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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

February 24th, 2021, 10:14 pm

How can we teach Cauchy to engineers and computer scientists? They don't teach that in school, Maybe it's because ML and and ANNs don't do Cauchy sequnces.

SGD: It's like getting a flat tire and fixing the problem by driving around the block a few times; maybe it will fix itself.

The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination.

I repeat for our CS friends and partners.

convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms.

And since we work in a complete RKHS space, it's bingo.
Last edited by Cuchulainn on February 24th, 2021, 10:29 pm, edited 2 times in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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http://www.datasim.nl
 
User avatar
Cuchulainn
Topic Author
Posts: 64438
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

February 24th, 2021, 10:21 pm

An ML/RKHS article with "Cauchy seq" inside

https://arxiv.org/pdf/0805.2368.pdf

Take heed.  A-priori in contrast to popular a-posteriori ML science.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl