ClassificationHard15 min50 pts

XOR Boundary

An XOR label over two features: no linear boundary beats 0.75 accuracy. Reach 0.90 on held-out rows with an engineered interaction.

Target

Accuracy ↑ 0.900

Baseline

0.750

Time

15 min

Points

Higher is better. Scored on 20 held-out rows.

Dataset

class 0
class 1

Each dot is one training row: feature 1 against feature 2, coloured by class.

Training split · 32 rows · 2 features. Scored by accuracy on 20 held-out rows; target ≥ 0.9, baseline 0.75.

What this lab teaches

XOR is the canonical proof that linear models need a feature: the product x1*x2 turns a non-separable label into a separable one.

The four-quadrant pattern

The 32 training rows arrive in mirrored quadruples at (+-x1, +-x2), with label 1 when the signs agree and label 0 when they differ - 16 ones and 16 zeros. The 20 hidden rows follow the same construction. Because the two positive regions sit on opposite diagonals, no straight boundary can isolate both without also catching a negative region.

Why 0.75 is the linear ceiling

Any line through the plane slices it into two half-planes; at best it captures three of the four quadrants, so a linear model tops out near 0.75 accuracy, which is the recorded baseline. The starter's constant majority vote (rate 0.5 is not greater than 0.5, so it predicts 0) scores 10/20 = 0.50 on the hidden rows.

The product feature

z = x1*x2 is positive exactly in the two label-1 quadrants and negative in the two label-0 quadrants, so the rule z > 0 scores 1.00 on the hidden rows. Logistic regression on [x1, x2, x1*x2] discovers the same boundary; standardize the rows after building the three columns so the product is on a comparable scale.

Common pitfalls

Adding x1-squared and x2-squared but not the interaction. Squares are identical across mirrored quadrants, so XOR stays invisible to the model.
Standardizing the raw features and then multiplying the standardized values without refitting the scale of the product column. The engineered feature's magnitude decides how much it can move the boundary.
Stopping at 0.75 because it beats chance. That is exactly the linear ceiling and the baseline; the target is 0.90 and only the interaction gets there.

Rules of the run

Define predict(train_X, train_y, test_X) and return exactly one value per test row.
Pure Python or numpy both run; no external data or network access.
Return probabilities or 0/1 labels; values are thresholded at 0.5.
The two raw features alone cannot reach the target.

Hint

Add x1 times x2 as a third feature (x1-squared and x2-squared can help too), standardize, and fit logistic regression.

Reference solution

Revealing doesn't affect your record — but try the hint first.

Timed run

15 min limit

50 points

not started · 15:00

The clock starts when you press start. Your predict(train_X, train_y, test_X) is then scored on 20 held-out rows within 15 minutes.

Starter codePython

def predict(train_X, train_y, test_X):
    # Baseline: predict the majority class.
    # Replace with an interaction model, then return one value per test row.
    rate = sum(train_y) / len(train_y)
    label = 1 if rate > 0.5 else 0
    return [label for _ in test_X]

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