients for simple computations, showcasing the simplicity and effectiveness of using computational nodes and the chain rule for gradient computation.', 'The application of the chain rule to find df over dy is demonstrated, with dq over dy equal to one, resulting in the effect of y on f being negative four.', 'The calculation of df over dq is explained as df over dq equals negative four, where z is negative four.', 'The gradient with respect to z is calculated as df over dz equals three, where q is equal to three.']}, {'end': 1732.019, 'segs': [{'end': 908.26, 'src': 'embed', 'start': 882.323, 'weight': 0, 'content': [{'end': 889.671, 'text': 'And then we also have these local gradients that we computed, the gradient of the immediate output of the node with respect to the inputs coming in.', 'start': 882.323, 'duration': 7.348}, {'end': 896.415, 'text': 'And so what happens during backprop is we have these, we start from the back of the graph right?', 'start': 890.792, 'duration': 5.623}, {'end': 899.376, 'text': 'And then we work our way from the end all the way back to the beginning.', 'start': 896.495, 'duration': 2.881}, {'end': 908.26, 'text': 'And when we reach each node, at each node we have the upstream gradients coming back with respect to the immediate output of the node.', 'start': 899.996, 'duration': 8.264}], 'summary': 'Backpropagation computes local gradients for each node in the graph.', 'duration': 25.937, 'max_score': 882.323, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k882323.jpg'}, {'end': 1108.959, 'src': 'embed', 'start': 1087.328, 'weight': 2, 'content': [{'end': 1097.154, 'text': "just keep track of this and then during backprop, as we're receiving numerical values of gradients coming from upstream, we just take what that is,", 'start': 1087.328, 'duration': 9.826}, {'end': 1104.618, 'text': 'multiply it by the local gradient, and then this is what we then send back to the connected nodes, the next nodes going backwards,', 'start': 1097.154, 'duration': 7.464}, {'end': 1108.959, 'text': 'without having to care about anything else besides these immediate surroundings.', 'start': 1104.938, 'duration': 4.021}], 'summary': 'During backprop, multiply numerical gradients by local gradient to send back to connected nodes.', 'duration': 21.631, 'max_score': 1087.328, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k1087328.jpg'}, {'end': 1290.219, 'src': 'heatmap', 'start': 1154.128, 'weight': 0.879, 'content': [{'end': 1159.749, 'text': 'we take the exponential, we add one and then finally we do one over this whole term.', 'start': 1154.128, 'duration': 5.621}, {'end': 1164.702, 'text': "And then here I've also filled in values of these.", 'start': 1162.079, 'duration': 2.623}, {'end': 1168.946, 'text': "So let's say, given values that we have for the w's and x's,", 'start': 1164.762, 'duration': 4.184}, {'end': 1173.951, 'text': 'we can make a forward pass and basically compute what the value is at every stage of the computation.', 'start': 1168.946, 'duration': 5.005}, {'end': 1183.961, 'text': "And here I've also written down here at the bottom the expressions for some derivatives that are going to be helpful later on.", 'start': 1176.734, 'duration': 7.227}, {'end': 1186.203, 'text': 'So same as we did before with the simple example.', 'start': 1184.021, 'duration': 2.182}, {'end': 1191.584, 'text': "Okay, so now, when we're going to do backprop through here right.", 'start': 1188.962, 'duration': 2.622}, {'end': 1196.268, 'text': "so again we're going to start at the very end of the graph, and so, here again,", 'start': 1191.584, 'duration': 4.684}, {'end': 1201.653, 'text': 'the gradient of the output with respect to the last variable is just one.', 'start': 1196.268, 'duration': 5.385}, {'end': 1202.294, 'text': "it's just trivial.", 'start': 1201.653, 'duration': 0.641}, {'end': 1206.929, 'text': 'And so now moving backwards one step.', 'start': 1203.807, 'duration': 3.122}, {'end': 1212.193, 'text': "So what's the gradient with respect to the input just before one over x?", 'start': 1206.99, 'duration': 5.203}, {'end': 1220.319, 'text': 'Well, so in this case we know that the upstream gradient that we have coming down is this red one.', 'start': 1212.974, 'duration': 7.345}, {'end': 1223.041, 'text': 'This is the upstream gradient that we have flowing down.', 'start': 1221.139, 'duration': 1.902}, {'end': 1225.603, 'text': 'And then now we need to find the local gradient.', 'start': 1223.781, 'duration': 1.822}, {'end': 1227.907, 'text': 'And the local gradient of this node.', 'start': 1226.504, 'duration': 1.403}, {'end': 1229.79, 'text': 'this node is one over x.', 'start': 1227.907, 'duration': 1.883}, {'end': 1237.607, 'text': 'so we have f of x equals one over x here in red, and the local gradient of this df over dx is equal to negative one over x squared.', 'start': 1229.79, 'duration': 7.817}, {'end': 1248.175, 'text': "Right. so here we're going to take negative one over x squared and plug in the value of x that we had during this forward pass 1.37,", 'start': 1239.229, 'duration': 8.946}, {'end': 1255.52, 'text': 'and so our final gradient with respect to this variable is going to be negative one over 1.37.', 'start': 1248.175, 'duration': 7.345}, {'end': 1259.303, 'text': 'squared times, one equals negative 0.53, right.', 'start': 1255.52, 'duration': 3.783}, {'end': 1267.942, 'text': "So moving back to the next node, we're gonna go through the exact same process.", 'start': 1263.979, 'duration': 3.963}, {'end': 1275.688, 'text': 'So here the gradient flowing from upstream is going to be negative 0.53.', 'start': 1268.623, 'duration': 7.065}, {'end': 1279.23, 'text': 'And here the local gradient, the node here is a plus one.', 'start': 1275.688, 'duration': 3.542}, {'end': 1290.219, 'text': 'And so now looking at our reference of derivatives at the bottom, we have that for a constant plus x, the local gradient is just one.', 'start': 1280.051, 'duration': 10.168}], 'summary': 'Explaining backpropagation through a computation graph, computing gradients and values.', 'duration': 136.091, 'max_score': 1154.128, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k1154128.jpg'}, {'end': 1581.05, 'src': 'embed', 'start': 1553.256, 'weight': 1, 'content': [{'end': 1564.365, 'text': 'all we did was plug in the values for each of these that we have and use a chain rule to numerically multiply this all the way backwards and get the gradients with respect to all of the variables.', 'start': 1553.256, 'duration': 