The Modern Mathematics of Deep Learning

Logistic 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: 1.2 Learning from examples · 2.2 Weight size and room to decide

Multiply a score by its binary label, change the sign, exponentiate, add one, and take the natural logarithm.

A surrogate loss gives a numerical score for a classification prediction.

Source formula example

{\mathcal{L}}^{\mathrm{surr}}(f,z)=\log\left(1+e^{-yf(x)}\right).

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

Symbols

$y$: A binary label −1 or +1.
$f(x)$: A real-valued prediction score.

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.

LabelScore

This example’s data flow

You choose

input.label = 1
input.score = 0

Calculated here

Result expression: { margin, loss: Math.log(1 + Math.exp(-margin)) }

margin = input.label * input.score Uses: input.label, input.score

Returns

Example result

margin: 0
loss: ≈0.6931

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

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

Source: Source expression