The Modern Mathematics of Deep Learning

Two normalized batch directions

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

The constant vector and the normalized batch vector each have unit length and are perpendicular.

Batch normalization removes these directions from the gradient.

Source formula example

\{\frac{\mathbf{1}}{\sqrt{m}},\frac{1}{\sqrt{m}}\mathcal{B}^{(0,1)}({b})\}

illustration · The displayed finite calculation does not prove the full gradient formula.

Symbols

$1$: Length-m vector of ones.
$m$: Batch size, at least 2.
$B⁽⁰,¹⁾(b)$: Centered batch divided by its standard deviation.

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

m = 3
batch = [-1, 0, 1]

Calculated here

Result expression: { one, normalized, oneNormSquared: dot(one, one), normalizedNormSquared: dot(normalized, normalized), dot: dot(one, normalized), }

one = [1, 1, 1].map((v) => v / Math.sqrt(m)) Uses: m
sigma = Math.sqrt(2 / 3)
normalized = batch.map((v) => v / sigma / Math.sqrt(m)) Uses: batch, sigma, m

Returns

Example result

one: array (3 items): [≈0.5774, ≈0.5774, ≈0.5774]
normalized: array (3 items): [≈-0.7071, 0, ≈0.7071]
oneNormSquared: 1
normalizedNormSquared: 1
dot: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented batch [−1,0,1] has positive variance and m = 3. The two directions are orthonormal for any nonconstant real batch.

Source: Source expression