11.109}, {'end': 1575.023, 'text': 'And so we can also fill out the gradients with respect to w1 and x1 here in exactly the same way.', 'start': 1568.416, 'duration': 6.607}, {'end': 1581.05, 'text': "And so one thing that I want to note is that, when we're creating these computational graphs,", 'start': 1575.644, 'duration': 5.406}], 'summary': 'Using the chain rule to calculate gradients for variables in computational graphs.', 'duration': 27.794, 'max_score': 1553.256, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k1553256.jpg'}], 'start': 856.281, 'title': 'Backpropagation in neural networks', 'summary': 'Explains backpropagation and local gradients, emphasizing the iterative process of computing gradients and the application of the chain rule, enabling numerical computation of gradients with respect to all variables.', 'chapters': [{'end': 1108.959, 'start': 856.281, 'title': 'Backpropagation and local gradients', 'summary': 'Explains the concept of backpropagation and local gradients, emphasizing on the iterative process of computing gradients with respect to the inputs and the chain rule, to propagate the gradients backwards through the neural network.', 'duration': 252.678, 'highlights': ['The process of backpropagation involves working from the end of the graph to the beginning, computing the gradients of the final loss with respect to the input values, and propagating these gradients back to the connected nodes using the chain rule.', 'Using the chain rule, the gradients of the loss function with respect to the input values are computed by multiplying the upstream gradients with the local gradients, facilitating the propagation of gradients through the network.', 'At each node, the local gradients are computed and used to multiply the numerical values of the gradients coming from upstream during backpropagation, simplifying the process of propagating gradients to the connected nodes.']}, {'end': 1732.019, 'start': 1111.28, 'title': 'Backpropagation in computational graphs', 'summary': 'Illustrates the process of backpropagation in a computational graph for a complex function, demonstrating the calculation of gradients with specific values and the application of the chain rule, enabling numerical computation of gradients with respect to all variables.', 'duration': 620.739, 'highlights': ['The process of backpropagation in a computational graph is illustrated for a complex function, demonstrating the calculation of gradients with specific values.', 'The application of the chain rule enables the numerical computation of gradients with respect to all variables, simplifying the process by plugging in specific values and multiplying backwards.', 'The concept of grouping nodes into more complex nodes, as long as the local gradient can be defined, is discussed, highlighting the flexibility in defining computational nodes.']}], 'duration': 875.738, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k856281.jpg', 'highlights': ['The process of backpropagation involves computing gradients of the final loss with respect to input values.', 'The application of the chain rule enables numerical computation of gradients with respect to all variables.', 'Local gradients are computed at each node and used to multiply the numerical values of the gradients during backpropagation.']}, {'end': 2109.772, 'segs': [{'end': 1797.887, 'src': 'embed', 'start': 1774.657, 'weight': 0, 'content': [{'end': 1782.66, 'text': 'I can go as simple as I need to to always be able to apply backprop and the chain rule and be able to compute all the gradients that I need.', 'start': 1774.657, 'duration': 8.003}, {'end': 1790.804, 'text': "And so this is something that you guys should think about when you're doing your homeworks, as, basically you know,", 'start': 1783.681, 'duration': 7.123}, {'end': 1793.085, 'text': "anytime you're having trouble finding gradients of something.", 'start': 1790.804, 'duration': 2.281}, {'end': 1797.887, 'text': 'just think about it as a computational graph, break it down into all of these parts and then use the chain rule.', 'start': 1793.085, 'duration': 4.802}], 'summary': 'Understanding computational graph and applying chain rule to compute gradients is crucial for backpropagation.', 'duration': 23.23, 'max_score': 1774.657, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k1774657.jpg'}, {'end': 1924.566, 'src': 'embed', 'start': 1896.858, 'weight': 1, 'content': [{'end': 1903.707, 'text': 'When we pass through this addition gate here, which had two branches coming out of it.', 'start': 1896.858, 'duration': 6.849}, {'end': 1907.172, 'text': 'it took the gradient, the upstream gradient, and it just distributed.', 'start': 1903.707, 'duration': 3.465}, {'end': 1909.936, 'text': 'it passed the exact same thing to both of the branches that were connected.', 'start': 1907.172, 'duration': 2.764}, {'end': 1914.3, 'text': "So here's a couple more that we can think about.", 'start': 1911.598, 'duration': 2.702}, {'end': 1918.763, 'text': "So what's a max gate look like?", 'start': 1914.76, 'duration': 4.003}, {'end': 1924.566, 'text': 'So we have a max gate here at the bottom right, where the inputs coming in are z and w.', 'start': 1918.783, 'duration': 5.783}], 'summary': 'Explanation of gradient distribution through an addition gate and the concept of a max gate.', 'duration': 27.708, 'max_score': 1896.858, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k1896858.jpg'}, {'end': 2089.48, 'src': 'heatmap', 'start': 2017.923, 'weight': 3, 'content': [{'end': 2028.83, 'text': "Okay, and so another one, what's a multiplication gate, which we saw earlier? Is there any interpretation of this? Okay, so.", 'start': 2017.923, 'duration': 10.907}, {'end': 2038.232, 'text': 'Okay, so the answer that was given is that the local gradient is basically just the value of the other variable.', 'start': 2032.591, 'duration': 5.641}, {'end': 2040.113, 'text': "Yeah, so that's exactly right.", 'start': 2038.993, 'duration': 1.12}, {'end': 2043.254, 'text': 'So we can think of this as a gradient switcher, right?', 'start': 2040.953, 'duration': 2.301}, {'end': 2049.614, 'text': 'A switcher and, I guess, a scaler where we take the upstream gradient and we scale it by the value of the other branch.', 'start': 2043.434, 'duration': 6.18}, {'end': 2065.17, 'text': 'Okay, and so one other thing to note is that when we have a place where one node is connected to multiple nodes, the gradients add up at this node.', 'start': 2055.347, 'duration': 