The Modern Mathematics of Deep Learning

From input batch to normalization

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

A batch of inputs is first transformed by the previous layer, then normalized using the resulting values.

This example makes the source notation or condition concrete.

Source formula example

{s}

illustration · This fixed batch does not show training or running-statistic behavior.

Symbols

$s$: Batch of inputs.
$b$: Previous-layer outputs over that batch.
$μb,σb$: Mean and standard deviation of b.

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.

Previous-layer bias

This example’s data flow

You choose

input.bias = 2

Initial values in code

s = [-1, 0, 1]

Calculated here

Result expression: { inputBatch: s, outputBatch: b, mean, variance }

b = s.map((x) => x + input.bias) Uses: s, input.bias
mean = b.reduce((a, v) => a + v, 0) / b.length Uses: b
variance = b.reduce((a, v) => a + (v - mean) ** 2, 0) / b.length Uses: b, mean

Returns

Example result

inputBatch: array (3 items): [-1, 0, 1]
outputBatch: array (3 items): [1, 2, 3]
mean: 2
variance: ≈0.6667

JavaScript

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

Result

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Technical statement

The invented previous layer maps each scalar input x to x+bias. The source figure uses s for the input batch and computes normalization statistics on the outputs.

Source: Source expression