neural networks part ii feed-forward neural networks
TRANSCRIPT
Neural Networks Part II
Feed-Forward Neural Networks
Previously on CSCI 4622
2
The Perceptron as a NN: Can learn linearly separable things like OR, AND, NAND, etc
The Multilevel Perceptron: Can learn non-linearly separable things like XOR
Layer 1 to 2:
Layer 2 to 3:
y =
⇢0 if wTx+ b 01 if wTx+ b > 0
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x1 0 1 0 1x2 0 0 1 1
x1 OR x2 0 1 1 1<latexit sha1_base64="UM8xfazAVtlDjQbsqgXmCAzAUIo=">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</latexit><latexit sha1_base64="UM8xfazAVtlDjQbsqgXmCAzAUIo=">AAACd3icbVHJTsMwEHXCHrYCx14sWhCnKumF5YTEhRuLKCA1VeW409bCcSJ7gqhCvoF/48aHcOGE2wYBhZFGfn5vxjOeiVIpDPr+m+POzS8sLi2veKtr6xubla3tW5NkmkOLJzLR9xEzIIWCFgqUcJ9qYHEk4S56OBvrd4+gjUjUDY5S6MRsoERfcIaW6lZevHrdCyMYCJUzrdmoyPkzt1Z4T92A7lPf+vcZhl44HNeiVm6W9FfIjGyvCE+ImNOLa1rQ74TgVwKoXlna9tKt1PyGPzH6FwQlqJHSLruV17CX8CwGhVwyY9qBn2LHPoiCSyi8MDOQMv7ABtC2ULEYTCefDK6ge5bp0X6irSukE/ZnRs5iY0ZxZCNjhkMzq43J/7R2hv2jTi5UmiEoPi3UzyTFhI63QHtCA0c5soBxLWyvlA+ZZhztrjw7hGD2y39Bq9k4bvhXzdrpSTmNZVIlu+SABOSQnJJzcklahJN3p+rUnT3nw6XuvnswDXWdMmeH/DI3+ATd3bFz</latexit><latexit sha1_base64="UM8xfazAVtlDjQbsqgXmCAzAUIo=">AAACd3icbVHJTsMwEHXCHrYCx14sWhCnKumF5YTEhRuLKCA1VeW409bCcSJ7gqhCvoF/48aHcOGE2wYBhZFGfn5vxjOeiVIpDPr+m+POzS8sLi2veKtr6xubla3tW5NkmkOLJzLR9xEzIIWCFgqUcJ9qYHEk4S56OBvrd4+gjUjUDY5S6MRsoERfcIaW6lZevHrdCyMYCJUzrdmoyPkzt1Z4T92A7lPf+vcZhl44HNeiVm6W9FfIjGyvCE+ImNOLa1rQ74TgVwKoXlna9tKt1PyGPzH6FwQlqJHSLruV17CX8CwGhVwyY9qBn2LHPoiCSyi8MDOQMv7ABtC2ULEYTCefDK6ge5bp0X6irSukE/ZnRs5iY0ZxZCNjhkMzq43J/7R2hv2jTi5UmiEoPi3UzyTFhI63QHtCA0c5soBxLWyvlA+ZZhztrjw7hGD2y39Bq9k4bvhXzdrpSTmNZVIlu+SABOSQnJJzcklahJN3p+rUnT3nw6XuvnswDXWdMmeH/DI3+ATd3bFz</latexit>
z2 = W 1x1 + b1
a2 = I(z2 > 0) = I(W 1x1 + b1 > 0)<latexit sha1_base64="bFJTMLtnFcASWTm+WeQFNfaTtV8=">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</latexit><latexit sha1_base64="bFJTMLtnFcASWTm+WeQFNfaTtV8=">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</latexit><latexit sha1_base64="bFJTMLtnFcASWTm+WeQFNfaTtV8=">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</latexit>
z3 = W 2a2 + b2
y = I(z3 > 0) = I(W 2a2 + b2 > 0)<latexit sha1_base64="ASe3O1wbpXNcLSB0W+Iovcd8mnM=">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</latexit><latexit sha1_base64="ASe3O1wbpXNcLSB0W+Iovcd8mnM=">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</latexit><latexit sha1_base64="ASe3O1wbpXNcLSB0W+Iovcd8mnM=">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</latexit>
b
b.V
The Multilayer Perceptron
3
String together lots of perceptron units
The Multilayer Perceptron
4
String together lots of perceptron units
Input Layer Hidden Layers Output Layer
FULLY- connect
FEED - FORWARD
MLP
=
-=
The Multilayer Perceptron
5
String together lots of perceptron units
Inputs, activities, activations, and outputs all live on nodes (neurons)
z = Wattsa= gcz )
The Multilayer Perceptron
6
String together lots of perceptron units
For simplicity: Think of inputs and outputs as activations too
The Multilayer Perceptron
7
String together lots of perceptron units
Each transition between layers has associated set of weights an biases
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The Multilayer Perceptron
8
String together lots of perceptron units
Store the weights connecting layers and in the matrix
is the weight from node j in layer to node i in layer
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yZ3 z3=¥2§2+ bhT T# rows
# cols
"
IN± :
= =
C)= = = =
The Multilayer Perceptron
9
String together lots of perceptron units
Store the biases for layer in the vector
is the bias feeding into node i in layer
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fbiases In layer 3
=
The Multilayer Perceptron
10
Question: What are the dimensions of each of the following?
