Deep Learning & Artificial Intelligence WS 2018/2019
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1 Deep Learning & Artificial Intelligence WS 2018/2019
2 Linear Regression
3 Model
4 Model
5 Error Function: Squared Error Has no special meaning except it makes gradients look nicer Prediction Ground truth / target
6 Objective Function with a single example with a set of examples
7 Objective Function Solution
8 Closed Form Solution
9 Closed Form Solution Fast to compute Only exists for some models and error functions Must be determined manually
10 Gradient Descent
11 Gradient Descent 1. Initialize at random 2. Compute error 3. Compute gradients w.r.t. parameters 4. Apply the above update rule 5. Go back to 2. and repeat until error does not decrease anymore
12 Computing Gradients
13 Computing Gradients Kronecker delta
14 Computing Gradients
15 Computing Gradients
16 Gradient Descent (Result) 1. Initialize at random 2. Compute error 3. Compute gradients w.r.t. parameters 4. Apply the above update rule 5. Go back to 2. and repeat until error does not decrease anymore
17 Probabilistic Interpretation Error term that captures unmodeled effects or random noise
18 Probabilistic Interpretation Error term that captures unmodeled effects or random noise
19 Probabilistic Interpretation Error term that captures unmodeled effects or random noise
20 Likelihood
21 Maximum Likelihood
22 Log-Likelihood
23 Maximum Log-Likelihood
24 Neural Networks & Backpropagation
25 Error Function Prediction Ground truth / target
26 Simple Fully-Connected Neural Network
27 Objective Function with a single example with a set of examples
28 Gradients: Towards Backpropagation
29 Gradients: Towards Backpropagation Number of neurons of the layer (excluding bias 1 )
30 Gradients: Towards Backpropagation
31 Gradients: Towards Backpropagation Can you do it for on your own?
32 Gradients: Towards Backpropagation
33 Gradients: Towards Backpropagation
34 Gradients: Towards Backpropagation
35 Gradients: Towards Backpropagation
36 Gradients: Towards Backpropagation
37 Gradients: Towards Backpropagation Can you do it for on your own?
38 Backpropagation Delta messages
39 Activation Functions & Vanishing Gradients
40 Common Activation Functions
41 Common Activation Functions Small or even tiny gradient
42 Vanishing Gradients Element-wise multiplication with small or even tiny gradients for each layer In a neural network with many layers, the gradients of the objective function w.r.t. the weights of a layer close to the inputs may become near zero! Gradient descent updates will starve
43 Weight Initialization
44 The Importance of Weight Initialization Simple CNN trained on MNIST for 12 epochs 10-batch rolling average of training loss Image Source:
45 The Importance of Weight Initialization Initialization with 0 values is ALWAYS WRONG! 0 here = everything is 0 = no error signal How to initialize properly?
46 Information Flow in a Neural Network Consider a network with... 5 hidden layers and 100 neurons per hidden layer the hidden layer activation function = identity function Let s omit the bias term for simplicity (commonly initialized with all 0 s).
47 Information Flow in a Neural Network Image Source:
48 Information Flow in a Neural Network What s the explanation for the previous image? One layer with some activation function and without the bias term:
49 Information Flow in a Neural Network
50 Information Flow in a Neural Network
51 Information Flow in a Neural Network (1) (2) (3) (1) tends to 0 when either (2) tends to 0 or (3) tends to 0. Preserve variance of activations throughout the network.
52 Information Flow in a Neural Network Variance approximation possible when pre-activation neurons are close to zero.
53 Variance Basic properties of variance for independent random variables with expected value = 0
54 Variance of Activations Random variables
55 Variance of Activations
56 Variance of Activations Variance preservation
57 Variance of Error Contribution
58 Variance of Error Contribution
59 Variance of Error Contribution assumption
60 Variance of Error Contribution Random variables
61 Variance of Error Contribution
62 Variance of Error Contribution Variance preservation
63 Glorot Initialization Glorot, X., & Bengio, Y. (2010). Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the 13th international conference on artificial intelligence and statistics (pp ).
64 Optimization Methods
65 Martens, J. (2010). Deep Learning via Hessian-Free Optimization. In Proceedings of the 27th International Conference on Machine Learning (pp ). Gradient Descent Too large learning rate zig-zag Too small learning rate starvation
66 Batch Gradient Descent Update based on the entire training data set Susceptible to converging to local minima Expensive and inefficient for large training data sets
67 Stochastic Gradient Descent (SGD) Update based on a single example More robust against local minima Noisy updates small learning rate
68 Mini-Batch Gradient Descent Update based on multiple examples More robust against local minima More stable than stochastic gradient descent Most common Often also called SGD despite multiple examples
69 Gradient Descent with Momentum Momentum dampens oscillations Gradient is computed before momentum is applied Typical momentum term:
70 Gradient Descent with Nesterov Momentum Gradient is computed after momentum is applied Anticipated update from momentum is used to include knowledge of momentum in the gradient Typically preferred over vanilla momentum
71 AdaGrad Adaptive (per-weight) learning rates Learning rates of frequently occurring features are reduced while learning rates of infrequent features remain large Monotonically decreasing learning rates Suited for sparse data Typical learning rate:
72 RMSProp Typical hyperparameters:
73 Adam Often used these days Typical hyperparameters:
74 Computation Graphs
75 Matrix-Vector Multiplication VECTOR float y SYMBOL TYPE data type symbolic variable MATMUL W MATRIX float x VECTOR float OPERATION
76 Indexing INDEXING A MATRIX float B i A i B MATRIX float VECTOR int
77 Graph Optimization SCALAR float z DIVIDE SCALAR float OPTIMIZATION x SCALAR float MULTIPLY x SCALAR float y SCALAR float
78 Automatic Differentiation SCALAR float y SCALAR float dy/dx SQUARE GRAD(y, x) MULTIPLY x SCALAR float 2 SCALAR float
79 Neural Network Layers VECTOR float z VECTOR float a TANH VECTOR float VECTOR float ADD z MATMUL LAYER OP DENSE W x b x MATRIX float VECTOR float VECTOR float VECTOR float
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