The Modern Mathematics of Deep Learning

Residual layer

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

Add the previous layer output to the activated new output. The two outputs must have matching shapes.

The direct addition carries earlier values forward.

Source formula example

\bar{\Phi}^{(\ell)}(x,\theta)=\varrho(\Phi^{(\ell)}(x,\theta))+\bar{\Phi}^{(\ell-1)}(x,\theta),

illustration · This is one scalar residual addition. It does not demonstrate a network-wide optimization guarantee.

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 outputNew pre-activation

This example’s data flow

You choose

input.previous = 2
input.newValue = 3

Calculated here

Result expression: { previous, activated: newValue, output: previous + newValue }

previous = input.previous Uses: input.previous
newValue = Math.max(0, input.newValue) Uses: input.newValue

Returns

Example result

previous: 2
activated: 3
output: 5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Source expression