The Modern Mathematics of Deep Learningexplained for non-mathematicians

Concepts / Generalization

A width limit and a kernel space

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 · 3.1 A rule based on distance · 3.2 Simple pieces, many layers · More source sections

The source studies a kernel limit as hidden widths increase and the corresponding reproducing-kernel Hilbert space.

For the toy network Φₙ(x)=n⁻¹ᐟ²∑aⱼReLU(wⱼx), initialize every aⱼ=0 and wⱼ=1. Its full parameter-gradient kernel is ReLU(x)ReLU(y) for every n.

The kernel space gives a function norm associated with the limiting kernel.

Source formula example

N1,…,NL−1→∞,(HK∞,∥⋅∥K∞)N_1,\ldots,N_{L-1}\to\infty,\quad(\mathcal H_{K^\infty},\|\cdot\|_{K^\infty})

illustration · This special deterministic initialization is not the source’s Gaussian initialization and does not prove a general infinite-width theorem. The source kernel space may be more complicated.

Symbols

K∞K^∞
The source limiting kernel.
HK∞ℋ_{K^∞}
The Hilbert space associated with that kernel.
nn
Finite hidden width in the toy.

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.n = 3
  • input.x = 2
  • input.y = 3
  • input.a = 2

Calculated here

Result expression: { width: n, featureX, featureY, kernel, functionValue: a * featureX, functionNorm: Math.abs(a), }

  • n = input.n Uses: input.n
  • x = input.x Uses: input.x
  • y = input.y Uses: input.y
  • a = input.a Uses: input.a
  • featureX = Math.max(0, x) Uses: x
  • featureY = Math.max(0, y) Uses: y
  • outputGradX = Array(n).fill(featureX / Math.sqrt(n)) Uses: n, featureX
  • outputGradY = Array(n).fill(featureY / Math.sqrt(n)) Uses: n, featureY

More expressions in code

Returns

Example result
width
3
featureX
2
featureY
3
kernel
6
functionValue
4
functionNorm
2

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

Result

Enable JavaScript to run this example.

Technical statement

At the toy initialization, gradients with respect to w vanish because all output weights aⱼ are zero. The remaining n gradient products sum to ReLU(x)ReLU(y), independent of n. For this rank-one kernel, functions are f(x)=a ReLU(x) and their kernel-space norm is |a|. This is a special example of the concepts, not the paper’s general limit.

Source: Kernel limit and its Hilbert space