Neural Network Back-propagation HYUNG IL KOO
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1 Neural Network Back-propagation HYUNG IL KOO
2 Hidden Layer Representations Backpropagation has an ability to discover useful intermediate representations at the hidden unit layers inside the networks which capture properties of the input spaces that are most relevant to learning the target function. When more layers of units are used in the network, more complex features can be invented. But the representations of the hidden layers are very hard to understand for humans.
3 Basic Math
4 Optimization Find x that minimizes f(x) If f(x) is differentiable, But, in many cases, solving the above equation is a still difficult problem.
5 Gradient descent f(x 1 ) y = f(x) f(x 2 ) x 1 x 2 x 3
6 Chain Rule Chain rule with a single variable Chain Rule (multiple variables) w = f x, y, z => dw = f dx + f dy + f dz dt x dt y dt z dt Δw f x Δx + f y Δy + f z Δz
7 Feed-forward neural network
8 Feed forward network example: 1 st layer w 11 w 12w21 +1 w 22 w 31 w 32 The 1st hidden layer Non-linear function
9 Feed forward network example: 2 nd layer u 11 u 21 u 22 u 12 u u 23 The 1st hidden layer +1 The 2 nd hidden layer
10 Layer 1 Layer 2 Soft Max Forward propagation The 1st hidden layer The 2 nd hidden layer Output
11 Layer 1 Layer 2 Soft Max Forward propagation matrix repr. The 1st hidden layer The 2 nd hidden layer Output
12 Back-propagation algorithm
13 Weight update method Parameter update : Gradient Descent ` Loss function (L) Learning rate
14 Layer 1 Layer 2 Soft Max Dataflow diagram VS The 1st hidden layer The 2 nd hidden layer Output Ground Truth
15 Soft Max Back-propagation step; Loss function Output Ground Truth Computing Loss function(l): ex) Cross entropy
16 Layer 1 Layer 2 Soft Max Layer 1 Layer 2 Soft Max Overview 1 0 The 1st hidden layer The 2 nd hidden layer Output Ground Truth Loss function
17 Layer 2 Soft Max Back-propagation; 2 nd layer The 1st hidden layer The 2 nd hidden layer Output Ground Truth Weight update the Layer 2 has to do Error propagation
18 Layer 2 Soft Max Back-propagation; 2 nd layer The 1st hidden layer The 2 nd hidden layer Output Ground Truth Weight update the Layer 2 has to do Error propagation
19 Error propagation (feed forward network) The 1st hidden layer The 2 nd hidden layer z 1 and z 2 are from its upper layer.
20 Weight updates (feed forward network) The 1st hidden layer The 2 nd hidden layer z 1 and z 2 are from its upper layer.
21 Layer 1 Layer 2 Soft Max Back propagation; 1 st layer The 1st hidden layer The 2 nd hidden layer Output Ground Truth the Layer 1 has to do Weight update w new ij = w old ij μ w ij Error propagation, Input update???
22 Weight updates w ij new = w ij old μ w ij (feed forward network) +1 The 1st hidden layer, and are y 1 y 2 y 3 from its upper layer
23 Block-based perspective
24 Basic Math Y= X AX Y+ΔY X + ΔX A X + ΔX = X AX + ΔX AX + X ΔX + ΔX AΔX X AX + X A + A ΔX = Y X Y A Y+ΔY X A + ΔA X = X AX + X ΔAX ΔY = X ΔAX = tr X ΔAX = tr(xx ΔA) Y A = XX
