Lecture 11: − Multi-layer Perceptron − Forward Pass − Backpropagation Aykut Erdem November 2016 Hacettepe University
Administrative • Assignment 2 due Nov. 10, 2016! • Midterm exam on Monday, Nov. 14, 2016 − You are responsible from the beginning till the end of this week − You can prepare and bring a full-page copy sheet (A4-paper, both sides) to the exam. • Assignment 3 will be out soon! − It is due December 1, 2016 − You will implement a 2-layer Neural Network 2
3 Last time… Linear classification slide by Fei-Fei Li & Andrej Karpathy & Justin Johnson
4 Last time… Linear classification slide by Fei-Fei Li & Andrej Karpathy & Justin Johnson
5 Last time… Linear classification slide by Fei-Fei Li & Andrej Karpathy & Justin Johnson
Interactive web demo time…. slide by Fei-Fei Li & Andrej Karpathy & Justin Johnson http://vision.stanford.edu/teaching/cs231n/linear-classify-demo/ 6
Last time… Perceptron x n x 1 x 2 x 3 . . . w n w 1 synaptic weights output X slide by Alex Smola f ( x ) = w i x i = h w, x i i 7
This Week • Multi-layer perceptron • Forward Pass • Backward Pass 8
Introduction 9
A brief history of computers 1970s 1980s 1990s 2000s 2010s Data 10 5 10 8 10 2 10 3 10 11 RAM ? 1MB 100MB 10GB 1TB CPU ? 10MF 1GF 100GF 1PF GPU deep kernel deep • Data grows nets methods nets at higher exponent • Moore’s law (silicon) vs. Kryder’s law (disks) slide by Alex Smola • Early algorithms data bound, now CPU/RAM bound 10
Not linearly separable data • Some datasets are not linearly separable ! - e.g. XOR problem • Nonlinear separation is trivial slide by Alex Smola 11
Addressing non-linearly separable data • Two options: - Option 1: Non-linear functions - Option 2: Non-linear classifiers slide by Dhruv Batra 12
Option 1 — Non-linear features • Choose non-linear features, e.g., - Typical linear features: w 0 + Σ i w i x i - Example of non-linear features: • Degree 2 polynomials, w 0 + Σ i w i x i + Σ ij w ij x i x j • Classifier h w ( x ) still linear in parameters w - As easy to learn - Data is linearly separable in higher dimensional spaces - Express via kernels slide by Dhruv Batra 13
Option 2 — Non-linear classifiers • Choose a classifier h w ( x ) that is non-linear in parameters w , e.g., - Decision trees, neural networks,… • More general than linear classifiers • But, can often be harder to learn (non-convex optimization required) • Often very useful (outperforms linear classifiers) • In a way, both ideas are related slide by Dhruv Batra 14
Biological Neurons • Soma (CPU) Cell body - combines signals • Dendrite (input bus) Combines the inputs from several other nerve cells • Synapse (interface) Interface and parameter store between neurons • Axon (cable) May be up to 1m long and will transport the slide by Alex Smola activation signal to neurons at di ff erent locations 15
Recall: The Neuron Metaphor • Neurons - accept information from multiple inputs, - transmit information to other neurons. • Multiply inputs by weights along edges • Apply some function to the set of inputs at each node slide by Dhruv Batra 16
Types of Neuron X 1 z = θ 0 + x i θ i θ 1 θ 0 i 1 y = θ 2 f ( ~ x, ✓ ) 1 + e − z 1 X θ D y = θ 0 + x i θ i θ 1 θ 0 i Linear Neuron θ 2 f ( ~ x, ✓ ) 1 θ D θ 1 θ 0 Logistic Neuron θ 2 f ( ~ x, ✓ ) • Potentially more. Requires a convex X z = θ 0 + x i θ i θ D loss function for gradient descent slide by Dhruv Batra i training. ⇢ 1 if z ≥ 0 Perceptron y = 17 0 otherwise
Limitation • A single “neuron” is still a linear decision boundary • What to do? • Idea: Stack a bunch of them together! slide by Dhruv Batra 18
Nonlinearities via Layers • Cascade neurons together • The output from one layer is the input to the next • Each layer has its own sets of weights y 1 i = k ( x i , x ) Kernels y 1 i ( x ) = σ ( h w 1 i , x i ) y 2 ( x ) = σ ( h w 2 , y 1 i ) Deep Nets slide by Alex Smola optimize all weights 19
Nonlinearities via Layers y 1 i ( x ) = σ ( h w 1 i , x i ) y 2 i ( x ) = σ ( h w 2 i , y 1 i ) y 3 ( x ) = σ ( h w 3 , y 2 i ) slide by Alex Smola 20