: ___________ , : __________ , : ___________ , : ______________
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0'
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4×3 3 3 1×3
Sigmoid Neurons
11
Next week we’ll see how we can learn the weights and the biases in a NN using SGD
This poses a problem with standard perceptron neurons with the Indicator activation function
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Sigmoid Neurons
12
Next week we’ll see how we can learn the weights and the biases in a NN using SGD
This poses a problem with standard perceptron neurons with the Indicator activation function
Q: What activation could we use instead?
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Sigmoid Neurons
13
Next week we’ll see how we can learn the weights and the biases in a NN using SGD
This poses a problem with standard perceptron neurons with the Indicator activation function
Q: What activation could we use instead?
A: How about a sigmoid activation?
Sigmoid Neurons
14
Is a sigmoid neuron all that different from a perceptron?
For a fixed input. ,
multiply weights and bias by a factor c
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=
Sigmoid Neurons
15
Is a sigmoid neuron all that different from a perceptron?
For a fixed input. ,
multiply weights and bias by a factor c
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Sigmoid Neurons
16
Is a sigmoid neuron all that different from a perceptron?
For a fixed input. ,
multiply weights and bias by a factor c
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Sigmoid Neurons
17
Is a sigmoid neuron all that different from a perceptron?
For a fixed input. ,
multiply weights and bias by a factor c
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Sigmoid Neurons
18
Is a sigmoid neuron all that different from a perceptron?
For a fixed input. ,
multiply weights and bias by a factor c
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sigm[c(wTx+ b)]<latexit sha1_base64="rdONF2gN4rQC1DnCURD3VIvFJe4=">AAACDXicbVBNT8JAEN3iF9Yv1KOXjUCCMSGFC3oj8eIRExCSUsl22cKG3bbZnSqk4Rd48a948aDGq3dv/htb4KDgSyZ5eW8mM/PcUHANlvVtZNbWNza3stvmzu7e/kHu8OhWB5GirEUDEaiOSzQT3Gct4CBYJ1SMSFewtju6Sv32PVOaB34TJiFzJBn43OOUQCL1csWC2QU2BiVjzQdyatNS3HU9/DC9a47xOXbPHLNg9nJ5q2zNgFdJZUHyaIFGL/fV7Qc0kswHKojWdsUKwYmJAk4Fm5rdSLOQ0BEZMDuhPpFMO/HsnSkuJkofe4FKygc8U39PxERqPZFu0ikJDPWyl4r/eXYE3oUTcz+MgPl0vsiLBIYAp9ngPleMgpgkhFDFk1sxHRJFKCQJpiFUll9eJa1q+bJs3VTz9doijSw6QaeohCqohuroGjVQC1H0iJ7RK3oznowX4934mLdmjMXMMfoD4/MHly2aBw==</latexit><latexit