Neural Networks II. Chen Gao. Virginia Tech Spring 2019 ECE-5424G / CS-5824

Transcription

1 Neural Networks II Chen Gao ECE-5424G / CS-5824 Virginia Tech Spring 2019

2 Neural Networks Origins: Algorithms that try to mimic the brain. What is this?

3 A single neuron in the brain Input Output Slide credit: Andrew Ng

4 An artificial neuron: Logistic unit x 0 x 1 x 2 Bias unit x = Output x 0 x 1 x 2 x 3 θ = h θ x = 1 θ 0 θ 1 θ 2 θ e θ x Weights Parameters x 3 Input Sigmoid (logistic) activation function Slide credit: Andrew Ng

5 Visualization of weights, bias, activation function range determined by g(.) bias b only change the position of the hyperplane Slide credit: Hugo Larochelle

6 Activation - sigmoid Squashes the neuron s preactivation between 0 and 1 Always positive Bounded Strictly increasing g x = e x Slide credit: Hugo Larochelle

7 Activation - hyperbolic tangent (tanh) Squashes the neuron s preactivation between -1 and 1 Can be positive or negative Bounded Strictly increasing g x = tanh x = ex e x e x + e x Slide credit: Hugo Larochelle

8 Activation - rectified linear(relu) Bounded below by 0 always non-negative Not upper bounded Tends to give neurons with sparse activities g x = relu x = max 0, x Slide credit: Hugo Larochelle

9 Activation - softmax For multi-class classification: we need multiple outputs (1 output per class) we would like to estimate the conditional probability p y = c x We use the softmax activation function at the output: g x = softmax x = ex 1 σ c e x c ex c σ c e x c Slide credit: Hugo Larochelle

10 Universal approximation theorem a single hidden layer neural network with a linear output unit can approximate any continuous function arbitrarily well, given enough hidden units Hornik, 1991 Slide credit: Hugo Larochelle

11 Neural network Multilayer x 0 x 1 x 2 a 0 a 1 a 2 Output h Θ x x 3 Layer 1 a 3 Layer 2 (hidden) Layer 3 Slide credit: Andrew Ng

12 Neural network x 0 x 1 x 2 x 3 a 0 a 1 a 2 a 3 h Θ x (1) (1) (1) (1) a 1 = g Θ10 x0 + Θ 11 x1 + Θ 12 x2 + Θ 13 x3 (1) (1) (1) (1) a 2 = g Θ20 x0 + Θ 21 x1 + Θ 22 x2 + Θ 23 x3 (j) a i = activation of unit i in layer j Θ j = matrix of weights controlling function mapping from layer j to layer j + 1 s j unit in layer j s j+1 units in layer j + 1 Size of Θ j? (1) (1) (1) (1) a 3 = g Θ30 x0 + Θ 31 x1 + Θ 32 x2 + Θ 33 x3 (1) (1) (1) h Θ (x) = g Θ 10 a0 + Θ11 a1 + Θ12 a2 + Θ13 a3 s j+1 (s j + 1) Slide credit: Andrew Ng

13 Neural network x 0 x = a 0 x 1 a 1 x 2 a 2 h Θ x x 0 x 1 x 2 x 3 z = Pre-activation z 1 z 2 z 3 x 3 a 3 Why do we need g(.)? (1) (1) (1) (1) a 1 = g Θ10 x0 + Θ 11 x1 + Θ 12 x2 + Θ 13 x3 = g(z 1 ) (1) (1) (1) (1) a 2 = g Θ20 x0 + Θ 21 x1 + Θ 22 x2 + Θ 23 x3 = g(z 2 ) (1) (1) (1) (1) a 3 = g Θ30 x0 + Θ 31 x1 + Θ 32 x2 + Θ 33 x3 = g(z 3 ) h Θ x = g Θ 2 10 a Θ 1 11 a Θ 1 12 a Θ a 3 = g(z (3) ) Slide credit: Andrew Ng

