Grundlagen der Künstlichen Intelligenz
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1 Grundlagen der Künstlichen Intelligenz Neural networks Daniel Hennes (WS 2017/18) University Stuttgart - IPVS - Machine Learning & Robotics 1
2 Today Logistic regression Neural networks Perceptron Multi-layer perceptron Backprop Deep learning Chapter
3 Linear classification Regression binary classification Decision boundary: line or surface that separates the classes Linear decision boundary is called linear separator dataset is said to be linearly separable Hard threshold: threshold(z) = 1 if z 0 and 0 otherwise x x 1 3
4 Linear classification Loss L(θ) = n (y i f θ (x i )) 2 = i=1 n ( yi threshold(θ T x i ) ) 2 i=1 Optimal parameters: θ = arg min L(θ) θ δl(θ) δθ = 0 Gradient is zero almost everywhere in weight space except at points θ T x i = 0 where it is undefined... But a simple update rule exist: perceptron learning rule θ θ + α ( y i threshold(θ T x i ) ) x i 4
5 Logistic regression Hard threshold: loss not differentiable prediction 1 or 0 (no measure of confidence) Logistic threshold: logistic(z) = 1 1+e z Prediction: 1 f θ (x i ) = 1 + e θt x i forms a soft boundary in the input space 0.5 probability for any input at the center of the boundary approaching 0 or 1 as we move away from the boundary Gradient of loss can be computed using the chain rule x x
6 Artificial neural networks inspiration neurons of > 20 types, synapses, 1ms-10ms cycle time Signals are noisy spike trains of electrical potential 6
7 McCulloch Pitts unit Output is a squashed linear function of the inputs: ( ) a i g(in i ) = g W j,i a j j A gross oversimplification of real neurons, but its purpose is to develop understanding of what networks of simple units can do 7
8 Activation functions (a) 1 if x t else 0, usually t := 0 and bias (b) sign(x) (c) 1/(1 + e x ) Changing the bias weight W 0,i moves the threshold location 8
9 Implementing logical functions 9
10 Network structures Feed-forward networks: single-layer perceptrons multi-layer perceptrons Feed-forward networks implement functions, have no internal state Recurrent networks: Hopfield networks have symmetric weights (W i,j = W j,i) g(x) = sign(x), a i = ±1; holographic associative memory Boltzmann machines use stochastic activation functions, MCMC in Bayes nets recurrent neural nets have directed cycles with delays = have internal state (like flip-flops), can oscillate etc. 10
11 Feed-forward example Feed-forward network = a parameterized family of nonlinear functions: a 5 = g(w 3,5 a 3 + W 4,5 a 4 ) = g(w 3,5 g(w 1,3 a 1 + W 2,3 a 2 ) + W 4,5 g(w 1,4 a 1 + W 2,4 a 2 )) Adjusting weights changes the function: so we do learning this way! 11
12 Single-layer perceptrons Perceptron output x x 2 Output units all operate separately no shared weights Adjusting weights moves the location, orientation, and steepness of cliff 12
13 Expressiveness of perceptrons Rosenblatt built a physical device, the Memistor, and used to give dramatic talks holding up the Memistor and describing it as the key to the future of mankind etc. Consider a perceptron with g = step function Can represent AND, OR, NOT, majority, etc., but not XOR Represents a linear separator in input space: W j x j > 0 or W x > 0 j Minsky & Papert (1969) pricked the neural network balloon 13
14 Perceptron learning Learn by adjusting weights to reduce error on training set The squared error for an example with input x and true output y is Optimization by gradient descent: L = (y h W (x)) 2 L = 2(y h W (x)) (y h W (x)) W j W j Simple weight update rule: = 2(y h W (x)) g (W x) W j W x = 2(y h W (x)) g (W x) x j W j W j + α (y h W (x)) g (W x) x j e.g., positive error increase network output increase weights on positive inputs, decrease on negative inputs 14
15 Multilayer perceptrons Layers are usually fully connected; numbers of hidden units typically chosen by hand 15
16 Expressiveness of MLPs All continuous functions with 2 layers, all functions with 3 layers h W (x 1, x 2 ) x x 2 h W (x 1, x 2 ) x x 2 Combine two opposite-facing threshold functions to make a ridge Combine two perpendicular ridges to make a bump Add bumps of various sizes and locations to fit any surface 16
17 Back-propagation learning Output layer: same as for single-layer perceptron where k = (y k a k )g (in k ) W j,k W j,k + α a j k Hidden layer: back-propagate the error from the output layer: j = g (in j ) k W j,k k Update rule for weights in hidden layer: W i,j W i,j + α a i j 17
18 Back-propagation derivation Gradient for loss L k at k-th output L k = (y k a k ) 2 W j,k W j,k L k W j,k = 2(y k a k ) a k W j,k = 2(y k a k ) g(in k) W j,k = 2(y k a k )g (in k ) in k W j,k = 2(y k a k )g (in k ) = 2(y k a k )g (in k )a j = 2a j k W j,k ( j W j,ka j ) W j,k W j,k + α a j k 18