sha1_base64="rdONF2gN4rQC1DnCURD3VIvFJe4=">AAACDXicbVBNT8JAEN3iF9Yv1KOXjUCCMSGFC3oj8eIRExCSUsl22cKG3bbZnSqk4Rd48a948aDGq3dv/htb4KDgSyZ5eW8mM/PcUHANlvVtZNbWNza3stvmzu7e/kHu8OhWB5GirEUDEaiOSzQT3Gct4CBYJ1SMSFewtju6Sv32PVOaB34TJiFzJBn43OOUQCL1csWC2QU2BiVjzQdyatNS3HU9/DC9a47xOXbPHLNg9nJ5q2zNgFdJZUHyaIFGL/fV7Qc0kswHKojWdsUKwYmJAk4Fm5rdSLOQ0BEZMDuhPpFMO/HsnSkuJkofe4FKygc8U39PxERqPZFu0ikJDPWyl4r/eXYE3oUTcz+MgPl0vsiLBIYAp9ngPleMgpgkhFDFk1sxHRJFKCQJpiFUll9eJa1q+bJs3VTz9doijSw6QaeohCqohuroGjVQC1H0iJ7RK3oznowX4934mLdmjMXMMfoD4/MHly2aBw==</latexit><latexit sha1_base64="rdONF2gN4rQC1DnCURD3VIvFJe4=">AAACDXicbVBNT8JAEN3iF9Yv1KOXjUCCMSGFC3oj8eIRExCSUsl22cKG3bbZnSqk4Rd48a948aDGq3dv/htb4KDgSyZ5eW8mM/PcUHANlvVtZNbWNza3stvmzu7e/kHu8OhWB5GirEUDEaiOSzQT3Gct4CBYJ1SMSFewtju6Sv32PVOaB34TJiFzJBn43OOUQCL1csWC2QU2BiVjzQdyatNS3HU9/DC9a47xOXbPHLNg9nJ5q2zNgFdJZUHyaIFGL/fV7Qc0kswHKojWdsUKwYmJAk4Fm5rdSLOQ0BEZMDuhPpFMO/HsnSkuJkofe4FKygc8U39PxERqPZFu0ikJDPWyl4r/eXYE3oUTcz+MgPl0vsiLBIYAp9ngPleMgpgkhFDFk1sxHRJFKCQJpiFUll9eJa1q+bJs3VTz9doijSw6QaeohCqohuroGjVQC1H0iJ7RK3oznowX4934mLdmjMXMMfoD4/MHly2aBw==</latexit>
Predictions: Forward Propagation
19
Suppose we’ve learned the weights in a NN with sigmoid neurons.
Given an input , how do we make a prediction? x<latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit>
s x. →× = y
e [ 0113
Yz →
Predictions: Forward Propagation
20
Suppose we’ve learned the weights in a NN with sigmoid neurons.
Given an input , how do we make a prediction? First, think about a simple network: x<latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit>
w'= [ Wi, Wiz) b '=[ b
'
,3
z2= W' Xtb
a2= g( wktb )
W' t
y
Predictions: Forward Propagation
21
Suppose we’ve learned the weights in a NN with sigmoid neurons.
Given an input , how do we make a prediction?
Re: where
x<latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit>
a`+1 = g(z`+1)<latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit><latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit><latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit>
z`+1 = W `a` + b`<latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit><latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit><latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit>
Ia
z2=W' A' tbl
- a2=g( Z' )
LT -*EH:i:gwizazitbn
Predictions: Forward Propagation
22
Suppose we’ve learned the weights in a NN with sigmoid neurons.
Given an input , how do we make a prediction?