9.823}, {'end': 2069.391, 'text': 'So, at these branches, using the multivariate chain rule,', 'start': 2065.831, 'duration': 3.56}, {'end': 2079.195, 'text': "we're just going to take the value of the upstream gradient coming back from each of these nodes and we'll add these together to get the total upstream gradient that's flowing back into this node.", 'start': 2069.391, 'duration': 9.804}, {'end': 2085.338, 'text': 'And you can see this from the multivariate chain rule and also thinking about this.', 'start': 2080.096, 'duration': 5.242}, {'end': 2089.48, 'text': "you can think about this that if you're going to change this node a little bit,", 'start': 2085.338, 'duration': 4.142}], 'summary': 'Multiplication gate acts as a gradient switcher and scaler, adding up gradients from multiple nodes. it scales upstream gradient by the value of the other branch.', 'duration': 51.468, 'max_score': 2017.923, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2017923.jpg'}], 'start': 1732.359, 'title': 'Computational graphs and backpropagation', 'summary': 'Discusses computational graphs and backpropagation, emphasizing the simplicity of computing gradients and providing insights into the intuitive interpretation of gradients for various gates, ensuring ease in handling complex expressions and homework assignments.', 'chapters': [{'end': 1793.085, 'start': 1732.359, 'title': 'Computational graphs and gradients', 'summary': 'Discusses the concept of computational graphs, emphasizing the simplicity of applying backpropagation and the chain rule to compute gradients, providing comfort in handling complex expressions and ensuring ease in finding gradients for homework assignments.', 'duration': 60.726, 'highlights': ['The concept of computational graphs simplifies the process of computing gradients, making it easier to apply backpropagation and the chain rule, providing comfort in handling complex expressions.', 'Understanding computational graphs can help in finding gradients for homework assignments, ensuring ease in dealing with complex expressions and computations.', 'The simplicity of representing expressions as computational graphs allows for easy application of backpropagation and the chain rule to compute gradients.']}, {'end': 2109.772, 'start': 1793.085, 'title': 'Backpropagation and computational graphs', 'summary': 'Explains the concept of backpropagation with computational graphs and provides insights into the intuitive interpretation of gradients, such as the sigmoid gate, add gate, max gate, and multiplication gate.', 'duration': 316.687, 'highlights': ['The add gate is a gradient distributor, passing the exact same gradient to both connected branches, which is evident in the case of the upstream gradient being one and the local gradient being 0.2, resulting in both branches receiving the same gradient value.', 'The max gate acts as a gradient router, distributing the full gradient to the branch with the maximum value and zero gradient to the other branch, demonstrated by the local gradient being 2 for the maximum value and 0 for the other branch.', 'The multiplication gate functions as a gradient switcher and scaler, where the local gradient is the value of the other variable, showcasing the upstream gradient scaled by the value of the other branch.', 'The concept of gradients adding up at nodes connected to multiple nodes is explained, illustrating the application of the multivariate chain rule and the impact of connected nodes on the total upstream gradient flowing back into the node.']}], 'duration': 377.413, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k1732359.jpg', 'highlights': ['Understanding computational graphs simplifies computing gradients and applying backpropagation.', 'The add gate distributes the exact same gradient to both connected branches.', 'The max gate routes the full gradient to the branch with the maximum value and zero gradient to the other branch.', 'The multiplication gate switches and scales gradients based on the value of the other branch.', 'Gradients add up at nodes connected to multiple nodes, illustrating the application of the multivariate chain rule.']}, {'end': 2931.541, 'segs': [{'end': 2194.349, 'src': 'embed', 'start': 2166.73, 'weight': 1, 'content': [{'end': 2173.593, 'text': "and what we've done here is just learn how to compute the gradients we need for arbitrarily complex functions.", 'start': 2166.73, 'duration': 6.863}, {'end': 2178.895, 'text': 'And so this is going to be useful when we talk about complex functions like neural networks later on.', 'start': 2174.513, 'duration': 4.382}, {'end': 2189.704, 'text': 'Yeah, so I can write this maybe on the board.', 'start': 2186.722, 'duration': 2.982}, {'end': 2194.349, 'text': 'Right. so basically,', 'start': 2192.527, 'duration': 1.822}], 'summary': 'Learned to compute gradients for complex functions, useful for neural networks.', 'duration': 27.619, 'max_score': 2166.73, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2166730.jpg'}, {'end': 2355.962, 'src': 'heatmap', 'start': 2232.981, 'weight': 3, 'content': [{'end': 2240.426, 'text': "it's going to take the effect of each of these intermediate variables right on our final output, F,", 'start': 2232.981, 'duration': 7.445}, {'end': 2247.69, 'text': 'and then compound each one with the local effect of our variable X on that intermediate value.', 'start': 2240.426, 'duration': 7.264}, {'end': 2251.681, 'text': "So yeah, it's basically just summing all these up together.", 'start': 2248.679, 'duration': 3.002}, {'end': 2263.708, 'text': "Okay, so now that we've done all these examples in the scalar case, we're going to look at what happens when we have vectors.", 'start': 2255.403, 'duration': 8.305}, {'end': 2269.151, 'text': 'So now for variables x, y, and z, instead of just being numbers, we have vectors for these.', 'start': 2264.328, 'duration': 4.823}, {'end': 2273.094, 'text': 'And so everything stays exactly the same, the entire flow.', 'start': 2269.872, 'duration': 3.222}, {'end': 2279.038, 'text': 'The only difference is that now our gradients are going to be Jacobian matrices, right?', 'start': 2273.434, 'duration': 5.604}, {'end': 2289.029, 'text': 'So these are now going to be matrices containing the derivative of each element of, for example, z with respect to each element of x.', 'start': 2279.078, 'duration': 9.951}, {'end': 2296.979, 'text': 'Okay, and so to you know.', 'start': 2293.877, 'duration': 3.102}, {'end': 2299.881, 'text': 'so give an example of something where