Backpropagation and Neural Networks part 1. Lecture 4-1
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1 Lecture 4: Backpropagation and Neural Networks part 1 Lecture 4-1
2 Administrative A1 is due Jan 20 (Wednesday). ~150 hours left Warning: Jan 18 (Monday) is Holiday (no class/office hours) Also note: Lectures are non-exhaustive. Read course notes for completeness. I ll hold make up office hours on Wed Jan20, Gates 259 Lecture 4-2
3 Where we are... scores function SVM loss data loss + regularization want Lecture 4-3
4 Optimization (image credits to Alec Radford) Lecture 4-4
5 Gradient Descent Numerical gradient: slow :(, approximate :(, easy to write :) Analytic gradient: fast :), exact :), error-prone :( In practice: Derive analytic gradient, check your implementation with numerical gradient Lecture 4-5
6 Computational Graph x * s (scores) hinge loss + L W R Lecture 4-6
7 Convolutional Network (AlexNet) input image weights loss Lecture 4-7
8 Neural Turing Machine input tape loss Lecture 4-8
9 Neural Turing Machine Lecture 4-9
10 e.g. x = -2, y = 5, z = -4 Lecture 4-10
11 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-11
12 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-12
13 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-13
14 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-14
15 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-15
16 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-16
17 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-17
18 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-18
19 e.g. x = -2, y = 5, z = -4 Chain rule: Want: Lecture 4-19
20 e.g. x = -2, y = 5, z = -4 Want: Lecture 4-20
21 e.g. x = -2, y = 5, z = -4 Chain rule: Want: Lecture 4-21
22 activations f Lecture 4-22
23 activations local gradient f Lecture 4-23
24 activations local gradient f gradients Lecture 4-24
25 activations local gradient f gradients Lecture 4-25
26 activations local gradient f gradients Lecture 4-26
27 activations local gradient f gradients Lecture 4-27
28 Another example: Lecture 4-28
29 Another example: Lecture 4-29
30 Another example: Lecture 4-30
31 Another example: Lecture 4-31
32 Another example: Lecture 4-32
33 Another example: Lecture 4-33
34 Another example: Lecture 4-34
35 Another example: Lecture 4-35
36 Another example: Lecture 4-36
37 Another example: (-1) * (-0.20) = 0.20 Lecture 4-37
38 Another example: Lecture 4-38
39 Another example: [local gradient] x [its gradient] [1] x [0.2] = 0.2 [1] x [0.2] = 0.2 (both inputs!) Lecture 4-39
40 Another example: Lecture 4-40
41 Another example: [local gradient] x [its gradient] x0: [2] x [0.2] = 0.4 w0: [-1] x [0.2] = -0.2 Lecture 4-41
42 sigmoid function sigmoid gate Lecture 4-42
43 sigmoid function sigmoid gate (0.73) * (1-0.73) = 0.2 Lecture 4-43
44 Patterns in backward flow add gate: gradient distributor max gate: gradient router mul gate: gradient switcher? Lecture 4-44
45 Gradients add at branches + Lecture 4-45
46 Implementation: forward/backward API Graph (or Net) object. (Rough psuedo code) Lecture 4-46
47 Implementation: forward/backward API x * z y (x,y,z are scalars) Lecture 4-47
48 Implementation: forward/backward API x * z y (x,y,z are scalars) Lecture 4-48
49 Example: Torch Layers Lecture 4-49
50 Example: Torch Layers = Lecture 4-50
51 Example: Torch MulConstant initialization forward() backward() Lecture 4-51
52 Example: Caffe Layers Lecture 4-52
53 Caffe Sigmoid Layer *top_diff (chain rule) Lecture 4-53
54 Gradients for vectorized code (x,y,z are now vectors) This is now the Jacobian matrix (derivative of each element of z w.r.t. each element of x) local gradient f gradients Lecture 4-54
55 Vectorized operations 4096-d input vector f(x) = max(0,x) (elementwise) 4096-d output vector Lecture 4-55
56 Vectorized operations Jacobian matrix 4096-d input vector f(x) = max(0,x) (elementwise) 4096-d output vector Q: what is the size of the Jacobian matrix? Lecture 4-56