Re: where
x<latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit>
a`+1 = g(z`+1)<latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit><latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit><latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit>
z`+1 = W `a` + b`<latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit><latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit><latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit>
Suppose the network has L layers
Intialize a1 = x<latexit sha1_base64="I5E+eit9kFybGhBmvQ+M07qRDjs=">AAACAHicbZC9TsMwFIUdfkv4C7AgsVi0SExV0qUwIFViYSwSoZXaUDmu01p1nMh2EFUUFl6FhQEQK4/BxtvgpBmg5UiWPp17r67v8WNGpbLtb2NpeWV1bb2yYW5ube/sWnv7tzJKBCYujlgkuj6ShFFOXEUVI91YEBT6jHT8yWVe79wTIWnEb9Q0Jl6IRpwGFCOlrYF1WDPTvh9AlN058AIW/JCZNXNgVe26XQguglNCFZRqD6yv/jDCSUi4wgxJ2XPsWHkpEopiRjKzn0gSIzxBI9LTyFFIpJcWF2TwRDtDGERCP65g4f6eSFEo5TT0dWeI1FjO13Lzv1ovUcGZl1IeJ4pwPFsUJAyqCOZxwCEVBCs21YCwoPqvEI+RQFjp0PIQnPmTF8Ft1M/r9nWj2mqWaVTAETgGp8ABTdACV6ANXIDBI3gGr+DNeDJejHfjY9a6ZJQzB+CPjM8f8FaUJA==</latexit><latexit sha1_base64="I5E+eit9kFybGhBmvQ+M07qRDjs=">AAACAHicbZC9TsMwFIUdfkv4C7AgsVi0SExV0qUwIFViYSwSoZXaUDmu01p1nMh2EFUUFl6FhQEQK4/BxtvgpBmg5UiWPp17r67v8WNGpbLtb2NpeWV1bb2yYW5ube/sWnv7tzJKBCYujlgkuj6ShFFOXEUVI91YEBT6jHT8yWVe79wTIWnEb9Q0Jl6IRpwGFCOlrYF1WDPTvh9AlN058AIW/JCZNXNgVe26XQguglNCFZRqD6yv/jDCSUi4wgxJ2XPsWHkpEopiRjKzn0gSIzxBI9LTyFFIpJcWF2TwRDtDGERCP65g4f6eSFEo5TT0dWeI1FjO13Lzv1ovUcGZl1IeJ4pwPFsUJAyqCOZxwCEVBCs21YCwoPqvEI+RQFjp0PIQnPmTF8Ft1M/r9nWj2mqWaVTAETgGp8ABTdACV6ANXIDBI3gGr+DNeDJejHfjY9a6ZJQzB+CPjM8f8FaUJA==</latexit><latexit sha1_base64="I5E+eit9kFybGhBmvQ+M07qRDjs=">AAACAHicbZC9TsMwFIUdfkv4C7AgsVi0SExV0qUwIFViYSwSoZXaUDmu01p1nMh2EFUUFl6FhQEQK4/BxtvgpBmg5UiWPp17r67v8WNGpbLtb2NpeWV1bb2yYW5ube/sWnv7tzJKBCYujlgkuj6ShFFOXEUVI91YEBT6jHT8yWVe79wTIWnEb9Q0Jl6IRpwGFCOlrYF1WDPTvh9AlN058AIW/JCZNXNgVe26XQguglNCFZRqD6yv/jDCSUi4wgxJ2XPsWHkpEopiRjKzn0gSIzxBI9LTyFFIpJcWF2TwRDtDGERCP65g4f6eSFEo5TT0dWeI1FjO13Lzv1ovUcGZl1IeJ4pwPFsUJAyqCOZxwCEVBCs21YCwoPqvEI+RQFjp0PIQnPmTF8Ft1M/r9nWj2mqWaVTAETgGp8ABTdACV6ANXIDBI3gGr+DNeDJejHfjY9a6ZJQzB+CPjM8f8FaUJA==</latexit>
for ` = 1, . . . , L� 1
z`+1 = W `a` + b`
a`+1 = g(z`+1)<latexit sha1_base64="U8+c+WU3EVMAaYX2L+qZSogv1T4=">AAACqnicbVFLb9QwEHbCq4TXFo5cLLZIi9qukl5KD5UqceGAUBEsu2i9XTnOZGvVsYM9QSxR/hw/gRv/BmeTA912ZFnffPPwzOe0VNJhHP8Nwjt3791/sPMwevT4ydNng93nX52prICJMMrYWcodKKlhghIVzEoLvEgVTNOrd218+gOsk0Z/wXUJi4KvtMyl4Oip5eD3XsRSWEldc2v5uqlVEzGEn2iLOjeWNpSBUqfJAWUqM+gO6IfDhDIWted7xbPNRWuW5vRXc1G32ftJQ0/p9KLFXYQ3nbPfuWnv3t6GX2uzGm33fhMx0Fk/b7QXLQfDeBxvjN4ESQ+GpLfz5eAPy4yoCtAoFHdunsQlLnxDlEKB379yUHJxxVcw91DzAtyi3mjd0NeeyWgrTW400g37f0XNC+fWReozC46XbjvWkrfF5hXmbxe11GWFoEX3UF4pioa2H0czaUGgWnvAhZV+ViouueUC/fe2IiTbK98Ek6PxyTj+dDQ8O+7V2CEvySsyIgk5JmfkPTknEyKCUfAxmAaz8DD8HH4L511qGPQ1L8g1C7N/TejNmg==</latexit><latexit