Deep Learning · TEE · section

R26 Q3a

Re examination February 2026 fully solved

Question

Question (10 marks). Train a perceptron for one epoch, η = 1, initial weights and bias 0, threshold 0. Samples: [−1,1,−1] → −1; [1,1,1] → 1; [−1,−1,1] → −1; [−1,1,1] → −1. Discuss limitations with examples.

Answer — step by step

Given

Train a perceptron for one epoch, η = 1, initial weights and bias 0, threshold 0.

Samples: [−1,1,−1] → −1.

[1,1,1] → 1.

[−1,−1,1] → −1.

[−1,1,1] → −1.

To find

Process samples in order for the stated epochs and report the final weights and bias.

Formula — which one and why

Target-based perceptron correction. This worked convention changes parameters only on a wrong prediction. Use the target itself as the signed multiplier; it differs from the binary error rule.

t is the bipolar target; η is the learning rate. Output is +1 above zero, −1 below zero and 0 at exactly zero.

\Delta w=\eta t x,\quad\Delta b=\eta t\quad\text{on a mismatch}LaTeX: \Delta w=\eta t x,\quad\Delta b=\eta t\quad\text{on a mismatch}

Yaad rakhne ka cue: Is convention mein galat hua toh target × input se correction.

Solution — step by step

State the threshold rule. Use output +1 if z>0, −1 if z<0, and 0 at z = 0. On a mismatch use Δw = η tx and Δb = ηt; on a match make no update. η = 1. A different rule at zero can change the learning trajectory. Write your convention explicitly.

Epoch 1, sample 1

Step 1: copy this sample and target.

x = [-1, 1, -1], target = -1.

Step 2: calculate the current net.

z = (0 × -1) + (0 × 1) + (0 × -1) + (0) = 0

Step 3: apply the threshold.

output = 0.

Step 4: calculate the update multiplier.

Update multiplier = -1

Step 5: calculate all weight changes.

Δw = -1 × [-1, 1, -1] = [1, -1, 1]

Step 6: calculate the bias change.

Δb = -1.

Step 7: carry the updated parameters forward.

New w = [1, -1, 1], new b = -1.

Epoch 1, sample 2

Step 1: copy this sample and target.

x = [1, 1, 1], target = 1.

Step 2: calculate the current net.

z = (1 × 1) + (-1 × 1) + (1 × 1) + (-1) = 0

Step 3: apply the threshold.

output = 0.

Step 4: calculate the update multiplier.

Update multiplier = 1

Step 5: calculate all weight changes.

Δw = 1 × [1, 1, 1] = [1, 1, 1]

Step 6: calculate the bias change.

Δb = 1.

Step 7: carry the updated parameters forward.

New w = [2, 0, 2], new b = 0.

Epoch 1, sample 3

Step 1: copy this sample and target.

x = [-1, -1, 1], target = -1.

Step 2: calculate the current net.

z = (2 × -1) + (0 × -1) + (2 × 1) + (0) = 0

Step 3: apply the threshold.

output = 0.

Step 4: calculate the update multiplier.

Update multiplier = -1

Step 5: calculate all weight changes.

Δw = -1 × [-1, -1, 1] = [1, 1, -1]

Step 6: calculate the bias change.

Δb = -1.

Step 7: carry the updated parameters forward.

New w = [3, 1, 1], new b = -1.

Epoch 1, sample 4

Step 1: copy this sample and target.

x = [-1, 1, 1], target = -1.

Step 2: calculate the current net.

z = (3 × -1) + (1 × 1) + (1 × 1) + (-1) = -2

Step 3: apply the threshold.

output = -1.

Step 4: calculate the update multiplier.

Update multiplier = 0

Step 5: calculate all weight changes.

Δw = 0 × [-1, 1, 1] = [0, 0, 0]

Step 6: calculate the bias change.

Δb = 0.

Step 7: carry the updated parameters forward.

New w = [3, 1, 1], new b = -1.

Final after exactly the requested epochs: w = [3, 1, 1], b = -1. A finite number of requested epochs does not itself prove convergence or correct classification of every sample.

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.

Key correction: recomputing each net with the latest weights gives net values 0,0,0,−2 and final w = [3,1,1], b = −1 under the stated bipolar rule. The supplied answer-key table contains inconsistent intermediate arithmetic.