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: 14mdemo planned

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Concept Structure

Weight Initialization: Xavier, He & muP

01Intuition

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02Math

Make the objects explicit and connect them with notation.

03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

2prerequisites

1next concepts

1related links

Learning map

Weight Initialization: Xavier, He & muP

BeforeLoss Landscapes, Sharpness & Flat MinimaNow3/4 sections readyTryUse the demo notes to predict the mechanism before moving on.NextScaling Laws & Emergent Abilities

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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

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concept:optimization/weight-initialization

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Conceptual Bridge

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Carry inLoss Landscapes, Sharpness & Flat Minima

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Work hereWeight Initialization: Xavier, He & muP

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

Carry outScaling Laws & Emergent Abilities

The next edge should feel earned: use the demo prediction here before following Scaling Laws & Emergent Abilities.

Test the linkUse the demo notes to predict the mechanism before moving on.Then continue to Scaling Laws & Emergent Abilities

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

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 2v \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

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Section prompt

No interactive demo yet for this concept. A good next step is a small "signal propagation" visualization that shows activation variance and gradient variance vs depth for different init choices.

No live visualization is registered for this concept yet.

The page still supports explanatory demo notes above; when a viz.tsx exists, it will render here without changing the route.

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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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

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Commit first

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

Visual Inquiry

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

3/4 stages readyDemo notes connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

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Commit first

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

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

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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

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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 demo notes to predict 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

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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?

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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 demo notes to predict the mechanism before moving on.

NextScaling Laws & Emergent Abilities

Intuitionready
Mathready
Codeready
Interactive Demoplanned

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 demo notes to predict 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

Leads To

Scaling Laws & Emergent Abilities

Learning Rate Schedules: Warmup, Decay & Cycling

Within this domain