The Modern Mathematics of Deep Learning

Probability expression

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.

Probability and expectation · Describe random variables and their averages under a law.

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: 5.2 When parameters move only a little

Divide width n by sample count m, round down, raise 1 − δ to that power, and subtract from 1.

This quantity appears in the source argument about the feature Gram matrix.

Source formula example

1-(1-\delta)^{\lfloor n/m\rfloor}

illustration · Calculating this number does not establish the positive-definiteness probability or its assumptions.

Symbols

$n$: Positive integer network width.
$m$: Positive integer number of examples.
$δ$: Probability parameter between zero and one.
$⌊·⌋$: Floor: round down to the greatest integer no larger than the value.

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.

WidthSample countδ

This example’s data flow

You choose

input.n = 4
input.m = 2
input.delta = 0.5

Calculated here

Result expression: { blocks, value }

blocks = Math.floor(input.n / input.m) Uses: input.n, input.m
value = 1 - (1 - input.delta) ** blocks Uses: input.delta, blocks

Returns

Example result

blocks: 2
value: 0.75

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy evaluates the displayed expression. The source uses a probability argument for collections of independently initialized feature vectors.

Source: Source expression