this is happening.', 'start': 2296.979, 'duration': 2.902}, {'end': 2303.664, 'text': "right, let's say that we have our input is going to now be a vector.", 'start': 2299.881, 'duration': 3.783}, {'end': 2312.386, 'text': "so let's say we have a 4096 dimensional input vector, and this is kind of a common size that you might see in convolutional neural networks later on.", 'start': 2303.664, 'duration': 8.722}, {'end': 2317.248, 'text': 'And our node is going to be an element-wise maximum.', 'start': 2313.587, 'duration': 3.661}, {'end': 2325.17, 'text': 'So we have f of x is equal to the maximum of x compared with zero element-wise.', 'start': 2317.508, 'duration': 7.662}, {'end': 2328.551, 'text': 'And then our output is going to be also a 4096 dimensional vector.', 'start': 2325.61, 'duration': 2.941}, {'end': 2336.128, 'text': "Okay, so in this case, what's the size of our Jacobian matrix?", 'start': 2333.246, 'duration': 2.882}, {'end': 2341.55, 'text': 'Remember, I said earlier, the Jacobian matrix is going to be like each row.', 'start': 2336.528, 'duration': 5.022}, {'end': 2348.014, 'text': "it's going to be partial derivatives, the matrix of partial derivatives of each dimension of the output with respect to each dimension of the input.", 'start': 2341.55, 'duration': 6.464}, {'end': 2355.962, 'text': "Okay, so the answer I heard was 4, 096 squared, and yeah, that's correct.", 'start': 2351.28, 'duration': 4.682}], 'summary': 'Discusses the impact of intermediate variables, gradients, and jacobian matrices in scalar and vector cases, with a specific example of a 4096-dimensional input vector and its corresponding jacobian matrix size.', 'duration': 56.048, 'max_score': 2232.981, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2232981.jpg'}, {'end': 2341.55, 'src': 'embed', 'start': 2317.508, 'weight': 0, 'content': [{'end': 2325.17, 'text': 'So we have f of x is equal to the maximum of x compared with zero element-wise.', 'start': 2317.508, 'duration': 7.662}, {'end': 2328.551, 'text': 'And then our output is going to be also a 4096 dimensional vector.', 'start': 2325.61, 'duration': 2.941}, {'end': 2336.128, 'text': "Okay, so in this case, what's the size of our Jacobian matrix?", 'start': 2333.246, 'duration': 2.882}, {'end': 2341.55, 'text': 'Remember, I said earlier, the Jacobian matrix is going to be like each row.', 'start': 2336.528, 'duration': 5.022}], 'summary': 'Function f of x outputs a 4096-dimensional vector. what is the size of the jacobian matrix?', 'duration': 24.042, 'max_score': 2317.508, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2317508.jpg'}, {'end': 2488.83, 'src': 'embed', 'start': 2463.677, 'weight': 2, 'content': [{'end': 2469.499, 'text': "Okay, so now we're going to go through a more concrete vectorized example of a computational graph.", 'start': 2463.677, 'duration': 5.822}, {'end': 2482.004, 'text': "So let's look at a case where we have the function f of x and w is equal to basically the L2 of w multiplied by x.", 'start': 2470.433, 'duration': 11.571}, {'end': 2488.83, 'text': "And so in this case, we're going to say x is n-dimensional and w is n by n.", 'start': 2482.004, 'duration': 6.826}], 'summary': 'Example of computational graph for f(x, w) = l2(w*x), where x is n-dimensional and w is n by n.', 'duration': 25.153, 'max_score': 2463.677, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2463677.jpg'}, {'end': 2757.257, 'src': 'heatmap', 'start': 2616.512, 'weight': 0.793, 'content': [{'end': 2620.733, 'text': "It's just gonna be two times our vector of q if we wanna write this out in vector form.", 'start': 2616.512, 'duration': 4.221}, {'end': 2624.574, 'text': 'And so what we get is that our gradient is 0.44 and 0.52, this vector.', 'start': 2621.334, 'duration': 3.24}, {'end': 2631.257, 'text': 'And so you can see that it just took q and it scaled it by two.', 'start': 2627.655, 'duration': 3.602}, {'end': 2633.399, 'text': 'Each element is just multiplied by two.', 'start': 2631.618, 'duration': 1.781}, {'end': 2640.303, 'text': 'So the gradient of a vector is always going to be the same size as the original vector.', 'start': 2634.739, 'duration': 5.564}, {'end': 2652.11, 'text': 'And each element of this gradient is going to, it means how much this particular element affects our final output of the function.', 'start': 2640.803, 'duration': 11.307}, {'end': 2658.496, 'text': "Okay, so now let's move one step backwards.", 'start': 2655.694, 'duration': 2.802}, {'end': 2666.401, 'text': "What's the gradient with respect to w? And so here again, we want to use the same concept of trying to apply the chain rule.", 'start': 2658.516, 'duration': 7.885}, {'end': 2670.424, 'text': 'So we want to compute our local gradient of q with respect to w.', 'start': 2666.601, 'duration': 3.823}, {'end': 2672.485, 'text': "And so let's look at this again element rise.", 'start': 2670.424, 'duration': 2.061}, {'end': 2681.311, 'text': "And if we do that, let's see what's the effect of each q, each element of q with respect to each element of w.", 'start': 2673.066, 'duration': 8.245}, {'end': 2683.653, 'text': 'So this is kind of the Jacobian that we talked about earlier.', 'start': 2681.311, 'duration': 2.342}, {'end': 2690.799, 'text': 'And if we look at this, in this multiplication, q is equal to w times x.', 'start': 2684.533, 'duration': 6.266}, {'end': 2694.062, 'text': "right, what's the?", 'start': 2690.799, 'duration': 3.263}, {'end': 2698.846, 'text': "let's see what's the derivative or the gradient of the first element of q.", 'start': 2694.062, 'duration': 4.784}, {'end': 2703.55, 'text': 'so our first element up top with respect to w11?', 'start': 2698.846, 'duration': 4.704}, {'end': 2704.972, 'text': 'So q1 with respect to w11..', 'start': 2703.61, 'duration': 1.362}, {'end': 2709.971, 'text': "What's that value? X one, exactly.", 'start': 2707.028, 'duration': 2.943}, {'end': 2721.864, 'text': 'Yeah, so we know that this is X one and we can write this out more generally of the gradient of QK with respect to, of WIJ is equal to XJ.', 'start': 2710.432, 'duration': 11.432}, {'end': 2731.104, 'text': 'Right. and then now, if we want to find the gradient with respect to of f, with respect to each wij.', 'start': 2723.7, 'duration': 7.404}, {'end': 2737.968, 'text': 'so, looking at these derivatives now, we can use this chain rule that we talked to earlier,', 'start': 2731.104, 'duration': 6.864}, {'end': 2749.795, 'text': 'where we basically compound df over dqk for each element of