eigsolve cotangent linear problem (1) returns unexpected result

[Yue-Zhengyuan]

I was playing with AD for t-J model, starting from simple update initialization. For test I use D = 8 and χ = 32 (which is possibly too small for D = 8). My AD setting is

fixedpoint(H, peps, env; optimizer_alg=(; maxiter=20), gradient_alg=(; iterscheme=:diffgauge))

During the optimization I get a lot of KrylovKit warnings:

[Warn] `eigsolve` cotangent linear problem (1) returns unexpected result: error = 2.957325495603906e-10

error is usually of order 1e-11 to 1e-9. The relevant code that gives this warning is
https://github.com/Jutho/KrylovKit.jl/blob/ff6d4752a863f617a8404910007b56c5b7943ed5/ext/KrylovKitChainRulesCoreExt/eigsolve.jl#L158-L161
May I ask what is implied by this warning?

BTW, I also have many "sensitive to gauge choice" warnings (#54):

[Warn] `eigsolve` cotangents sensitive to gauge choice: (|Δgauge| = 1.4551915228366852e-11)

|Δgauge| in my case is usually of order 1e-12 to 1e-10.

[Yue-Zhengyuan]

@pbrehmer @lkdvos Do you know how to interpret this warning?

[pbrehmer]

This warning shows up since in iterscheme=:diffgauge the gauge fixing routine is also being differentiated during the AD backwards pass of a CTMRG iteration. This gauge fixing routine works by computing the leading eigenvector of a transfer matrix consisting of a CTM edge tensor and a random MPS. Precisely when the AD framework differentiates through these eigenvalue decompositions, the gauge warnings might appear.

When computing the backwards derivative of eigenvalue (or singular value) decompositions, one has to account for the gauge degree of freedom these decompositions have. With a proper cost function these gauge terms should drop out, so one would like to get warned if these terms are actually finite (i.e. if |Δgauge| is finite).

Now, if |Δgauge| is finite but really small like in your case, our current feeling is that probably this is unproblematic and perhaps gets projected out anyway at the end - in these cases these warnings can probably be ignored. I would also think that these warnings will occasionally appear even with large enough chi.

In other cases though, these warnings might actually show quite large gauge dependency and then typically the optimization won't work. I have not yet understood when these warnings will appear and in which cases they actually point to problematic behavior. I do think that they tend to show up more frequently when optimizing PEPS with symmetric spaces.