The Modern Mathematics of Deep Learning

A scaled sigmoid approaches a step

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: 1.2.2 Can the network express the answer?

For a sigmoid, increasing positive a makes the output approach 1 when x>0 and 0 when x<0. At x=0, the output remains ρ(b).

The toy uses the logistic sigmoid and b=0, so the boundary value is 1/2.

This limit is used in the source universal-approximation argument.

Source formula example

\varrho(ax+b)\to\mathds{1}_{(0,\infty)}(x)+\varrho(b)\mathds{1}_{\{0\}}(x)

approximation · The code evaluates a finite scale. It does not compute the limit or prove the source claim for every sigmoid.

Symbols

$a$: Positive scale tending to infinity.
$b$: A fixed real offset; zero in the toy.
$ρ$: A sigmoid 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.

Positive scale ax

This example’s data flow

You choose

input.a = 4
input.x = 1

Calculated here

Result expression: { value, limit }

value = sigmoid(input.a * input.x) Uses: sigmoid, input.a, input.x
limit = input.x > 0 ? 1 : input.x === 0 ? sigmoid(0) : 0 Uses: input.x, sigmoid

Returns

Example result

value: ≈0.982
limit: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The strict-positive indicator excludes zero. The separate singleton indicator keeps the correct value ρ(b) at x=0. The source uses this pointwise limit while developing the discriminatory property of sigmoid activations.

Source: Sigmoid limit