# R24 Q4a

Subject: Deep Learning

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/#tee/read/14/tee-c14-s8

Canonical: https://deep-learning-study-room.vercel.app/read/dl/tee/section/tee-c14-s8/

## Question

**Question (10 marks).** Train a perceptron for 2 epochs with η = 1, initial weights and bias 0, threshold 0. Samples in order: [1,1,1,1] → 1; [−1,1,−1,−1] → 1; [1,1,1,−1] → −1; [1,−1,−1,1] → −1. Bias input is 1.

## Answer — step by step

### Given

Train a perceptron for 2 epochs with η = 1, initial weights and bias 0, threshold 0.

Samples in order: [1,1,1,1] → 1.

[−1,1,−1,−1] → 1.

[1,1,1,−1] → −1.

[1,−1,−1,1] → −1.

Bias input is 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.



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



**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, 1], target = 1.

**Step 2: calculate the current net.**

z = (0 × 1) + (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, 1, 1]

**Step 6: calculate the bias change.**

Δb = 1.

**Step 7: carry the updated parameters forward.**

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

### Epoch 1, sample 2

**Step 1: copy this sample and target.**

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

**Step 2: calculate the current net.**

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

**Step 3: apply the threshold.**

output = -1.

**Step 4: calculate the update multiplier.**

Update multiplier = 1

**Step 5: calculate all weight changes.**

Δw = 1 × [-1, 1, -1, -1] = [-1, 1, -1, -1]

**Step 6: calculate the bias change.**

Δb = 1.

**Step 7: carry the updated parameters forward.**

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

### Epoch 1, sample 3

**Step 1: copy this sample and target.**

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

**Step 2: calculate the current net.**

z = (0 × 1) + (2 × 1) + (0 × 1) + (0 × -1) + (2) = 4

**Step 3: apply the threshold.**

output = 1.

**Step 4: calculate the update multiplier.**

Update multiplier = -1

**Step 5: calculate all weight changes.**

Δw = -1 × [1, 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, 1], new b = 1.

### Epoch 1, sample 4

**Step 1: copy this sample and target.**

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

**Step 2: calculate the current net.**

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

**Step 3: apply the threshold.**

output = 1.

**Step 4: calculate the update multiplier.**

Update multiplier = -1

**Step 5: calculate all weight changes.**

Δw = -1 × [1, -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, 2, 0, 0], new b = 0.

### Epoch 2, sample 1

**Step 1: copy this sample and target.**

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

**Step 2: calculate the current net.**

z = (-2 × 1) + (2 × 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, 1, 1]

**Step 6: calculate the bias change.**

Δb = 1.

**Step 7: carry the updated parameters forward.**

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

### Epoch 2, sample 2

**Step 1: copy this sample and target.**

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

**Step 2: calculate the current net.**

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

**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, -1] = [0, 0, 0, 0]

**Step 6: calculate the bias change.**

Δb = 0.

**Step 7: carry the updated parameters forward.**

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

### Epoch 2, sample 3

**Step 1: copy this sample and target.**

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

**Step 2: calculate the current net.**

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

**Step 3: apply the threshold.**

output = 1.

**Step 4: calculate the update multiplier.**

Update multiplier = -1

**Step 5: calculate all weight changes.**

Δw = -1 × [1, 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, 2, 0, 2], new b = 0.

### Epoch 2, sample 4

**Step 1: copy this sample and target.**

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

**Step 2: calculate the current net.**

z = (-2 × 1) + (2 × -1) + (0 × -1) + (2 × 1) + (0) = -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, 1] = [0, 0, 0, 0]

**Step 6: calculate the bias change.**

Δb = 0.

**Step 7: carry the updated parameters forward.**

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

**Final after exactly the requested epochs:**  w = **[-2, 2, 0, 2]**, b = **0**. A finite number of requested epochs does not itself prove convergence or correct classification of every sample.