14 Neural network x 0 x = a 0 x 1 a 1 x 2 a 2 h Θ x x 0 x 1 x 2 x 3 z = Pre-activation z 1 z 2 z 3 x 3 a 3 a 1 = g(z1 ) a 2 = g(z2 ) a 3 = g(z3 ) h Θ x = g(z (3) ) z = Θ (1) x = Θ (1) a (1) a = g(z ) Add a 0 = 1 z (3) = Θ a h Θ x = a (3) = g(z (3) ) Slide credit: Andrew Ng

15 Flow graph - Forward propagation X z a (3) h Θ x a z (3) W (1) b (1) W b How do we evaluate our prediction? z = Θ (1) x = Θ (1) a (1) a = g(z ) Add a 0 = 1 z (3) = Θ a h Θ x = a (3) = g(z (3) )

16 Cost function Logistic regression: Neural network: Slide credit: Andrew Ng

17 Gradient computation Need to compute: Slide credit: Andrew Ng

18 Gradient computation Given one training example x, y a (1) = x z = Θ (1) a (1) a = g(z ) (add a 0 ) z (3) = Θ a a (3) = g(z (3) (3) ) (add a 0 ) z (4) = Θ (3) a (3) a (4) = g z 4 = h Θ x Slide credit: Andrew Ng

19 Gradient computation: Backpropagation Intuition: δ j (l) = error of node j in layer l For each output unit (layer L = 4) δ (4) = a (4) y δ (3) = δ (4) δ(4) δ(4) a z (3) = (4) z (4) a (3) δ(4) a (4) z (4) a (3) z (3) = 1 *Θ 3 T δ (4). g z 4. g (z (3) ) z (3) = Θ a a (3) = g(z (3) ) z (4) = Θ (3) a (3) a (4) = g z 4 Slide credit: Andrew Ng

20 Backpropagation algorithm Training set x (1), y (1) x (m), y (m) Set Θ (1) = 0 For i = 1 to m Set a (1) = x Perform forward propagation to compute a (l) for l = 2.. L use y (i) to compute δ (L) = a (L) y (i) Compute δ (L 1), δ (L 2) δ Θ (l) = Θ (l) a (l) δ (l+1) Slide credit: Andrew Ng

21 Activation - sigmoid Partial derivative g x = g x 1 g x g x = e x Slide credit: Hugo Larochelle

22 Activation - hyperbolic tangent (tanh) Partial derivative g x = 1 g x 2 g x = tanh x = ex e x e x + e x Slide credit: Hugo Larochelle

23 Activation - rectified linear(relu) Partial derivative g x = 1 x > 0 g x = relu x = max 0, x Slide credit: Hugo Larochelle

24 Initialization For bias Initialize all to 0 For weights Can t initialize all weights to the same value we can show that all hidden units in a layer will always behave the same need to break symmetry Recipe: U[-b, b] the idea is to sample around 0 but break symmetry Slide credit: Hugo Larochelle

25 Putting it together Pick a network architecture No. of input units: Dimension of features No. output units: Number of classes Reasonable default: 1 hidden layer, or if >1 hidden layer, have same no. of hidden units in every layer (usually the more the better) Grid search Slide credit: Hugo Larochelle

26 Putting it together Early stopping Use a validation set performance to select the best configuration To select the number of epochs, stop training when validation set error increases Slide credit: Hugo Larochelle

27 Other tricks of the trade Normalizing your (real-valued) data Decaying the learning rate as we get closer to the optimum, makes sense to take smaller update steps mini-batch can give a more accurate estimate of the risk gradient Momentum can use an exponential average of previous gradients Slide credit: Hugo Larochelle

28 Dropout Idea: «cripple» neural network by removing hidden units each hidden unit is set to 0 with probability 0.5 hidden units cannot co-adapt to other units hidden units must be more generally useful Slide credit: Hugo Larochelle