The Modern Mathematics of Deep Learning

A shallow 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: 3.2 Simple pieces, many layers · 3.3 Other ways to measure possibility

Each hidden unit applies ReLU to a linear function of x. Multiply each hidden output by its output weight, add them, and add the output bias.

This form shows how a two-layer network constructs a piecewise-linear function.

Source formula example

\Phi(x)=\sum_{i=1}^{n}w_{i}^{(2)}{\varrho_{R}}(w_{i}^{(1)}x+b^{(1)}_{i})+b^{(2)},\quad x\in{\mathbb{R}},

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

Symbols

$n$: Positive integer number of hidden units.
$w_i^{(1)},b_i^{(1)}$: First-layer weight and bias for unit i.
$w_i^{(2)},b^{(2)}$: Output weight for unit i and output bias.

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

firstWeights = [1, -1]
hiddenBiases = [0, 1]
outputWeights = [1, 2]
outputBias = 1

Calculated here

Result expression: { hidden, output: hidden.reduce((sum, value, i) => sum + value * outputWeights[i], outputBias), }

hidden = firstWeights.map((weight, i) => Math.max(0, weight * input.x + hiddenBiases[i]), ) Uses: firstWeights, input.x, hiddenBiases

Returns

Example result

hidden: array (2 items): [2, 0]
output: 3

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