The Modern Mathematics of Deep Learning

Residual difference

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.

Residual networks · Describe residual layer updates and their continuous analogue.

Convolution, recurrent and residual architectures are not part of this linear penguin model.

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

Subtract the previous state from the new state to recover the added activated value.

This identity relates a residual layer to a discrete update.

Source formula example

\bar{\Phi}^{(\ell)}-\bar{\Phi}^{(\ell-1)}=\varrho(\Phi^{(\ell)})\eqqcolon h(\ell,\Phi^{(\ell)}),

exact · A single discrete update does not prove an ODE approximation.

Symbols

$Φ̄⁽ˡ⁾$: Residual network output at layer ℓ.
$Φ⁽ˡ⁾$: Layer preactivation.
$ϱ$: Activation.
$h$: Activated increment.

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.

Previous state

This example’s data flow

You choose

input.previous = 2

Calculated here

Result expression: { next, difference: next - previous, activated }

previous = input.previous Uses: input.previous
activated = Math.max(0, 3)
next = previous + activated Uses: previous, activated

Returns

Example result

next: 5
difference: 3
activated: 3

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy uses one scalar and ReLU.

Source: Source expression