Deep Learning · TEE · section
R25 Q4b · Bipolar AND perceptron
February 2025 re-exam · complete solved paper
Question
Question. Implement AND with bipolar inputs/targets and learning rate1. (10 marks)
Answer — step by step
Given
Truth table: (−1,−1)→−1;(−1,+1)→−1;(+1,−1)→−1;(+1,+1)→+1. η=1.
To find
A trained separating perceptron.
Formula: which one and why
Initial weights/bias and zero-threshold tie rule are unspecified. Choose w=(0,0),b=0; output+1ifnet≥0,else−1. Use error update w←w+η(t−y)x,b←b+η(t−y). A sign-only misclassification update differs by a factor2 but can also learn the gate.
LaTeX: z=w^Tx+b,\quad w\leftarrow w+(t-y)x,\quad b\leftarrow b+(t-y)Step 1. Run all examples in the declared order
| Epoch | x | Target | Old parameters | Net | Output | t−y | New parameters |
|---|---|---|---|---|---|---|---|
| 1 | [-1, -1] | -1 | [0, 0]; b=0 | 0 | 1 | -2 | [2, 2]; b=-2 |
| 1 | [-1, 1] | -1 | [2, 2]; b=-2 | -2 | -1 | 0 | [2, 2]; b=-2 |
| 1 | [1, -1] | -1 | [2, 2]; b=-2 | -2 | -1 | 0 | [2, 2]; b=-2 |
| 1 | [1, 1] | 1 | [2, 2]; b=-2 | 2 | 1 | 0 | [2, 2]; b=-2 |
| 2 | [-1, -1] | -1 | [2, 2]; b=-2 | -6 | -1 | 0 | [2, 2]; b=-2 |
| 2 | [-1, 1] | -1 | [2, 2]; b=-2 | -2 | -1 | 0 | [2, 2]; b=-2 |
| 2 | [1, -1] | -1 | [2, 2]; b=-2 | -2 | -1 | 0 | [2, 2]; b=-2 |
| 2 | [1, 1] | 1 | [2, 2]; b=-2 | 2 | 1 | 0 | [2, 2]; b=-2 |
Step 2. Verify every truth-table row with final parameters
| x | Net | Output | Target |
|---|---|---|---|
| [-1, -1] | -6 | -1 | -1 |
| [-1, 1] | -2 | -1 | -1 |
| [1, -1] | -2 | -1 | -1 |
| [1, 1] | 2 | 1 | 1 |
Final answer
One valid learned solution: w=[2, 2], b=-2. Last full epoch makes no changes; all four rows are correct.
Source / convention note. Initialization, order and threshold convention explicitly supplied as assumptions; multiple equivalent separating weights exist.