19 Back-propagation derivation L k W i,j = 2(y k a k ) a k W i,j = 2(y k a k ) g(in k) W i,j = 2(y k a k )g (in k ) in k W i,j = 2 k W i,j = 2 k W j,k a j W i,j = 2 k W j,k g(in j) W i,j = 2 k W j,k g (in j) inj W i,j = 2 k W j,k g (in j) = 2 k W j,k g (in j)a i W i,j ( i W i,ja i j = g (in j) W j,k k W i,j W i,j + α a i j k ) ( j W j,ka j ) 19
20 Back-propagation learning At each epoch, sum gradient updates for all examples and apply Training curve Total error on training set Number of epochs Typical problems: slow convergence, local minima MLPs are quite good for complex pattern recognition tasks, but resulting hypotheses cannot be understood easily 20
21 Handwritten digit recognition MNIST dataset 3-nearest-neighbor: 2.4% error unit MLP: = 1.6% error LeNet: unit MLP: 0.9% error Current best: 0.21% error 21
22 Deep learning Deep Learning Ian Goodfellow, Yoshua Bengio and Aaron Courville Following slides are heavily based on and include figures from the book. 22
23 Representation matters CHAPTER 1. INTRODUCTION Figure 1.1: Example of different representations: suppose we want to separate two categories of data by drawing a line between them in a scatterplot. In the plot on the left, we represent some data using Cartesian coordinates, and the task is impossible. In the plot on the right, we represent the data with polar coordinates and the task becomes simple23to
24 Deep learning & representation learning Representation learning Set of methods that allows a machine to be fed with raw data and to automatically discover the representation needed for detection or classification Deep learning Representation learning with multiple levels of representations, obtained by composing simple but nonlinear modules that each transform the representation at one level, into a higher, slightly more abstract one LeCun,
25 Output Output Output Mapping from features Output Mapping from features Mapping from features Additional layers of more abstract features Handdesigned program Handdesigned features Features Simple features Input Input Input Input Rule-based systems Classic machine learning Representation learning Deep learning Figure 1.5: Flowcharts showing how the different parts of an AI system relate to each 25
26 CAR PERSON ANIMAL Output (object identity) 3rd hidden layer (object parts) 2nd hidden layer (corners and contours) 1st hidden layer (edges) Visible layer (input pixels) 26
27 Computational Graphs Computational graphs Element Set + + Element Set Logistic Regression Logistic Regression w 1 x 1 w 2 x 2 w x Depth: longest path from Figure input to1.3 output computational step conceptual step Google s Tensorflow, Keras, Facebook s Caffe2, (Py)Torch (Goodfellow 2016) 27
28 Growing datasets (big data) gure 1.8: Dataset sizes have increased greatly over time. In the early 1900s, statistic died datasets using hundreds or thousands of manually compiled measurements (Ga 0; Gosset, 1908; Anderson, 1935; Fisher, 1936). In the 1950s through 1980s, the pion 28
29 Deeper networks igure 1.11: Since the introduction of hidden units, artificial neural networks have doubled Perceptron (1) GPU-accelerated CNN (11) size roughly every 2.4 years. Biological neural network sizes from Wikipedia ( 2015). MLP for speech recognition (6) GPU-accelerated MLP (15) LeNet-5 (8) Multi-GPU CNN (18) Deep belief network (10) GoogLeNet (20) 29
30 Deep vs. shallow networks Deeper networks tend to perform better Not just because they have more parameters Shallow networks overfit at 20 million Deeper models benefit from having over 60 million Suggest that deep models express a useful preference over the space of learnable functions (e.g. hierarchical representation) 30
31 Weight sharing: convolution 31
32 Solving object recognition Figure 1.12: Since deep networks reached the scale necessary to compete in the ImageNet Large ImageNet Scale Visual Large Recognition Scale Visual Challenge, Recognition they havechallenge consistently won the competition every year, > and 10 yielded million lower images and hand-annotated lower error rates (what each objects time. Data are from depicted?) Russakovsky et al. ( 2014b > ) 1and million He etimages al. ( 2015 incl. ). bounding boxes (where are the objects?) Deep networks have consistently won the competition since
33 Adversarial examples Adversarial examples are generated to trick the neural net Adding a small perturbation to the input imperceivable by the human eye yet profound effect on classification and confidence! 33
34 Adversarial examples 34
35 Summary Linear classification perceptron learning rule Perceptrons (one-layer networks) insufficiently expressive Multi-layer networks are sufficiently expressive; can be trained by gradient descent, i.e., error back-propagation Deep learning is representation learning with multiple layers of abstraction Deep learning success through: 1. more data 2. computational resources 3. weight sharing 4. easy to use frameworks 35
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