The Modern Mathematics of Deep Learning

Two-layer ReLU network

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: 6.5 Removing connections

Add output-weighted ReLU hidden features.

This network is the starting point for the source pruning argument.

Source formula example

\Phi_{a}(\cdot,\theta)=\sum_{i=1}^{n}w^{(2)}_{i}{\varrho_{R}}(\langle(w_{i}^{(1)},\cdot\rangle+b_{i}^{(1)})),

exact · Evaluating one network does not establish a pruning guarantee.

Symbols

$wᵢ⁽¹⁾,bᵢ⁽¹⁾$: Incoming weight and bias of hidden unit i.
$wᵢ⁽²⁾$: Output weight of unit i.
$n$: Positive number of hidden units.
$ϱR$: ReLU activation.

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.

Input

This example’s data flow

You choose

input.x = 1

Calculated here

Result expression: { hidden, output: 2 * hidden[0] - hidden[1] }

x = input.x Uses: input.x
hidden = [Math.max(0, x), Math.max(0, -x + 1)] Uses: x

Returns

Example result

hidden: array (2 items): [1, 0]
output: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented model has two units and scalar input.

Source: Source expression