The Modern Mathematics of Deep Learning

Neural tangent kernel at parameters

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.

Kernels and linearization · Relate parameter gradients or features to kernel matrices.

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: 2.1 A view through similarity

The neural tangent kernel is the dot product of the parameter gradients at two inputs. Its subscript records which parameter vector is used.

It compares how network outputs change when parameters change.

Source formula example

K_{\theta}

illustration · This example uses Φ(x,θ)=θ₁x+θ₂, whose kernel happens not to depend on θ; general networks can change their kernel as parameters change.

Symbols

$Kθ$: The neural tangent kernel at θ.
$Θ₀$: Initial parameters.
$θ(t)$: Parameters at time t.

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

x1 = 2
x2 = 3

Calculated here

Result expression: { gradient1, gradient2, kernel: gradient1.reduce((s, v, i) => s + v * gradient2[i], 0), }

gradient1 = [x1, 1] Uses: x1
gradient2 = [x2, 1] Uses: x2

Returns

Example result

gradient1: array (2 items): [2, 1]
gradient2: array (2 items): [3, 1]
kernel: 7

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. This example uses Φ(x,θ)=θ₁x+θ₂, whose kernel happens not to depend on θ; general networks can change their kernel as parameters change.

Source: Inline notation