🧭 Business Decision Classifier — Logistic Regression Live

Because logistic regression's decision boundary — the set of points where p = 0.5 — is always exactly a straight line (w1·x + w2·y + b = 0), or a hyperplane in higher dimensions. When the two outcome classes genuinely overlap (as in the "Overlapping" preset) or the true separating pattern is curved or non-linear, no choice of w1, w2, b can push the loss to zero. Training still converges — it finds the single straight line that minimises total cross-entropy — it just can't achieve perfect accuracy on data that isn't linearly separable.

The learning rate η scales every gradient step: w ← w – η·∂L/∂w. Because the cross-entropy loss surface for logistic regression is a smooth convex bowl, too large an η causes updates to overshoot the minimum, making the loss curve oscillate or diverge instead of settling. Too small an η makes each step barely move the weights, so training crawls toward convergence and wastes far more steps than necessary — though, unlike in deep networks, it will eventually get there since there is only one minimum to find.

For each training example, the error term is e = p – y (predicted probability minus true label). The gradients are ∂L/∂w1 = e·x, ∂L/∂w2 = e·y, and ∂L/∂b = e, averaged across the mini-batch. These fall directly out of applying the chain rule through the log-loss and the sigmoid — the same mechanics as backpropagation, just for a network with no hidden layers. Each update step subtracts η times these averaged gradients from w1, w2 and b.

Every pixel of the plot corresponds to a hypothetical (engagement score, support tickets) pair. A forward pass computes the model's predicted probability p at that point; the pixel is tinted red if p < 0.5 (predicted high risk) or blue if p ≥ 0.5 (predicted low risk), with opacity scaled to |p – 0.5| × 2 so confidently-classified regions look vivid and the area near the boundary fades out. The solid line traces exactly where p = 0.5 — the linear decision boundary itself.

"Separable" places the two clusters far apart with low variance, so a line can achieve near-100% accuracy — useful for watching clean, fast convergence. "Overlapping" moves the clusters closer with more spread, so some points will always sit on the wrong side of any straight line — showing the loss plateauing above zero, which is expected and correct behaviour, not a bug. "Noisy" keeps the separable layout but randomly flips roughly 12% of labels, mimicking real-world label noise and showing how a few mislabelled points can permanently limit achievable accuracy.