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[ci][docs] fix link checking action by switching from linkchecker to lychee and update some links (#7027)

* Correct spelling

Most changes were in comments, and there were a few changes to literals for log output.

There were no changes to variable names, function names, IDs, or functionality.

* Clarify a phrase in a comment
Co-authored-by: James Lamb <jaylamb20@gmail.com>

* Clarify a phrase in a comment
Co-authored-by: James Lamb <jaylamb20@gmail.com>

* Clarify a phrase in a comment
Co-authored-by: James Lamb <jaylamb20@gmail.com>

* Correct spelling

Most are code comments, but one case is a literal in a logging message.

There are a few grammar fixes too.
Co-authored-by: James Lamb <jaylamb20@gmail.com>

* fix warnings

* Apply suggestions from code review

* Update feature_group.h

* Update feature_group.h

* Update src/treelearner/serial_tree_learner.cpp

* Update multiclass_metric.hpp

* Update auc_mu metric to use data weights if provided

* Calculate class sizes and total weights in Init so we only do it once

* Fix lint error

* Empty commit to trigger CI jobs

This reverts commit 656d2676.

* Fix bug where small values of max_bin cause crash.

* Revert "Fix bug where small values of max_bin cause crash."

This reverts commit fe5c8e2547057c1fa5750bcddd359dd7708fab4b.

* Add auc-mu multiclass metric.

* Fix bug where scores are equal.

* Merge.

* Change name to auc_mu everywhere (instead of auc-mu).

* Fix comparison between signed and unsigned int.

* Change name to AUC-mu in docs and output messages.

* Improve test.

* Use prefix increment.

* Update R package.

* Fix style issues.

* Tidy up test code.

* Read all lines first then process.

* Allow passing AUC-mu weights directly as a list in parameters.

* Remove unused code, improve example and docs.

* fix many cpp lint errors

* indent

* fix bug

* fix more

* fix gpu

* more fixes

* Implement top-k multiclass error metric. Add new parameter top_k_threshold.

* Add test for multiclass metrics

* Make test less sensitive to avoid floating-point issues.

* Change tabs to spaces.

* Fix problem with test in Python 2. Refactor to use np.testing. Decrease number of training rounds so loss is larger and easier to compare.

* Move multiclass tests into test_engine.py

* Change parameter name from top_k_threshold to multi_error_top_k.

* Fix top-k error metric to handle case where scores are equal. Update tests and docs.

* Change name of top-k metric to multi_error@k.

* Change tabs to spaces.

* Fix formatting.

* Fix minor issues in docs.

* added all necessary includes - fixed build/include_what_you_use error

* fixed the order of includes (build/include_order)

* fix include

* reduce dependency on header file

* fix build

* [WIP] refine config

* [wip] ready for the auto code generate

* auto generate config codes

* use with to open file

* fix bug

* fix pylint

* fix bug

* fix pylint

* fix bugs.

* tmp for failed test.

* fix tests.

* added nthreads alias

* added new aliases from new config.h

* fixed duplicated alias

* refactored parameter_generator.py

* added new aliases from config.h and removed remaining old names

* fix bugs & some miss alias

* added aliases

* add more descriptions.

* add comment.

* refine prediction logic.

* fix test.

* fix out_len in training score of Dart.

* improve predict speed for high dimension data.

* many refactors.

* remove multi_loglossova.

* fix tests.

* avoid using lambda function.

* fix some format.

* reduce branching.

1. merge python-package
2. add dump model to json
3. fix bugs
4. clean code with pylint
5. update python examples

* RAII for utils, application and c_api(partical)

* raii for class in include folder

* raii for application and boosting

* raii for dataset and dataset loader

* raii for dense bin and parser

* RAII refactor for almost all classes

* RAII for c_api

* clean code

* refine repeated code

* Decouple the "sigmoid" between objective and boosting.

* change std::vector<bool> back to std::vector<char> due to concurrence problem

* slight reduce some memory cost

Support multiclass classification (#53)