The Modern Mathematics of Deep Learning

Sigmoid limits

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.

Functions and derivatives · Describe maps and how their outputs change.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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?

A sigmoidal activation approaches 1 as x increases without bound and 0 as x decreases without bound.

The limit describes the activation far from zero.

Source formula example

\varrho(x)\to 1

illustration · The finite evaluations show the direction of change for logistic sigmoid; finite values do not themselves prove the limits.

Symbols

$ϱ$: A sigmoidal function.
$x$: 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.

This example’s data flow

Calculated here

Result expression: { negative: sigmoid(-2), positive: sigmoid(2) }

Returns

Example result

negative: ≈0.1192
positive: ≈0.8808

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. The finite evaluations show the direction of change for logistic sigmoid; finite values do not themselves prove the limits.

Source: Inline notation