I have a weightlifting spreadsheet with weight on the vertical axis and reps on the horizontal axis. The value of each cell is the estimated 1 rep max if I accomplish that lift. In theory if my e1RM is 100kg then I can lift any permutation of (weight,reps) that have the same e1RM. This is akin to knowing Pareto Frontier of my current strength.
I use conditional formatting to color cells according to the probability that I can lift them—if I lifted 50kg for 10 reps then I can definitely do 50kg for 9 reps, so that cell is green. But if e1RM(50,10) > e1RM(40,15) then I can probably do that too so it's light green. The visualization naturally becomes Pareto-like.
If I'm feeling strong I can aim for higher weight, lower reps. Or if I'm feeling weak I can close out a (weight, reps) that's below my current e1RM but I haven't accomplished yet. The end result is that I'm always "accomplishing" some sort of PR no matter how I feel.
I call this e1RM Bingo.
jerkstate 1 hours ago [-]
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)
cman1444 3 hours ago [-]
Could you please share this spreadsheet? I would really love to have my own version of this.
kachnuv_ocasek 54 minutes ago [-]
Just copy-paste that description to Claude and have it create the spreadsheet.
joncrane 3 hours ago [-]
This is a cool way to gamify weightlifting. Cheers!
godwinson__4-8 2 hours ago [-]
Indeed, GP should take a spin at turning into an app. Could be worthwhile to have Claude take a first stab at a MVP.
If pursued, good luck!
jerkstate 57 minutes ago [-]
If this is something you are interested in, I did make a mobile friendly SPA similar to this: https://curvefit.app
deadbabe 18 minutes ago [-]
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.
wollowollo 1 hours ago [-]
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.
jerkstate 1 hours ago [-]
> high reps are better for hypertrophy work
Some nuance here: the latest research shows that proximity to failure is the main hypertrophy driver regardless of load and rep count; high rep count makes proximity to failure harder to gauge; so high load/low reps close to failure is probably better for hypertrophy (there are other good reasons to do higher reps/lower load work though)
bob1029 53 minutes ago [-]
The most effective (difficult) training regimens usually avoid the middle of the distribution. You generally want to be operating at the extremes with some rotation schedule or duty cycle. High intensity interval training is an example of this philosophy that occurs within a single workout session.
If you want the most 'optimal' form of this (aka, hell on earth), you should purchase a rowing machine. Being able to engage with very aggressive, full-body exercise every single day without exceptions is almost like cheating biology. You can maintain a 2-3x VO2 max premium over your peers with very little risk of injury.
Aachen 6 hours ago [-]
Misread the title and got excited about a Pareto font, that is, the best possible font (presumably: distinct l/I, O/0, scores within error margins of the top readability and reading speed scores, widely available, etc.)
Maybe in vein but did anyone already figure this one out? The closest I got was PT sans, open-licensed commissioned by the Russian ministry for communication (I found it surprising that a country that doesn't use Latin script made the best font!), but it's not widely shipped so you need to figure out how to include font files whenever you want to use it
gadders 5 hours ago [-]
I thought it was some sort of Italian activism group.
"The Pareto Front today claimed responsiblity for...."
lucaslazarus 3 minutes ago [-]
You mean the People’s front of Pareto!
orthoxerox 5 hours ago [-]
...20% of the attacks causing 80% of the casualties?
mdnahas 3 hours ago [-]
As someone with an Italian grandmother and both a CS and Econ degree, I got a great laugh out of this joke! Bravo!
pphysch 1 hours ago [-]
No!! That is the Front of Pareto[1], a totally different group. Our Pareto Front only causes 20% of the casualties.
I didn't know fonts can have options. Another learning curve on how to enable that in Latex/Html/Libreoffice/anywhere else I use fonts ^^'. But still helpful to know about!
ss02 disambiguation seems to be the one I'd be wanting to turn on, with tnum for monospace numbers being a good option as well that I hadn't even realised I wanted from a font!
airstrike 34 minutes ago [-]
Yeah, those features are awesome! Sadly support for them is quite lacking in applications...
