The Modern Mathematics of Deep Learning

Bounded losses

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?

The condition ℒ(f,Z) ∈ [0,1] means every considered loss lies between zero and one, including the endpoints.

A bounded loss supports the stated probability analysis.

Source formula example

\mathcal{L}(f,Z)\in[0,1]

illustration · Checking finitely many values does not prove a uniform bound over all outcomes.

Symbols

$ℒ$: The loss.
$f$: A prediction function.
$Z$: A random example.

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

losses = [0, 0.25, 1]

Calculated here

Result expression: { losses, bounded: losses.every((v) => v >= 0 && v <= 1) }

Returns

Example result

losses: array (3 items): [0, 0.25, 1]
bounded: true

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. Checking finitely many values does not prove a uniform bound over all outcomes.

Source: Inline notation