The Modern Mathematics of Deep Learning

Network diagram values

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.

Model classes and structure · Describe families of rules, their size and structural properties.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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: 1.2 Learning from examples

The diagram has three inputs, hidden layers with four and six units, and one output. Each pre-activation is a weighted sum plus bias, and each hidden activation applies ϱ.

Following each layer explains the labels on the diagram.

Source formula example

x_{1}

illustration · The figure specifies layer widths; these weights, zero biases, inputs and ReLU activation are invented examples.

Symbols

$a$: Architecture with widths (3,4,6,1) and activation ϱ.
$xᵢ$: The i-th input coordinate.
$W⁽ℓ⁾, b⁽ℓ⁾$: Weights and biases at layer ℓ.
$Φ⁽ℓ⁾ᵢ$: The i-th pre-activation at layer ℓ.
$Φ̄⁽ℓ⁾ᵢ$: The corresponding hidden activation.
$Φₐ$: The network output.
$ϱ$: The activation function.

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

x = [1, 2, 3]
W1 = [ [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1], ]
W2 = [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [1, 1, 0, 0], [0, 1, 1, 0], ]
W3 = [[1, 1, 1, 1, 1, 1]]

Calculated here

Result expression: { x, pre1, hidden1, pre2, hidden2, output }

pre1 = apply(W1, x) Uses: apply, W1, x
hidden1 = pre1.map((v) => Math.max(0, v)) Uses: pre1
pre2 = apply(W2, hidden1) Uses: apply, W2, hidden1
hidden2 = pre2.map((v) => Math.max(0, v)) Uses: pre2
output = apply(W3, hidden2) Uses: apply, W3, hidden2

Returns

Example result

x: array (3 items): [1, 2, 3]
pre1: array (4 items): [1, 2, 3, 6]
hidden1: array (4 items): [1, 2, 3, 6]
pre2: array (6 items): [1, 2, 3, 6, …]
hidden2: array (6 items): [1, 2, 3, 6, …]
output: array (1 items): [20]

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 figure specifies layer widths; these weights, zero biases, inputs and ReLU activation are invented examples.

Source: Inline notation