57 Vectorized operations Jacobian matrix 4096-d input vector f(x) = max(0,x) max(0,x) (elementwise) (elementwise) Q: what is the size of the Jacobian matrix? [4096 x 4096!] 4096-d output vector Q2: what does it look like? Lecture 4-57
58 Vectorized operations in practice we process an entire minibatch (e.g. 100) of examples at one time: d input vectors f(x) = max(0,x) max(0,x) (elementwise) (elementwise) d output vectors i.e. Jacobian would technically be a [409,600 x 409,600] matrix :\ Lecture 4-58
59 Assignment: Writing SVM/Softmax Stage your forward/backward computation! margins E.g. for the SVM: Lecture 4-59
60 Summary so far - - neural nets will be very large: no hope of writing down gradient formula by hand for all parameters backpropagation = recursive application of the chain rule along a computational graph to compute the gradients of all inputs/parameters/intermediates implementations maintain a graph structure, where the nodes implement the forward() / backward() API. forward: compute result of an operation and save any intermediates needed for gradient computation in memory backward: apply the chain rule to compute the gradient of the loss function with respect to the inputs. Lecture 4-60
61 Lecture 4-61
62 Neural Network: without the brain stuff (Before) Linear score function: Lecture 4-62
63 Neural Network: without the brain stuff (Before) Linear score function: (Now) 2-layer Neural Network Lecture 4-63
64 Neural Network: without the brain stuff (Before) Linear score function: (Now) 2-layer Neural Network x 3072 W1 h 100 W2 s 10 Lecture 4-64
65 Neural Network: without the brain stuff (Before) Linear score function: (Now) 2-layer Neural Network x 3072 W1 h 100 W2 s 10 Lecture 4-65
66 Neural Network: without the brain stuff (Before) Linear score function: (Now) 2-layer Neural Network or 3-layer Neural Network Lecture 4-66
67 Full implementation of training a 2-layer Neural Network needs ~11 lines: Lecture 4-67
68 Assignment: Writing 2layer Net Stage your forward/backward computation! Lecture 4-68
69 Lecture 4-69
70 Lecture 4-70
71 Lecture 4-71
72 Lecture 4-72
73 sigmoid activation function Lecture 4-73
74 Lecture 4-74
75 Be very careful with your Brain analogies: Biological Neurons: - Many different types - Dendrites can perform complex nonlinear computations - Synapses are not a single weight but a complex non-linear dynamical system - Rate code may not be adequate [Dendritic Computation. London and Hausser] Lecture 4-75
76 Leaky ReLU max(0.1x, x) Activation Functions Sigmoid Maxout tanh ReLU tanh(x) ELU max(0,x) Lecture 4-76
77 Neural Networks: Architectures 2-layer Neural Net, or 1-hidden-layer Neural Net 3-layer Neural Net, or 2-hidden-layer Neural Net Fully-connected layers Lecture 4-77
78 Example Feed-forward computation of a Neural Network We can efficiently evaluate an entire layer of neurons. Lecture 4-78
79 Example Feed-forward computation of a Neural Network Lecture 4-79
80 Setting the number of layers and their sizes more neurons = more capacity Lecture 4-80
81 Do not use size of neural network as a regularizer. Use stronger regularization instead: (you can play with this demo over at ConvNetJS: edu/people/karpathy/convnetjs/demo/classify2d.html) Lecture 4-81
82 Summary - we arrange neurons into fully-connected layers - the abstraction of a layer has the nice property that it allows us to use efficient vectorized code (e.g. matrix multiplies) - neural networks are not really neural - neural networks: bigger = better (but might have to regularize more strongly) Lecture 4-82
83 Next Lecture: More than you ever wanted to know about Neural Networks and how to train them. Lecture 4-83
84 inputs x outputs y complex graph reverse-mode differentiation (if you want effect of many things on one thing) for many different x forward-mode differentiation (if you want effect of one thing on many things) for many different y Lecture 4-84
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