Machine Learning for Large-Scale Data Analysis and Decision Making A. Neural Networks Week #6

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1 Machine Learning for Large-Scale Data Analysis and Decision Making A Neural Networks Week #6

2 Today Neural Networks A. Modeling B. Fitting C. Deep neural networks Today s material is (adapted) from Joelle Pineau s slides 2

3 Neural Networks Flexible model class Highly-non linear models Good for regression/classification/density estimation Models behind Deep Learning Historical aspects 3

4 Recall Linear Classification x 2 x 1

5 Recall Linear Classification y(x) =w x + w 0 Decision (w x + w 0 ) > 0 = (w x + w 0 ) < 0 = x 2 x 1

6 Recall Linear Classification y(x) =w x + w 0 Decision (w x + w 0 ) > 0 = (w x + w 0 ) < 0 = x 2 x 1

7 What if data is not linearly separable? Exclusive OR (XOR) x 2 x 1

8 What if data is not linearly separable? Exclusive OR (XOR) x 2 Use the joint decision of several linear classifier? x 1

9 What if data is not linearly separable? Exclusive OR (XOR) x 2 Use the joint decision of several linear classifier? x 2 w x + w 0 w x + w 0 x 1 x 1

10 Combining models (w x + w 0 ) > 0 = f(x): (w x + w 0 ) < 0 = x 2 w x + w 0 (w f (x): x + w 0 ) > 0 = (w x + w 0 ) < 0 = w x + w 0 x 1

11 Combining models (w x + w 0 ) > 0 = f(x): (w x + w 0 ) < 0 = f(x) = and f (x) = = x 2 w x + w 0 (w f (x): x + w 0 ) > 0 = (w x + w 0 ) < 0 = f(x) = and f (x) = = f(x) = and f (x) = = w x + w 0 x 1

12 Combining models (w x + w 0 ) > 0 = f(x): (w x + w 0 ) < 0 = f(x) = and f (x) = = x 2 w x + w 0 (w f (x): x + w 0 ) > 0 = (w x + w 0 ) < 0 = f(x) = and f (x) = = f(x) = and f (x) = = w x + w 0 x 1 1. Evaluate each model 2.Combine the output of models

13 Combining models (w x + w 0 ) > 0 = f(x): (w x + w 0 ) < 0 = f(x) = and f (x) = = x 2 w x + w 0 (w f (x): x + w 0 ) > 0 = (w x + w 0 ) < 0 = f(x) = and f (x) = = f(x) = and f (x) = = w x + w 0 x 1 1. Evaluate each model 2.Combine the output of models f (x) =threshold(w f(x) f (x) + w 0 )

14 Combining model (graphical view) f(x) f (x) f (x) 7

15 Combining model (graphical view) x 1 f(x) f (x) x 2 f (x) 7

16 Combining model (graphical view) x 1 f(x) f (x) {, } x 2 f (x) 7

17 Combining model (graphical view) Neural Network x 1 f(x) f (x) {, } x 2 f (x) 7

18 Combining model (graphical view) Neural Network x 1 f(x) f (x) {, } x 2 f (x) Perceptron/ Neuron 7

19 Feed-forward neural network Input Layer Hidden Layer(s) Output Layer 1. An input layer Its size is the number of inputs + 1 x 1 2. A set of hidden layer(s) Its size is a hyper-parameter x An output layer Its size is the number of outputs A scalar for simple classification problems Can be a vector 8

20 Feed-forward neural network Input Layer Hidden Layer(s) Output Layer 1. An input layer Its size is the number of inputs + 1 x 1 2. A set of hidden layer(s) Its size is a hyper-parameter x An output layer Its size is the number of outputs A scalar for simple classification problems Can be a vector 8

21 Feed-forward neural network Input Layer Hidden Layer(s) Output Layer x 1 Each arrow denotes a connection A signal associated with a weight Each node is the weighted sum of its input followed by a nonlinear activation x 2 σ( i w i2 o 1 i) Connections go left to right 1 σ( i w i1 x i ) No connections within a layer No backward connections (recurrent) 9

22 Feed-forward neural network Input Layer Hidden Layer(s) Output Layer x 1 Each arrow denotes a connection A signal associated with a weight Each node is the weighted sum of its input followed by a nonlinear activation x 2 σ( i w i2 o 1 i) Connections go left to right 1 σ( i w i1 x i ) No connections within a layer No backward connections (recurrent) 9

23 Compute a prediction (forward pass) Input Layer Hidden Layer(s) Output Layer x 1 x 2 2. σ( i w i2 o 1 i) 1 1. σ( i w i1 x i )

24 Compute a prediction (forward pass) Input Layer Hidden Layer(s) Output Layer x 1 x 2 2. σ( i w i2 o 1 i) 1 1. σ( i w i1 x i )

25 Fitting a neural network How do we estimate the model s parameters? No-closed form solution Gradient-based optimization Threshold functions are not differentiable Replace by sigmoid (inverse logit). A soft threshold. 11

26 Fitting a neural network How do we estimate the model s parameters? No-closed form solution Gradient-based optimization Threshold functions are not differentiable Replace by sigmoid (inverse logit). A soft threshold. sigmoid(a) := ( 1 1 exp( a) ) logit(a) := log ( a 1 a ) 11

