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Re78: Recap of Gradients and Partial Derivatives (AIMA4e pp. 119-122)

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An overview of Re70-Re76. The airport problem; squares vs. sums of distances; the six-dimensional solution space for three airports, many cities; the gradient as `slope' toward the best solution; the two-dimensional solution space for one airport, two cities; vector calculus as `pile of numbers' approach; local- vs. global-best solutions; the partial derivatives that make up the gradient, a total derivative; solving the two-cities-one-airport problem by gradient descent.

Air date: Sunday, 11th Dec. 2022, 11:00 PM Eastern/US.

We're focusing on the math and code of AIMA4e^1 right now, December 2022. This is in service of our plan to deep-dive the book from Jan.-Jun., 2023.

The airport problem from p. 120: Where to locate three airports amongst several cities in Romania such that the sum of the squares of the straight-line distances between each city and its nearest airport (which is a good way of measuring how well the airports have been placed) is minimized?

Re70: Gradients and Partial Derivatives Part 1 (AIMA4e pp. 119-122)^2

The math of `Local Search in Continuous Spaces'. The three airports and the six-dimensional vector that represents their coordinates and therefore solutions to the problem; why the sums of squares of the distances reflects better and worse airport locations; how the gradient is like a `slope' in the space of better and worse solutions; the gap between passive and active knowledge and a demonstration of confidence being calibrated.

Re71: Gradients and Partial Derivatives Part 2 (AIMA4e pp. 119-122)^3

Put the airport problem first. A simple, two-city-one-airport version of the problem; how vectors and calculus can represent this real-world problem as a `pile of numbers'; the airport coordinates as independent variables; the objective function as dependent variable; the gradient as tool for improving our first guess-hypothesis; local vs. global solution optimization.

Re72: Gradients and Partial Derivatives Part 3 (AIMA4e pp. 119-122)^4

Be in the math. The four key equations that define our problem and solutions--the solution guesses (vectors), the objective function of the solution guesses (our `score' or `cost' to be minimized), the gradient vector (the `slope' toward better and worse solutions), and the six partial derivatives that combine to make up our gradient vector (a six-dimensional `slope').

Re73: Gradients and Partial Derivatives Part 4 (AIMA4e pp. 119-122)^5

The limits that define our gradient. Discussion of how the gradient will indicate the `direction' of improving our first guess; full expansion of the gradient for the three-airport problem into six equations of partial derivatives and limits, each with respect to a different independent variable (airport partial coordinate).

Re74: Gradients and Partial Derivatives Part 5 (AIMA4e pp. 119-122)^6

Bringing the algebra back down to numbers. Calculating squares of distances on the map of Romania; a first look at the algebra of calculating the objective function given the coordinates of cities and airports.

Re75: Gradients and Partial Derivatives Part 6 (AIMA4e pp. 119-122)^7

Can we please just place an airport? A first airport guess for the two-city version of the problem; calculating the objective function value given that first guess and the coordinates of our two cities.

Re76: Gradients and Partial Derivatives Part 7 (AIMA4e pp. 119-122)^8

Moving the airport to improve its value. Mapping one, two and three guesses as directed by gradient descent calculations; the lucky-unlucky choice of our first airport location; the hand-math algebraic manipulations and the spreadsheet-math iterations that show the behavior of the partial derivatives in the limit; the common-sense basis of our confidence in the solution provided by gradient descent.

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References

Retraice (2022/12/04). Re70: Gradients and Partial Derivatives Part 1 (AIMA4e pp. 119-122). retraice.com. https://www.retraice.com/segments/re70 Retrieved 5th Dec. 2022.

Retraice (2022/12/05). Re71: Gradients and Partial Derivatives Part 2 (AIMA4e pp. 119-122). retraice.com. https://www.retraice.com/segments/re71 Retrieved 6th Dec. 2022.

Retraice (2022/12/06). Re72: Gradients and Partial Derivatives Part 3 (AIMA4e pp. 119-122). retraice.com. https://www.retraice.com/segments/re72 Retrieved 7th Dec. 2022.

Retraice (2022/12/07). Re73: Gradients and Partial Derivatives Part 4 (AIMA4e pp. 119-122). retraice.com. https://www.retraice.com/segments/re73 Retrieved 8th Dec. 2022.

Retraice (2022/12/08). Re74: Gradients and Partial Derivatives Part 5 (AIMA4e pp. 119-122). retraice.com. https://www.retraice.com/segments/re74 Retrieved 9th Dec. 2022.

Retraice (2022/12/09). Re75: Gradients and Partial Derivatives Part 6 (AIMA4e pp. 119-122). retraice.com. https://www.retraice.com/segments/re75 Retrieved 10th Dec. 2022.

Retraice (2022/12/10). Re76: Gradients and Partial Derivatives Part 7 (AIMA4e pp. 119-122). retraice.com. https://www.retraice.com/segments/re76 Retrieved 11th Dec. 2022.

Russell, S., & Norvig, P. (2020). Artificial Intelligence: A Modern Approach. Pearson, 4th ed. ISBN: 978-0134610993. Searches: https://www.amazon.com/s?k=978-0134610993 https://www.google.com/search?q=isbn+978-0134610993 https://lccn.loc.gov/2019047498

Footnotes

^1 Russell & Norvig (2020). ^2 Retraice (2022/12/04). ^3 Retraice (2022/12/05). ^4 Retraice (2022/12/06). ^5 Retraice (2022/12/07). ^6 Retraice (2022/12/08). ^7 Retraice (2022/12/09). ^8 Retraice (2022/12/10).