The Modern Mathematics of Deep Learning

Expectation of a threshold feature

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: 4.2 Approximating with random pieces

The indicator is 1 when the inner product exceeds the threshold and 0 otherwise. Its expectation is the probability of that event.

The toy uses Xi = −1 or +1 equally likely and a fixed threshold 1/2.

A probability can be represented as an expected indicator.

Source formula example

{\mathbbm{E}}\big[\mathds{1}_{(0,\infty)}(\langle\Xi,\cdot\rangle-\widetilde{\Xi})\big]

exact · The expectation is exact for this two-outcome toy distribution. An independently drawn copy has the same distribution, not necessarily the same realized value. This does not approximate an arbitrary distribution or establish membership in a generalized Barron space.

Symbols

$Ξ$: A random direction vector; scalar in the toy.
$Ξ̃$: A random threshold; fixed in the toy.
$x$: An input vector.

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.

Input x

This example’s data flow

You choose

input.x = 1

Initial values in code

directions = [-1, 1]
threshold = 0.5

Calculated here

Result expression: { pairs, probabilities: [0.5, 0.5], indicators, expectation: (indicators[0] + indicators[1]) / 2, }

indicators = directions.map((direction) => direction * input.x - threshold > 0 ? 1 : 0, ) Uses: directions, input.x, threshold
pairs = directions.map((direction) => [direction, threshold]) Uses: directions, threshold

Returns

Example result

pairs: array (2 items): [array (2 items): [-1, 0.5], array (2 items): [1, 0.5]]
probabilities: array (2 items): [0.5, 0.5]
indicators: array (2 items): [0, 1]
expectation: 0.5

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