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Logistic Regression

A linear score, squashed into a probability and trained on cross-entropy. Watch the boundary swing into place, read its weights as odds, and find the one thing it can never draw.

Builds onLoss Functions Linear Regression

Trained in your browser · seeded example points

Act 1 of 7 · Scoresstep 1 of 21

Dashed: the boundary z = 0. Arrow: w, pointing towards orange.

Speed

30 points with two features each, in two classes — blue (0) and orange (1). Logistic regression scores every point with one linear formula, z = w₁x₁ + w₂x₂ + b, and calls z = 0 the boundary. This first guess, w = (2.0, 2.2) and b = −0.2, is a bad one on purpose.

p = \sigma(2.00\,x_1 + 2.20\,x_2 - 0.20), \quad L = 1.1724, \quad \text{accuracy} = 40\%

The whole model

Score — z = w·x + b, a linear function of the features.
Probability — p = σ(z); the boundary is z = 0, where p = 0.5.
Loss — mean −ln P(true class), the binary cross-entropy.
Gradient — mean (p − y)·x: each point pushes by how wrong it is.

It is a neural network with no hidden layer: one neuron, sigmoid output.

A common misconception

“The sigmoid is a curve, so logistic regression can draw curved boundaries.”

The sigmoid bends the probabilities, never the boundary. This page trained it for 2000 steps on 200 points in two rings: 58% accuracy, where always guessing the bigger class scores 50%. On 200 points split by a line, the same code reaches 92%.

Check yourself

Question 1 of 3

A logistic-regression model gives a point the score z = 0. What probability does it assign to class 1?

Logistic Regression

A linear score, squashed into a probability and trained on cross-entropy. Watch the boundary swing into place, read its weights as odds, and find the one thing it can never draw.

Builds onLoss Functions Linear Regression

Trained in your browser · seeded example points

Act 1 of 7 · Scoresstep 1 of 21

Dashed: the boundary z = 0. Arrow: w, pointing towards orange.

Speed

p = \sigma(2.00\,x_1 + 2.20\,x_2 - 0.20), \quad L = 1.1724, \quad \text{accuracy} = 40\%

The whole model

Score — z = w·x + b, a linear function of the features.
Probability — p = σ(z); the boundary is z = 0, where p = 0.5.
Loss — mean −ln P(true class), the binary cross-entropy.
Gradient — mean (p − y)·x: each point pushes by how wrong it is.

It is a neural network with no hidden layer: one neuron, sigmoid output.

A common misconception

“The sigmoid is a curve, so logistic regression can draw curved boundaries.”

Check yourself

Question 1 of 3

A logistic-regression model gives a point the score z = 0. What probability does it assign to class 1?

DSAverse

Logistic Regression

See also

Logistic Regression

See also