Add StrongStationarityProperty

[PierreQuinton]

• Change matrix generation in _inputs.py to be able to generate strongly stationary matrices, strictly weakly stationary matrices and non weakly-stationary matrices
• Change typical matrices to contain the newly added types of matrices
• Rename a few things in _inputs.py to make lines shorter
• Increase tolerance of some existing tests (LinearUnderScalingProperty and test_equivalence_mean of CAGrad) to make them pass on the new typical_matrices
• Add StrongStationarityProperty in _property_testers.py
• Make Constant, DualProj, Mean, Random, Sum and UPGrad extend StrongStationarityProperty.

Note that MGDA should not and does not pass these tests. CAGrad should pass these tests but does not (probably due to some implementation issue of CAGrad). AlignedMTL passes the tests but it's hard to tell if it should in theory. For a few other aggregators, we did not try the property tester, and we do not know if they have the property in theory.
EDIT: this might have changed after changing the dimensions of the matrices that we test.

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[ValerianRey]

On CUDA, 2 non-stationarity tests fail for UPGrad and 1 for MGDA. On CPU, everything passes. Specifically, this assertion:

assert norm > 1e-03

fails, because the norm is 2.2e-4 and 5.5e-5 for 2 matrices with UPGrad, and 8.9e-9 for one matrix with MGDA.

[ValerianRey]

Docstrings of all three new classes could be improved IMO.

[ValerianRey]

Shouldn't many more aggregators have this property? You told me a few months ago that those aggregators were supposed to be strongly stationary:
✅ mean
❌ MGDA
✅ Dualproj
🤷🏻‍♂️ PCGrad
🤷🏻‍♂️ GradDrop
✅ IMTL-G
❌ CAGrad
✅ RGW
✅ NashMTL

[PierreQuinton]

I'll go over all aggregator theoretically to assess this property, I think from memory that you are correct

[PierreQuinton]

An aggregator is said to be:

• Weakly stationary if it returns $0$ on and only on matrices that are weak stationary.
• Strongly stationary if it returns $0$ on and only on matrices that are strongly stationary.

Note that some aggregators may be neither. Typically, when there is a pref vector, for a positive pref vector we can have strong, for a non-negative pref vector we can only have weak and for an arbitrary pref vector we have no guarantee.

Weak stationary aggregators:

• MGDA

Strong stationary aggregators:

• UPGrad (depends on pref vector)
• Mean
• Constant (depends on pref vector)
• DualProj (depends on pref vector)
• Random
• Sum

Neither:

• Aligned-MTL
• CAGrad
• GradDrop
• IMTL-G
• PCGrad

Unkown:

• Krum
• Nash-MTL
• Trimmed Mean

For ConFIG, the behavior is not specified on (any) stationary matrices, so it is very implementation dependent, testing should therefore be enough.

[ValerianRey]

• Maybe we should entirely drop the concept of weak stationarity: it's not interesting enough to justify having a property for this.
• Then, I would define Strongly stationary to be: "if it returns 0 only on matrices that are strongly stationary." (without the "on and"). The reason is that the converse of this (the "on and") is implied by the non-conflicting property.