The Modern Mathematics of Deep Learning

Margin surrogate loss

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.

Individual losses · Give one observed example a loss score.

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: 2.2 Weight size and room to decide

For positive γ, this loss is 1 below zero, decreases linearly to zero on [0, γ], and is zero above γ.

A margin loss gives a smooth transition around the classification boundary.

Source formula example

\ell_{\gamma}(t)\coloneqq\mathds{1}_{(-\infty,\gamma]}(t)-\tfrac{t}{\gamma}\mathds{1}_{[0,\gamma]}(t),\quad t\in{\mathbb{R}}.

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

Symbols

$t$: A real margin.
$γ$: A positive margin threshold.

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.

Margin tγ

This example’s data flow

You choose

input.gamma = 1
input.margin = 0.5

Calculated here

Result expression: { loss: indicator1 - (t / input.gamma) * indicator2 }

t = input.margin Uses: input.margin
indicator1 = t <= input.gamma ? 1 : 0 Uses: t, input.gamma
indicator2 = t >= 0 && t <= input.gamma ? 1 : 0 Uses: t, input.gamma

Returns

Example result

loss: 0.5

JavaScript

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

Result

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Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Source expression