The Modern Mathematics of Deep Learning

Minimum training 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.

Risk and assessment · Distinguish loss on a sample from expected loss.

The fitted rule is evaluated on 102 held-out observations, not used for fitting.

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

Multiply each linear prediction by its label, then take the smallest result. For binary labels ±1, a positive margin means every listed example is classified correctly.

The smallest margin records the least confident training example.

Source formula example

\gamma(\theta)=\min_{i\in[m]}y^{(i)}\langle\theta,x^{(i)}\rangle

exact · The result is exact for these finite toy inputs, apart from ordinary computer rounding.

Symbols

$y^{(i)}$: Binary label, either −1 or +1 in this example.
$θ$: Real-valued linear model weights.

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.

Weight

This example’s data flow

You choose

input.weight = 2

Initial values in code

xs = [1, -2]
labels = [1, -1]

Calculated here

Result expression: { margins, minimum: Math.min(...margins) }

margins = xs.map((x, i) => labels[i] * input.weight * x) Uses: xs, labels, input.weight

Returns

Example result

margins: array (2 items): [2, 4]
minimum: 2

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

This computes the minimum over the displayed finite training set. A positive training margin does not guarantee performance on new data.

Source: Margin