The Modern Mathematics of Deep Learning

An SGD convergence bound

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: 1.2.1 Finding better settings

Compare expected objective gap to 1/√K for a case where the objective is constant.

This example makes the source statement computable on a finite model.

Source formula example

{\mathbbm{E}}[r(\bar{\Theta})]-r(\theta^{*})\leq\frac{1}{\sqrt{K}},

illustration · This degenerate exact case does not prove the SGD convergence theorem.

Symbols

$r$: Differentiable convex objective on the unit ball.
$Θ̄$: Random output of the SGD procedure.
$θ*$: A minimizer in the unit ball.
$K$: Positive iteration count.

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.

Iterations K

This example’s data flow

You choose

input.steps = 4

Initial values in code

expectedObjective = 0
minimumObjective = 0

Calculated here

Result expression: { expectedGap: expectedObjective - minimumObjective, bound: 1 / Math.sqrt(input.steps), }

Returns

Example result

expectedGap: 0
bound: 0.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy has r=0 on the unit ball, initialization 0, all stochastic directions 0, and θ*=0. The source theorem assumes differentiable convex r, starts at zero, step sizes K⁻¹ᐟ², directions of Euclidean norm at most 1 almost surely, and the output rule in Algorithm 1.

Source: Source expression