The Modern Mathematics of Deep Learning

Activation functions

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 · 1.2.2 Can the network express the answer? · 1.2.3 Does it work on new examples? · More source sections

An activation function maps a real number to a real number. ReLU is one example: it returns the input if positive and zero otherwise.

A network can apply an activation to each entry.

Source formula example

\varrho\colon{\mathbb{R}}\to{\mathbb{R}}

illustration · ReLU is one activation; the source also considers other activation functions.

Symbols

$ϱ$: An activation function from ℝ to ℝ.
$ℝ$: The real numbers.

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, 0, 2]

Calculated here

Result expression: { x, activated: x.map((v) => Math.max(0, v)) }

Returns

Example result

x: array (3 items): [-1, 0, 2]
activated: array (3 items): [0, 0, 2]

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. ReLU is one activation; the source also considers other activation functions.

Source: Inline notation