Machine Learning

Logistic Regression

Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood.

status: publishedimportance: criticaldifficulty 3/5math: undergraduateread: 17mlive demo

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Concept Structure

Logistic Regression

01Intuition

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02Math

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03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

3prerequisites

2next concepts

3related links

Learning map

Logistic Regression

BeforeLinear Regression & Least SquaresNow4/4 sections readyTryManipulate one control and predict the visible change.NextClassification Metrics, Thresholds, and Calibration

Object flow

4/4 sections readyAsk about this Research room

ConceptLogistic RegressionMachine Learning EquationLogistic Regression equation 1Exact equation object CodeLogistic Regression code witness 1Exact code witness DemoLogistic Regression interactive demoVisualization object ClaimBinary logistic regression maps a linear score through a sigmoid and...Exact claim check SourceCS229 Lecture Notes: Logistic RegressionExact source object

ConceptLogistic RegressionMachine Learning

2 sources attachedLocal snapshot ready

concept:machine-learning/logistic-regression

Codewitness nearby Predictbefore reveal Roomobject handoff

Conceptual Bridge

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Carry inLinear Regression & Least Squares

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Work hereLogistic Regression

Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood.

Carry outClassification Metrics, Thresholds, and Calibration

The next edge should feel earned: use the demo prediction here before following Classification Metrics, Thresholds, and Calibration.

Test the linkManipulate one control and predict the visible change.Then continue to Classification Metrics, Thresholds, and Calibration

01IntuitionStart with the picture, metaphor, or geometric mechanism.02MathMake the objects explicit and connect them with notation.03CodeMirror the equations with runnable implementation details.04Interactive DemoManipulate the mechanism and watch the idea respond.

Intuition

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Section prompt

You are here because many AI systems make categorical decisions from scores: a classifier scores labels, a language model scores tokens, and DPO scores preference pairs with a logistic loss. Logistic regression is the smallest honest version of that pattern.

Before this, know linear regression, maximum likelihood, and cross-entropy. By the end, you should be able to explain what a logit is, why the sigmoid creates a probability, and why the gradient for one example is $q-y$ .

Linear regression predicts a number on the whole real line. Classification needs a probability between $0$ and $1$ . Logistic regression keeps the linear score, then bends it through the sigmoid:

q=\sigma(z).

The score $z$ is the logit. It can be any real number. The probability $q$ is constrained to the interval $(0,1)$ .

The decision boundary is where the model is exactly unsure:

q=0.5.

For a one-dimensional model $z=wx+b$ , this happens when $wx+b=0$ . Changing $w$ changes the steepness and orientation. Changing $b$ shifts the boundary.

The most common misconception is that logistic regression is "linear regression with a squashed output." The output is squashed, but the training objective is different: it is Bernoulli likelihood, not squared error.

Another subtle caveat: the number $q$ is a model probability, not a promise that the fitted classifier is calibrated on future data. And with perfectly separable training data, unregularized logistic regression may keep increasing the weight norm instead of reaching a finite maximum-likelihood solution.

Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

Section prompt

Equation 1z=w^{\mathsf T}x+b,Equation 2q=P_\theta(Y=1\mid x)=\sigma(z)=\frac{1}{1+e^{-z}}.

For one example with features $x\in\mathbb R^d$ and binary label $y\in\{0,1\}$ , define the linear score

z=w^{\mathsf T}x+b,

where $w\in\mathbb R^d$ and $b\in\mathbb R$ . The predicted probability of label $1$ is

q=P_\theta(Y=1\mid x)=\sigma(z)=\frac{1}{1+e^{-z}}.

The Bernoulli likelihood for this example is

P_\theta(y\mid x)=q^y(1-q)^{1-y}.

The negative log-likelihood is

\ell(z,y)=-y\log q-(1-y)\log(1-q).

This is binary cross-entropy, also called Bernoulli negative log-likelihood. Differentiating this loss through the sigmoid gives the compact logit gradient:

\frac{\partial \ell}{\partial z}=q-y.

That single expression is the training signal. If $y=1$ and $q$ is too small, then $q-y<0$ , so gradient descent increases the logit. If $y=0$ and $q$ is too large, then $q-y>0$ , so gradient descent lowers the logit.

