I remarked on this claim while the paper was in review when I was asked to give some feedback on the paper [it’s still under official review I think]. Some of the earlier proposed metrics are in fact full metrics in the relevant sense, provided they have full coverage on the reward space. e.g. STARC distances are metrics on the quotient space of reward functions by the equivalences, aren’t they, if the distance metric has full coverage? (Which is the same as what this project-then-take-angle metric is.) In particular, I think the angle relates 1-1 with the L2 distance on a suitably canonicalised unit ball from STARC/EPIC.
I still think this is an important point, and I’ve been thinking there should be a bloggy write-up of the maths in this area on LW/AF! Maybe you (or I, or Jacek, or Charlie, or Joar, or whoever...) could make that happen.
The original EPIC definition, and the STARC defs, can be satisfied while yielding only a pseudometric on the quotient space. But they also include many full (quotient) metrics, and the (kinda default?) L2 choice (assuming full-support weighting) yields a full metric.
I remarked on this claim
while the paper was in reviewwhen I was asked to give some feedback on the paper [it’s still under official review I think]. Some of the earlier proposed metrics are in fact full metrics in the relevant sense, provided they have full coverage on the reward space. e.g. STARC distances are metrics on the quotient space of reward functions by the equivalences, aren’t they, if the distance metric has full coverage? (Which is the same as what this project-then-take-angle metric is.) In particular, I think the angle relates 1-1 with the L2 distance on a suitably canonicalised unit ball from STARC/EPIC.You’re right. For some reason, I thought EPIC was a pseudometric on the quotient space and not on the full reward space.
I think this makes the thing I’m saying much less useful.
I still think this is an important point, and I’ve been thinking there should be a bloggy write-up of the maths in this area on LW/AF! Maybe you (or I, or Jacek, or Charlie, or Joar, or whoever...) could make that happen.
The original EPIC definition, and the STARC defs, can be satisfied while yielding only a pseudometric on the quotient space. But they also include many full (quotient) metrics, and the (kinda default?) L2 choice (assuming full-support weighting) yields a full metric.