CSE446: Neural Networks Winter 2015

Transcription

1 CSE446: Neural Networks Winter 2015 Luke Ze<lemoyer Slides adapted from Carlos Guestrin

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4 Human Neurons Switching Gme ~ second Number of neurons ConnecGons per neuron Scene recognigon Gme 0.1 seconds Number of cycles per scene recognigon? 100 à much parallel computagon!

5 Perceptron as a Neural Network g This is one neuron: Input edges x 1... x n, along with basis The sum is represented graphically Sum passed through an acgvagon funcgon g

6 1 0.9 Sigmoid Neuron g Just change g! Why would be want to do this? Notice new output range [0,1]. What was it before? Look familiar?

7 OpGmizing a neuron We train to minimize sum- squared error f(g(x)) = f (g(x))g (x) x l w i = j [y j g(w 0 + i w i x j i )] w i g(w 0 + i w i x j i ) w i g(w 0 + i w i x j i )=xj i w i g(w 0 + i w i x j i )=xj i g (w 0 + i w i x j i ) Solution just depends on g : derivative of activation function!

8 Re- deriving the perceptron update g For a specific, incorrect example: w = w + y*x (our familiar update!)

9 Sigmoid units: have to differengate g g (x) =g(x)(1 g(x))

10 Aside: Comparison to logisgc regression P(Y X) represented by: Learning rule MLE:

11 Perceptron, linear classificagon, Boolean funcgons: x i {0,1} Can learn x 1 x 2? x 1 + x 2 Can learn x 1 x 2? g x 1 + x 2 Can learn any conjuncgon or disjuncgon? x x n (n- 0.5) + x x n Can learn majority? (- 0.5*n) + x x n What are we missing? The dreaded XOR!, etc.

12 Going beyond linear classificagon Solving the XOR problem y = x 1 XOR x 2 = (x 1 x 2 ) (x 2 x 1 ) v 1 = (x 1 x 2 ) = x 1 - x 2 v 2 = (x 2 x 1 ) = x 2 - x 1 y = v 1 v 2 = v 1 +v 2 1 x 1 x v 1 v y

13 Hidden layer Single unit: 1- hidden layer: No longer convex funcgon!

14 Example data for NN with hidden layer Carlos Guestrin

15 Learned weights for hidden layer

16 Learning the weights

17 Learning an encoding

18 NN for images Carlos Guestrin

19 Weights in NN for images

20 1- hidden layer: Forward propagagon Compute values lel to right 1. Inputs: x 1,, x n 2. Hidden: v 1,, v n 3. Output: y 1 x 1 x 2 1 v 1 v 2 y

21 Gradient descent for 1- hidden layer v j k = g Dropped w 0 to make derivation simpler w i k x i i out(x) =g w k v j k k out(x) w k = v j k g k w k v j k Gradient for last layer same as the single node case, but with hidden nodes v as input!

22 Gradient descent for 1- hidden layer Dropped w 0 to make derivation simpler f(g(x)) = f (g(x))g (x) x out(x) w k i = g w k g( k i w k i x i ) w k i g wi k x i i For hidden layer, two parts: Normal update for single neuron Recursive computation of gradient on output layer

23 MulGlayer neural networks Inference and Learning: Forward pass: left to right, each hidden layer in turn Gradient computation: right to left, propagating gradient for each node Forward Gradient

24 Forward propagagon predicgon Recursive algorithm Start from input layer Output of node V k with parents U 1,U 2, :

25 Back- propagagon learning Just gradient descent!!! Recursive algorithm for compugng gradient For each example Perform forward propagagon Start from output layer Compute gradient of node V k with parents U 1,U 2, Update weight w i k Repeat (move to preceding layer)

26 Back- propagagon pseudocode IniGalize all weights to small random numbers UnGl convergence, do: For each training example x,y: 1. Forward propagagon, compute node values V k 2. For each output unit o (with labeled output y): δ o = V o (1- V o )(y- V o ) 3. For each hidden unit h: δ h = V h (1- V h ) Σ k in output(h) w h,k δ k 4. Update each network weight w i,j from node i to node j w i,j = w i,j + ηδ j x i,j

27 Convergence of backprop Perceptron leads to convex opgmizagon Gradient descent reaches global minima MulGlayer neural nets not convex Gradient descent gets stuck in local minima SelecGng number of hidden units and layers = fuzzy process NNs have made a HUGE comeback in the last few years!!! Neural nets are back with a new name!!!! Deep belief networks Huge error reducgon when trained with lots of data on GPUs

28 Overfiung in NNs Are NNs likely to overfit? Yes, they can represent arbitrary funcgons!!! Avoiding overfiung? More training data Fewer hidden nodes / be<er topology RegularizaGon Early stopping

29 Object RecogniGon Slides from Jeff Dean at Google

30 Number DetecGon Slides from Jeff Dean at Google

31 Acoustic Modeling for Speech Recognition label Close collaboration with Google Speech team Trained in <5 days on cluster of 800 machines 30% reduction in Word Error Rate for English! ( biggest single improvement in 20 years of speech research ) Launched in 2012 at time of Jellybean release of Android Slides from Jeff Dean at Google

32 2012-era Convolutional Model for Object Recognition Softmax to predict object class Fully-connected layers Convolutional layers! (same weights used at all! spatial locations in layer)!! Convolutional networks developed by! Yann LeCun (NYU) Layer 7... Layer 1 Input Basic architecture developed by Krizhevsky, Sutskever & Hinton (all now at Google).! Won 2012 ImageNet challenge with 16.4% top-5 error rate Slides from Jeff Dean at Google

33 2014-era Model for Object Recognition Module with 6 separate! convolutional layers 24 layers deep! Developed by team of Google Researchers:! Won 2014 ImageNet challenge with 6.66% top-5 error rate Slides from Jeff Dean at Google

34 Good Fine-grained Classification hibiscus dahlia Slides from Jeff Dean at Google

35 Good Generalization Both recognized as a meal Slides from Jeff Dean at Google

36 Sensible Errors snake dog Slides from Jeff Dean at Google

37 Works in practice for real users. Slides from Jeff Dean at Google

38 Works in practice for real users. Slides from Jeff Dean at Google

39 What you need to know about neural Perceptron: networks RelaGonship to general neurons MulGlayer neural nets RepresentaGon DerivaGon of backprop Learning rule Overfiung