For a dataset with rows $x_i\in\mathbb R^d$ , labels $y_i\in\{0,1\}$ , and probabilities $q_i=\sigma(w^{\mathsf T}x_i+b)$ , the average loss is

L(w,b)=\frac1n\sum_{i=1}^n\left[-y_i\log q_i-(1-y_i)\log(1-q_i)\right].

The gradients are

\nabla_w L=\frac1n X^{\mathsf T}(q-y),

and

\frac{\partial L}{\partial b}=\frac1n\sum_{i=1}^n(q_i-y_i).

Here $X\in\mathbb R^{n\times d}$ , $q,y\in\mathbb R^n$ , and $w\in\mathbb R^d$ . The shape mirrors linear regression, but the residual-like term is probability error $q-y$ , not numeric target error $Xw-y$ .

Code

Keep the implementation aligned with the notation so the algorithm is legible.

Section prompt

Code witness 1import numpy as np # Binary logistic regression in one feature. # Shapes: X is (n, 2), y is (...python

import numpy as np

# Binary logistic regression in one feature.
# Shapes: X is (n, 2), y is (n,), w is (2,).
x = np.array([-3, -2, -1, 0, 1, 2, 3], dtype=float)
y = np.array([0, 0, 0, 0, 1, 1, 1], dtype=float)
X = np.column_stack([np.ones_like(x), x])
w = np.array([-0.2, 0.7])

def sigmoid(z):
    return 1 / (1 + np.exp(-z))

def loss_and_grad(w):
    z = X @ w
    q = sigmoid(z)
    eps = 1e-12
    loss = -np.mean(y * np.log(q + eps) + (1 - y) * np.log(1 - q + eps))
    grad = X.T @ (q - y) / y.size
    return loss, grad, q

for step in range(60):
    loss, grad, q = loss_and_grad(w)
    w = w - 0.5 * grad

loss, grad, q = loss_and_grad(w)
print("bias, slope:", np.round(w, 3))
print("probabilities:", np.round(q, 3))
print("loss:", round(loss, 4))
print("gradient:", np.round(grad, 6))
print("decision boundary x:", round(-w[0] / w[1], 3))

The code mirrors the math: z = X @ w creates logits, q = sigmoid(z) creates probabilities, and X.T @ (q - y) / n is the gradient.

Interactive Demo

Use direct manipulation to connect the explanation to a moving system.

Section prompt

Choose a point and inspect its current probability before revealing the logit-gradient direction. Predict whether training wants to raise that point's logit, lower it, or leave it mostly unchanged.

The curve shows $q=\sigma(wx+b)$ over one feature. Points above the midline are positive examples; points below are negative examples. The reveal marks the selected point's contribution $q-y$ and reports the average loss. Move the slope and bias to see how the same data can be underconfident, overconfident, or misclassified.

Live Concept Demo

Explore Logistic Regression

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difficulty 3/5undergraduatecode-aligned

Demo Prediction Checkpoint

Manipulate one control and predict the visible change.

Commit to what Logistic Regression should make visible before reading the result.

After The First Pass

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Once the invariant is visible in the intuition, math, code, and demo, use these panels to inspect the mechanism visually, check source support, practice the idea, and attach a grounded research question.

Mechanism Storyboard

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Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood.

Prediction open01 / Intuition

Prediction lens

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Commit first

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Visual Inquiry

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Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood.

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Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

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Commit first

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Source Grounding

Canonical references for the mechanism on this page.

course-notes · 2023CS229 Lecture Notes: Logistic RegressionStanford CS229

Curriculum source for logistic regression as a Bernoulli GLM trained by likelihood.

Open source

book · 2016Deep LearningGoodfellow, Bengio, and Courville

Curriculum source for logits, sigmoid/softmax classifiers, and cross-entropy training.

Open source

Claim Review

Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood.

Status1 substantive review recorded

Claims without a substantive review badge still need exact source-support review.

Sources2 references

cs229-logistic-regression, goodfellow-2016-deep-learning

Witnesses4 local objects

Use equation, code, and demo objects to check whether the source support is operational.

