Neural networks Feedforward neural network - activation function September Abstract6, 2012 [email protected] September 6, 2012 6 iversité de Sherbrooke September 6, 2012 Septe ARTIFICIAL NEURON Math for my slides “Feedforward neural network”. Abst [email protected] September 6, 2012 Abstract 2 Abstract P Math for my slides “Feedforward neu Topics: connection weights, bias,>activation function Abstract x) = bMath + for w x = b + w x my slides “Feedforward neural network”. i i Abstract Abstrac i Math for my slides “Feedforward neural network”. • Neuron September 6, 2012 P pre-activation (or input activation): > Math for my slides “Feedforward neural network”. P Abstract P Math for my slides “Feedforward neural network”. • a(x) = b + w x = b + w x i i P i “Feedforward neural n > Math for my slides (x) = g(a(x)) = g(b + w x ) > •• a(x) = b+ + iwwi ixxi i==b bi++wiwix x P Math for my slides “Feedforwar a(x) = b P i P Math for my slides “Feedforward neural network”. >> •P a(x) = b= + b•+h(x) = b + w x P P• a(x) wi= x = b + w x g(a(x)) = g(b + > i wi xi ) i wiix i i P P > h(x) = g(a(x)) =g(b g(b++ i>w w•xiix)a(x) x b w w = b + w x = b + w x i) h(x) = g(a(x)) = 1 •• d 1 d i i P i i • a(x) = b + w x = b + w P i i i • a(x) = b + w x = b + w x i Abstract i i • Neuron (output)i activation • h(x) = g(a(x)) = g(b + w x ) • h(x) = g(a(x)) = g(b + w x i i i i )P i i • x x b w w 1 d 1 d P P ••• xxh(x) b w w11 =wwdg(b x= 11 x d g(a(x)) d + • h(x) = g(a(x)) = g(b + w x ) • h(x) = g(a(x)) = g(b + w x ) i i i i 1 i dforward neural network”. i i wi • x1• xx1d xbd wb 1 ww 1 d wd • w w ••• w x1 xd b w 1 w d • x x • x x 1 d 1 d • w > • w +• w arex the connection weights • { ••• {{w • w • { • w • P • { ·) b is the neuron bias • g(·) b w x ) i i { isbbcalled the activation function g(·) ••••ig(·) • g(·) b • { • g(·) •b { (x) = g(a(x)) ... 3 ACTIVATION FUNCTION Math for my slides “Feedforward neural n Topics: linear activation function • Performs squashing • Not no input very interesting... Abstract • a(x) = b + P > w i xi = b + w x P • h(x) = g(a(x)) = g(b + i wi xi ) i • x1 xd b w 1 w d • w • { • g(a) = a 1 Abstract ACTIVATION Math for myFUNCTION slides “Feedforward neural network”. P > Topics: sigmoid activation•function a(x) = b + i wi xi = b + w x • Squashes the neuron’s P pre-activation between • h(x) = g(a(x)) = g(b + i wi xi ) 0 and 1 • Always positive • Bounded • Strictly increasing • x1 xd b w 1 w d • w • { • g(a) = a • g(a) = sigm(a) = 1 1+exp( a) exp(a) exp( a) exp(2a) 1 4 Math for my slides “Feedforward neural network”. P ACTIVATION FUNCTION • a(x) = b + w x =b+w x i > i i P function Topics: hyperbolic tangent (‘‘tanh’’) activation • h(x) = g(a(x)) = g(b + wx) • Squashes the neuron’s • xbetween 1 xd b w 1 w d pre-activation -1 and 1 • w • Can be positive or negative • { i i i • Bounded • g(a) = a • Strictly increasing • g(a) = sigm(a) = • g(a) = tanh(a) = 1 1+exp( a) exp(a) exp( a) exp(a)+exp( a) = exp(2a) 1 exp(2a)+1 5 6 ACTIVATION FUNCTION • x1 xd b w 1 w d • w Topics: rectified linear activation function below by 0 • { (always non-negative) • Bounded • Not upper bounded • Strictly • Tends increasing • g(a) = a • g(a) = sigm(a) = to give neurons with sparse activities • g(a) = tanh(a) = 1 1+exp( a) exp(a) exp( a) exp(a)+exp( a) • g(a) = max(0, a) • g(a) = reclin(a) = max(0, a) = exp(2a) 1 exp(2a)+1
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