q, with dqk over wij for each element of wij.', 'start': 2737.968, 'duration': 11.827}, {'end': 2757.257, 'text': 'So we find the effect of each element of w on each element of q, and sum this across all q.', 'start': 2751.115, 'duration': 6.142}], 'summary': 'The transcript discusses gradients, chain rule, and jacobian for vector calculations.', 'duration': 140.745, 'max_score': 2616.512, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2616512.jpg'}, {'end': 2860.506, 'src': 'heatmap', 'start': 2783.392, 'weight': 4, 'content': [{'end': 2789.936, 'text': 'And remember, the important thing is always to check the gradient with respect to a variable should have the same shape as the variable.', 'start': 2783.392, 'duration': 6.544}, {'end': 2794.759, 'text': 'So this is something really useful in practice to sanity check.', 'start': 2790.396, 'duration': 4.363}, {'end': 2799.522, 'text': "Once you've computed what your gradient should be, check that this is the same shape as your variable.", 'start': 2794.859, 'duration': 4.663}, {'end': 2810.054, 'text': 'Because, again, the element, each element of your gradient, is quantifying how much that element is contributing to your,', 'start': 2802.289, 'duration': 7.765}, {'end': 2811.715, 'text': 'is affecting your final output.', 'start': 2810.054, 'duration': 1.661}, {'end': 2819.659, 'text': 'Yeah The bold size, oh, the bold size one is an indicator function.', 'start': 2813.796, 'duration': 5.863}, {'end': 2823.621, 'text': "So this is saying that it's just one if k equals i.", 'start': 2820.019, 'duration': 3.602}, {'end': 2835.772, 'text': "Okay, so, let's see, so we've done that, and so now just, let's see, one more example.", 'start': 2827.946, 'duration': 7.826}, {'end': 2840.355, 'text': 'Now our last thing we need to find is the gradient with respect to qi.', 'start': 2836.252, 'duration': 4.103}, {'end': 2854.484, 'text': 'So here, if we compute the partial derivatives, we can see that dqk over dxi is equal to wki right, using the same same way as we did it for w,', 'start': 2840.636, 'duration': 13.848}, {'end': 2860.506, 'text': 'and then again we can just use the chain rule and get the total expression for that.', 'start': 2854.484, 'duration': 6.022}], 'summary': 'Validate gradient shape and quantify element contribution in practice.', 'duration': 77.114, 'max_score': 2783.392, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2783392.jpg'}], 'start': 2112.678, 'title': 'Backpropagation, gradients, and jacobian matrix', 'summary': 'Explains computing gradients using backpropagation and the chain rule, applying gradients for parameter updates, transitioning from scalar to vector variables, calculating jacobian matrices, and vectorized computational graph examples. it highlights the efficient computation of a diagonal 4096 by 4096 jacobian matrix and emphasizes the importance of checking gradient shapes as a sanity check.', 'chapters': [{'end': 2289.029, 'start': 2112.678, 'title': 'Backpropagation and gradients', 'summary': 'Explains how to compute gradients for complex functions using backpropagation and the chain rule, and how to apply the gradients for updating parameters. it also discusses the transition from scalar to vector variables and the change in gradients to jacobian matrices.', 'duration': 176.351, 'highlights': ['The chapter explains how to compute gradients for complex functions using backpropagation and the chain rule, and how to apply the gradients for updating parameters.', 'It discusses the transition from scalar to vector variables and the change in gradients to Jacobian matrices.', 'The chain rule is used to calculate the effect of each intermediate variable on the final output and compound it with the local effect of the variable on that intermediate value.', 'When dealing with vector variables, the gradients become Jacobian matrices containing the derivatives of each element of the output with respect to each element of the input.']}, {'end': 2931.541, 'start': 2293.877, 'title': 'Computational graph and jacobian matrix', 'summary': 'Covers the computation of jacobian matrix in a computational graph, showing that in the example of a 4096-dimensional input vector, the jacobian matrix is 4096 by 4096, but due to its diagonal element-wise structure, it can be efficiently computed, and then delves into a vectorized example of a computational graph involving the l2 norm and backpropagation, showcasing the calculation of gradients with respect to intermediate variables and the application of chain rule in computing local gradients, emphasizing the importance of checking the gradient shape as a sanity check.', 'duration': 637.664, 'highlights': ['The Jacobian matrix for a 4096-dimensional input vector is 4096 by 4096, but due to its diagonal element-wise structure, it can be efficiently computed, making it practical despite its large size.', 'The example of a computational graph involving the L2 norm and backpropagation demonstrates the calculation of gradients with respect to intermediate variables and the application of chain rule in computing local gradients.', 'Emphasizing the importance of checking the gradient shape as a sanity check to ensure that each element of the gradient quantifies its contribution to the final output.']}], 'duration': 818.863, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2112678.jpg', 'highlights': ['The Jacobian matrix for a 4096-dimensional input vector is 4096 by 4096, but due to its diagonal element-wise structure, it can be efficiently computed, making it practical despite its large size.', 'The chapter explains how to compute gradients for complex functions using backpropagation and the chain rule, and how to apply the gradients for updating parameters.', 'The example of a computational graph involving the L2 norm and backpropagation demonstrates the calculation of gradients with respect to intermediate variables and the application of chain rule in computing local gradients.', 'It discusses the transition from scalar to vector variables and the change in gradients to Jacobian matrices.', 'Emphasizing the importance of checking the gradient shape as a sanity check to ensure that each element of the gradient quantifies its contribution to the final output.', 'The chain rule is used to calculate the effect of each intermediate variable on the final output and compound it with the local effect of the variable on that intermediate value.', 'When dealing with vector variables, the gradients become Jacobian matrices containing the derivatives of each element of the output with respect to each element of the input.']