4 ms·

I am training for a marathon, and, as I increase both by distance and pace, I am always excited when I have a "Pareto run": a run along the Pareto frontier of m

I am training for a marathon, and, as I increase both by distance and pace, I am always excited when I have a "Pareto run": a run along the Pareto frontier of me trying to maximize distance and speed. When explaining it to some coworkers, I stumbled on a fairly intuitive explanation: "I've run farther before, and I've run faster before, but I've never run _this_ far, _this fast." There was some pushback about why not just call it a PR (personal record), but I would only use that term for fixed distances (1mi, 5k, 10k, etc.) or a consistent route that I've run many times before. Nobody would say "I set my 7.40 mile PR today." More importantly, it misses the comparison to all farther (and faster) runs—it's not exciting to set a 5k PR just because you've barely run that distance before, and the pace is actually slower that a 10k you've done. (Had a Pareto run of 7.40 miles @ 6:28/mi last week!)

reedf1 2mo ago
The cycling equivalent is your power curve, i.e. the longest you've held a power for a certain time interval.
froxtrot 2mo ago
Not relevant to pareto, but that's a really fun way to look at running. Not quite as fast as your pareto run shows, but I'll definitely keep that metric back of mind to keep the psyche high for running.
voidhorse 2mo ago
So, while it's true that your runs with high speed and distance when both are considered are Pareto points, your max speed run and max distance run overall are also Pareto points. So calling these high distance+speed runs "Pareto" doesn't actually distinguish them completely from other runs. A point is Pareto so long as it is non-dominated—that is, you're not looking for dominating points, you're looking for points that are "no worse" than all others, in all criteria, when you consider that point as a reference. So your Pareto Runs are indeed Pareto points. However, your run with your fastest possible speed, even if your distance was really bad, is also still a Pareto efficient point. (Taking >= as more efficient here) By definition, the point A is Pareto if there is no point B such that in all criteria, B >= A, and for at least one criteria B > A. Take the run with the best speed. It is Pareto because we cannot find a single point B that satisfies both of these conditions. Your "Pareto Run" doesn't satisfy this set of conditions because it is worse in terms of speed, even if it has better distance than the max speed point. The only way your Pareto runs would be the only Pareto points in your record is if they simultaneously hit maxima for distance and speed when compared to all other points. So, for them to be the sole Pareto point, the clause ""I've run farther before, and I've run faster before..." would have to be false! The point would have to break both your all time records to be the solitary Pareto point. With running, because of how speed and distance are related this will basically never happen. The definition of Pareto efficiency is essentially negative in nature--it's not about finding specific dominating points, it's about finding points that are not dominated by any others on any criterion, period. All criteria are weighted equally in the search for Pareto points. It doesn't build in any weighting like considering maximum across criteria as "better" than points that only maximize one criteria. For a "biobjective" problem like your runs, the Pareto set will always contain the points (MAX, -) and (-, MAX)--they may not be unique over the criteria but there will always be at least one representative for each, I believe.
- speedstyle 2mo ago
  They never said it was the only Pareto run? Just a new one, improving the overall Pareto front
  - voidhorse 2mo ago
    Ugh, you're right! I read too quickly. Please disregard me OP. I'll leave the comment here just in case it clarifies things for poor readers such as myself lol.

reedf1 2mo ago

The cycling equivalent is your power curve, i.e. the longest you've held a power for a certain time interval.

froxtrot 2mo ago

Not relevant to pareto, but that's a really fun way to look at running. Not quite as fast as your pareto run shows, but I'll definitely keep that metric back of mind to keep the psyche high for running.

voidhorse 2mo ago

So, while it's true that your runs with high speed and distance when both are considered are Pareto points, your max speed run and max distance run overall are also Pareto points. So calling these high distance+speed runs "Pareto" doesn't actually distinguish them completely from other runs. A point is Pareto so long as it is non-dominated—that is, you're not looking for dominating points, you're looking for points that are "no worse" than all others, in all criteria, when you consider that point as a reference. So your Pareto Runs are indeed Pareto points. However, your run with your fastest possible speed, even if your distance was really bad, is also still a Pareto efficient point. (Taking >= as more efficient here) By definition, the point A is Pareto if there is no point B such that in all criteria, B >= A, and for at least one criteria B > A. Take the run with the best speed. It is Pareto because we cannot find a single point B that satisfies both of these conditions. Your "Pareto Run" doesn't satisfy this set of conditions because it is worse in terms of speed, even if it has better distance than the max speed point. The only way your Pareto runs would be the only Pareto points in your record is if they simultaneously hit maxima for distance and speed when compared to all other points. So, for them to be the sole Pareto point, the clause ""I've run farther before, and I've run faster before..." would have to be false! The point would have to break both your all time records to be the solitary Pareto point. With running, because of how speed and distance are related this will basically never happen. The definition of Pareto efficiency is essentially negative in nature--it's not about finding specific dominating points, it's about finding points that are not dominated by any others on any criterion, period. All criteria are weighted equally in the search for Pareto points. It doesn't build in any weighting like considering maximum across criteria as "better" than points that only maximize one criteria. For a "biobjective" problem like your runs, the Pareto set will always contain the points (MAX, -) and (-, MAX)--they may not be unique over the criteria but there will always be at least one representative for each, I believe.

speedstyle 2mo ago
They never said it was the only Pareto run? Just a new one, improving the overall Pareto front
- voidhorse 2mo ago
  Ugh, you're right! I read too quickly. Please disregard me OP. I'll leave the comment here just in case it clarifies things for poor readers such as myself lol.