Replying separately(rather than editing), to make sure this comment gets seen.
It is worth noting that although UGC may not be true, weaker versions of UGC get many similar results. So for example, one can get most results of UGC if one believes that NP ⇐ P^U/poly where U is a unique games oracle. Then one would get similar results under the assumption that the polynomial hierarchy does not collapse. Also if one instead of believing that UGC is NP-complete one believes that UGC has no polynomial time algorithm then you get some of the same results or very similar results.
Since many people don’t assign that high a probability to UGC, it may be worth asking what evidence should cause us to update to assign a higher probability to UGC (short of a proof). There seem to be four obvious lines of evidence:
First, showing that some problems expected to be NP-intermediate (e.g. factoring, graph isomorphism, ring isomorphism) can be reduced to unique games. (Remark: the obvious candidate here would be graph isomorphism. I’m not aware of such a reduction but my knowledge of the literature here is not very good.) This would be strong evidence for at least some of the weaker versions of UGC, and thus evidence for UGC.
Second, showing that the attempted methods to solve unique games fail for intrinsic reasons that cannot be overcome by those methods. Right now, the two noteworthy methods are Sums of Squares and QAOA. If there are fundamental barriers to variants of either method working that would make UGC substantially more plausible.
Third, proving more of the inapproximation results implied by UGC by methods independent of UGC. There’s been some progress on this, but it may turn out that the inapproximations implied by UGC are the correct bounds even as UGC is false. At the same time, even as these sorts of results are proved, they make UGC less directly useful for trying to actually estimate whether recursive self-improvement can substantially occur.
Fourth, it may be possible to prove not that some unexpected collapse occurs when given access to a unique games oracle that would be unlikely unless UGC is true. Obvious candidates would be something like NP < co-NP^U, or NP < P^U/poly.
Replying separately(rather than editing), to make sure this comment gets seen.
It is worth noting that although UGC may not be true, weaker versions of UGC get many similar results. So for example, one can get most results of UGC if one believes that NP ⇐ P^U/poly where U is a unique games oracle. Then one would get similar results under the assumption that the polynomial hierarchy does not collapse. Also if one instead of believing that UGC is NP-complete one believes that UGC has no polynomial time algorithm then you get some of the same results or very similar results.
Since many people don’t assign that high a probability to UGC, it may be worth asking what evidence should cause us to update to assign a higher probability to UGC (short of a proof). There seem to be four obvious lines of evidence:
First, showing that some problems expected to be NP-intermediate (e.g. factoring, graph isomorphism, ring isomorphism) can be reduced to unique games. (Remark: the obvious candidate here would be graph isomorphism. I’m not aware of such a reduction but my knowledge of the literature here is not very good.) This would be strong evidence for at least some of the weaker versions of UGC, and thus evidence for UGC.
Second, showing that the attempted methods to solve unique games fail for intrinsic reasons that cannot be overcome by those methods. Right now, the two noteworthy methods are Sums of Squares and QAOA. If there are fundamental barriers to variants of either method working that would make UGC substantially more plausible.
Third, proving more of the inapproximation results implied by UGC by methods independent of UGC. There’s been some progress on this, but it may turn out that the inapproximations implied by UGC are the correct bounds even as UGC is false. At the same time, even as these sorts of results are proved, they make UGC less directly useful for trying to actually estimate whether recursive self-improvement can substantially occur.
Fourth, it may be possible to prove not that some unexpected collapse occurs when given access to a unique games oracle that would be unlikely unless UGC is true. Obvious candidates would be something like NP < co-NP^U, or NP < P^U/poly.