Add start_sigma to ADVI 2

[markusschmaus]

Recreation of #6096 after failing test (fixed in latest commit).

Introduces the start_sigma argument which allows to set a starting value for the sigmas of mean field approximation in ADVI. I am using the ordering property to create the mapping between the variables and the flat 1d array required by ADVI.

See also:
https://discourse.pymc.io/t/quality-of-life-improvements-to-advi/10254

Checklist

• Explain important implementation details 👆
• Make sure that the pre-commit linting/style checks pass.
• Link relevant issues (preferably in nice commit messages)
• Are the changes covered by tests and docstrings?
• Fill out the short summary sections 👇

Major / Breaking Changes

• None

Bugfixes / New features

• Allow start_sigma being given to fit.
• For ASVGD forward the key word arguments start and random_seed to the constructor of the default approximation FullRank instead of sending them to super().__init__ which can't process them
• Enable _iterate_with_loss to run with n=0 by giving an appropriate logger message. This allows to check initialization values, which is useful for testing, but could also be useful for debugging a user model

Docs / Maintenance

• The documentation for the start argument was outdated, it claimed type Point, but it gets passed through to a function that requires StartDict. Since users are probably unfamiliar with this custom type (Dict[Union[Variable, str], Union[np.ndarray, Variable, str]]), I decided to also mention the most relevant and most commonly used subtype dict[str, np.ndarray].

[codecov]

Codecov Report

Merging #6132 (762a6ca) into main (2b39a0c) will decrease coverage by 8.65%.
The diff coverage is 81.25%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #6132      +/-   ##
==========================================
- Coverage   92.91%   84.25%   -8.66%     
==========================================
  Files         101      101              
  Lines       20929    20941      +12     
==========================================
- Hits        19446    17644    -1802     
- Misses       1483     3297    +1814     

Impacted Files	Coverage Δ
pymc/variational/inference.py	84.97% <57.14%> (-1.13%)	⬇️
pymc/variational/approximations.py	87.67% <100.00%> (+0.34%)	⬆️
pymc/tests/tuning/test_scaling.py	0.00% <0.00%> (-100.00%)	⬇️
pymc/tests/tuning/test_starting.py	0.00% <0.00%> (-100.00%)	⬇️
pymc/tests/distributions/test_bound.py	0.00% <0.00%> (-100.00%)	⬇️
pymc/tests/distributions/test_censored.py	0.00% <0.00%> (-100.00%)	⬇️
pymc/tests/distributions/test_transform.py	0.00% <0.00%> (-100.00%)	⬇️
pymc/tests/distributions/test_truncated.py	0.00% <0.00%> (-100.00%)	⬇️
pymc/tests/distributions/test_simulator.py	0.00% <0.00%> (-99.51%)	⬇️
pymc/tests/distributions/test_dist_math.py	0.00% <0.00%> (-99.26%)	⬇️
... and 16 more

[junpenglao]

Thank you!

[markusschmaus]

• For ASVGD forward the key word arguments start and random_seed to the constructor of the default approximation FullRank instead of sending them to super().__init__ which can't process them

For the record, this is what caused the test to fail since it expected super().__init__ to raise a TypeError when it receives a start parameter.