Optimization

Weight Initialization: Xavier, He & muP

How to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP.

status: reviewimportance: importantdifficulty 3/5math: undergraduateread: 14mprediction probe

Back to Optimization Next: Scaling Laws & Emergent Abilities

Concept Structure

Weight Initialization: Xavier, He & muP

01Intuition

Start with the picture, metaphor, or geometric mechanism.

02Math

Make the objects explicit and connect them with notation.

03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Commit a prediction against the code witness.

2prerequisites

1next concepts

1related links

Learner Contract

What this page should let you do.

You are here becauseHow to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP.

This Optimization concept is the current object: keep the same invariant visible across Intuition, Math, Code, Interactive Demo.

Before thisLoss Landscapes, Sharpness & Flat Minima

2 prerequisites listed; refresh them before leaning on the math or code.

Loss Landscapes, Sharpness & Flat Minima Neural Tangent Kernel (NTK) & Infinite-Width Limits (review)

By the end4/4 sections ready | code witness expected | prediction probe

Explain the mechanism, trace the main notation, and answer the prediction probe against the code witness.

Do this firstIntuition

Read the intuition before the notation; the math should name a mechanism you already felt.

Then go nextScaling Laws & Emergent Abilities

Follow this edge after making one prediction here; the next page should reuse the result, not restart the route.

Test the linkUse the prediction probe to commit to the mechanism before moving on.Then continue to Scaling Laws & Emergent Abilities

Claim/source review status

Claim review not recorded

No structured claim checks yet; source metadata is not enough to establish support.Metadata-derived; review may be AI-assisted. Not a human certification.

Claims0/0 reviewed

Sources0 cited

Codeattached

Demoplanned

Reviewednot recorded

Updatedpage 2026-06-29

Object flow

4/4 sections readyAsk about this Research room

ConceptWeight Initialization: Xavier, He & muPOptimization EquationWeight Initialization: Xavier, He & muP equation 1Exact equation object CodeWeight Initialization: Xavier, He & muP code witness 1Exact code witness ClaimWeight Initialization: Xavier, He & muP central claimMechanism claim EquationWeight Initialization: Xavier, He & muP math objectsEquation and notation CodeWeight Initialization: Xavier, He & muP code witnessRunnable code object

ConceptWeight Initialization: Xavier, He & muPOptimization

Local snapshot ready

concept:optimization/weight-initialization

Codewitness nearby Predictbefore reveal Roomobject handoff

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 DemoCommit a prediction against the code witness.

Intuition

Build the mental picture first so the rest of the page has something to attach to.

Section prompt

Initialization is the starting geometry of learning.

If weights are too small, signals shrink as they pass through layers and gradients vanish. If weights are too large, activations and gradients explode and training becomes unstable.

Good initializations aim to keep the "scale" of information roughly constant as it flows forward (activations) and backward (gradients).

Two classics:

Xavier/Glorot: good for tanh/sigmoid-like activations.
He/Kaiming: good for ReLU-like activations (because ReLU zeros out about half of inputs).

Modern scaling work adds another layer: you want hyperparameters to transfer as width changes. muP is one way to parameterize networks so you can tune on a small model and scale up with fewer surprises.

Math

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

Section prompt

Equation 1 $\mathrm{Var}(h_j) \approx n_{in}\,v\,\sigma^2.$ Equation 2 $v \approx \frac{1}{n_{in}}.$

Consider a linear layer $h = Wx$ where $x\in\mathbb R^{n_{in}}$ has i.i.d. components with $\mathrm{Var}(x_i)=\sigma^2$ . Assume $W_{ij}$ are i.i.d. with mean 0 and variance $\mathrm{Var}(W_{ij}) = v$ .

Then each output coordinate is a sum of $n_{in}$ terms, so:

\mathrm{Var}(h_j) \approx n_{in}\,v\,\sigma^2.

To keep variance stable across layers ( $\mathrm{Var}(h_j) \approx \sigma^2$ ), set:

v \approx \frac{1}{n_{in}}.

This motivates Xavier-like scaling. For ReLU, roughly half the mass is zeroed, so to keep variance stable you use about twice the variance:

\mathrm{Var}(W_{ij}) \approx \frac{2}{n_{in}}.

muP (Maximal Update Parameterization) extends this idea to updates: scale parameters and learning rates with width so that update magnitudes stay comparable across model sizes.