Tabular numbers are awesome!
__s 4 hours ago [-]
Trouble with fonts is sometimes monospace good, sometimes monodpace bad
Anyways I'll namedrop Iosevka as perfect monospace font for working on 13" laptop
throw-the-towel 3 hours ago [-]
Seconding Iosevka, it's great for code! And you can get a version without ligatures.
adornKey 5 hours ago [-]
I'd also be interested in the Pareto Front of Fonts. That would be the final font collection - to rule them all.
peesem 4 hours ago [-]
look at Atkinson Hyperlegible? commissioned by the Braille foundation for low-vision readers which means it's very readable
arduanika 4 hours ago [-]
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.
ChatGPT 5.6 Luna on the right (cheaper) cover most of the frontier, with a point for Deepseek flash, and higher performance overlapping heavily between 5.6 Sol and Fable.
That DeepSeek point will probably move back towards Luna as deepseek announced a "significant" price increase coming to their API [1], which kind of demonstrates that beating the Pareto frontier is where the difficulty actually is).
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.
bob1029 42 minutes ago [-]
I think Luna might be just small enough to provide some kind of stepwise improvement in how it is hosted.
Going from 81GB of weights to 79GB of weights can mean a 50% reduction in GPU capacity required.
If you can fit a model in just one GPU (or rack) as opposed to across an entire datacenter, the latency gains can be substantial too. If you can reduce token latency by half, that would double the amount of customers you could support.
CodeIsTheEnd 5 hours ago [-]
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 1 hours ago [-]
The cycling equivalent is your power curve, i.e. the longest you've held a power for a certain time interval.
froxtrot 4 hours 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.
shermantanktop 2 hours ago [-]
I’m sadly twitchy when I hear “Pareto” - having endured numerous middle managers suggesting they can deliver 80% of the scope in 20% of the time (unrelated to the frontier topic here). Do that at each level of an org and the nonsense multiples rapidly.
The 80/20 “rule,” as far as I know, is meant to be descriptive after the fact. It can’t be used as a planning assumption. To be fair to those managers, they don’t really mean to be rigorous. They are just trying to justify cutting scope.
MarkusQ 2 hours ago [-]
Managers trying to justify _cutting_ scope...
Is your planet accepting immigrants? I think I'd like it there
nonameiguess 30 minutes ago [-]
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.
bob1029 4 hours ago [-]
Pareto front sounds like an interesting way to optimize, but it suffers from the curse of dimensionality just like anything else.
As the number of objectives (dimensions) increases, the number of samples you need to cover the frontier increases exponentially. You will very rarely find solutions that actually dominate other solutions in many practical optimization scenarios. With 2 dimensions you have a 25% chance of domination. With 10 dimensions it's a .098% chance.
The most useful cases I've seen tend to occur where we just optimize for two things at once. The chances of domination are high, it's easy to visualize and very efficient to implement. As we get into higher dimensional spaces, things get weird really fast.
peri-cl 4 hours ago [-]
> "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
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
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.
krapht 4 hours ago [-]
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.
Question: in auto racing, could one have a Pareto Front balancing single lap pace (qualifying optimization) and race pace (pace over an entire stint of e.g. 20+ laps)?
transitorykris 3 hours ago [-]
You’d be looking at fuel level affecting choices of ride height, brake bias, etc. Possibly changes in line and distance travelled too. But, I’m curious how psychology can fit in here (or not. How do you measure it?). Driver concentration and confidence are important when trimming out aero or other adjustments for single lap flyers.
cpa 2 hours ago [-]
At $JOB, I use the Pareto frontier all the time.
If one option is at least as good on every relevant dimension and better on one, just pick it. That's not really a trade-off, and it shouldn't need escalation. Eg, if two SaaS tools cost the same and have similar support, but one fits your use case better, you choose that one. Otherwise, you just suck at your job!
The interesting decisions only start once you're already on the frontier, where getting more of one thing means giving up something else. If the better tool costs 50% more, now you're trading capability against cost, and that may need sign-off.
