The Modern Mathematics of Deep Learning

ReLU and positive scaling

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: 2.2 Weight size and room to decide · 3.3 Other ways to measure possibility

For a positive scale λ, scaling before ReLU gives the same answer as scaling after ReLU.

This property is used when studying norms and margins.

Source formula example

{\varrho_{R}}(\lambda x)=\lambda{\varrho_{R}}(x)

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

Symbols

$λ$: A positive real number.
$x$: A real number.

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.

InputPositive scale

This example’s data flow

You choose

input.scale = 3
input.x = 2

Calculated here

Result expression: { before: Math.max(0, input.scale * input.x), after: input.scale * Math.max(0, input.x), }

Returns

Example result

before: 6
after: 6

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