}, {'end': 3878.967, 'segs': [{'end': 2963.877, 'src': 'embed', 'start': 2935.467, 'weight': 2, 'content': [{'end': 2944.89, 'text': "Okay, so the way that we've been thinking about this is like a really modularized implementation where, in our computational graph,", 'start': 2935.467, 'duration': 9.423}, {'end': 2950.032, 'text': 'we look at each node locally and we compute the local gradients and chain them, with upstream gradients coming down.', 'start': 2944.89, 'duration': 5.142}, {'end': 2954.854, 'text': 'And so you can think of this as basically a forward and a backwards API.', 'start': 2950.572, 'duration': 4.282}, {'end': 2963.877, 'text': 'In the forward pass, we implement a function computing the output of this node, and then in the backwards pass, we compute the gradient.', 'start': 2954.914, 'duration': 8.963}], 'summary': 'Modularized implementation computes local gradients in computational graph for forward and backward api.', 'duration': 28.41, 'max_score': 2935.467, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2935467.jpg'}, {'end': 3023.412, 'src': 'embed', 'start': 2993.191, 'weight': 3, 'content': [{'end': 2997.658, 'text': 'We can iterate through all of these gates and just call forward on each of the gates.', 'start': 2993.191, 'duration': 4.467}, {'end': 3005.331, 'text': 'And we just want to do this in topologically sorted order so we process all the inputs coming into a node before we process that node.', 'start': 2998.408, 'duration': 6.923}, {'end': 3007.292, 'text': 'And then going backwards.', 'start': 3005.891, 'duration': 1.401}, {'end': 3013.054, 'text': "we're just going to then go through all of the gates in this reverse sorted order and then call backwards on each of these gates.", 'start': 3007.292, 'duration': 5.762}, {'end': 3023.412, 'text': 'Okay. and so if we look at then the implementation for a particular gate, so for example this multiply gate,', 'start': 3016.164, 'duration': 7.248}], 'summary': 'Iterate through gates in topologically sorted order for processing inputs and calling forward/backwards on each gate.', 'duration': 30.221, 'max_score': 2993.191, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2993191.jpg'}, {'end': 3113.766, 'src': 'embed', 'start': 3088.405, 'weight': 1, 'content': [{'end': 3096.752, 'text': "Okay, so if you look at a lot of deep learning frameworks and libraries, you'll see that they exactly follow this kind of modularization.", 'start': 3088.405, 'duration': 8.347}, {'end': 3105.64, 'text': "So, for example, CAFE is a popular deep learning framework and you'll see, if you go look through the CAFE source code,", 'start': 3097.453, 'duration': 8.187}, {'end': 3113.766, 'text': "you'll get to some directory that says layers and in layers, which are basically computational nodes, usually layers might be slightly more.", 'start': 3105.64, 'duration': 8.126}], 'summary': "Deep learning frameworks use modularization, like cafe's layers directory.", 'duration': 25.361, 'max_score': 3088.405, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k3088405.jpg'}, {'end': 3662.91, 'src': 'embed', 'start': 3630.137, 'weight': 0, 'content': [{'end': 3639.901, 'text': 'So now W1 can be many different kinds of templates, right? And then W2 now basically is a weighted sum of all of these templates.', 'start': 3630.137, 'duration': 9.764}, {'end': 3646.384, 'text': 'So now it allows you to weight together multiple templates in order to get the final score for a particular class.', 'start': 3640.902, 'duration': 5.482}, {'end': 3662.91, 'text': 'Right. so okay.', 'start': 3662.17, 'duration': 0.74}], 'summary': 'W1 can be multiple templates, w2 is a weighted sum, enabling multiple templates for final score.', 'duration': 32.773, 'max_score': 3630.137, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k3630137.jpg'}, {'end': 3885.052, 'src': 'embed', 'start': 3860.153, 'weight': 4, 'content': [{'end': 3865.82, 'text': 'The entire implementation of a two layer neural network is actually really simple, it can just be done in 20 lines.', 'start': 3860.153, 'duration': 5.667}, {'end': 3871.812, 'text': "And so you'll get some practice with this in assignment two, writing out all of these parts.", 'start': 3867.083, 'duration': 4.729}, {'end': 3874.583, 'text': 'And okay.', 'start': 3873.542, 'duration': 1.041}, {'end': 3878.967, 'text': "so now that we've sort of seen what neural networks are as a function, right, like you know,", 'start': 3874.583, 'duration': 4.384}, {'end': 3885.052, 'text': "we hear people talking a lot about how there's biological inspirations for neural networks, and so,", 'start': 3878.967, 'duration': 6.085}], 'summary': 'Implementing a two-layer neural network can be done in 20 lines, providing practice for assignment two.', 'duration': 24.899, 'max_score': 3860.153, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k3860153.jpg'}], 'start': 2935.467, 'title': 'Implementing modularized deep learning', 'summary': 'Discusses implementing modularized deep learning, emphasizing modularized backward and forward api, modularization in deep learning, and understanding neural networks, with an emphasis on computational graph implementation and hierarchical computation.', 'chapters': [{'end': 3086.524, 'start': 2935.467, 'title': 'Modularized backward and forward api', 'summary': 'Discusses a modularized implementation of a computational graph where each node locally computes gradients and chains them, with the implementation of forward and backward passes through topologically sorted gates, emphasizing the importance of caching values in the forward pass and applying the chain rule in the backward pass.', 'duration': 151.057, 'highlights': ['The chapter discusses a modularized implementation of a computational graph where each node locally computes gradients and chains them.', 'The implementation of forward and backward passes through topologically sorted gates is emphasized.', 'Emphasizes the importance of caching values in the forward pass and applying the chain rule in the backward pass.']}, {'end': 3300.606, 'start': 3088.405, 'title': 'Modularization in deep learning', 'summary': 'Discusses the modularization in deep learning frameworks, emphasizing the use of computational nodes, back propagation, and the computational graph way of thinking.', 'duration': 212.201, 'highlights': ['The chapter discusses the modularization in deep learning frameworks, emphasizing the use of computational nodes, back propagation, and the computational graph way of thinking.', 'Back propagation is highlighted as a core technique for getting gradients, involving a recursive application of the chain rule.', 'The deep learning frameworks follow a modularization approach, using computational nodes like sigmoid, convolution, and argmax, and implementing forward and backward passes.']