Basically, everyone should be able to get to the frontier on their own. Coordination and arbitration at higher levels of the org / between different departments should happen on the frontier, where the trade-offs involve several people or teams.
lorey 4 hours ago [-]
Found this to display the optimal LLM choice while building evalry. It's such a useful tool, not only for thinking about it, but for visualization, too.
Nice, this is exactly what I use for the multi-objective optimizer on a quantum network simulator I'm building — scoring topologies on fidelity/latency/success rate tradeoffs.
matsemann 4 hours ago [-]
My thesis many years ago was on multi-objective optimization using evolutionary algorithms (in my profile), and maintaining a wide pareto front was what most algorithms (like NSGA-II) were attempting. If all individuals cluster around a small area (in for instance a weight/strength tradeoff), you will quickly get stuck. So should select solutions to keep for further search along the whole front (for instance some solution that is very strong but unfortunately also very heavy). Maybe keep some of them as candidates even if worse (not part of the pareto front), just to keep that part of the search space alive and avoid local optima.
Of course, what's hard anyways when you have a good set of solutions that are pareto optimal, is to then choose between them. Especially as the dimensions (objectives) grow. In my example we can end up with many variants of strength/weight trade-offs that each are optimal, which one to choose?
denismenace 6 hours ago [-]
I'm assuming you must have discovered this through the OpenRouter LLM performance graphs.
I think the one from hugging face is much clearer, albeit its an arena metric and a bit tricky to find the pareto view. Look for the the top Navigation Bar
(Agent Chat Code Image Video). Chat -> (dropdown) Text -> (Side panel) View as Pareto.
We used to just call that efficiency. Overusage of "pareto frontier" annoys me almost as much people talking about "electrons" instead of just saying electricity or power.
stevefan1999 4 hours ago [-]
I wonder why LLM love this word so much. Same as mint, seam, tier.
amingilani 4 hours ago [-]
A seam is a place where you can alter behavior in your program without editing in
that place
“Chapter 4: The Seam Model”, Michael C. Feathers, Working Effectively with Legacy Code
voidhorse 4 hours ago [-]
One nuance that people sometimes miss is that pareto optimality in the continuous case and discrete case are distinct. Using continuous case algorithms on discrete feasible set optimization problems will make you miss the interior optimal points--only extremal/supported points on the positive orthant hull are identified by the continuous algos.
Matthias Ehrgott's books on multicriteria optimization explain Pareto efficiency very well without sacrificing rigor. I think they do a better job than this article.
bhanu786 6 hours ago [-]
may, anyone explain what is this
chriswarbo 5 hours ago [-]
If we have a set of things (e.g. language models) and some measures we care about (e.g. cost, speed, whether weights are open, scores for a few benchmarks, etc.), then some of those things will be "pareto optimal" (see below) and some won't. The "pareto front" is the subset that is pareto optimal.
Some thing is "pareto optimal" when there isn't another thing that's AT LEAST AS GOOD in ALL measures, and BETTER in at least one way. For example, if we say there are no ties (for simplicity), then the cheapest language model is pareto optimal; the fastest model is pareto optimal; those which score highest on each benchmark are pareto optimal; and so on.
Tradeoffs can also be pareto optimal: for example, if the cheapest model is also slow, then there will be more pareto optimal models which are "cheapest for their speed"; and so on for other tradeoffs (e.g. fastest that achieves a certain benchmark score; cheapest model with open weights; etc.).
If you're making a decision about which thing to choose, you only need to care about those in the pareto front (since, by definition, anything that's not pareto optimal is objectively worse on at least one measure).
