Surely one should think of this as a vector in a space with more dimensions than 1.
In your equation you can just 1,000,000x magnitude and it will move in the “positive direction”.
In the real world you can become a billionaire from selling toothbrushes and still be “overtaken” by a guy who wrote one blog post that happened to be real dang good
I made a drawing but lw won’t allow adding it on phone I think
Or, worse, if most directions are net negative and you have to try quite hard to find one which is positive, almost everyone optimizing for magnitude will end up doing harm proportional to how much they optimize magnitude.
True, and even more, if optimizing for impact or magnitude has Goodhart effects, of various types, then even otherwise good directions are likely to be ruined by pushing on them too hard. (In large part because it seems likely that the space we care about is not going to have linear divisions into good and bad, there will be much more complex regions, and even when pointed in a directino that is locally better, pushing too far is possible, and very hard to predict from local features even if people try, which they mostly don’t.)
Why would we not have the “Direction” component standardized to have unit norm?
I think what the OP is getting at is that the space of endeavors has a bunch of privileged directions of high impact, and your impact depends on (1) how good your aim is and (2) how hard you shoot. So it’d be something like magnitude times the sum of cosine similarities with each high-impact vector; or perhaps just the magnitude if we use the high-impact vectors as the basis.
Also, “Magnitude” is probably the wrong term for the component in question; it seems to mean “how much you achieve”, but that’s actually what “Impact” is measuring! And indeed, impact is very much a function of the direction in which you’re going. “Magnitude” should instead be “Effort” or “Short-Term Profit” or something.
(Yes, I truly believe that nitpicking this toy model is the best use of my time right now.)
I think the point wasn’t having a unit norm, it was that impact wasn’t defined as directional, so we’d need to remove the dimensionality from a multidimensionally defined direction.
So to continue the nitpicking, I’d argue impact = || Magnitude * Direction ||, or better, ||Impact|| = Magnitude * Direction, so that we can talk about size of impact. And that makes my point in a different comment even clearer—because almost by assumption, the vast majority of those with large impact are pointed in net-negative directions, unless you think either a significant proportion of directions are positive, or that people are selecting for it very strongly, which seems not to be the case.
Surely one should think of this as a vector in a space with more dimensions than 1.
In your equation you can just 1,000,000x magnitude and it will move in the “positive direction”.
In the real world you can become a billionaire from selling toothbrushes and still be “overtaken” by a guy who wrote one blog post that happened to be real dang good
I made a drawing but lw won’t allow adding it on phone I think
Or, worse, if most directions are net negative and you have to try quite hard to find one which is positive, almost everyone optimizing for magnitude will end up doing harm proportional to how much they optimize magnitude.
True, and even more, if optimizing for impact or magnitude has Goodhart effects, of various types, then even otherwise good directions are likely to be ruined by pushing on them too hard. (In large part because it seems likely that the space we care about is not going to have linear divisions into good and bad, there will be much more complex regions, and even when pointed in a directino that is locally better, pushing too far is possible, and very hard to predict from local features even if people try, which they mostly don’t.)
I was thinking something potentially similar. This is super nitpicky, but the better equation would be impact = Magnitude * ||Direction||
Why would we not have the “Direction” component standardized to have unit norm?
I think what the OP is getting at is that the space of endeavors has a bunch of privileged directions of high impact, and your impact depends on (1) how good your aim is and (2) how hard you shoot. So it’d be something like magnitude times the sum of cosine similarities with each high-impact vector; or perhaps just the magnitude if we use the high-impact vectors as the basis.
Also, “Magnitude” is probably the wrong term for the component in question; it seems to mean “how much you achieve”, but that’s actually what “Impact” is measuring! And indeed, impact is very much a function of the direction in which you’re going. “Magnitude” should instead be “Effort” or “Short-Term Profit” or something.
(Yes, I truly believe that nitpicking this toy model is the best use of my time right now.)
I think the point wasn’t having a unit norm, it was that impact wasn’t defined as directional, so we’d need to remove the dimensionality from a multidimensionally defined direction.
So to continue the nitpicking, I’d argue impact = || Magnitude * Direction ||, or better, ||Impact|| = Magnitude * Direction, so that we can talk about size of impact. And that makes my point in a different comment even clearer—because almost by assumption, the vast majority of those with large impact are pointed in net-negative directions, unless you think either a significant proportion of directions are positive, or that people are selecting for it very strongly, which seems not to be the case.