[ci][docs] fix link checking action by switching from linkchecker to lychee and update some links

[StrikerRUS]

All our recent link check jobs are failing: https://github.com/microsoft/LightGBM/actions/workflows/linkchecker.yml. Last successful job was 5 months ago.

The most errors are about 429 Rate Limit GitHub urls, but others are quite random ones. It looks like our current linkchecker isn't reliable enough.

Initially I thought to use linkinator but then found lychee. This tool looks quite simple to use but powerful.

Latest linkchecker results:

That's it. 1058 links in 1076 URLs checked. 3 warnings found. 4 errors found.
Stopped checking at 2025-09-11 08:54:22+000 (48 minutes, 31 seconds)

New lychee results:

🔍 2188 Total (in 27s) ✅ 2167 OK 🚫 0 Errors 👻 21 Excluded

I left some inline comments below to help understand some non obvious moments.

[jameslamb]

Thanks for all the hard work! This is awesome, I'm glad to see so many broken links fixed and outdated documentation removed.

I totally support switching to lychee. I've been using it in pydistcheck for a little while and really liking it.

Would you consider (can be a follow-up) using https://github.com/lycheeverse/lychee-action instead?

Benefits of this:

• could remove more code from test-docs.sh
• it's very fast (no need for conda, for example)

Here's an example of a setup like that: jameslamb/pydistcheck#312

I'm approving this PR because I'd be happy to see it merged as-is. Everything I've suggested could be follow-up PRs, if you agree with the suggestinos.

[jameslamb]

I think we should also add this to lychee.toml: https://lychee.cli.rs/recipes/excluding-paths/

The more that gets put into that file, the easier it'd be to run this interactively during development.

[StrikerRUS]

@jameslamb Thanks a lot for reviewing this! Nice to hear that you are already familiar with lychee 😃.

Would it be OK for you if I push changes for GitHub token and paths exclusion in this PR, and move this to GitHub Action in a follow-up PR?

[jameslamb]

Would it be OK for you if I push changes for GitHub token and paths exclusion in this PR, and move this to GitHub Action in a follow-up PR?

Yes absolutely!

[StrikerRUS]

@jameslamb I've pushed changes I promised you in #7027 (comment). Could you please check them when have time? Please notice I have to add stackoverflow site to ignore list due to the following errors:

Issues found in 3 inputs. Find details below.

[docs/_build/html/FAQ.html]:
[403] https://stackoverflow.com/questions/18085571/pip-install-error-setup-script-specifies-an-absolute-path | Rejected status code (this depends on your "accept" configuration): Forbidden

[docs/_build/html/Parameters-Tuning.html]:
[403] https://stats.stackexchange.com/questions/317073/explanation-of-min-child-weight-in-xgboost-algorithm | Rejected status code (this depends on your "accept" configuration): Forbidden

[README.md]:
[403] https://stackoverflow.com/questions/ask?tags=lightgbm | Rejected status code (this depends on your "accept" configuration): Forbidden
[403] https://stackoverflow.com/questions/tagged/lightgbm?sort=votes | Rejected status code (this depends on your "accept" configuration): Forbidden

🔍 2208 Total (in 26s) ✅ 2183 OK 🚫 4 Errors 👻 21 Excluded

Now results look like the following:

🔍 2208 Total (in 26s) ✅ 2183 OK 🚫 0 Errors 👻 25 Excluded

[jameslamb]

This is looking great... all of my suggestions have been addressed and I'm happy to see this only takes around 2 minutes to run (build link) compared to nearly an hour for the linkcheck-based job (build link)!!!

[StrikerRUS]

... I'm happy to see this only takes around 2 minutes to run (build link) compared to nearly an hour for the linkcheck-based job (build link)!!!

Yeah, thanks to GitHub token lychee mechanism we don't need to manually slow down link checking to avoid 429 Rate limit errors.