The Modern Mathematics of Deep Learningexplained for non-mathematicians

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Hypothesis sets

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.

Model classes and structure Β· Describe families of rules, their size and structural properties.

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 Γ— 2; Held-out evaluation: 102 Β· #58 Β· 40.6 mm, 193 mm. Known answer: AdΓ©lie; Prediction: AdΓ©lie Β· [β‰ˆ0.7383, β‰ˆ0.2444, β‰ˆ0.01737].

Used in these source sections: 1.2 Learning from examples Β· 1.2.2 Can the network express the answer? Β· 1.2.3 Does it work on new examples? Β· More source sections

A hypothesis set 𝓕 contains the allowed prediction functions. f ∈ 𝓕 says the chosen rule is one of them.

Learning chooses a rule from a specified class.

Source formula example

FβŠ‚M(X,Y){\mathcal{F}}\subset{\mathcal{M}}({\mathcal{X}},{\mathcal{Y}})

illustration Β· The arrays list outputs on the two finite inputs; the source allows more general function classes.

Symbols

F\mathcal{F}
A set of measurable functions from 𝒳 to 𝒴.
ff
One function in the hypothesis set.
M(𝒳,𝒴)\mathcal{M}(𝒳,𝒴)
The measurable functions from inputs to outputs.

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

  • domain = [0, 1]
  • hypotheses = [ [0, 0], [0, 1], ]

Calculated here

Result expression: { domain, hypotheses, chosen, belongs: hypotheses.includes(chosen) }

  • chosen = hypotheses[1] Uses: hypotheses

Returns

Example result
domain
array (2 items): [0, 1]
hypotheses
array (2 items): [array (2 items): [0, 0], array (2 items): [0, 1]]
chosen
array (2 items): [0, 1]
belongs
true

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 arrays list outputs on the two finite inputs; the source allows more general function classes.

Source: Inline notation