27 Fit the parameters (w) (backward pass) Input Layer Hidden Layer(s) Output Layer x 1 x 2 (y ŷ) 2 (y ŷ) 2 w j = (y f( i w io i )) 2 w j = (y f( i w i f( j w jx j ) i ) 2 w j 1 12

28 Fit the parameters (w) (backward pass) Input Layer Hidden Layer(s) Output Layer Derive a gradient wrt the parameters (w) x 1 x 2 (y ŷ) 2 (y ŷ) 2 w j = (y f( i w io i )) 2 w j = (y f( i w i f( j w jx j ) i ) 2 w j 1 12

29 Fit the parameters (w) (backward pass) Input Layer Hidden Layer(s) Output Layer Derive a gradient wrt the parameters (w) x 1 x 2 (y ŷ) 2 (y ŷ) 2 w j = (y f( i w io i )) 2 w j = (y f( i w i f( j w jx j ) i ) 2 w j 1 In practice the order of the computation is important. 12

30 Fit the parameters (w) (backward pass) Input Layer Hidden Layer(s) Output Layer Derive a gradient wrt the parameters (w) x 1 x 2 (y ŷ) 2 (y ŷ) 2 w j = (y f( i w io i )) 2 w j = (y f( i w i f( j w jx j ) i ) 2 w j 1 In practice the order of the computation is important. For MLP you back-propagate the gradient starting from the output node(s) and heading toward the input. 12

31 Gradient descent No closed-form formula Repeat the following steps (until convergence): 1. Calculate a gradient 2. Apply the update w t ij w t+1 ij = w t ij α wt ij Stochastic gradient descent One example at a time Batch gradient descent All examples at a time 13

32 What can an MLP learn? 1. A single unit (neuron) Linear classifier + non-linear output 2. A network with a single hidden layer Any continuous function (but may require exponentially many hidden units as a function of the inputs) 3. A network with two (or more) hidden layers Any function can be approximated with arbitrary accuracy. 14

33 Hyper-parameters & Other practical aspects

34 Activation functions Non-linear functions that transform the weighted sum of the inputs σ( i w i1 x i ) Rectified Linear units (Relu) Can shutoff units Efficient gradients Faster training (empirical)

35 Output units Easy to model different types/distributions of output variables Binary (Bernoulli) Sigmoid unit σ(a) = 1 1 +exp( a) Categorical (Multinoulli/Categorical) Softmax units Continuous (Gaussian) softmax(a) i = exp(a i) j exp(a j) Identity identity(a) =a 17

36 Regularization Weight decay on the parameters L2 penalty on the parameters Early stopping of the optimization procedure Monitor the validation loss and terminate when it stops improving Number of hidden layers and hidden units per layer 18

37 Momentum w t+1 ij = w t ij (α wt ij + β wt 1 ij ) Pro: Can allow you to jump over small local optima Pro: Goes faster through flat areas by using acquired speed Con: Can also jump over global optima Con: One more hyper-parameter to tune More advanced adaptive steps: adagrad, adam 19

38 Where do NNs shine Input is high-dimensional discrete or real-valued Output is discrete or real valued, or a vector of values Possibly noisy data Form of target function is unknown Human interpretability is not important The computation of the output based on the input has to be fast 20

39 Neural Networks takeaways (Highly) non-linear models Can learn to order/rank inputs easily Scale to very large datasets Very flexible models Composed of simple units (neurons) Adapt to different types of data May require fiddling with model architecture + optimization hyperparameters Standardizing data can be very important 21

40 How do we solve more complex tasks? Complex task: Rich input that maps to rich outputs Perception (e.g., visual recognition, speech understanding) Related tasks: Language understanding, question/ answering, dialogue 22

41 Neural networks Good properties for such tasks Flexible Can be combined Specific modules for specific subtasks that connect to others Can be fitted to large datasets Community bias 23

42 Deep neural networks Several layers of hidden nodes Parameters at different levels of representation Very young field with little theoretical understanding 24

43 [Figures from Yoshua Bengio]

44 Different architectures General neural networks with many layers Specific architectures for specific problems Images: Convolutional neural networks Text (varying input sizes): Recurrent neural networks 26

45 One example: recurrent neural networks (RNNs) Input Layer Hidden Layer(s) Output Layer x 1 x 2 1

46 One example: recurrent neural networks (RNNs) Input Layer Hidden Layer(s) Output Layer x 1 x 2 1

47 One example: recurrent neural networks (RNNs)

48 One example: recurrent neural networks (RNNs)

49 One example: recurrent neural networks (RNNs) The Brown Dog

50 One example: recurrent neural networks (RNNs) y t+1 The Brown Dog

51 One example: recurrent neural networks (RNNs) y t 1 y t y t+1 The Brown Dog

52 RNNs for machine translation [

53 RNNs takeaways (I) Can be used to learn from varying-length input Typically for discrete data (e.g., words) Can be used both for predictions and for representations 31

54 RNNs takeaways (II) Can easily be used as modules inside more complex systems Current applications: Natural language understanding (Q&A, Dialogs), machine translation Very active research field Still difficult to learn very long sequences (reading a book) Not available in scikit-learn 32

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