The Modern Mathematics of Deep Learning

Neural tangent kernel

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

Compute the gradient of each output with respect to all parameters, then take their dot product.

For the toy linear predictor Φ(x,θ) = θx, the parameter gradient is x.

This kernel compares how two outputs respond to parameter changes.

Source formula example

K_{\theta}\colon{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}},\quad K_{\theta}(x_{1},x_{2})=\big(\nabla_{\theta}\Phi(x_{1},\theta)\big)^{T}\nabla_{\theta}\Phi(x_{2},\theta).

illustration · The example uses a scalar linear model. General networks require their full parameter gradients.

Symbols

$K_θ$: Kernel at parameter vector θ.
$∇_θΦ$: Gradient of network output with respect to its parameters.

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.

First inputSecond input

This example’s data flow

You choose

input.x1 = 2
input.x2 = 3

Calculated here

Result expression: { gradient1, gradient2, kernel: gradient1 * gradient2 }

gradient1 = input.x1 Uses: input.x1
gradient2 = input.x2 Uses: input.x2

Returns

Example result

gradient1: 2
gradient2: 3
kernel: 6

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

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

Source: Source expression