Substantively reviewedBinary logistic regression maps a linear score through a sigmoid and trains by Bernoulli negative log-likelihood, whose logit gradient is predicted probability minus label.Claim metadata: source checked

CS229 supports the sigmoid hypothesis, Bernoulli likelihood, and log-likelihood gradient; Goodfellow supports sigmoid as a Bernoulli-parameter map and logistic regression as likelihood-trained classification. Together they support the page's NLL/BCE and q-y gradient convention.

Sources: CS229 Lecture Notes: Logistic Regression, Deep LearningThis source check covers binary logistic regression, not multinomial softmax regression, empirical calibration guarantees, cost-sensitive thresholding, or regularized solutions under perfect separation.A bounded review summary is present; still check caveats and exact source scope.

Checked CS229 section 2.1 and Goodfellow chapters 3/5. CS229 gives sigmoid, Bernoulli likelihood, log-likelihood, and gradient ascent (y-h)x; negating gives the page's BCE/NLL q-y descent signal. Goodfellow supports sigmoid as a Bernoulli parameter and logistic regression as likelihood-trained classification.

Reviewer: codex+gpt-pro-prior; reviewed 2026-06-28

source-span-cs229-logistic-regression source-span-goodfellow-2016-deep-learning math-object-1 math-object-2 code-witness-1 interactive-demo

Source support candidates

course-notes 2023CS229 Lecture Notes: Logistic Regression

Curriculum source for logistic regression as a Bernoulli GLM trained by likelihood.

book 2016Deep Learning

Curriculum source for logits, sigmoid/softmax classifiers, and cross-entropy training.

Mechanism witnesses

Equation 1

z=w^{\mathsf T}x+b,

Equation 2

q=P_\theta(Y=1\mid x)=\sigma(z)=\frac{1}{1+e^{-z}}.

Code witness 1import numpy as np # Binary logistic regression in one feature. # Shapes: X is (n, 2), y is (...Demo stateLive mechanism probe

Practice Loop

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Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood.

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Logistic Regression.

Hint 1

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Hint 2

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Hint 3

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Your answer canvas

Local checks

Claim

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Mechanism

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Bridge

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Misconception check

Object research drawerClose

ConceptLogistic RegressionMachine Learning

Code witness comparisonLogistic Regression code witness 1x = np.array([-3, -2, -1, 0, 1, 2, 3], dtype=float)Prediction before revealLogistic Regression interactive demoManipulate one control and predict the visible change.

Grounded room questionWhat is the smallest example that makes Logistic Regression click without losing the math?Local snapshot ready

Research Room

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conceptMachine Learning

Logistic Regression

Anchored question

What is the smallest example that makes Logistic Regression click without losing the math?

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Evidence to inspect

Source ids to inspect: cs229-logistic-regression, goodfellow-2016-deep-learning
Definition, prerequisite, and contrast concept links
The equation or code witness that makes the concept operational
One demo state that shows the invariant instead of a slogan

What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Logistic Regression Object key: concept:machine-learning/logistic-regression Context: Machine Learning Anchor id: concept/concept-notebook/machine-learning/logistic-regression Open question: What is the smallest example that makes Logistic Regression click without losing the math? Evidence to inspect: - Source ids to inspect: cs229-logistic-regression, goodfellow-2016-deep-learning - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action.

Open source object

concept/concept-notebook/machine-learning/logistic-regression
concept:machine-learning/logistic-regression

Learning Map

Before / Now / Try / Next

BeforeLinear Regression & Least Squares

NowIntuition → Math → Code → Demo

TryManipulate one control and predict the visible change.

NextClassification Metrics, Thresholds, and Calibration

Intuitionready
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Codeready
Interactive Demoready

Object Companion

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Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood.

Your question

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Context prompt

You are my AI learning companion for Continuous Function. Current context: Machine Learning concept. Learning surface: Logistic Regression. What this page says: Logistic regression turns a linear score into a probability, then learns a classification boundary by Bernoulli likelihood. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

Domain

Machine Learning

machine-learningclassificationlikelihoodlogits

Prerequisites

Linear Regression & Least Squares Maximum Likelihood Cross-Entropy

Leads To

Classification Metrics, Thresholds, and Calibration Direct Preference Optimization

Gradient Descent Label Smoothing & Soft Targets RLHF: Reward Modeling + KL-Regularized Policy Optimization

Within this domain