The Modern Mathematics of Deep Learning

Batch normalization as an identity

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

Choosing the shift as the batch mean and the scale as the batch standard deviation returns the input.

The shift and scale allow the normalization layer to recover its input.

Source formula example

\mathrm{BN}_{{b}}^{(\beta,\gamma)}(y)=y

illustration · The displayed source division requires nonzero standard deviation.

Symbols

$μb$: Batch mean.
$σb$: Positive batch standard deviation.
$β$: Shift, set to μb.
$γ$: Scale, set to σb.
$y$: Real input value.

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.

Value

This example’s data flow

You choose

input.y = 3

Initial values in code

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

Calculated here

Result expression: { mean, sigma, y, result }

sigma = Math.sqrt(2 / 3)
y = input.y Uses: input.y
result = (sigma * (y - mean)) / sigma + mean Uses: sigma, y, mean

Returns

Example result

mean: 2
sigma: ≈0.8165
y: 3
result: 3

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The scalar batch is [1,2,3], with nonzero standard deviation. The identity holds componentwise for the source vector-valued version.

Source: Source expression