The Modern Mathematics of Deep Learning

The marked risk identity

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.

Source assumptions and notation · Interpret symbols and assumptions in a source statement.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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?

The (*) marks equalities justified using the earlier squared-loss identity. It is an annotation rather than a new variable.

The annotation identifies where the identity is applied.

Source formula example

(*)

illustration · The example checks the identity under one finite squared-loss distribution; the marker in the source refers to the general lemma.

Symbols

$(*)$: A marker referring to the earlier risk identity.
$f*$: The Bayes-optimal predictor.

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

Initial values in code

target = [0, 2]
probabilities = [0.5, 0.5]
optimal = 1
prediction = 2

Calculated here

Result expression: { excessRisk: risk(prediction) - risk(optimal), squaredDifference: (prediction - optimal) ** 2, }

Returns

Example result

excessRisk: 1
squaredDifference: 1

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 example checks the identity under one finite squared-loss distribution; the marker in the source refers to the general lemma.

Source: Inline notation