The Modern Mathematics of Deep Learning

Binary classification 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 · 1.2.3 Does it work on new examples?

For a binary label y and signed prediction f(x), this loss is 1 when their product is negative and 0 otherwise. When both are ±1, it also equals (1 − yf(x))/2.

This measures whether a signed binary prediction is incorrect.

Source formula example

{\mathcal{L}}(f,(x,y))=\mathds{1}_{(-\infty,0)}(yf(x))

illustration · The second formula requires both prediction and label to be −1 or 1; zero is not counted as a negative product.

Symbols

$x$: An input.
$y$: A binary label.
$f(x)$: A signed prediction.
$ℒ$: The loss.

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

y = -1
prediction = 1

Calculated here

Result expression: { y, prediction, indicator, binaryFormula: (1 - y * prediction) / 2 }

indicator = y * prediction < 0 ? 1 : 0 Uses: y, prediction

Returns

Example result

y: -1
prediction: 1
indicator: 1
binaryFormula: 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 second formula requires both prediction and label to be −1 or 1; zero is not counted as a negative product.

Source: Inline notation