The Modern Mathematics of Deep Learning

A class of loss functions

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.

Generalization and risk bounds · Relate sample behavior to population risk under stated assumptions.

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: 1.2.3 Does it work on new examples?

𝒢 contains the functions z ↦ ℒ(f,z) obtained by applying the loss to each model f in the hypothesis class.

This class allows a uniform comparison of losses across models.

Source formula example

{\mathcal{G}}

illustration · This finite class uses two constant predictors and two targets; it does not establish a generalization bound.

Symbols

$𝒢$: The class of loss functions induced by the model class.
$f$: A prediction function.
$Z⁽ⁱ⁾$: A random data example.
$ℒ$: The loss function.

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

targets = [0, 1]
predictions = [0, 1]

Calculated here

Result expression: { targets, predictions, losses }

losses = predictions.map((f) => targets.map((y) => (f - y) ** 2)) Uses: predictions, targets

Returns

Example result

targets: array (2 items): [0, 1]
predictions: array (2 items): [0, 1]
losses: array (2 items): [array (2 items): [0, 1], array (2 items): [1, 0]]

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. This finite class uses two constant predictors and two targets; it does not establish a generalization bound.

Source: Inline notation