The Modern Mathematics of Deep Learning

A feature is an output-weight 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.

Features · Transform inputs into features used by a rule.

The project scales bill and flipper measurements using training-row means and scales; its feature vector has two entries.

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: 5.2 When parameters move only a little

When the hidden features are fixed, the derivative of the output with respect to an output weight equals that unit’s feature.

These quantities connect network parameters to loss and optimization.

Source formula example

\big(\nabla_{\theta^{(2)}}\Phi(x,\Theta)\big)_{k}=\varrho\left(\left\langle\Theta^{(1)}_{k},\begin{bmatrix}x\\ 1\end{bmatrix}\right\rangle\right),\quad x\in{\mathbb{R}}^{d},\ k\in[n].

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$Θ⁽¹⁾ₖ$: Incoming weight and bias vector of unit k.
$Θ⁽²⁾$: Output weights.
$x$: Input vector in ℝᵈ.
$ϱ$: Activation function.
$k$: Hidden-unit index.

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

This example’s data flow

You choose

input.x = 1

Calculated here

Result expression: { feature, derivative: feature }

feature = Math.max(0, input.x + 1) Uses: input.x

Returns

Example result

feature: 2
derivative: 2

JavaScript

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

Result

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Technical statement

The hidden unit is ReLU(x + 1). Its output weight appears linearly; this derivative is exact at every input.

Source: Source expression