Machine Learning

Multinomial Logistic Regression

Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class.

status: publishedimportance: criticaldifficulty 3/5math: undergraduateread: 18mlive demo

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Concept Structure

Multinomial Logistic Regression

01Intuition

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02Math

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03Code

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04Interactive Demo

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2prerequisites

1next concepts

3related links

Learning map

Multinomial Logistic Regression

BeforeLogistic RegressionNow4/4 sections readyTryManipulate one control and predict the visible change.NextScaled Dot-Product Attention & Transformer Layers

Object flow

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ConceptMultinomial Logistic RegressionMachine Learning EquationMultinomial Logistic Regression equation 1Exact equation object CodeMultinomial Logistic Regression code witness 1Exact code witness DemoMultinomial Logistic Regression interactive demoVisualization object ClaimFor one softmax-classification example, the gradient of negative log-...Exact claim check SourceCS229 Lecture Notes: Softmax RegressionExact source object

ConceptMultinomial Logistic RegressionMachine Learning

2 sources attachedLocal snapshot ready

concept:machine-learning/multinomial-logistic-regression

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Conceptual Bridge

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Carry inLogistic Regression

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Work hereMultinomial Logistic Regression

Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class.

Carry outScaled Dot-Product Attention & Transformer Layers

The next edge should feel earned: use the demo prediction here before following Scaled Dot-Product Attention & Transformer Layers.

Test the linkManipulate one control and predict the visible change.Then continue to Scaled Dot-Product Attention & Transformer Layers

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 a language model, image classifier, and attention mechanism all need the same basic move: turn a list of scores into a distribution. Binary logistic regression has one logit. Softmax regression has one logit per class.

Before this, know logistic regression and cross-entropy. By the end, you should be able to explain why softmax probabilities sum to one, why adding the same constant to every logit changes nothing, and why the logit gradient is $q_k-\mathbf 1[k=y]$ .

Imagine three labels: cat, dog, and bird. The model gives each label a score. A score by itself is not a probability because scores can be negative, large, or unnormalized. Softmax exponentiates the scores and normalizes them:

q_k=\frac{\exp(z_k)}{\sum_j \exp(z_j)}.

The highest score gets the largest probability, but every class competes through the denominator. Raising one logit steals probability mass from the others.

Softmax also has a useful invariance. If you add the same number $c$ to every logit, every exponential is multiplied by $\exp(c)$ , and that factor cancels:

\operatorname{softmax}(z+c\mathbf 1)=\operatorname{softmax}(z).

That is why stable implementations subtract the maximum logit before exponentiating. It changes the numbers used internally, not the distribution.

Math

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Section prompt

Equation 1z=Wx+b\in\mathbb R^K.Equation 2q_k=P_\theta(Y=k\mid x)=\frac{\exp(z_k)}{\sum_{j=1}^K \exp(z_j)}.

For one input $x\in\mathbb R^d$ and $K$ classes, collect class weights in $W\in\mathbb R^{K\times d}$ and biases in $b\in\mathbb R^K$ . The logit vector is

z=Wx+b\in\mathbb R^K.

The predicted class distribution is

q_k=P_\theta(Y=k\mid x)=\frac{\exp(z_k)}{\sum_{j=1}^K \exp(z_j)}.

If the observed class is $y\in\{1,\dots,K\}$ , the likelihood is $q_y$ . The negative log-likelihood is

\ell(z,y)=-\log q_y.

Substituting the softmax gives the log-sum-exp form:

\ell(z,y)=-z_y+\log\sum_{j=1}^K \exp(z_j).

Now differentiate with respect to one logit $z_k$ :

\frac{\partial \ell}{\partial z_k} = \frac{\exp(z_k)}{\sum_j \exp(z_j)}-\mathbf 1[k=y] = q_k-\mathbf 1[k=y].

That is the multi-class version of the logistic-regression signal $q-y$ . The correct class has a negative gradient unless it already has probability near $1$ , so gradient descent raises its logit. Incorrect classes have positive gradients, so gradient descent lowers their logits.

For a batch with design matrix $X\in\mathbb R^{n\times d}$ , one-hot labels $Y\in\mathbb R^{n\times K}$ , probabilities $Q\in\mathbb R^{n\times K}$ , and weights $W\in\mathbb R^{d\times K}$ , one common convention writes logits as

Z=XW+\mathbf 1b^{\mathsf T}.

The average-loss gradients are

\nabla_W L=\frac1n X^{\mathsf T}(Q-Y),

and

\nabla_b L=\frac1n\sum_{i=1}^n (Q_i-Y_i).

The notation differs by whether examples are rows or columns, but the invariant is stable: softmax plus cross-entropy creates a probability-error vector over classes.

Code

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

Section prompt

Code witness 1import numpy as np # Three-class softmax regression on tiny 2D points. # Shapes: X is (n, 3)...python

import numpy as np

# Three-class softmax regression on tiny 2D points.
# Shapes: X is (n, 3) with bias column, W is (3, K).
X_raw = np.array([[-2, 0], [-1, 1], [0, -1], [1, 1], [2, 0], [1, -2]], float)
y = np.array([0, 0, 1, 1, 2, 2])
K = 3
X = np.column_stack([np.ones(len(X_raw)), X_raw])
Y = np.eye(K)[y]
W = np.zeros((X.shape[1], K))

def softmax(z):
    z = z - z.max(axis=1, keepdims=True)
    exp_z = np.exp(z)
    return exp_z / exp_z.sum(axis=1, keepdims=True)

for step in range(120):
    logits = X @ W
    Q = softmax(logits)
    grad = X.T @ (Q - Y) / len(X)
    W -= 0.7 * grad

Q = softmax(X @ W)
loss = -np.mean(np.log(Q[np.arange(len(y)), y] + 1e-12))
print("class probabilities:")
print(np.round(Q, 3))
print("loss:", round(loss, 4))
print("first example logit gradient:", np.round(Q[0] - Y[0], 3))
print("row sums:", np.round(Q.sum(axis=1), 6))

The code mirrors the math: softmax normalizes logits row by row, Q - Y is the probability-error matrix, and X.T @ (Q - Y) / n is the weight gradient.

