The Modern Mathematics of Deep Learning

Risk along a weight interpolation

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

Compare loss at two endpoints and a parameter interpolated between them.

A path with controlled loss helps analyze the loss landscape.

Source formula example

\max\{{r}(\theta),{r}(\theta_{\min})\}

illustration · This simple convex quadratic does not establish a low-loss path for a general network.

Symbols

$r$: Risk as a function of network parameters.
$θ$: One endpoint parameter vector.
$θmin$: A minimum used as the other endpoint in the source.

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

Initial values in code

first = 0
minimum = 1

Calculated here

Result expression: { weight, endpointRisks: [risk(first), risk(minimum)], pathRisk: risk(weight), endpointMaximum: Math.max(risk(first), risk(minimum)), }

t = input.t Uses: input.t
weight = (1 - t) * first + t * minimum Uses: t, first, minimum

Returns

Example result

weight: 0.5
endpointRisks: array (2 items): [1, 0]
pathRisk: 0.25
endpointMaximum: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This invented scalar quadratic has endpoint θ = 0, a minimum θmin = 1, and the interpolated midpoint θ = 0.5. The source constructs particular paths using fixed hidden features and a minimum; the toy illustrates the endpoint maximum.

Source: Source expression