Deep Learning · TEE · section

R23 Q5b

Re examination January 2023 fully solved

Question

Question (10 marks). Justify multiple ANN layers. Design a fraudulent/non-fraudulent transaction classifier for 10,000 samples with 4 features; choose layers/units and count parameters.

Answer — step by step

Given

10,000 samples; 4 features per sample; two target classes: fraudulent/non-fraudulent.

Hidden layer sizes are design choices, not fixed in the question.

To find

Explain multiple layers, propose an appropriate classifier and count parameters for that explicit design.

Formula — which one and why

Dense-layer parameters. Every output neuron receives every input, so multiply inputs by outputs for weights. Add one bias per receiving neuron. Sum separate layers.

n_in = incoming features; n_out = receiving neurons. The input layer itself has no learned receiving weights or biases.

$P=(n_{\mathrm{in}}+1)n_{\mathrm{out}}$ LaTeX: P=(n_{\mathrm{in}}+1)n_{\mathrm{out}}

Yaad rakhne ka cue: Har output ke liye saare inputs plus ek bias.

Solution — step by step

A single perceptron forms a weighted sum and applies a threshold. Its decision boundary is linear: w·x+b = 0. It can learn linearly separable patterns such as AND and OR under suitable learning conditions, but cannot represent XOR with one linear separator.

Opposite corners for XOR cannot be split by one straight line Tap the diagram to enlarge.

For binary AND, w₁ = w₂ = 1 and b = −1.5 give preactivations −1.5, −0.5, −0.5 and 0.5 for 00, 10, 01 and 11. A step that returns 1 at nonnegative inputs produces 0,0,0,1.

For XOR, use h₁ = step(x₁+x₂−0.5) for OR and h₂ = step(x₁+x₂−1.5) for AND. Then XOR = step(h₁−2h₂−0.5). For inputs 00,01,10,11, the hidden pairs are (0,0), (1,0), (1,0), (1,1), producing 0,1,1,0. XNOR = step(−h₁+2h₂+0.5), producing 1,0,0,1. This is a multilayer construction, not a claim that a single perceptron solves XOR.

A multilayer feedforward network adds hidden nonlinear features before the final boundary. Without hidden nonlinearities, adding linear layers still produces one affine map.

Chosen design: 4 → 8 → 4 → 1 with ReLU hidden layers, one sigmoid output and binary cross-entropy. Hidden widths are explicit design assumptions.

Architecture: 4 → 8 → 4 → 1. Each non-input neuron has one bias.

Layer 4 → 8

Step 1: count connections.

$4(8)=32$ LaTeX: 4(8)=32

Step 2: count receiving-neuron biases.

There are 8 receiving neurons, so 8 biases.

Step 3: add weights and biases.

$32+8=40$ LaTeX: 32+8=40

Layer 8 → 4

Step 1: count connections.

$8(4)=32$ LaTeX: 8(4)=32

Step 2: count receiving-neuron biases.

There are 4 receiving neurons, so 4 biases.

Step 3: add weights and biases.

$32+4=36$ LaTeX: 32+4=36

Layer 4 → 1

Step 1: count connections.

$4(1)=4$ LaTeX: 4(1)=4

Step 2: count receiving-neuron biases.

There are 1 receiving neurons, so 1 biases.

Step 3: add weights and biases.

$4+1=5$ LaTeX: 4+1=5

Total = 81. Input values and activations are not trainable parameters.

Use stratified train/validation/test partitions; handle class imbalance using appropriate training weights and evaluate precision/recall or a suitable decision threshold alongside accuracy. More layers are useful only if they learn needed nonlinear structure; the problem statement does not prescribe one uniquely correct architecture.

Verify input (0,0)

Step 1: calculate hidden OR preactivation.

$z_1=0+0-0.5=-0.5$ LaTeX: z_1=0+0-0.5=-0.5

Step 2: apply its threshold.

$h_1=0$ LaTeX: h_1=0

Step 3: calculate hidden AND preactivation.

$z_2=0+0-1.5=-1.5$ LaTeX: z_2=0+0-1.5=-1.5

Step 4: apply its threshold.

$h_2=0$ LaTeX: h_2=0

Step 5: calculate XOR output preactivation.

Step 6: apply the XOR threshold.

$y_{XOR}=0$ LaTeX: y_{XOR}=0

Step 7: calculate XNOR output preactivation.

Step 8: apply the XNOR threshold.

$y_{XNOR}=1$ LaTeX: y_{XNOR}=1

Verify input (0,1)

Step 1: calculate hidden OR preactivation.

$z_1=0+1-0.5=0.5$ LaTeX: z_1=0+1-0.5=0.5

Step 2: apply its threshold.

$h_1=1$ LaTeX: h_1=1

Step 3: calculate hidden AND preactivation.

$z_2=0+1-1.5=-0.5$ LaTeX: z_2=0+1-1.5=-0.5

Step 4: apply its threshold.

$h_2=0$ LaTeX: h_2=0

Step 5: calculate XOR output preactivation.

Step 6: apply the XOR threshold.

$y_{XOR}=1$ LaTeX: y_{XOR}=1

Step 7: calculate XNOR output preactivation.

Step 8: apply the XNOR threshold.

$y_{XNOR}=0$ LaTeX: y_{XNOR}=0

Verify input (1,0)

Step 1: calculate hidden OR preactivation.

$z_1=1+0-0.5=0.5$ LaTeX: z_1=1+0-0.5=0.5

Step 2: apply its threshold.

$h_1=1$ LaTeX: h_1=1

Step 3: calculate hidden AND preactivation.

$z_2=1+0-1.5=-0.5$ LaTeX: z_2=1+0-1.5=-0.5

Step 4: apply its threshold.

$h_2=0$ LaTeX: h_2=0

Step 5: calculate XOR output preactivation.

Step 6: apply the XOR threshold.

$y_{XOR}=1$ LaTeX: y_{XOR}=1

Step 7: calculate XNOR output preactivation.

Step 8: apply the XNOR threshold.

$y_{XNOR}=0$ LaTeX: y_{XNOR}=0

Verify input (1,1)

Step 1: calculate hidden OR preactivation.

$z_1=1+1-0.5=1.5$ LaTeX: z_1=1+1-0.5=1.5

Step 2: apply its threshold.

$h_1=1$ LaTeX: h_1=1

Step 3: calculate hidden AND preactivation.

$z_2=1+1-1.5=0.5$ LaTeX: z_2=1+1-1.5=0.5

Step 4: apply its threshold.

$h_2=1$ LaTeX: h_2=1

Step 5: calculate XOR output preactivation.

Step 6: apply the XOR threshold.

$y_{XOR}=0$ LaTeX: y_{XOR}=0

Step 7: calculate XNOR output preactivation.

Step 8: apply the XNOR threshold.

$y_{XNOR}=1$ LaTeX: y_{XNOR}=1

Final answer: XOR = [0,1,1,0]; XNOR = [1,0,0,1] for 00,01,10,11. The hidden nonlinear units make this possible.