The Modern Mathematics of Deep Learning

Two-unit 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: 1.3 The questions ahead

Two hidden units each apply ReLU to a weighted input plus a bias. The output is their weighted sum.

This small network illustrates how parameters determine a prediction.

Source formula example

\Phi(x,\theta)=\theta_{1}{\varrho_{R}}(\theta_{3}x+\theta_{5})+\theta_{2}{\varrho_{R}}(\theta_{4}x+\theta_{6}),\qquad\theta\in{\mathbb{R}}^{6},\quad x\in{\mathbb{R}},

exact · The result is exact for these finite toy inputs, apart from ordinary computer rounding.

Symbols

$θ$: Six real parameters.
$x$: A real input.

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 x

This example’s data flow

You choose

input.x = 2

Initial values in code

theta = [1, 2, 1, -1, 0, 1]

Calculated here

Result expression: { units, output: theta[0] * units[0] + theta[1] * units[1] }

units = [ Math.max(0, theta[2] * input.x + theta[4]), Math.max(0, theta[3] * input.x + theta[5]), ] Uses: theta, input.x

Returns

Example result

units: array (2 items): [2, 0]
output: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Source expression