CS 7643: Deep Learning
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1 CS 7643: Deep Learning Topics: Computational Graphs Notation + example Computing Gradients Forward mode vs Reverse mode AD Dhruv Batra Georgia Tech
2 Administrativia HW1 Released Due: 09/22 PS1 Solutions Coming soon (C) Dhruv Batra 2
3 Project Goal Chance to try Deep Learning Combine with other classes / research / credits / anything You have our blanket permission Extra credit for shooting for a publication Encouraged to apply to your research (computer vision, NLP, robotics, ) Must be done this semester. Main categories Application/Survey Compare a bunch of existing algorithms on a new application domain of your interest Formulation/Development Formulate a new model or algorithm for a new or old problem Theory Theoretically analyze an existing algorithm (C) Dhruv Batra 3
4 Administrativia Project Teams Google Doc DaWlc9zsmfKMyuGS39JAn9dpeXhhQ/edit#gid=0 Project Title 1-3 sentence project summary TL;DR Team member names + GT IDs (C) Dhruv Batra 4
5 Recap of last time (C) Dhruv Batra 5
6 How do we compute gradients? Manual Differentiation Symbolic Differentiation Numerical Differentiation Automatic Differentiation Forward mode AD Reverse mode AD aka backprop (C) Dhruv Batra 6
7 Computational Graph Any DAG of differentiable modules is allowed! (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 7
8 Directed Acyclic Graphs (DAGs) Exactly what the name suggests Directed edges No (directed) cycles Underlying undirected cycles okay (C) Dhruv Batra 8
9 Directed Acyclic Graphs (DAGs) Concept Topological Ordering (C) Dhruv Batra 9
10 Directed Acyclic Graphs (DAGs) (C) Dhruv Batra 10
11 Computational Graphs Notation #1 f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) (C) Dhruv Batra 11
12 Computational Graphs Notation #2 f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) (C) Dhruv Batra 12
13 Example f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) + sin( ) * x 1 x 2 (C) Dhruv Batra 13
14 Logistic Regression as a Cascade Given a library of simple functions Compose into a complicate function log 1 1+e w x w x (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato, Yann LeCun 14
15 Forward mode vs Reverse Mode Key Computations (C) Dhruv Batra 15
16 Forward mode AD g 16
17 Reverse mode AD g 17
18 Example: Forward mode AD f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) + sin( ) * x 1 x 2 (C) Dhruv Batra 18
19 Example: Forward mode AD f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) ẇ 3 = ẇ 1 +ẇ 2 + ẇ 1 = cos(x 1 )ẋ 1 ẇ 2 =ẋ 1 x 2 + x 1 ẋ 2 sin( ) * ẋ 1 ẋ 1 ẋ 2 x 1 x 2 (C) Dhruv Batra 19
20 Example: Forward mode AD f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) ẇ 3 = ẇ 1 +ẇ 2 + ẇ 1 = cos(x 1 )ẋ 1 ẇ 2 =ẋ 1 x 2 + x 1 ẋ 2 sin( ) * ẋ 1 ẋ 1 ẋ 2 x 1 x 2 (C) Dhruv Batra 20
21 Example: Reverse mode AD f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) + sin( ) * x 1 x 2 (C) Dhruv Batra 21
22 Example: Reverse mode AD f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) w 3 =1 + w 1 = w 3 w 2 = w 3 sin( ) * x 1 = w 1 cos(x 1 ) x 1 = w 2 x 2 x 2 = w 2 x 1 x 1 x 2 (C) Dhruv Batra 22
23 Forward Pass vs Forward mode AD vs Reverse Mode AD + f(x 1,x 2 )=x 1 x 2 +sin(x 1 ) sin( ) * x 1 x 2 ẇ 3 = ẇ 1 +ẇ 2 + w 3 =1 + ẇ 1 = cos(x 1 )ẋ 1 ẇ 2 =ẋ 1 x 2 + x 1 ẋ 2 w 1 = w 3 w 2 = w 3 sin( ) ẋ 1 ẋ 1 ẋ 2 * sin( ) x 1 = w 1 cos(x 1 ) x 1 = w 2 x 2 x 2 = w 2 x 1 * x 1 x 2 x 1 x 2 (C) Dhruv Batra 23
24 Forward mode vs Reverse Mode What are the differences? Which one is more memory efficient (less storage)? Forward or backward? (C) Dhruv Batra 24
25 Forward mode vs Reverse Mode What are the differences? Which one is more memory efficient (less storage)? Forward or backward? Which one is faster to compute? Forward or backward? (C) Dhruv Batra 25
26 Plan for Today (Finish) Computing Gradients Forward mode vs Reverse mode AD Patterns in backprop Backprop in FC+ReLU NNs Convolutional Neural Networks (C) Dhruv Batra 26
27 Patterns in backward flow Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
28 Patterns in backward flow Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
29 Patterns in backward flow add gate: gradient distributor Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
30 Patterns in backward flow add gate: gradient distributor Q: What is a max gate? Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
31 Patterns in backward flow add gate: gradient distributor max gate: gradient router Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
32 Patterns in backward flow add gate: gradient distributor max gate: gradient router Q: What is a mul gate? Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
33 Patterns in backward flow add gate: gradient distributor max gate: gradient router mul gate: gradient switcher Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
34 Gradients add at branches + Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
35 Duality in Fprop and Bprop FPROP BPROP SUM + COPY + (C) Dhruv Batra 35
36 Modularized implementation: forward / backward API Graph (or Net) object (rough psuedo code) 36 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
37 Modularized implementation: forward / backward API x * z y (x,y,z are scalars) 37 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
38 Modularized implementation: forward / backward API x * z y (x,y,z are scalars) 38 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
39 Example: Caffe layers Caffe is licensed under BSD 2-Clause 39 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
40 Caffe Sigmoid Layer * top_diff (chain rule) Caffe is licensed under BSD 2-Clause 40 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
41 (C) Dhruv Batra 41
42 (C) Dhruv Batra 42
43 Key Computation in DL: Forward-Prop (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato, Yann LeCun 43
44 Key Computation in DL: Back-Prop (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato, Yann LeCun 44