sha1_base64="U8+c+WU3EVMAaYX2L+qZSogv1T4=">AAACqnicbVFLb9QwEHbCq4TXFo5cLLZIi9qukl5KD5UqceGAUBEsu2i9XTnOZGvVsYM9QSxR/hw/gRv/BmeTA912ZFnffPPwzOe0VNJhHP8Nwjt3791/sPMwevT4ydNng93nX52prICJMMrYWcodKKlhghIVzEoLvEgVTNOrd218+gOsk0Z/wXUJi4KvtMyl4Oip5eD3XsRSWEldc2v5uqlVEzGEn2iLOjeWNpSBUqfJAWUqM+gO6IfDhDIWted7xbPNRWuW5vRXc1G32ftJQ0/p9KLFXYQ3nbPfuWnv3t6GX2uzGm33fhMx0Fk/b7QXLQfDeBxvjN4ESQ+GpLfz5eAPy4yoCtAoFHdunsQlLnxDlEKB379yUHJxxVcw91DzAtyi3mjd0NeeyWgrTW400g37f0XNC+fWReozC46XbjvWkrfF5hXmbxe11GWFoEX3UF4pioa2H0czaUGgWnvAhZV+ViouueUC/fe2IiTbK98Ek6PxyTj+dDQ8O+7V2CEvySsyIgk5JmfkPTknEyKCUfAxmAaz8DD8HH4L511qGPQ1L8g1C7N/TejNmg==</latexit><latexit sha1_base64="U8+c+WU3EVMAaYX2L+qZSogv1T4=">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</latexit>
Predictions: Forward Propagation
23
Suppose we’ve learned the weights in a NN with sigmoid neurons.
Given an input , how do we make a prediction?
Re: where
x<latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit><latexit sha1_base64="x2qbD6BmvRnY3pXmkJNsBDx/pfo=">AAAB8nicbVBNT8JAEJ3iF9Yv1KOXjWDiibRc0BuJF4+YWCGhDdkuW9iw3Ta7WyNp+BtePKjx6q/x5r9xCz0o+JJJXt6bycy8MOVMacf5tiobm1vbO9Vde2//4PCodnzyoJJMEuqRhCeyH2JFORPU00xz2k8lxXHIaS+c3hR+75FKxRJxr2cpDWI8FixiBGsj+Q0798MIPc3thj2s1Z2mswBaJ25J6lCiO6x9+aOEZDEVmnCs1MB1Uh3kWGpGOJ3bfqZoiskUj+nAUIFjqoJ8cfMcXRhlhKJEmhIaLdTfEzmOlZrFoemMsZ6oVa8Q//MGmY6ugpyJNNNUkOWiKONIJ6gIAI2YpETzmSGYSGZuRWSCJSbaxFSE4K6+vE68VvO66dy16p12mUYVzuAcLsGFNnTgFrrgAYEUnuEV3qzMerHerY9la8UqZ07hD6zPH/Qij9I=</latexit>
a`+1 = g(z`+1)<latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit><latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit><latexit sha1_base64="g67eZfvd/mlQV0FqveJvn/8s8Hs=">AAACFXicbZDLSsNAFIYn9VbjLerSzWArVIol6aa6EApuXFYwttDUMplO2sHJhZmJUEOewo2v4saFilvBnW/jJM1CW38Y+PnOOZw5vxsxKqRpfmulpeWV1bXyur6xubW9Y+zu3Ygw5pjYOGQh77lIEEYDYksqGelFnCDfZaTr3l1k9e494YKGwbWcRmTgo3FAPYqRVGhonFT1xHE9iNLbxCGM1a0UnsNxLYcPBYR1aKXHelUfGhWzYeaCi8YqTAUU6gyNL2cU4tgngcQMCdG3zEgOEsQlxYykuhMLEiF8h8akr2yAfCIGSX5WCo8UGUEv5OoFEub090SCfCGmvqs6fSQnYr6Wwf9q/Vh6p4OEBlEsSYBni7yYQRnCLCM4opxgyabKIMyp+ivEE8QRlirJLARr/uRFYzcbZw3zqllpt4o0yuAAHIIasEALtMEl6AAbYPAInsEreNOetBftXfuYtZa0YmYf/JH2+QOGnJv3</latexit>