}, {'end': 3878.967, 'start': 3300.606, 'title': 'Understanding neural networks', 'summary': 'Explains the concept of neural networks as a class of functions where simpler functions are stacked on top of each other in a hierarchical way to create a more complex nonlinear function, allowing for multiple stages of hierarchical computation and deeper networks of arbitrary depth.', 'duration': 578.361, 'highlights': ['Neural networks are a class of functions where simpler functions are stacked on top of each other in a hierarchical way to create a more complex nonlinear function, allowing for multiple stages of hierarchical computation (e.g., stacking linear transformations, adding nonlinear functions in between, and creating deeper networks of arbitrary depth).', 'The concept of multiple layer networks allows for the creation of multiple templates for different classes, enabling the combination and weighting of these templates to compute the final score for a particular class, addressing the limitation of having only one template for each class.', 'The implementation of a two-layer neural network is relatively simple, requiring only around 20 lines of code, and the concept of forward pass, backward passes, and chain rule to compute gradients is essential for training neural networks.']}], 'duration': 943.5, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k2935467.jpg', 'highlights': ['The concept of multiple layer networks allows for the creation of multiple templates for different classes, enabling the combination and weighting of these templates to compute the final score for a particular class, addressing the limitation of having only one template for each class.', 'The chapter discusses the modularization in deep learning frameworks, emphasizing the use of computational nodes, back propagation, and the computational graph way of thinking.', 'The chapter discusses a modularized implementation of a computational graph where each node locally computes gradients and chains them.', 'The implementation of forward and backward passes through topologically sorted gates is emphasized.', 'The implementation of a two-layer neural network is relatively simple, requiring only around 20 lines of code, and the concept of forward pass, backward passes, and chain rule to compute gradients is essential for training neural networks.']}, {'end': 4436.351, 'segs': [{'end': 4024.658, 'src': 'embed', 'start': 3994.269, 'weight': 2, 'content': [{'end': 3998.671, 'text': 'We get this value of this output and we pass it down to the connecting neurons.', 'start': 3994.269, 'duration': 4.402}, {'end': 4005.366, 'text': 'So, if you look at this, this is actually you can think about this in a very similar way, right?', 'start': 4001.303, 'duration': 4.063}, {'end': 4007.907, 'text': "Like you know, these are what's.", 'start': 4005.426, 'duration': 2.481}, {'end': 4012.99, 'text': 'the signals coming in are kind of the connected at synapses, right?', 'start': 4007.907, 'duration': 5.083}, {'end': 4015.172, 'text': 'The synapse connecting the multiple neurons.', 'start': 4013.01, 'duration': 2.162}, {'end': 4019.395, 'text': 'The dendrites are integrating all of these.', 'start': 4015.872, 'duration': 3.523}, {'end': 4024.658, 'text': "they're integrating all of this information together in the cell body and then we have the output carried on.", 'start': 4019.395, 'duration': 5.263}], 'summary': 'Neural network processes input signals through synapses and dendrites for output.', 'duration': 30.389, 'max_score': 3994.269, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k3994269.jpg'}, {'end': 4067.96, 'src': 'embed', 'start': 4041.389, 'weight': 0, 'content': [{'end': 4047.455, 'text': "And we've talked about examples like sigmoid activation function and different kinds of nonlinearities.", 'start': 4041.389, 'duration': 6.066}, {'end': 4050.056, 'text': 'And so sort of.', 'start': 4049.216, 'duration': 0.84}, {'end': 4061.559, 'text': 'one kind of loose analogy that you can draw is that these nonlinearities can represent something sort of like the firing or spiking rate of the neurons.', 'start': 4050.056, 'duration': 11.503}, {'end': 4067.96, 'text': 'Where neurons transmit signals to connected neurons using kind of these discrete spikes.', 'start': 4061.939, 'duration': 6.021}], 'summary': 'Nonlinear functions represent neuron firing rates in neural networks.', 'duration': 26.571, 'max_score': 4041.389, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k4041389.jpg'}, {'end': 4237.662, 'src': 'embed', 'start': 4187.529, 'weight': 1, 'content': [{'end': 4194.853, 'text': "And so there's all these, it's kind of a much more complex thing than what we're dealing with.", 'start': 4187.529, 'duration': 7.324}, {'end': 4202.216, 'text': "There's references, for example, this dendritic computation that you can look at if you're interested in this topic.", 'start': 4195.793, 'duration': 6.423}, {'end': 4212.98, 'text': 'But in practice, we can sort of see how it may resemble a neuron at this very high level, but neurons are in practice much more complicated than that.', 'start': 4203.596, 'duration': 9.384}, {'end': 4220.421, 'text': "Okay, so we talked about how there's many different kinds of activation functions that could be used.", 'start': 4215.495, 'duration': 4.926}, {'end': 4222.804, 'text': "There's the ReLU that I mentioned earlier.", 'start': 4220.682, 'duration': 2.122}, {'end': 4232.476, 'text': "And we'll talk about all of these different kinds of activation functions in much more detail later on choices of these activation functions that you might want to use.", 'start': 4222.824, 'duration': 9.652}, {'end': 4237.662, 'text': "And so we'll also talk about different kinds of neural network architectures.", 'start': 4234.058, 'duration': 3.604}], 'summary': 'Discussion of complex neural network components and activation functions, with more details to come later.', 'duration': 50.133, 'max_score': 4187.529, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k4187529.jpg'}, {'end': 4436.351, 'src': 'embed', 'start': 4390.902, 'weight': 4, 'content': [{'end': 4398.065, 