The Modern Mathematics of Deep Learning

A ReLU derivative identity

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.

Functions and derivatives · Describe maps and how their outputs change.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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

Away from zero, ReLU has derivative 1 for positive inputs and 0 for negative inputs. Multiplying that derivative by x gives the original ReLU value.

This identity is used in the source discussion of adjacent-layer balance.

Source formula example

\varrho(x)=\varrho^{\prime}(x)x

exact · The classical derivative of ReLU is not defined at x=0, so the code rejects that point. The identity also holds for other activation functions satisfying the stated condition.

Symbols

$ρ^{\prime}$: Derivative of the activation where it exists.

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.

Nonzero x

This example’s data flow

You choose

input.x = 2

Calculated here

Result expression: { value, derivative, derivativeTimesX: derivative * x }

x = input.x Uses: input.x
value = Math.max(0, x) Uses: x
derivative = x > 0 ? 1 : 0 Uses: x

Returns

Example result

value: 2
derivative: 1
derivativeTimesX: 2

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: Activation derivative identity