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ET May 25 FN

Correct Marks : 3 Max. Selectable Options : 0 (3) Question Label : Multiple Select Question Consider a feedforward neural network with one hidden layer trained using backpropagation for a binary classification task. The network has the following architecture:

• Input layer with 15 neurons

• Hidden layer with 25 neurons

• Output layer with 1 neuron

Assume that:

• The activation function for both the hidden and output layers is the sigmoid function, $\sigma (z)$$\sigma(z)$,

• The derivative of the sigmoid function is ${\sigma }^{\prime }(z)=\sigma (z)(1-\sigma (z))$$\sigma'(z) = \sigma(z)(1 - \sigma(z))$,

• The loss function used is binary cross-entropy (BCE),

• ${\hat{y}}^{(i)}$$\hat{y}^{(i)}$ is the output for the $i$$i$-th example, $y^{(i)}$$y^{(i)}$ is the true label, and $h_j^{(i)}$$h_j^{(i)}$ is the output of the $j$$j$-th hidden neuron for the $i$$i$-th example,

• There are no biases in the network.

Let $w_j$$w_j$ represent the weight connecting the $j$$j$th neuron of the hidden layer to the output neuron. Identify the correct weight update rule $\mathrm{\Delta }{w}_j$$\Delta w_j$ for each type of gradient descent: Vanilla Gradient Descent (GD), mini-Batch Gradient Descent (BGD), and Stochastic Gradient Descent (SGD). (Recall: with BCE + sigmoid, the gradient w.r.t. the pre-activation simplifies to ${\hat{y}}^{(i)}-y^{(i)}$$\hat{y}^{(i)} - y^{(i)}$.

Options :

6406534764456. ✔ GD: $\mathrm{\Delta }{w}_j=-\eta \cdot \frac{1}{N}\sum _{i=1}^N({\hat{y}}^{(i)}-y^{(i)})h_j^{(i)}$$\Delta w_j = -\eta \cdot \frac{1}{N} \sum_{i=1}^{N} (\hat{y}^{(i)} - y^{(i)}) h_j^{(i)}$

6406534764457. ✔ BGD: $\mathrm{\Delta }{w}_j=-\eta \cdot \frac{1}{B}\sum _{i=1}^B({\hat{y}}^{(i)}-y^{(i)})h_j^{(i)}$$\Delta w_j = -\eta \cdot \frac{1}{B} \sum_{i=1}^{B} (\hat{y}^{(i)} - y^{(i)}) h_j^{(i)}$

6406534764458. ✔ SGD: $\mathrm{\Delta }{w}_j=-\eta \cdot ({\hat{y}}^{(i)}-y^{(i)})h_j^{(i)}$$\Delta w_j = -\eta \cdot (\hat{y}^{(i)} - y^{(i)}) h_j^{(i)}$

6406534764459. ✖ GD: $\mathrm{\Delta }{w}_j=-\eta \cdot \frac{1}{N}\sum _{i=1}^N({\hat{y}}^{(i)}-y^{(i)}){\sigma }^{\prime }\left(\sum _{k=1}^{25}{w}_k{h}_k^{(i)}\right)h_j^{(i)}$$\Delta w_j = -\eta \cdot \frac{1}{N} \sum_{i=1}^{N} (\hat{y}^{(i)} - y^{(i)}) \sigma' \left( \sum_{k=1}^{25} w_k h_k^{(i)} \right) h_j^{(i)}$

6406534764460. ✖ BGD: $\mathrm{\Delta }{w}_j=-\eta \cdot \frac{1}{N}\sum _{i=1}^N({\hat{y}}^{(i)}-y^{(i)})h_j^{(i)}$$\Delta w_j = -\eta \cdot \frac{1}{N} \sum_{i=1}^{N} (\hat{y}^{(i)} - y^{(i)}) h_j^{(i)}$

6406534764461. ✖ SGD: $\mathrm{\Delta }{w}_j=-\eta \cdot \frac{1}{B}\sum _{i=1}^B({\hat{y}}^{(i)}-y^{(i)})h_j^{(i)}$$\Delta w_j = -\eta \cdot \frac{1}{B} \sum_{i=1}^{B} (\hat{y}^{(i)} - y^{(i)}) h_j^{(i)}$