45 Jacobian of ReLU 4096-d input vector f(x) = max(0,x) (elementwise) 4096-d output vector Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
46 Jacobian of ReLU 4096-d input vector Q: what is the size of the Jacobian matrix? f(x) = max(0,x) (elementwise) 4096-d output vector 46 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
47 Jacobian of ReLU 4096-d input vector Q: what is the size of the Jacobian matrix? [4096 x 4096!] f(x) = max(0,x) (elementwise) 4096-d output vector 47 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
48 Jacobian of ReLU 4096-d input vector Q: what is the size of the Jacobian matrix? [4096 x 4096!] f(x) = max(0,x) (elementwise) 4096-d output vector in practice we process an entire minibatch (e.g. 100) of examples at one time: i.e. Jacobian would technically be a [409,600 x 409,600] matrix :\ Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
49 Jacobian of ReLU 4096-d input vector Q: what is the size of the Jacobian matrix? [4096 x 4096!] f(x) = max(0,x) (elementwise) 4096-d output vector Q2: what does it look like? Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
50 Jacobians of FC-Layer (C) Dhruv Batra 50
51 Jacobians of FC-Layer (C) Dhruv Batra 51
52 Jacobians of FC-Layer (C) Dhruv Batra 52
53 Convolutional Neural Networks (without the brain stuff) Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
54 Fully Connected Layer Example: 200x200 image 40K hidden units ~2B parameters!!! - Spatial correlation is local - Waste of resources + we have not enough training samples anyway.. Slide Credit: Marc'Aurelio Ranzato 54
55 Locally Connected Layer Example: 200x200 image 40K hidden units Filter size: 10x10 4M parameters Note: This parameterization is good when input image is registered (e.g., face recognition). Slide Credit: Marc'Aurelio Ranzato 55
56 Locally Connected Layer STATIONARITY? Statistics is similar at different locations Example: 200x200 image 40K hidden units Filter size: 10x10 4M parameters Note: This parameterization is good when input image is registered (e.g., face recognition). Slide Credit: Marc'Aurelio Ranzato 56
57 Convolutional Layer Share the same parameters across different locations (assuming input is stationary): Convolutions with learned kernels Slide Credit: Marc'Aurelio Ranzato 57
58 Convolutions for mathematicians (C) Dhruv Batra 58
59 "Convolution of box signal with itself2" by Convolution_of_box_signal_with_itself.gif: Brian Ambergderivative work: Tinos (talk) - Convolution_of_box_signal_with_itself.gif. Licensed under CC BY-SA 3.0 via Commons - th_itself2.gif (C) Dhruv Batra 59
60 Convolutions for computer scientists (C) Dhruv Batra 60
61 Convolutions for programmers (C) Dhruv Batra 61
62 Convolution Explained (C) Dhruv Batra 62
63 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 63
64 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 64
65 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 65
66 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 66
67 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 67
68 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 68
69 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 69
70 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 70
71 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 71
72 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 72
73 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 73
74 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 74
75 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 75
76 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 76
77 Convolutional Layer (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 77
78 Convolutional Layer Mathieu et al. Fast training of CNNs through FFTs ICLR 2014 (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 78
79 Convolutional Layer * = (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 79
80 Convolutional Layer Learn multiple filters. E.g.: 200x200 image 100 Filters Filter size: 10x10 10K parameters (C) Dhruv Batra Slide Credit: Marc'Aurelio Ranzato 80
81 Fully Connected Layer 32x32x3 image -> stretch to 3072 x 1 input activation x 3072 weights 1 10 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
82 Fully Connected Layer 32x32x3 image -> stretch to 3072 x 1 input activation x 3072 weights number: the result of taking a dot product between a row of W and the input (a 3072-dimensional dot product) Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
83 Convolutional Layer 83
84 Convolutional Layer 84
85 Convolution Layer 32x32x3 image -> preserve spatial structure 32 height 3 32 depth width Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
86 Convolution Layer 32x32x3 image 5x5x3 filter 32 Convolve the filter with the image i.e. slide over the image spatially, computing dot products 3 32 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
87 Convolution Layer 32x32x3 image Filters always extend the full depth of the input volume 5x5x3 filter 32 Convolve the filter with the image i.e. slide over the image spatially, computing dot products 3 32 Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
88 Convolution Layer 32 32x32x3 image 5x5x3 filter number: the result of taking a dot product between the filter and a small 5x5x3 chunk of the image (i.e. 5*5*3 = 75-dimensional dot product + bias) Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
89 Convolution Layer 32 32x32x3 image 5x5x3 filter activation map 28 convolve (slide) over all spatial locations Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
90 Convolution Layer consider a second, green filter 32 32x32x3 image 5x5x3 filter activation maps 28 convolve (slide) over all spatial locations Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
91 For example, if we had 6 5x5 filters, we ll get 6 separate activation maps: activation maps Convolution Layer We stack these up to get a new image of size 28x28x6! Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n
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