The Modern Mathematics of Deep Learning

Smoothed margin

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.

Optimization theory · Study convergence and loss geometry under 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: 2.3 The path matters

The smoothed margin rβ(θ) is −log(r(βθ))/β for the exponential loss, with β positive.

The source studies its relation to the minimum training margin.

Source formula example

r_{\beta}

illustration · The example has one positive-margin sample; the general formula averages over all samples.

Symbols

$r$: Empirical exponential risk.
$β$: A positive scaling parameter.
$θ$: The parameter vector.
$rβ$: The smoothed margin.

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

beta = 2
theta = 1

Calculated here

Result expression: { beta, theta, risk, smoothedMargin: -Math.log(risk) / beta }

risk = Math.exp(-beta * theta) Uses: beta, theta

Returns

Example result

beta: 2
theta: 1
risk: ≈0.1353
smoothedMargin: 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 has one positive-margin sample; the general formula averages over all samples.

Source: Inline notation