The Modern Mathematics of Deep Learning

Two connected parameter paths

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.

Optimization theory · Study convergence and loss geometry under assumptions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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: 5.1 The shape of training loss

A path assigns a parameter value to each interpolation value. Two paths can meet at a shared parameter.

The source constructs connected paths to analyze loss between parameter settings.

Source formula example

\gamma_{1}

illustration · These invented scalar paths do not construct the source’s feature-duplication path or establish its loss bound.

Symbols

$γ₁,γ₂$: Paths in the network parameter space.
$t$: Interpolation parameter in [0,1] for this teaching example.

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.

Interpolation

This example’s data flow

You choose

input.t = 0.5

Calculated here

Result expression: { gamma1, gamma2, sharedEndpoint: 1 }

t = input.t Uses: input.t
gamma1 = (1 - t) * 0 + t * 1 Uses: t
gamma2 = (1 - t) * 1 + t * 2 Uses: t

Returns

Example result

gamma1: 0.5
gamma2: 1.5
sharedEndpoint: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy has γ₁(t) = t and γ₂(t) = 1+t. They meet because γ₁(1) = γ₂(0) = 1. The source paths depend on network parameters and construction assumptions.

Source: Source expression