Pareto optimality does not compare one measure against another: something that's 10000x slower can still be pareto optimal, if it's 1% cheaper than the alternatives. To pick a "best" thing, you could give a weight/importance to each measure, and combine them into an overall score: but that's subjective, and might vary between people and tasks. In contrast, focusing on the pareto front is a way to ignore those things that will never be the best, regardless of weighting.
matsemann 4 hours ago [-]
I honestly think the wikipedia article is too complicated. My own image example here as an another attempt to explain: https://imgur.com/a/5ZQIJDb
Mapping the cost of something (like an algorithm), and the time it takes (so lower is better for both). 1, 3 and 5 are all optimal in their own sense. No one is strictly better than the other, just different tradeoffs you have to choose yourself. However, you would never choose 2, because for a lower cost you could get the same result choosing 3. Same with 4, 6 and 7, they all have something that's both faster and at the same time just as cheap you could choose.
A pareto front is a bit like the classical "fast, cheap, good, choose 2". There are always tradeoffs, but if something is both slow, expensive and not better than something that's faster and cheaper, it's a bad choice, and thus not on the "pareto front".
felixguendling 6 hours ago [-]
If you optimize one criterion, it's simple: lowest is best or highest is best.
If you optimize multiple criteria, all optimal trade offs between any of the selected criteria are "best" in some way.
continuational 6 hours ago [-]
When you have a tradeoff between two parameters, which points dominate the others in the sense that you can't choose another point without getting less of one of the parameters.
bhanu786 6 hours ago [-]
thanks, may you tell me where we can use them?
joshka 5 hours ago [-]
The current thing that comes up regularly is choosing an LLM setup.
isoprophlex 6 hours ago [-]
"what's the family of optimal choices when you have multiple dimensions to rank on?"
Say a race vehicle has acceleration, top speed as defining parameters. Some are slow but accelerate hard, others need a long time to reach very high top speeds. Others are in between, or just flat out bad at both.
The pareto frontier is the set of vehicles that are best: pick one from the frontier and you can be sure that for it's given top speed, none accelerate faster. And vice versa, pick one with a given acceletation and you are sure none have a better top speed
ChrisMarshallNY 6 hours ago [-]
Basically, prioritization.
It’s really that simple.
Eschew obfuscation.
Lerc 6 hours ago [-]
Almost the complete opposite of prioritisation.
The Pareto points are where you sacrifice the least of anything to get the most of everything.
There's the saying about buying computers. Good, Cheap, Fast, pick any two. That's where you would prioritise.
If someone makes something that better, cheaper, and faster, or even pretty close to the best on two of those and clearly better on the other. It's a Pareto point.
Over time computers are getting better, cheaper and faster (software notwithstanding). The leading edge of that advance of all of the things is the Pareto front.
jrrv 4 hours ago [-]
Given this in the TFA
> a Pareto front represents the set of solutions where no solution outperforms any other solution in the set at every objective
I do not believe you are correct when you say
> something that better, cheaper, and faster, or even pretty close to the best on two of those and clearly better on the other. It's a Pareto point.
Since that would outperform on every objective
GP's point that it's prioritisation does not seem incorrect to me. Prioritisation involves considering trade-offs of various approaches and deciding which aspects & attributes to optimise for, at the expense of others.
Lerc 1 hours ago [-]
[dead]
ChrisMarshallNY 2 hours ago [-]
I’m not sure how that’s the opposite of prioritizing, but if I’m wrong, then I’ll happily admit it.
We choose our items/workflows/technologies/whatever, so we get the best/most efficient/most effective/whatever, across the widest possible set.
Sounds like prioritizing, to me, but I’m just a dumb hick, so I suppose I can be wrong.
Lerc 1 hours ago [-]
Prioritising is when you choose something over another. A drag racer prioritises time to travel a quarter mile.
Going for the Pareto is when you elect not to prioritise. It is explicitly deciding to not choose one property over another ant to keep everything as much as you can.
stevefan1999 4 hours ago [-]
No, it's more about maximizing the utility function and finding the "shield" where you'd start getting diminishing return beyond that
bhanu786 6 hours ago [-]
you have confused me
jethkl 4 hours ago [-]
[dead]
tug2024 2 hours ago [-]
[dead]
hnnbxu2nwi 2 hours ago [-]
The lesson landed
Rendered at 17:16:20 GMT+0000 (Coordinated Universal Time) with Vercel.
