The Modern Mathematics of Deep Learning

A nonzero activation derivative

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: 6.2 Shortcuts between layers

Some activation functions have an input where their derivative is nonzero.

This example makes the source notation or condition concrete.

Source formula example

\varrho^{\prime}(\lambda)\neq 0

illustration · One sigmoid value does not establish the condition for all activations.

Symbols

$ϱ$: Activation function.
$λ$: Real input.
$ϱ^{\prime}(λ)$: Derivative at λ.

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.

Input λ

This example’s data flow

You choose

input.lambda = 0

Calculated here

Result expression: { activation, derivative, nonzero: derivative !== 0 }

lambda = input.lambda Uses: input.lambda
activation = 1 / (1 + Math.exp(-lambda)) Uses: lambda
derivative = activation * (1 - activation) Uses: activation

Returns

Example result

activation: 0.5
derivative: 0.25
nonzero: true

JavaScript

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

Result

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Technical statement

The sigmoid activation is differentiable, and at λ=0 its derivative is 1/4. The source discusses a nonzero derivative condition when motivating local behavior of layers.

Source: Source expression