Pareto Front

I wrote this app as a SPA! It uses a curve formulation similar to Brzycki, except I added a “shape” parameter (an exponent gamma between 0 and 1) that slopes the 1rm downwards at the right side.

My main finding for “pick whatever weight you want today” was that picking a lot of different weights made the curve less identifiable, so my latest iteration encourages you to pick a ladder for a few sentinel exercises per mesocycle in order to improve the statistical power. In addition, strength improves more quickly at >80% of 1RM, and hypertrophy depends on proximity to failure, so if you pick a lower weight, you really need to go to failure, which burns you out for the rest of your session, where leaving 1-2 reps in reserve is probably sufficient for hypertrophy and leaves a lot more gas in the tank for the rest of the session. Definitely open to suggestion/discussion here.

https://curvefit.app (it runs on Cloudflare free tier, so I won’t have to start running ads or charging until I hit a couple thousand users)

Could you please share this spreadsheet? I would really love to have my own version of this.

I wouldn't call this a Pareto Frontier, but simple isolines through a 2d function.

There are weight x rep combinations that have a e1rm of 80kg, 85kg, 90kg, and so on.

These are just equal elevation contours through the e1rm(x, y) function.

The Pareto concept doesn't require that we calculate a function of all the dimensions and find contours; that sort of thing is not involved.

But we could apply it here like this. Suppose we conduct a weight lifting contest as follows: contestants can lift any weight any number of times, and record the weight and reps.

Then, how do we rank the results to find a winner, or winners? We have multiple dimensions, not a single dimension like "seconds to run 10 km".

We can find the Pareto front set of the performances by eliminating all that have been dominated. A lift is dominated if another lift is no worse (no less weight, and no fewer reps), and strictly better: eight the weight is higher, or there are more reps, or both.

We then end up with undominated winners, e.g. there could be three like this: { (100kg, 1), (80kg, 2), (70kg, 5) } but (70kg, 4) would not belong, due to being dominated by the third one, and (90kg, 1) would not due to being dominated by the first. The middle one is not dominated by either: though it's less weight than the 100kg, it is more reps, and though it is fewer reps than the 70kg, it is more weight.

Given the Pareto front set, if we want to determine a single winner, we need a function to reduce the parameters to a single value. (The function should be such that if we included the eliminated losers under that function, none of them would emerge winner over the Pareto front set). This e1rm function looks like it fits the bill.

If we have this function, we don't need the Pareto concept; we just run all the results through the function and pick the contestant(s) that maximize it.

This is a cool way to gamify weightlifting. Cheers!

Respectfully, it’s nothing new. Weightlifting industry has known this concept forever, it’s often just expressed as charts rather than graphs, as it is easier to interpret.

But they go even a step further, they extend into 3 dimensions to also add body weight as a variable. So your graph would really have to be a 3D volume. Because different levels of body weight have different capabilities.

Respectfully, that's a cool illustration of the idea of xRMs etc but is missing the whole point of programming for higher or lower reps. E.g. lower reps are more stressful / higher cost of recovery but more strength-specific; high reps are better for hypertrophy work. But then, any well designed program will have you working across a range of rep ranges and so on.

Please don't make an app based on this.

I thought it was some sort of Italian activism group.

"The Pareto Front today claimed responsiblity for...."

Trouble with fonts is sometimes monospace good, sometimes monodpace bad

Anyways I'll namedrop Iosevka as perfect monospace font for working on 13" laptop

Inter with optional open type features turned on?

https://rsms.me/inter/

look at Atkinson Hyperlegible? commissioned by the Braille foundation for low-vision readers which means it's very readable

No, it's far from the best possible font, but to its credit, it gets most aspects of typography right by just focusing on the ~1/5 of the requirements that actually really count.

I'd also be interested in the Pareto Front of Fonts. That would be the final font collection - to rule them all.

lol same :D

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.

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

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.

I've been wondering if OpenAI make Luna artificially cheap to get people into their eco system.

I think it's great and hope the price can stay the same.

> "As we get into higher dimensional spaces, things get weird really fast."

The geometric problem of computing a d-dimensional Pareto set of cardinality n

https://en.wikipedia.org/wiki/Maxima_of_a_point_set

has a truly weird property not covered by the computational complexity discussion on that page. It says there's an algorithm achieving O(n log(n)^(d-3) log log n), which is true and also a lie. The algorithm that achieves that asymptotic form is a galactic algorithm; and not an ordinary one in the sense of "has a large constant multiplicative factor", but one with this property (I've never found any other algorithm which exhibits it):

The runtime is within a bounded constant factor of n^2, for all n up to some critical N whose size is exponential in d (I think it was exactly 2^d or something).

I.e. the runtime has "two shapes": it's purely quadratic up to a galactically-large constant, and thereafter has a transition into to a slower function. The asymptotic version in the textbooks isn't achievable in the real world (for all but very small dimension).

There's an elementary proof using generating functions.

edit to add: If anyone's curious about it, a simplified version of the recurrence relation that's enough to exhibit this behavior (you can instantly see it if you graph this numerically) is

    f(n,d=0) = 1
    f(n=1,d) = 1
    f(n,d)   = n + 2f(⌊n/2⌋, d) + 2f(⌊n/2⌋, d-1)

The curse of dimensionality times the reality that good metrics are elusive or themselves a bit cursed. Many outcomes you're engineering or product-managing toward are quite squishy, hard to define, and hard to evaluate. "Easy to use" or "can be used within 10 minutes" or "cleans up this current order form" are easy to state but hard to rate and/or hard to actionably implement as metrics.

I've built large, deep product evaluation frameworks, and it is 100% of the time a running argument with stakeholders, inside and out, "well you should have measured it this way" or "I think we should be targeting X not Y" or "why didn't you consider Z in the metric??"

The Pareto Front in practice is squishy, fuzzy, and often quite moist and moldy.

One I spent a few months working on was pathfinding for trucks. The goal is to find dominant solutions over {shortest time, lowest cost (tolls + fuel), avg road speed variance - traffic sensitivity} and then return 3-4 routes that are equal distance from each other in this dimensional space for users to pick from.

As you say, the most useful things happen in low-dimensional spaces.

That's the "Pareto Principle" whereas the frontier is talking about Pareto efficiency. They have the same name because both were first developed by the economist Vilfredo Pareto, but they're not actually otherwise related.

Using the 80/20 rule to plan, is like that other old saw "Half the money I spend on advertising is wasted. The trouble is, I don't know which half"