I use conditional formatting to color cells according to the probability that I can lift them—if I lifted 50kg for 10 reps then I can definitely do 50kg for 9 reps, so that cell is green. But if e1RM(50,10) > e1RM(40,15) then I can probably do that too so it's light green. The visualization naturally becomes Pareto-like.
If I'm feeling strong I can aim for higher weight, lower reps. Or if I'm feeling weak I can close out a (weight, reps) that's below my current e1RM but I haven't accomplished yet. The end result is that I'm always "accomplishing" some sort of PR no matter how I feel.
I call this e1RM Bingo.
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)
If pursued, good luck!
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.
Please don't make an app based on this.
Some nuance here: the latest research shows that proximity to failure is the main hypertrophy driver regardless of load and rep count; high rep count makes proximity to failure harder to gauge; so high load/low reps close to failure is probably better for hypertrophy (there are other good reasons to do higher reps/lower load work though)
If you want the most 'optimal' form of this (aka, hell on earth), you should purchase a rowing machine. Being able to engage with very aggressive, full-body exercise every single day without exceptions is almost like cheating biology. You can maintain a 2-3x VO2 max premium over your peers with very little risk of injury.
Maybe in vein but did anyone already figure this one out? The closest I got was PT sans, open-licensed commissioned by the Russian ministry for communication (I found it surprising that a country that doesn't use Latin script made the best font!), but it's not widely shipped so you need to figure out how to include font files whenever you want to use it
"The Pareto Front today claimed responsiblity for...."
[1] - http://montypython.50webs.com/scripts/Life_of_Brian/8.htm
https://rsms.me/inter/
ss02 disambiguation seems to be the one I'd be wanting to turn on, with tnum for monospace numbers being a good option as well that I hadn't even realised I wanted from a font!
Tabular numbers are awesome!
Anyways I'll namedrop Iosevka as perfect monospace font for working on 13" laptop
ChatGPT 5.6 Luna on the right (cheaper) cover most of the frontier, with a point for Deepseek flash, and higher performance overlapping heavily between 5.6 Sol and Fable.
That DeepSeek point will probably move back towards Luna as deepseek announced a "significant" price increase coming to their API [1], which kind of demonstrates that beating the Pareto frontier is where the difficulty actually is).
[1] https://www.bloomberg.com/news/articles/2026-08-06/deepseek-...
I think it's great and hope the price can stay the same.
Going from 81GB of weights to 79GB of weights can mean a 50% reduction in GPU capacity required.
If you can fit a model in just one GPU (or rack) as opposed to across an entire datacenter, the latency gains can be substantial too. If you can reduce token latency by half, that would double the amount of customers you could support.
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!)
The 80/20 “rule,” as far as I know, is meant to be descriptive after the fact. It can’t be used as a planning assumption. To be fair to those managers, they don’t really mean to be rigorous. They are just trying to justify cutting scope.
Is your planet accepting immigrants? I think I'd like it there
As the number of objectives (dimensions) increases, the number of samples you need to cover the frontier increases exponentially. You will very rarely find solutions that actually dominate other solutions in many practical optimization scenarios. With 2 dimensions you have a 25% chance of domination. With 10 dimensions it's a .098% chance.
The most useful cases I've seen tend to occur where we just optimize for two things at once. The chances of domination are high, it's easy to visualize and very efficient to implement. 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
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.
As you say, the most useful things happen in low-dimensional spaces.
If one option is at least as good on every relevant dimension and better on one, just pick it. That's not really a trade-off, and it shouldn't need escalation. Eg, if two SaaS tools cost the same and have similar support, but one fits your use case better, you choose that one. Otherwise, you just suck at your job!
The interesting decisions only start once you're already on the frontier, where getting more of one thing means giving up something else. If the better tool costs 50% more, now you're trading capability against cost, and that may need sign-off.
Basically, everyone should be able to get to the frontier on their own. Coordination and arbitration at higher levels of the org / between different departments should happen on the frontier, where the trade-offs involve several people or teams.
