The Modern Mathematics of Deep Learning

Empirical minimizers

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.

Learning and parameter updates · Select a rule or update parameters using a learning rule.

The current objective supplies a gradient; training updates the model parameters.

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: 1.1 A note on symbols · 1.2 Learning from examples · 1.3 The questions ahead

An empirical minimizer is an allowed function with the smallest average loss on the training sample.

It selects the best training loss within a class.

Source formula example

\widehat{f}_{{s}}\in{\mathcal{F}}

illustration · The finite example has two candidates; a minimum need not exist in every infinite class.

Symbols

$f̂_{s}$: A minimizer of empirical risk for sample s.
$\mathcal{F}$: The class of allowed functions.
$s$: The training sample.

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.

This example’s data flow

Initial values in code

candidates = [ { name: "a", losses: [1, 1] }, { name: "b", losses: [0, 0] }, ]

Calculated here

Result expression: { risks, selected: candidates[index].name }

risks = candidates.map( (c) => c.losses.reduce((a, b) => a + b, 0) / c.losses.length, ) Uses: candidates
index = risks.indexOf(Math.min(...risks)) Uses: risks

Returns

Example result

risks: array (2 items): [1, 0]
selected: b

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. The finite example has two candidates; a minimum need not exist in every infinite class.

Source: Inline notation