'text': "And so that's basically all there is to kind of the main idea of what's a neural network.", 'start': 4390.902, 'duration': 7.163}, {'end': 4410.431, 'text': 'Okay, so just to summarize, we talked about how we can arrange neurons into these computations, right, of fully connected or linear layers.', 'start': 4401.106, 'duration': 9.325}, {'end': 4417.195, 'text': 'This abstraction of a layer has a nice property that we can use very efficient vectorized code to compute all of these.', 'start': 4411.612, 'duration': 5.583}, {'end': 4426.623, 'text': "We also talked about how it's important to keep in mind that neural networks do have some analogy and loose inspiration from biology,", 'start': 4418.155, 'duration': 8.468}, {'end': 4428.264, 'text': "but they're not really neural.", 'start': 4426.623, 'duration': 1.641}, {'end': 4430.926, 'text': "I mean, this is a pretty loose analogy that we're making.", 'start': 4428.324, 'duration': 2.602}, {'end': 4435.05, 'text': "And next time we'll talk about convolutional neural networks.", 'start': 4431.807, 'duration': 3.243}, {'end': 4436.351, 'text': 'Okay, thanks.', 'start': 4435.911, 'duration': 0.44}], 'summary': 'Neural networks use efficient vectorized code for computations, with loose inspiration from biology.', 'duration': 45.449, 'max_score': 4390.902, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k4390902.jpg'}], 'start': 3878.967, 'title': 'Analogies between neural networks and biological neurons', 'summary': 'Discusses the loose analogies between artificial neural networks and biological neurons, emphasizing similarities in signal processing, integration, and activation functions, cautioning against oversimplified comparisons. it also covers different activation functions, neural network architectures, computation process with matrix multiplication and non-linearities, and efficiency of vectorized code.', 'chapters': [{'end': 4212.98, 'start': 3878.967, 'title': 'Analogies between neural networks and biological neurons', 'summary': 'Discusses the loose analogies between artificial neural networks and biological neurons, emphasizing the similarities in signal processing, integration, and activation functions, while also highlighting the complexity and differences between the two systems, cautioning against oversimplified comparisons.', 'duration': 334.013, 'highlights': ['Neurons and artificial nodes share similarities in signal processing, integration, and activation functions, with examples such as w times x plus b computation and ReLU non-linearity, emphasizing loose analogies (Relevance: 5)', 'Caution is advised in making brain analogies, as biological neurons are significantly more complex than their artificial counterparts, involving complex dendritic computations and variable firing rates (Relevance: 4)', 'Integration of signals in dendrites and the passing of output to downstream neurons in biological neurons can be loosely compared to the computational graph and transmission of output in artificial neural networks, providing insight into the inspirations behind artificial neural network designs (Relevance: 3)']}, {'end': 4436.351, 'start': 4215.495, 'title': 'Neural network architectures overview', 'summary': 'Discussed different activation functions, neural network architectures, and the computation process with matrix multiplication and non-linearities. it also highlighted the efficiency of vectorized code for neural network computations and the loose analogy with biology.', 'duration': 220.856, 'highlights': ['The chapter discussed different activation functions, neural network architectures, and the computation process with matrix multiplication and non-linearities.', 'It highlighted the efficiency of vectorized code for neural network computations.', 'It mentioned the loose analogy of neural networks with biology.']}], 'duration': 557.384, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d14TUNcbn1k/pics/d14TUNcbn1k3878967.jpg', 'highlights': ['Neurons and artificial nodes share similarities in signal processing, integration, and activation functions, with examples such as w times x plus b computation and ReLU non-linearity, emphasizing loose analogies (Relevance: 5)', 'Caution is advised in making brain analogies, as biological neurons are significantly more complex than their artificial counterparts, involving complex dendritic computations and variable firing rates (Relevance: 4)', 'Integration of signals in dendrites and the passing of output to downstream neurons in biological neurons can be loosely compared to the computational graph and transmission of output in artificial neural networks, providing insight into the inspirations behind artificial neural network designs (Relevance: 3)', 'The chapter discussed different activation functions, neural network architectures, and the computation process with matrix multiplication and non-linearities.', 'It highlighted the efficiency of vectorized code for neural network computations.', 'It mentioned the loose analogy of neural networks with biology.']}], 'highlights': ['Students at Stanford University are allocated $100 in credits to use for Google Cloud for their assignments and projects, with emails expected to be received this week.', 'The computational graph represents any function, aiding in backpropagation (5)', 'Neurons and artificial nodes share similarities in signal processing, integration, and activation functions, with examples such as w times x plus b computation and ReLU non-linearity, emphasizing loose analogies (Relevance: 5)', 'Understanding computational graphs simplifies computing gradients and applying backpropagation.', 'The process of backpropagation involves computing gradients of the final loss with respect to input values.', 'The add gate distributes the exact same gradient to both connected branches.', 'The max gate routes the full gradient to the branch with the maximum value and zero gradient to the other branch.', 'The multiplication gate switches and scales gradients based on the value of the other branch.', 'The concept of multiple layer networks allows for the creation of multiple templates for different classes, enabling the combination and weighting of these templates to compute the final score for a particular class, addressing the limitation of having only one template for each class.', 'The Jacobian matrix for a 4096-dimensional input vector is 4096 by 4096, but due to its diagonal element-wise structure, it can be efficiently computed, making it practical despite its large size.']}