Code

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

Section prompt

Code witness 1import numpy as np rs = np.random.RandomState(0) def run(depth=20, width=512, scale=1.0, relu...python

import numpy as np

rs = np.random.RandomState(0)

def run(depth=20, width=512, scale=1.0, relu=True):
    x = rs.randn(width)
    vars = []
    for _ in range(depth):
        W = rs.randn(width, width) * (scale / np.sqrt(width))
        x = W @ x
        if relu: x = np.maximum(x, 0)
        vars.append(float(x.var()))
    return vars

for name, scale in [("too small", 0.3), ("xavier-ish", 1.0), ("too large", 3.0)]:
    v = run(scale=scale)
    print(name, "var@layer1", round(v[0], 3), "var@layer20", round(v[-1], 3))

Interactive Demo

Use the authored prediction probe to test the mechanism without a live widget.

Section prompt

Prediction check: before running the witness, rank too small, xavier-ish, and too large by their expected layer-20 variance. Then change relu=False and see how the activation choice changes the initialization scale that feels stable.

After The First Pass

Turn the concept into an inspected object.

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

See the idea move before the page explains it

How to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP.

Demo notes open01 / Intuition

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Weight Initialization: Xavier, He & muP should make visible.

Visual Inquiry

Make the image answer a mathematical question

How to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP.

4/4 stages readyDemo notes connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Weight Initialization: Xavier, He & muP easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

Source gapNo canonical references are listed yet.

Add source metadata before treating this page as source-grounded.

Claim Review

How to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP.

StatusSubstantive claim review pending

Source IDs and witness objects are attached for review; they are not proof by themselves.

SourcesNo references

Add source metadata before claiming support.

Witnesses3 local objects

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

Source support candidates

No structured source note is attached yet.

Mechanism witnesses

Equation 1

\mathrm{Var}(h_j) \approx n_{in}\,v\,\sigma^2.

Equation 2

v \approx \frac{1}{n_{in}}.

Code witness 1import numpy as np rs = np.random.RandomState(0) def run(depth=20, width=512, scale=1.0, relu...

Practice Loop

Try the idea before it explains itself

How to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP.

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Weight Initialization: Xavier, He & muP.

Hint 1

Reveal when your model needs a nudge.

Hint 2

Reveal when your model needs a nudge.

Hint 3

Reveal when your model needs a nudge.

Your answer canvas

Local checks

Claim

A concrete answer is on the canvas.

Mechanism

The answer names why the claim should hold.

Bridge

It touches the page context or a neighboring idea.

Misconception check

Object research drawerClose

ConceptWeight Initialization: Xavier, He & muPOptimization

Code witness comparisonWeight Initialization: Xavier, He & muP code witness 1rs = np.random.RandomState(0)Prediction before revealWeight Initialization: Xavier, He & muP predictionUse the prediction probe to commit to the mechanism before moving on.

Grounded room questionWhat is the smallest example that makes Weight Initialization: Xavier, He & muP click without losing the math?Local snapshot ready

Research Room

Attach the question to an exact object

Pick the concept, equation, source, code witness, claim, misconception, or demo state before asking for help. The handoff stays grounded to that object.

Next local actionNo local draft saved yet

Open the draft below to save one note and next action in this browser.

conceptOptimization

Weight Initialization: Xavier, He & muP

Anchored question

What is the smallest example that makes Weight Initialization: Xavier, He & muP click without losing the math?

Local action draftNo local draft saved yetExpand only when ready to capture one local next action

Local action draft

This draft stays locally in this browser for concept:optimization/weight-initialization.

Draft noteNext action

No local draft saved.

Evidence to inspect

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 - Weight Initialization: Xavier, He & muP Object key: concept:optimization/weight-initialization Context: Optimization Anchor id: concept/concept-notebook/optimization/weight-initialization Open question: What is the smallest example that makes Weight Initialization: Xavier, He & muP click without losing the math? Evidence to inspect: - 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/optimization/weight-initialization
concept:optimization/weight-initialization

Learning Map

Before / Now / Try / Next

BeforeLoss Landscapes, Sharpness & Flat Minima

NowIntuition → Math → Code → Demo

TryUse the prediction probe to commit to the mechanism before moving on.

NextScaling Laws & Emergent Abilities

Intuitionready
Mathready
Codeready
Interactive Demoready

Object Companion

Ask beside the selected object

How to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP.

Your question

GoalComfortStyleStuck on

Context prompt

You are my AI learning companion for Continuous Function. Current context: Optimization concept. Learning surface: Weight Initialization: Xavier, He & muP. What this page says: How to pick weight scales so activations/gradients stay stable: Xavier/Glorot, He/Kaiming, and width-scaling ideas like muP. Current section: Intuition, math, code, and interactive demo. Suggested next step: Use the prediction probe to commit to the mechanism before moving on.. 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

Optimization

optimizationinitializationmupstability

Prerequisites

Loss Landscapes, Sharpness & Flat Minima Neural Tangent Kernel (NTK) & Infinite-Width Limits (review)

Leads To

Scaling Laws & Emergent Abilities

Learning Rate Schedules: Warmup, Decay & Cycling

Within this domain