z`+1 = W `a` + b`<latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit><latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit><latexit sha1_base64="fy0SaoVScwpHyfIH2HDC2W+LVfg=">AAACJHicbVDLSsNAFJ34rPEVdelmsBUEoSTdVEGh4MZlBWMLTVom00k7dDIJMxOhhvyMG3/FjQsrLtz4LSZpFtp6YeA87uXOPV7EqFSm+aWtrK6tb2xWtvTtnd29fePg8EGGscDExiELRddDkjDKia2oYqQbCYICj5GON7nJ/c4jEZKG/F5NI+IGaMSpTzFSmTQwrmp64ng+fEr7iUMYg+fQSuE17PQLVngo7ZdWQb051Wv6wKiadbMouAysElRBWe2BMXOGIY4DwhVmSMqeZUbKTZBQFDOS6k4sSYTwBI1IL4McBUS6SXFlCk8zZQj9UGSPK1iovycSFEg5DbysM0BqLBe9XPzP68XKv3ATyqNYEY7ni/yYQRXCPDI4pIJgxaYZQFjQ7K8Qj5FAWGXB5iFYiycvA7tRv6ybd41qq1mmUQHH4AScAQs0QQvcgjawAQbP4BW8g5n2or1pH9rnvHVFK2eOwJ/Svn8A1M+h7w==</latexit>
Suppose the network has L layers
Intialize
If classification and sigmoid neuron output
then predict based on
a1 = x<latexit sha1_base64="I5E+eit9kFybGhBmvQ+M07qRDjs=">AAACAHicbZC9TsMwFIUdfkv4C7AgsVi0SExV0qUwIFViYSwSoZXaUDmu01p1nMh2EFUUFl6FhQEQK4/BxtvgpBmg5UiWPp17r67v8WNGpbLtb2NpeWV1bb2yYW5ube/sWnv7tzJKBCYujlgkuj6ShFFOXEUVI91YEBT6jHT8yWVe79wTIWnEb9Q0Jl6IRpwGFCOlrYF1WDPTvh9AlN058AIW/JCZNXNgVe26XQguglNCFZRqD6yv/jDCSUi4wgxJ2XPsWHkpEopiRjKzn0gSIzxBI9LTyFFIpJcWF2TwRDtDGERCP65g4f6eSFEo5TT0dWeI1FjO13Lzv1ovUcGZl1IeJ4pwPFsUJAyqCOZxwCEVBCs21YCwoPqvEI+RQFjp0PIQnPmTF8Ft1M/r9nWj2mqWaVTAETgGp8ABTdACV6ANXIDBI3gGr+DNeDJejHfjY9a6ZJQzB+CPjM8f8FaUJA==</latexit><latexit sha1_base64="I5E+eit9kFybGhBmvQ+M07qRDjs=">AAACAHicbZC9TsMwFIUdfkv4C7AgsVi0SExV0qUwIFViYSwSoZXaUDmu01p1nMh2EFUUFl6FhQEQK4/BxtvgpBmg5UiWPp17r67v8WNGpbLtb2NpeWV1bb2yYW5ube/sWnv7tzJKBCYujlgkuj6ShFFOXEUVI91YEBT6jHT8yWVe79wTIWnEb9Q0Jl6IRpwGFCOlrYF1WDPTvh9AlN058AIW/JCZNXNgVe26XQguglNCFZRqD6yv/jDCSUi4wgxJ2XPsWHkpEopiRjKzn0gSIzxBI9LTyFFIpJcWF2TwRDtDGERCP65g4f6eSFEo5TT0dWeI1FjO13Lzv1ovUcGZl1IeJ4pwPFsUJAyqCOZxwCEVBCs21YCwoPqvEI+RQFjp0PIQnPmTF8Ft1M/r9nWj2mqWaVTAETgGp8ABTdACV6ANXIDBI3gGr+DNeDJejHfjY9a6ZJQzB+CPjM8f8FaUJA==</latexit><latexit sha1_base64="I5E+eit9kFybGhBmvQ+M07qRDjs=">AAACAHicbZC9TsMwFIUdfkv4C7AgsVi0SExV0qUwIFViYSwSoZXaUDmu01p1nMh2EFUUFl6FhQEQK4/BxtvgpBmg5UiWPp17r67v8WNGpbLtb2NpeWV1bb2yYW5ube/sWnv7tzJKBCYujlgkuj6ShFFOXEUVI91YEBT6jHT8yWVe79wTIWnEb9Q0Jl6IRpwGFCOlrYF1WDPTvh9AlN058AIW/JCZNXNgVe26XQguglNCFZRqD6yv/jDCSUi4wgxJ2XPsWHkpEopiRjKzn0gSIzxBI9LTyFFIpJcWF2TwRDtDGERCP65g4f6eSFEo5TT0dWeI1FjO13Lzv1ovUcGZl1IeJ4pwPFsUJAyqCOZxwCEVBCs21YCwoPqvEI+RQFjp0PIQnPmTF8Ft1M/r9nWj2mqWaVTAETgGp8ABTdACV6ANXIDBI3gGr+DNeDJejHfjY9a6ZJQzB+CPjM8f8FaUJA==</latexit>
for ` = 1, . . . , L� 1
z`+1 = W `a` + b`
a`+1 = g(z`+1)<latexit sha1_base64="U8+c+WU3EVMAaYX2L+qZSogv1T4=">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</latexit><latexit sha1_base64="U8+c+WU3EVMAaYX2L+qZSogv1T4=">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</latexit><latexit sha1_base64="U8+c+WU3EVMAaYX2L+qZSogv1T4=">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</latexit>
