The Modern Mathematics of Deep Learning

Normalize a whole batch

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.

Batch normalization · Describe batch normalization and its derivatives.

The project uses a linear rule with two features. The fitted weights produce scores or normalized mass.

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: 6.4 Adjusting the scale

Normalize every batch entry using the same batch mean and standard deviation, then scale and shift.

A batch map lets us study how batch entries affect one another.

Source formula example

\mathcal{B}^{(\beta,\gamma)}({b})=(\mathrm{BN}_{{b}}^{(\beta,\gamma)}(y^{(1)}),\dots,\mathrm{BN}_{{b}}^{(\beta,\gamma)}(y^{(m)})),\quad{b}=(y^{(i)})_{i=1}^{m}\in{\mathbb{R}}^{m},

exact · This fixed batch does not explain training or inference behavior with running averages.

Symbols

$b$: Batch of m real values.
$β,γ$: Real shift and scale.
$B$: Map from the complete batch to normalized batch.
$m$: Positive batch size.

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.

ScaleShift

This example’s data flow

You choose

input.gamma = 1
input.beta = 0

Initial values in code

batch = [1, 2, 3]
mean = 2

Calculated here

Result expression: { batch, output: batch.map((v) => (input.gamma * (v - mean)) / sigma + input.beta), }

sigma = Math.sqrt(2 / 3)

Returns

Example result

batch: array (3 items): [1, 2, 3]
output: array (3 items): [≈-1.225, 0, ≈1.225]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The batch variance must be positive for the displayed formula.

Source: Source expression