DEEP LEARNING IN PYTHON. The need for optimization
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1 DEEP LEARNING IN PYTHON The need for optimization
2 A baseline neural network Input 2 Hidden Layer 5 2 Output Actual Value of Target: 3 Error: Actual - Predicted = 4
3 A baseline neural network Input 2 Hidden Layer 5 3 Output Actual Value of Target: 3 Error: Actual - Predicted = 0
4 Predictions with multiple points Making accurate predictions gets harder with more points At any set of weights, there are many values of the error corresponding to the many points we make predictions for
5 Loss function Aggregates errors in predictions from many data points into single number Measure of model s predictive performance
6 Squared error loss function Prediction Actual Error Squared Error Total Squared Error: 50 Mean Squared Error: 50
7 Loss function Loss function Weight Weight2
8 Loss function Lower loss function value means a better model Goal: Find the weights that give the lowest value for the loss function Gradient descent
9 Gradient descent Imagine you are in a pitch dark field Want to find the lowest point Feel the ground to see how it slopes Take a small step downhill Repeat until it is uphill in every direction
10 Gradient descent steps Start at random point Until you are somewhere flat: Find the slope Take a step downhill
11 Optimizing a model with a single weight Loss(w) Minimum value w
12 DEEP LEARNING IN PYTHON Let s practice!
13 DEEP LEARNING IN PYTHON Gradient descent
14 Gradient descent Loss(w) w
15 Gradient descent If the slope is positive: Going opposite the slope means moving to lower numbers Subtract the slope from the current value Too big a step might lead us astray Solution: learning rate Update each weight by subtracting learning rate * slope
16 Slope calculation example Actual Target Value = 0 To calculate the slope for a weight, need to multiply: Slope of the loss function w.r.t value at the node we feed into The value of the node that feeds into our weight Slope of the activation function w.r.t value we feed into
17 Slope calculation example Actual Target Value = 0 To calculate the slope for a weight, need to multiply: Slope of the loss function w.r.t value at the node we feed into The value of the node that feeds into our weight Slope of the activation function w.r.t value we feed into
18 Slope calculation example Actual Target Value = 0 Slope of mean-squared loss function w.r.t prediction: 2 * (Predicted Value - Actual Value) = 2 * Error 2 * -4
19 Slope calculation example Actual Target Value = 0 To calculate the slope for a weight, need to multiply: Slope of the loss function w.r.t value at the node we feed into The value of the node that feeds into our weight Slope of the activation function w.r.t value we feed into
20 Slope calculation example Actual Target Value = 0 To calculate the slope for a weight, need to multiply: Slope of the loss function w.r.t value at the node we feed into The value of the node that feeds into our weight Slope of the activation function w.r.t value we feed into
21 Slope calculation example Actual Target Value = 0 To calculate the slope for a weight, need to multiply: Slope of the loss function w.r.t value at the node we feed into The value of the node that feeds into our weight Slope of the activation function w.r.t value we feed into
22 Slope calculation example Actual Target Value = 0 2 * -4 * 3-24 If learning rate is 0.0, the new weight would be 2-0.0(-24) = 2.24
23 Network with two inputs affecting prediction 3 2 4
24 Code to calculate slopes and update weights In []: import numpy as np In [2]: weights = np.array([, 2]) In [3]: input_data = np.array([3, 4]) In [4]: target = 6 In [5]: learning_rate = 0.0 In [6]: preds = (weights * input_data).sum() In [7]: error = preds - target In [8]: print(error) 5
25 Code to calculate slopes and update weights In [9]: gradient = 2 * input_data * error In [0]: gradient Out[0]: array([30, 40]) In []: weights_updated = weights - learning_rate * gradient In [2]: preds_updated = (weights_updated * input_data).sum() In [3]: error_updated = preds_updated - target In [4]: print(error_updated) -2.5
26 DEEP LEARNING IN PYTHON Let s practice!
27 DEEP LEARNING IN PYTHON Backpropagation
28 Backpropagation Allows gradient descent to update all weights in neural network (by getting gradients for all weights) Comes from chain rule of calculus Important to understand the process, but you will generally use a library that implements this Prediction Error
29 Backpropagation process Trying to estimate the slope of the loss function w.r.t each weight Do forward propagation to calculate predictions and errors
30 Backpropagation process ReLU Activation Function Actual Target Value =
31 Backpropagation process ReLU Activation Function Actual Target Value = 0 Error =
32 Backpropagation process Go back one layer at a time Gradients for weight is product of:. Node value feeding into that weight 2. Slope of loss function w.r.t node it feeds into 3. Slope of activation function at the node it feeds into
33 ReLU Activation Function
34 Backpropagation process Need to also keep track of the slopes of the loss function w.r.t node values Slope of node values are the sum of the slopes for all weights that come out of them
35 DEEP LEARNING IN PYTHON Let s practice!
36 DEEP LEARNING IN PYTHON Backpropagation in practice
37 Backpropagation ReLU Activation Function Actual Target Value = 0 Error = Top weight s slope = * 6 Bottom weight s slope = 3 * 6
38 Backpropagation
39 Calculating slopes associated with any weight Gradients for weight is product of:. Node value feeding into that weight 2. Slope of activation function for the node being fed into 3. Slope of loss function w.r.t output node
40 Backpropagation Current Weight Value Gradient
41 Backpropagation: Recap Start at some random set of weights Use forward propagation to make a prediction Use backward propagation to calculate the slope of the loss function w.r.t each weight Multiply that slope by the learning rate, and subtract from the current weights Keep going with that cycle until we get to a flat part
42 Stochastic gradient descent It is common to calculate slopes on only a subset of the data ( batch ) Use a different batch of data to calculate the next update Start over from the beginning once all data is used Each time through the training data is called an epoch When slopes are calculated on one batch at a time: stochastic gradient descent
43 DEEP LEARNING IN PYTHON Let s practice
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