aL<latexit sha1_base64="Ss4ZKzF5gA4Bkaib8XGhoLnTgm0=">AAAB+XicbVBNT8JAEJ3iF9avokcvG8HEE2m5oDcSLx48YCJCApVsly1s2H5kd6shtT/Fiwc1Xv0n3vw3bqEHBV8yyct7M5mZ58WcSWXb30ZpbX1jc6u8be7s7u0fWJXDOxklgtAOiXgkeh6WlLOQdhRTnPZiQXHgcdr1ppe5332gQrIovFWzmLoBHofMZwQrLQ2tSs1MB56PcHafXmfIrJlDq2rX7TnQKnEKUoUC7aH1NRhFJAloqAjHUvYdO1ZuioVihNPMHCSSxphM8Zj2NQ1xQKWbzk/P0KlWRsiPhK5Qobn6eyLFgZSzwNOdAVYTuezl4n9eP1H+uZuyME4UDclikZ9wpCKU54BGTFCi+EwTTATTtyIywQITpdPKQ3CWX14lnUb9om7fNKqtZpFGGY7hBM7AgSa04Ara0AECj/AMr/BmPBkvxrvxsWgtGcXMEfyB8fkDwYOR4A==</latexit><latexit sha1_base64="Ss4ZKzF5gA4Bkaib8XGhoLnTgm0=">AAAB+XicbVBNT8JAEJ3iF9avokcvG8HEE2m5oDcSLx48YCJCApVsly1s2H5kd6shtT/Fiwc1Xv0n3vw3bqEHBV8yyct7M5mZ58WcSWXb30ZpbX1jc6u8be7s7u0fWJXDOxklgtAOiXgkeh6WlLOQdhRTnPZiQXHgcdr1ppe5332gQrIovFWzmLoBHofMZwQrLQ2tSs1MB56PcHafXmfIrJlDq2rX7TnQKnEKUoUC7aH1NRhFJAloqAjHUvYdO1ZuioVihNPMHCSSxphM8Zj2NQ1xQKWbzk/P0KlWRsiPhK5Qobn6eyLFgZSzwNOdAVYTuezl4n9eP1H+uZuyME4UDclikZ9wpCKU54BGTFCi+EwTTATTtyIywQITpdPKQ3CWX14lnUb9om7fNKqtZpFGGY7hBM7AgSa04Ara0AECj/AMr/BmPBkvxrvxsWgtGcXMEfyB8fkDwYOR4A==</latexit><latexit sha1_base64="Ss4ZKzF5gA4Bkaib8XGhoLnTgm0=">AAAB+XicbVBNT8JAEJ3iF9avokcvG8HEE2m5oDcSLx48YCJCApVsly1s2H5kd6shtT/Fiwc1Xv0n3vw3bqEHBV8yyct7M5mZ58WcSWXb30ZpbX1jc6u8be7s7u0fWJXDOxklgtAOiXgkeh6WlLOQdhRTnPZiQXHgcdr1ppe5332gQrIovFWzmLoBHofMZwQrLQ2tSs1MB56PcHafXmfIrJlDq2rX7TnQKnEKUoUC7aH1NRhFJAloqAjHUvYdO1ZuioVihNPMHCSSxphM8Zj2NQ1xQKWbzk/P0KlWRsiPhK5Qobn6eyLFgZSzwNOdAVYTuezl4n9eP1H+uZuyME4UDclikZ9wpCKU54BGTFCi+EwTTATTtyIywQITpdPKQ3CWX14lnUb9om7fNKqtZpFGGY7hBM7AgSa04Ara0AECj/AMr/BmPBkvxrvxsWgtGcXMEfyB8fkDwYOR4A==</latexit>
Other Popular Activation Functions
24
perceptron sigmoid
tanh ReLu
Other Popular Activation Functions
25
perceptron sigmoid
tanh ReLu
Different Probs – Different Architectures
26
What if we want to do regression instead of classification?
Different Probs – Different Architectures
27
What if we want to do classification with more than two classes?
Different Probs – Different Architectures
28
What if we want to do classification with more than two classes?
←IsPEDESTRIAN
• 95 ⇐ PREDICT PED
•
25.00÷Then
Different Probs – Different Architectures
29
What if we want to do classification with more than two classes?
Different Probs – Different Architectures
30
What if we want to do classification with more than two classes?
Loss Functions
31
How do we know if we’re doing well?
Single Output – Squared Loss
Regression or Classification
L 1 w , b. × , y )
→ FI 1 yi - a ;)2
Loss Functions
32
