The Modern Mathematics of Deep Learningexplained for non-mathematicians

Concepts / Generalization

Linearizing a network

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.

Prediction rules and layers · Describe or apply prediction rules, layers and parameters.

The project uses a linear rule with two features. The fitted weights produce scores or normalized mass.

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: 2.1 A view through similarity

Take the original output at the starting parameters and add the dot product of its parameter gradient with the parameter change.

The toy model uses Φ(x,θ) = θ²x and start Θ₀ = 1.

A linearized model approximates the change in a network near its starting parameters.

Source formula example

Φlin(⋅,θ)≔Φ(⋅,Θ0)+⟨∇θΦ(⋅,Θ0),θ−Θ0⟩,\Phi^{\mathrm{lin}}(\cdot,\theta)\coloneqq\Phi(\cdot,\Theta_{0})+\langle\nabla_{\theta}\Phi(\cdot,\Theta_{0}),\theta-\Theta_{0}\rangle,

approximation · This computes the first-order approximation of a particular smooth scalar model. The linearized value generally differs from the original model away from the starting parameter.

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

You choose

  • input.x = 2
  • input.theta = 1.5

Initial values in code

  • start = 1

Calculated here

Result expression: { startOutput, gradient, linearized, actual }

  • x = input.x Uses: input.x
  • startOutput = start ** 2 * x Uses: start, x
  • gradient = 2 * start * x Uses: start, x
  • linearized = startOutput + gradient * (input.theta - start) Uses: startOutput, gradient, input.theta, start
  • actual = input.theta ** 2 * x Uses: input.theta, x

Returns

Example result
startOutput
2
gradient
4
linearized
4
actual
4.5

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

Result

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Technical statement

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

Source: Source expression