The Modern Mathematics of Deep Learning

An optimization objective

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.

Objectives · Combine losses and penalties into a criterion for parameters or rules.

Project predictions and known answers produce losses. Classification uses cross-entropy and α/2 times squared non-bias weights; Ridge uses normalized-mass MSE and α times all squared coefficients, including the intercept.

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: 1.2.1 Finding better settings · 5.1 The shape of training loss · 6.4 Adjusting the scale

The function r maps parameters to the real objective value being optimized.

An optimizer uses objective values and derivatives to choose parameter updates.

Source formula example

r

illustration · The example uses a quadratic objective; the source discusses more general objectives.

Symbols

$r$: An objective function.
$θ$: A parameter, scalar in this 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

theta = 2

Calculated here

Result expression: { theta, objective: r(theta), gradient: 2 * theta }

Returns

Example result

theta: 2
objective: 4
gradient: 4

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 unless the source values are stated. The example uses a quadratic objective; the source discusses more general objectives.

Source: Inline notation