How do we know if we’re doing well?
Multiple Outputs – Squared Loss
Classification
: ;§ " jcxi . a tip
° Ettenat
Loss Functions
33
How do we know if we’re doing well?
Single Output – Cross-Entropy
Regression or Classification
,§g{yi log ( at )+ Ctyi) high .and
Loss Functions
34
How do we know if we’re doing well?
Multiple Outputs – SoftMax
Classification
Training a Feed-Forward NN
35
So how do we train such a monstrosity?
Training a Feed-Forward NN
36
So how do we train such a monstrosity?
Pretty much the usual way
Define a loss function: Loss(x,y;W,b) = L(W,b)<latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit><latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit><latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit>
Training a Feed-Forward NN
37
So how do we train such a monstrosity?
Pretty much the usual way
Define a loss function:
Then we just do Stochastic Gradient Descent:
No problem right …
Loss(x,y;W,b) = L(W,b)<latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit><latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit><latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit>
w`ij w`
ij � ⌘@L
@w`ij
, b`i b`i � ⌘@L
@b`i<latexit sha1_base64="Tt/xv9DSrRGujanZVTzkBNfJboY=">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</latexit><latexit sha1_base64="Tt/xv9DSrRGujanZVTzkBNfJboY=">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</latexit><latexit sha1_base64="Tt/xv9DSrRGujanZVTzkBNfJboY=">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</latexit>
Training a Feed-Forward NN
38
So how do we train such a monstrosity?
Pretty much the usual way
Define a loss function:
Then we just do Stochastic Gradient Descent:
No problem right …
Turns out those partial derivatives are not so easy to compute
This is where Back Propagation comes in. It will require:
Calculus Linear Algebra An Iron Stomach
Loss(x,y;W,b) = L(W,b)<latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit><latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit><latexit sha1_base64="ZoZ7AfgQ8QBFd053ZFPsCDUUTXs=">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</latexit>
w`ij w`
ij � ⌘@L
@w`ij
, b`i b`i � ⌘@L
@b`i<latexit sha1_base64="Tt/xv9DSrRGujanZVTzkBNfJboY=">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</latexit><latexit sha1_base64="Tt/xv9DSrRGujanZVTzkBNfJboY=">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</latexit><latexit sha1_base64="Tt/xv9DSrRGujanZVTzkBNfJboY=">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</latexit>
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