Example: Which LLM gives me the best ELI5 explanations for a given price. https://evalry.com/benchmarks/explain-like-i-m-5-321
Of course, what's hard anyways when you have a good set of solutions that are pareto optimal, is to then choose between them. Especially as the dimensions (objectives) grow. In my example we can end up with many variants of strength/weight trade-offs that each are optimal, which one to choose?
https://artificialanalysis.ai/#intelligence-comparison-tabs
https://huggingface.co/spaces/lmarena-ai/arena-leaderboard
“Chapter 4: The Seam Model”, Michael C. Feathers, Working Effectively with Legacy Code
Matthias Ehrgott's books on multicriteria optimization explain Pareto efficiency very well without sacrificing rigor. I think they do a better job than this article.
Some thing is "pareto optimal" when there isn't another thing that's AT LEAST AS GOOD in ALL measures, and BETTER in at least one way. For example, if we say there are no ties (for simplicity), then the cheapest language model is pareto optimal; the fastest model is pareto optimal; those which score highest on each benchmark are pareto optimal; and so on.
Tradeoffs can also be pareto optimal: for example, if the cheapest model is also slow, then there will be more pareto optimal models which are "cheapest for their speed"; and so on for other tradeoffs (e.g. fastest that achieves a certain benchmark score; cheapest model with open weights; etc.).
If you're making a decision about which thing to choose, you only need to care about those in the pareto front (since, by definition, anything that's not pareto optimal is objectively worse on at least one measure).
Pareto optimality does not compare one measure against another: something that's 10000x slower can still be pareto optimal, if it's 1% cheaper than the alternatives. To pick a "best" thing, you could give a weight/importance to each measure, and combine them into an overall score: but that's subjective, and might vary between people and tasks. In contrast, focusing on the pareto front is a way to ignore those things that will never be the best, regardless of weighting.
Mapping the cost of something (like an algorithm), and the time it takes (so lower is better for both). 1, 3 and 5 are all optimal in their own sense. No one is strictly better than the other, just different tradeoffs you have to choose yourself. However, you would never choose 2, because for a lower cost you could get the same result choosing 3. Same with 4, 6 and 7, they all have something that's both faster and at the same time just as cheap you could choose.
A pareto front is a bit like the classical "fast, cheap, good, choose 2". There are always tradeoffs, but if something is both slow, expensive and not better than something that's faster and cheaper, it's a bad choice, and thus not on the "pareto front".
Say a race vehicle has acceleration, top speed as defining parameters. Some are slow but accelerate hard, others need a long time to reach very high top speeds. Others are in between, or just flat out bad at both.
The pareto frontier is the set of vehicles that are best: pick one from the frontier and you can be sure that for it's given top speed, none accelerate faster. And vice versa, pick one with a given acceletation and you are sure none have a better top speed
It’s really that simple.
Eschew obfuscation.
The Pareto points are where you sacrifice the least of anything to get the most of everything.
There's the saying about buying computers. Good, Cheap, Fast, pick any two. That's where you would prioritise.
If someone makes something that better, cheaper, and faster, or even pretty close to the best on two of those and clearly better on the other. It's a Pareto point.
Over time computers are getting better, cheaper and faster (software notwithstanding). The leading edge of that advance of all of the things is the Pareto front.
> a Pareto front represents the set of solutions where no solution outperforms any other solution in the set at every objective
I do not believe you are correct when you say
> something that better, cheaper, and faster, or even pretty close to the best on two of those and clearly better on the other. It's a Pareto point.
Since that would outperform on every objective
GP's point that it's prioritisation does not seem incorrect to me. Prioritisation involves considering trade-offs of various approaches and deciding which aspects & attributes to optimise for, at the expense of others.
We choose our items/workflows/technologies/whatever, so we get the best/most efficient/most effective/whatever, across the widest possible set.
Sounds like prioritizing, to me, but I’m just a dumb hick, so I suppose I can be wrong.
Going for the Pareto is when you elect not to prioritise. It is explicitly deciding to not choose one property over another ant to keep everything as much as you can.