The Modern Mathematics of Deep Learning

Trajectory initialization scales

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.

Initialization · Describe initial parameters and their distributions.

The current objective supplies a gradient; training updates the model parameters.

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 source compares two normal initializations. It states baseline weight variance 1/n and bias variance 1, and scaled variances σ²/n and σ².

Initialization scales affect the distribution of network layers.

Source formula example

\Theta_{1}

illustration · The PDF states a σ^ℓ scaling claim, but its printed nonzero bias variances do not generally imply that scaling beyond the first layer. This example computes the stated variance values, not the claim.

Symbols

$Θ₁$: Baseline random initialization.
$Θσ$: Scaled random initialization.
$n$: Hidden width.
$σ$: A positive scale.
$ℓ$: Layer index.

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.

This example’s data flow

Initial values in code

n = 20
sigma = 2
layer = 2

Calculated here

Result expression: { n, sigma, baselineWeightVariance: 1 / n, baselineBiasVariance: 1, scaledWeightVariance: (sigma * sigma) / n, scaledBiasVariance: sigma * sigma, claimedFactor: sigma ** layer, }

Returns

Example result

n: 20
sigma: 2
baselineWeightVariance: 0.05
baselineBiasVariance: 1
scaledWeightVariance: 0.2
scaledBiasVariance: 4
claimedFactor: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. The PDF states a σ^ℓ scaling claim, but its printed nonzero bias variances do not generally imply that scaling beyond the first layer. This example computes the stated variance values, not the claim.

Source: Inline notation