The Modern Mathematics of Deep Learning

Network parameter collection

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.

Prediction rules and layers · Describe or apply prediction rules, layers and parameters.

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: 1.2 Learning from examples

Each layer has a weight matrix and bias vector; together they form the network parameter collection.

Their shapes determine how inputs flow through a network.

Source formula example

\theta=(\theta^{(\ell)})_{\ell=1}^{L}=((W^{(\ell)},b^{(\ell)}))_{\ell=1}^{L}\in\bigtimes_{\ell=1}^{L}({\mathbb{R}}^{N_{\ell}\times N_{\ell-1}}\times{\mathbb{R}}^{N_{\ell}})\cong{\mathbb{R}}^{P(N)}

illustration · The vectorization order is a choice; the source’s equivalence concerns parameter-space dimensions.

Symbols

$θ$: All layer weights and biases.
$W⁽ˡ⁾,b⁽ˡ⁾$: Real weight matrix and bias vector in layer ℓ.
$Nₗ$: Number of nodes in layer ℓ.
$P(N)$: Total number of scalar parameters.

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.

First weight

This example’s data flow

You choose

input.w = 1

Calculated here

Result expression: { layers, count }

layers = [ { weights: [ [input.w, 2], [3, 4], ], bias: [0, 1], }, { weights: [[1, 2]], bias: [0] }, ] Uses: input.w
count = layers.reduce( (sum, l) => sum + l.weights.reduce((n, row) => n + row.length, 0) + l.bias.length, 0, ) Uses: layers

Returns

Example result

layers: array (2 items): [object (2 items), object (2 items)]
count: 9

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy widths are (2,2,1), giving (2×2+2)+(1×2+1) = 9 scalars.

Source: Source expression