The Modern Mathematics of Deep Learning

A nonconvex training objective

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.

Optimization theory · Study convergence and loss geometry under assumptions.

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: 1.3 The questions ahead

The paper gives two parameter vectors for the same two-unit ReLU network. Both fit the two training points exactly, but their midpoint has training loss 1/2.

A midpoint with higher loss than both endpoints shows this parameter-space objective is not convex.

Source formula example

\widehat{\mathcal{R}}_{{s}}(\Phi(\cdot,\theta))=\widehat{\mathcal{R}}_{{s}}(\Phi(\cdot,\bar{\theta}))=0\quad\text{and}\quad\widehat{\mathcal{R}}_{{s}}\left(\Phi(\cdot,(\theta+\bar{\theta})/2)\right)=\frac{1}{2},

exact · The result is exact for these finite toy inputs, apart from ordinary computer rounding.

Symbols

$θ,θ̄$: Two six-dimensional parameter vectors from the source example.
$s$: The two source data points (−1,0) and (1,1).

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.

Blend t

This example’s data flow

You choose

input.t = 0.5

Initial values in code

theta = [1, -1, 1, 1, 1, 0]
other = [-1, 1, 1, 1, 0, 1]
xs = [-1, 1]
ys = [0, 1]

Calculated here

Result expression: { first: evaluate(theta), second: evaluate(other), blend: evaluate(blend) }

blend = theta.map((value, i) => (1 - input.t) * value + input.t * other[i]) Uses: theta, input.t, other

Returns

Example result

first: predictions: array (2 items): [0, 1] · risk: 0
second: predictions: array (2 items): [0, 1] · risk: 0
blend: predictions: array (2 items): [0, 0] · risk: 0.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source uses θ=(1,−1,1,1,1,0), θ̄=(−1,1,1,1,0,1), and mean squared loss on the two displayed examples. The vectors swap the hidden units. At their midpoint the two output weights are zero, giving the zero predictor.

Source: Nonconvex example