Logistic Regression: Sigmoid Decision Boundary

Model weights

Weight w₁ (feature x₁) 0.00

Weight w₂ (feature x₂) 0.00

Bias b 0.00

Live readouts

Accuracy

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Training steps

Cross-entropy loss

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p(x) = σ(0.00·x₁ + 0.00·x₂ + 0.00)

How it works

Every point is a 2-feature sample (x₁, x₂) with a true label y ∈ {0, 1} (blue = 0, red = 1). Logistic regression scores it with a linear combination and squashes the score into a probability with the sigmoid function:

z(x) = w₁x₁ + w₂x₂ + b
p(x) = σ(z) = 1 / (1 + e^(-z))

The rippled surface above the point cloud is exactly p(x) plotted as a height field over the (x₁, x₂) plane — low and blue where the model predicts class 0, high and red where it predicts class 1. The white contour on the floor is the decision boundary, the line z(x) = 0 where p = 0.5 exactly.

Quality is measured with the binary cross-entropy loss, averaged over all n points:

L = -(1/n) Σ [ yᵢ·log(pᵢ) + (1-yᵢ)·log(1-pᵢ) ]

Train step performs one real batch gradient-descent update — it computes the loss gradient with respect to each weight and moves against it:

∂L/∂w₁ = (1/n) Σ (pᵢ - yᵢ)·x₁ᵢ
∂L/∂w₂ = (1/n) Σ (pᵢ - yᵢ)·x₂ᵢ
∂L/∂b  = (1/n) Σ (pᵢ - yᵢ)
w ← w - η·∂L/∂w   (learning rate η = 0.5)

Drag the sliders yourself to feel why gradient descent is needed at all: a good boundary tilts the surface so blue points sit low and red points sit high — Auto-train repeats the step above until the loss stops falling.

Class 0 point Class 1 point Probability surface p(x)

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