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 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)
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.
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)
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.
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.
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
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!
Huh, I'm positive this was one of the first I considered (or perhaps it was a different dyslexia font) but I can't find the reason why I would have preferred PT Sans over this. Each has some pros and cons: PT Sans has same-width digits; Hyperlegible has a nicer-looking Q (imo); PT Sans has a normal-looking zero (not struck); Hyperlegible has a normal-looking 'fi' combination (no ligature that merges the two characters into a new one); Hyperlegible has a 6 distinct from an inverted 9; PT Sans needs a bit less horizontal space. The font specifically designed for accessibility should be the easy winner if all else is equal anyway
They apparently released Hyperlegible Next in 2025 which, flipping between tabs on Google Fonts (since the original website doesn't show the fonts), is nearly identical but has five new weight settings (nobody should imo ever use thin fonts though, it noticeably harms readability for me and my sight is only the tiniest bit below normal vision, but ok it's an option) and improved kerning (the original font had extremely little space between 'll', for example)
The 2025 version sadly doesn't ship with my version of TexLive, but the original (from 2020) already does so that makes it easy to use as well! Cool stuff, thanks for the tip :)
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.
What are the 4/5ths that you don't like, and which one(s) would you recommend instead? (I'm mostly interested/focussed on the sans variant but also happy to hear about serif and mono)
It's not that I don't like them. It's more just that some of these design considerations are not worth the effort to fret over, and if you're smart about your time, doing 1/5 of the work will get you something like ~4/5 of the results.
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!)
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.
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 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.
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.
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.
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.
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.
I don't think power law distributions are the same thing. However, actually, in different fields, power law distributions are descriptive enough you can use them as targets for abstract criteria.
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.
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.
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)?
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.
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.
First mention in chats for me was September 2016. Not sure how that topic came up but LLMs aren't the only way to get there. It's also a rather obvious principle to come up with, at least I'm pretty sure I was doing it without thinking I should name the concept of picking the option that best fits your requirements
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.
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.
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?
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.
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.
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.
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".
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.
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.
"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
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.
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.
I read it as finding the point of maximum effectiveness. The point at which the most is done for the most.
Getting to that point can be calculated (in some cases), but I suspect most folks get there by trial and error. Finding out what is effective, and what is not, and choosing what is effective, over what is not, until there's no longer a choice. That often becomes tribal knowledge, and is handed down. There's always someone trying to improve it, and when they figure it out, that gets added to the tribal knowledge. Basically, that's how nature does it, so there's some serious prior art. Natural Selection is brutal prioritization.
In Morocco, they used to announce the end of the Ramadan fast, by holding up a black thread and a white thread, and waiting until they could not tell the difference.
Then, they'd fire a cannon, and everybody would dig into some awesome soup. Sort of the same thing.
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.
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)
Great - commenting to refer to this
Could you please share this spreadsheet? I would really love to have my own version of this.
Just copy-paste that description to Claude and have it create the spreadsheet.
This is a cool way to gamify weightlifting. Cheers!
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!
If this is something you are interested in, I did make a mobile friendly SPA similar to this: https://curvefit.app
Clever name
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.
> 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)
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.
Respectfully, that's missing the point of the comment. It's a fun thing to hit PRs, not everything needs a 'well actually'
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.
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
I thought it was some sort of Italian activism group.
"The Pareto Front today claimed responsiblity for...."
You mean the People’s front of Pareto!
splitters!
...20% of the attacks causing 80% of the casualties?
As someone with an Italian grandmother and both a CS and Econ degree, I got a great laugh out of this joke! Bravo!
No!! That is the Front of Pareto[1], a totally different group. Our Pareto Front only causes 20% of the casualties.
[1] - http://montypython.50webs.com/scripts/Life_of_Brian/8.htm
Inter with optional open type features turned on?
https://rsms.me/inter/
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!
Yeah, those features are awesome! Sadly support for them is quite lacking in applications...
Tabular numbers are awesome!
I feel like there should be a GUI tool to bake in any alternates into the font itself.
Ok, I'll make it!
Trouble with fonts is sometimes monospace good, sometimes monodpace bad
Anyways I'll namedrop Iosevka as perfect monospace font for working on 13" laptop
Seconding Iosevka, it's great for code! And you can get a version without ligatures.
