The Modern Mathematics of Deep Learning

The printed initialization scaling

Penguin learning project

The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV

This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.

Model classes and structure · Describe families of rules, their size and structural properties.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.

Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].

Used in these source sections: 3.3 Other ways to measure possibility

The paper prints a scaling-in-distribution claim when every weight and bias standard deviation is multiplied by σ. For two hidden layers with nonzero Gaussian biases, a second-moment calculation gives different answers.

At σ=2, n=1, and input x=0, the actual second moment is 6; scaling the baseline second-layer output by σ² gives second moment 12.

The mismatch matters because the next printed trajectory-length statement relies on this scaling claim.

Source formula example

\bar{\Phi}^{(\ell)}(\cdot,\Theta_{\sigma})\sim\sigma^{\ell}\bar{\Phi}^{(\ell)}(\cdot,\Theta_{1}),

illustration · The second-moment calculation refutes the printed general activation-distribution scaling in this case. It does not by itself refute the separate expected trajectory-length equality. The zero-bias curve is a deterministic special case, not the Gaussian source experiment.

Symbols

$σ$: Positive factor multiplying standard deviations in the stated initialization.
$Θ_1,Θ_σ$: Baseline and scaled random network parameters.
$ℓ$: Layer index; the counterexample uses ℓ=2.

Source formula example

This source example uses its own inputs and assumptions. Editing it does not change the project.

Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.

Scale σ

This example’s data flow

You choose

input.sigma = 2

Initial values in code

zeroBiasBaselineLength = 1

Calculated here

Result expression: { actualSecondMoment, claimedScaledSecondMoment, momentsMatch: actualSecondMoment === claimedScaledSecondMoment, zeroBiasBaselineLength, zeroBiasScaledLength, }

sigma = input.sigma Uses: input.sigma
actualSecondMoment = sigma ** 2 / 2 + sigma ** 4 / 4 Uses: sigma
claimedScaledSecondMoment = (3 * sigma ** 4) / 4 Uses: sigma
zeroBiasScaledLength = sigma ** 2 Uses: sigma

Returns

Example result

actualSecondMoment: 6
claimedScaledSecondMoment: 12
momentsMatch: false
zeroBiasBaselineLength: 1
zeroBiasScaledLength: 4

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The PDF states that independent centered Gaussian weights have variances 1/n and σ²/n, while biases have variances 1 and σ². For n=1, two hidden layers and x=0, symmetry gives E[ReLU(Z)²]=Var(Z)/2 for centered Gaussian Z. Conditioning on the first activation yields σ²/2+σ⁴/4 for the scaled second activation. The baseline second moment is 3/4, and multiplying that output by σ² would give 3σ⁴/4. They differ at σ=2. The next source statement claims expected curve length σ^ℓc, but this counterexample to the preceding distribution identity does not establish whether that separate length statement fails. As a contrast, the code also gives a zero-bias deterministic curve t↦t through two scalar layers of weight σ: its length is σ² times the baseline length. This is only a special case and is not a proposed correction.

Source: Printed initialization scaling