Interactive Demo

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Section prompt

Set three logits and choose the true class. Before revealing the gradient, predict the shape of the update: does one wrong class nearly match the true-class upward push, is the wrong probability mass split across wrong classes, or are all updates already small?

The reveal shows the invariant that is easy to miss: for one example, the true-class upward update has magnitude $1-q_y$ , while all wrong-class downward updates sum to the same amount. A single wrong class can nearly tie the true-class push, but it cannot be larger unless the demo has changed the object being measured. The shared-shift slider adds the same value to every logit. Watch the raw scores move while the probabilities and gradients stay fixed. That small invariant is the bridge from classical softmax regression to stable neural-network implementations.

Live Concept Demo

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difficulty 3/5undergraduatecode-aligned

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After The First Pass

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Mechanism Storyboard

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Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class.

Prediction open01 / Intuition

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Visual Inquiry

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Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class.

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Source Grounding

Canonical references for the mechanism on this page.

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

Curriculum source for softmax regression and multi-class likelihood.

Open source

bookDive into Deep Learning: Softmax RegressionZhang, Lipton, Li, and Smola

Curriculum source for logits, softmax probabilities, and implementation-oriented multi-class classification.

Open source

Claim Review

Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class.

Status1 substantive review recorded

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

Sources2 references

cs229-softmax-regression, d2l-softmax-regression

Witnesses4 local objects

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Substantively reviewedFor one softmax-classification example, the gradient of negative log-likelihood with respect to each logit is predicted probability minus the one-hot label.Claim metadata: source checked

CS229 gives softmax logits, NLL/cross-entropy, and derivative phi_i - 1{y=i}; D2L independently derives softmax-cross-entropy derivative softmax(o)_j - y_j and the rowwise vectorized model.

Sources: CS229 Lecture Notes: Softmax Regression, Dive into Deep Learning: Softmax RegressionThis supports single-label softmax classification mechanics, not calibrated probabilities, multi-label classification, attention semantics, or any claim that one wrong class can exceed the total true-class push.A bounded review summary is present; still check caveats and exact source scope.

Checked CS229 section 2.3 and D2L section 4.1: both support softmax probabilities, one-hot labels, cross-entropy/NLL, and the logit derivative as predicted probability minus the label indicator. Prior GPT Pro blocker about the wrong-class update was resolved by teaching true-class push equals total wrong-class pull.

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

source-span-cs229-softmax-regression source-span-d2l-softmax-regression math-object-1 math-object-2 code-witness-1 interactive-demo

Source support candidates

course-notes 2023CS229 Lecture Notes: Softmax Regression

Curriculum source for softmax regression and multi-class likelihood.

bookDive into Deep Learning: Softmax Regression

Curriculum source for logits, softmax probabilities, and implementation-oriented multi-class classification.

Mechanism witnesses

Equation 1

z=Wx+b\in\mathbb R^K.

Equation 2

q_k=P_\theta(Y=k\mid x)=\frac{\exp(z_k)}{\sum_{j=1}^K \exp(z_j)}.

Code witness 1import numpy as np # Three-class softmax regression on tiny 2D points. # Shapes: X is (n, 3)...Demo stateLive mechanism probe

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Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class.

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ConceptMultinomial Logistic RegressionMachine Learning

Code witness comparisonMultinomial Logistic Regression code witness 1X_raw = np.array([[-2, 0], [-1, 1], [0, -1], [1, 1], [2, 0], [1, -2]], float)Prediction before revealMultinomial Logistic Regression interactive demoManipulate one control and predict the visible change.

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

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conceptMachine Learning

Multinomial Logistic Regression

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Evidence to inspect

Source ids to inspect: cs229-softmax-regression, d2l-softmax-regression
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What would resolve this

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Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Multinomial Logistic Regression Object key: concept:machine-learning/multinomial-logistic-regression Context: Machine Learning Anchor id: concept/concept-notebook/machine-learning/multinomial-logistic-regression Open question: What is the smallest example that makes Multinomial Logistic Regression click without losing the math? Evidence to inspect: - Source ids to inspect: cs229-softmax-regression, d2l-softmax-regression - 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/multinomial-logistic-regression
concept:machine-learning/multinomial-logistic-regression

Learning Map

Before / Now / Try / Next

BeforeLogistic Regression

NowIntuition → Math → Code → Demo

TryManipulate one control and predict the visible change.

NextScaled Dot-Product Attention & Transformer Layers

Intuitionready
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Codeready
Interactive Demoready

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Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class.

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Context prompt

You are my AI learning companion for Continuous Function. Current context: Machine Learning concept. Learning surface: Multinomial Logistic Regression. What this page says: Softmax regression turns one score per class into a normalized probability distribution and learns by moving probability mass toward the observed class. 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-learningclassificationsoftmaxlogits

Prerequisites

Logistic Regression Cross-Entropy

Leads To

Scaled Dot-Product Attention & Transformer Layers

Label Smoothing & Soft Targets Knowledge Distillation: Learning from Teachers Cross-Entropy

Within this domain