25 Block-based representation X Y Z Y = WX + b Z = φ(y) X Y Z X = W Y Y = diag φ Z Z W b = Y = Y X
26 Soft Max Forward propagation (block-based representation) Layer 1 Layer 2 Output Layer 1 Layer 2
27 Soft Max Backward propagation; 2 nd layer VS Layer 1 Layer 2 Output Ground Truth Error propagation
28 Soft Max Backward propagation; 2 nd layer VS Layer 1 Layer 2 Error propagation Output Ground Truth VS
29 Soft Max Backward propagation; 2 nd layer VS Layer 1 Layer 2 Output Ground Truth Error propagation Weight update
30 Soft Max Backward propagation; 1 st layer VS Layer 1 Layer 2 Error propagation Output Ground Truth
31 Soft Max Backward propagation; 1 st layer VS Layer 1 Layer 2 Error propagation Weight update Output Ground Truth
32 Input Optimization while fixing all weights
33 Input update (feed forward network) +1 The 1st hidden layer y 1, y 2 and 3 are from its upper layer.
34 Output Loss Building
35 I new = I old μ I Output Loss Building
36 Output Loss Building
37 Inceptionism: Going Deeper into Neural Networks
38 Inceptionism: Going Deeper into Neural Networks
39 DEEP INSIDE CONVOLUTION NETWORKS Inputs maximizing class score
40 Inputs maximizing class score S c I λ I 2 Objective Function (to be maximized) K. Simonyan, A. Vedaldi, A. Zisserman, Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps, ICLR Workshop 2014
41 Inputs maximizing class score S c I λ I 2 Objective Function (to be maximized) Goose class
42 Maximizing class score S c I λ I 2 Objective Function (to be maximized) Goose class
43 Inputs maximizing class score dumbbell cup dalmatian bell pepper lemon husky
44 Inputs maximizing class score computer keyboard kit fox limousine Washing machine goose ostrich
45 DEEP INSIDE CONVOLUTION NETWORKS Saliency visualization
46 Saliency visualization Linear score model for class c: w : importance of corresponding pixels of I for class c
47 Saliency visualization Dog Class Objective Function
48 Saliency visualization Dog Class Differentiation Objective Function S c (I 0 ) I 0 =
49 Saliency visualization I 0 = saliency map
50 yacht dog monkey washing machine cow building
51 A NEURAL ALGORITHM OF ARTISTIC STYLE
52 Loss Gatys, Leon A., Alexander S. Ecker, and Matthias Bethge. "A neural algorithm of artistic style." arxiv preprint arxiv: (2015).
53 Loss
54
55 Artistic style
56 Artistic style
57 Backups
58 Vector-by-scalar
59 Scalar-by-vector
60 Vector-by-vector
61 Scalar-by-matrix
62 Why W = X Y? We want to find W satisfying: ΔL = tr from Y = WX ΔL = Y ΔY. W ΔW. [Intuitive derivation] ΔY = ΔWX ΔL = W Y ΔY = ΔWX = tr Y = X Y Y ΔWX = tr X Y ΔW
63 X Y Z V Y = WX Z = Y + b V = φ(z) X Y Z V X = W Y Y = Z Z = diag φ Z V W = Y X b = Z
64 Proof Element-wise operation y 1 y 2 y n y i = z i φ z i Y y 1 y 2 y n = diag φ Z Z z 1 = φ(y 1 ) z 2 = φ(y 2 ) z n = φ(y n ) z 1 z 2 z n
65 Proof Y = W X L Y Y Y W X L Y W X X = W Y = X y = WX L tr y WX L Y Y W W L WX = tr Y W = Y Y WX = tr X Y W X
66 New Layer Design
67 Layer 1 Layer m Layer N Soft Max New layer addition α 1 β 1 α 2 α 3 β 2 β 3 The 1st layer A newly added layer The Nth layer Output
68 Layer m Layer m New layer design Forward pass Compute output from input Backward pass Compute the derivatives w.r.t. data Update weights?? dl dl α 1 β 1 dα 1 dl dβ 1 dl α 2 β 2 dα 2 dl dβ 2 dl α 3 β 3 dα 3 dβ 3 Forward pass Backward pass
69 Max pooling layer Example: Max pooling
70 max max Derivatives of max For forward pass For backward pass x dl dx = dl dz dz dx y z dl dy = dl dz dz dy dl dz Forward pass Backward pass
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