Representational Power • Neural network with at least one hidden layer is a universal approximator (can represent any function). Proof in: Approximation by Superpositions of Sigmoidal Function, Cybenko, paper slide by Raquel Urtasun, Richard Zemel, Sanja Fidler • The capacity of the network increases with more hidden units and more hidden layers 21
A simple example • Consider a neural network 0 1 2 3 4 5 6 7 8 9 with two layers of neurons. - neurons in the top layer represent known shapes. - neurons in the bottom layer represent pixel intensities. • A pixel gets to vote if it has ink on it. - Each inked pixel can vote for several di ff erent shapes. 𝑦 � • The shape that gets the 𝑦 � ¡ ¡𝑔(∑𝑥 � 𝑦 � ) slide by Geoffrey Hinton 𝑦 � most votes wins. … … 𝑦 � 22 �
How to display the weights 1 2 3 4 5 6 7 8 9 0 The input image Give each output unit its own “map” of the input image and display the weight coming from each pixel in the location of that pixel in the map. Use a black or white blob with the area representing the slide by Geoffrey Hinton magnitude of the weight and the color representing the sign. 23
How to learn the weights 1 2 3 4 5 6 7 8 9 0 The image Show the network an image and increment the weights from active pixels to the correct class. Then decrement the weights from active pixels to whatever class the network guesses. slide by Geoffrey Hinton 24
1 2 3 4 5 6 7 8 9 0 The image slide by Geoffrey Hinton 25
1 2 3 4 5 6 7 8 9 0 The image slide by Geoffrey Hinton 26
1 2 3 4 5 6 7 8 9 0 The image slide by Geoffrey Hinton 27
1 2 3 4 5 6 7 8 9 0 The image slide by Geoffrey Hinton 28
1 2 3 4 5 6 7 8 9 0 The image slide by Geoffrey Hinton 29
The learned weights 1 2 3 4 5 6 7 8 9 0 The image The details of the learning algorithm will be explained later. slide by Geoffrey Hinton 30
Why insu ffi cient • A two layer network with a single winner in the top layer is equivalent to having a rigid template for each shape. - The winner is the template that has the biggest overlap with the ink. • The ways in which hand-written digits vary are much too complicated to be captured by simple template matches of whole shapes. - To capture all the allowable variations of a digit we need to learn the features that it is composed of. slide by Geoffrey Hinton 31
Multilayer Perceptron • Layer Representation y y i = W i x i W 4 x i +1 = σ ( y i ) x4 W 3 • (typically) iterate between linear mapping Wx and x3 nonlinear function W 2 • Loss function l ( y, y i ) x2 to measure quality of W 1 estimate so far x1 slide by Alex Smola 32
Forward Pass 33
Forward Pass: What does the Network Compute? • Output of the network can be written as: D X X h j ( x ) = f ( v j 0 + x i v ji ) i =1 J slide by Raquel Urtasun, Richard Zemel, Sanja Fidler X o k ( x ) = g ( w k 0 + h j ( x ) w kj ) j =1 (j indexing hidden units, k indexing the output units, D number of inputs) • Activation functions f , g : sigmoid/logistic, tanh, or rectified linear (ReLU) 1 + exp( − z ) , tanh ( z ) = exp( z ) − exp( − z ) 1 σ ( z ) = exp( z ) + exp( − z ) , ReLU ( z ) = max(0 , z ) 34
Forward Pass in Python • Example code for a forward pass for a 3-layer network in Python: slide by Raquel Urtasun, Richard Zemel, Sanja Fidler • Can be implemented e ffi ciently using matrix operations • Example above: W 1 is matrix of size 4 × 3, W 2 is 4 × 4. What about biases and W 3 ? 35 [http://cs231n.github.io/neural-networks-1/]
Special Case • What is a single layer (no hiddens) network with a sigmoid act. function? slide by Raquel Urtasun, Richard Zemel, Sanja Fidler • Network: 1 o k ( x ) = 1 + exp( − z k ) J X = w k 0 + z k x j w kj j =1 • Logistic regression! 36
Example • Classify image of handwritten digit (32x32 pixels): 4 vs non-4 !vs.!all)?! ust!all!the!parameters! ,!to! • How would you build your network? ut!this!is!a!complicated! • For example, use one hidden layer and the sigmoid activation function: slide by Raquel Urtasun, Richard Zemel, Sanja Fidler 1 o k ( x ) = 1 + exp( − z k ) J X = w k 0 + h j ( x ) w kj z k j =1 • How can we train the network, that is, adjust all the parameters w ? 37
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