The Modern Mathematics of Deep Learning

A function defined by expectation

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.

Probability and expectation · Describe random variables and their averages under a law.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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: 4.3 Structure from an equation

For each x, Yₓ is a random value. The output g(x) is its expectation.

In this toy Yₓ takes x−1 and x+1 with equal probability, so its expectation equals x.

Many stochastic representations express a function as an average.

Source formula example

g(x)={\mathbbm{E}}[Y_{x}],\quad x\in{\mathbb{R}}^{d},

exact · The toy computes an expectation exactly because its complete finite distribution is specified. A sample mean generally differs from the expectation.

Symbols

$Y_x$: A random variable whose distribution can depend on x.
$x$: A real d-vector; the toy is one-dimensional.

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.

x

This example’s data flow

You choose

input.x = 2

Initial values in code

probabilities = [0.5, 0.5]

Calculated here

Result expression: { values, probabilities, expectation }

values = [input.x - 1, input.x + 1] Uses: input.x
expectation = values.reduce((sum, value, i) => sum + value * probabilities[i], 0) Uses: values, probabilities

Returns

Example result

values: array (2 items): [1, 3]
probabilities: array (2 items): [0.5, 0.5]
expectation: 2

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