That looks... swingy! Took a second to get used to it but I think I like the refreshing take for a monospace font :) https://fonts.google.com/specimen/Iosevka+Charon+Mono
look at Atkinson Hyperlegible? commissioned by the Braille foundation for low-vision readers which means it's very readable
Huh, I'm positive this was one of the first I considered (or perhaps it was a different dyslexia font) but I can't find the reason why I would have preferred PT Sans over this. Each has some pros and cons: PT Sans has same-width digits; Hyperlegible has a nicer-looking Q (imo); PT Sans has a normal-looking zero (not struck); Hyperlegible has a normal-looking 'fi' combination (no ligature that merges the two characters into a new one); Hyperlegible has a 6 distinct from an inverted 9; PT Sans needs a bit less horizontal space. The font specifically designed for accessibility should be the easy winner if all else is equal anyway
They apparently released Hyperlegible Next in 2025 which, flipping between tabs on Google Fonts (since the original website doesn't show the fonts), is nearly identical but has five new weight settings (nobody should imo ever use thin fonts though, it noticeably harms readability for me and my sight is only the tiniest bit below normal vision, but ok it's an option) and improved kerning (the original font had extremely little space between 'll', for example)
The 2025 version sadly doesn't ship with my version of TexLive, but the original (from 2020) already does so that makes it easy to use as well! Cool stuff, thanks for the tip :)
I'd also be interested in the Pareto Front of Fonts. That would be the final font collection - to rule them all.
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.
What are the 4/5ths that you don't like, and which one(s) would you recommend instead? (I'm mostly interested/focussed on the sans variant but also happy to hear about serif and mono)
It's not that I don't like them. It's more just that some of these design considerations are not worth the effort to fret over, and if you're smart about your time, doing 1/5 of the work will get you something like ~4/5 of the results.
...so it is the Pareto font indeed!
lol same :D
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!)
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.
You can see the Pareto Frontier well in DeepSWE's chart here - https://deepswe.datacurve.ai/
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'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.
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.
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.
> "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
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.
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.
Managers trying to justify _cutting_ scope...
Is your planet accepting immigrants? I think I'd like it there
I don't think power law distributions are the same thing. However, actually, in different fields, power law distributions are descriptive enough you can use them as targets for abstract criteria.
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.
https://www.mayerowitz.io/blog/mario-meets-pareto
https://news.ycombinator.com/item?id=49195231
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.
A very useful concept to know!
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)?
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.
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.
Example: Which LLM gives me the best ELI5 explanations for a given price. https://evalry.com/benchmarks/explain-like-i-m-5-321
I'm assuming you must have discovered this through the OpenRouter LLM performance graphs.
First mention in chats for me was September 2016. Not sure how that topic came up but LLMs aren't the only way to get there. It's also a rather obvious principle to come up with, at least I'm pretty sure I was doing it without thinking I should name the concept of picking the option that best fits your requirements
artificialanalysis for me.
https://artificialanalysis.ai/#intelligence-comparison-tabs
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.
https://huggingface.co/spaces/lmarena-ai/arena-leaderboard
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.
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?
I wonder why LLM love this word so much. Same as mint, seam, tier.
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
Simple idea, big payoff
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.
Now I want a hat with the Agnostic Front logo but that says Pareto Front.
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.
may, anyone explain what is this
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.
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".
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.
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.
thanks, may you tell me where we can use them?
The current thing that comes up regularly is choosing an LLM setup.
"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
Basically, prioritization.
It’s really that simple.
Eschew obfuscation.
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.
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.
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.
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.
I read it as finding the point of maximum effectiveness. The point at which the most is done for the most.
Getting to that point can be calculated (in some cases), but I suspect most folks get there by trial and error. Finding out what is effective, and what is not, and choosing what is effective, over what is not, until there's no longer a choice. That often becomes tribal knowledge, and is handed down. There's always someone trying to improve it, and when they figure it out, that gets added to the tribal knowledge. Basically, that's how nature does it, so there's some serious prior art. Natural Selection is brutal prioritization.
In Morocco, they used to announce the end of the Ramadan fast, by holding up a black thread and a white thread, and waiting until they could not tell the difference.
Then, they'd fire a cannon, and everybody would dig into some awesome soup. Sort of the same thing.
No, it's more about maximizing the utility function and finding the "shield" where you'd start getting diminishing return beyond that
you have confused me
The lesson landed