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CSE 5160 Machine Learning (Spring 2021) Assignment #4 (Due on April 23, 2021)

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Given a neural network, the structure is shown below. Each neuron in the neural network uses the logistic or sigmoid function 𝑔(𝑧) = !

!” $!” as activation function. 𝑧%

[‘] is the output of the

linear part of 𝑗)* neuron in layer 𝑙; π‘Ž% [‘] is the output of the activation part of 𝑗)* neuron in layer 𝑙.

1. [20 points] (Forward propagation) Given a training example (οΏ½βƒ—οΏ½, 𝑦), π‘₯ ∈ ℝ+, what is the

output of the neural network 𝑦.?

2. [50 points] (Backpropagation) The loss function is defined by logistic loss function 𝐿(𝑦.,𝑦) =

βˆ’[π‘¦π‘™π‘œπ‘”π‘¦. + (1 βˆ’ 𝑦)π‘™π‘œπ‘”(1 βˆ’ 𝑦.)] . Please derive the partial derivatives of loss function with

respect to parameters in the stochastic gradient descent update rules, that is, derive ,- ,.[$]

and

,- ,/[$]

, 𝑙 = 1,2,3.

π‘₯! π‘₯0

π‘₯+

….

…. ….

….

𝑧! [!] π‘Ž!

[!]

𝑧0 [!] π‘Ž0

[!]

𝑧1 [!] π‘Ž1

[!]

𝑧! [0] π‘Ž!

[0]

𝑧0 [0] π‘Ž0

[0]

𝑧! [1] π‘Ž!

[1]

𝑦.

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