Suppose I’m making plans to visit my parents for the holidays. I go online, and I book plane tickets. I do not worry about how I’m going to get to the airport - indeed, that question probably wouldn’t even enter my mind until a few days before the flight. Why not? Well, without thinking through any particular plan, I expect getting to the airport to be easy. There are many ways to get there (train, Uber/Lyft, drive & park) all of which I’ve used before and any of which would be fine.

This is relaxation-based search. I have a planning problem: visit my parents for the holidays. I’m searching for a plan which satisfies all my constraints: arrive at my parents’ house ~2-3 days before Christmas, arrive at the airport an hour before the plane takes off, don’t spend lots of time en route, don’t spend too much money, and many many other constraints (most of which are implicit in my head). In order to search efficiently, I relax the problem: I ignore many constraints which I expect will be easy to satisfy, like arriving at the airport on time or packing my bag. I throw out all but the hardest-to-satisfy constraints, which in this case mostly involve the plane tickets. I solve the “relaxed” search problem, with just the hardest constraints, then go back and tweak the plan to handle other constraints later.

Why This Matters: High-Dimensional Worlds

Our world is high-dimensional. Even a trip to visit my parents for the holidays involves hundreds of choices: dates, modes of transport, packing list, how early to arrive at the airport, etc. Multiply them all out, and we have an exponentially vast space of possibilities.

And that’s just a trip to visit my parents. Imagine the possibilities involved in designing a website or a scientific experiment!

When searching in spaces this large, we cannot afford to test every possibility. We are forced to rule out vast swaths of the search space without even looking at them. Relaxation-based search is one of the most general algorithms we know of which can do this.

The key is that we can usually express the problem-space using constraints which each depend on only a few dimensions. For instance, if I want to arrive at the airport an hour before the plane takes off, that constraint only involves two dimensions: my arrival time at the airport, and the takeoff time of the flight. It does not directly depend on what time I wake up, whether I pack a toothbrush, my parents’ plans, cost of the plane tickets, etc, etc.

So, if I can relax the problem to just a few of the most-difficult constraints, hopefully those constraints will involve a relatively small number of variables, and I can search through that smaller space more efficiently. When booking flights, for instance, I mostly just think about flight date, flight time, and cost (and maybe airline, sometimes, as a tiebreaker). I don’t worry about all the other dimensions, like packing my bag, getting to the airport, eating, buying presents for my parents, etc, etc.

Then, holding my solution for the most-taut constraints constant, I can go handle the other constraints. I plan around my flights to decide when to set my alarm, what/when to eat, and how to get to the airport. Each of those, in turn, only involves a few variables at a time, while holding the rest of the solution constant.

Tautness And Slackness

Imagine that I mistakenly think that the plane flight is an easy part of my holiday plans. I show up at the airport on December 22nd, find a service desk, and ask what flights are available.

This is likely to go poorly.

One crucial element to make relaxation-based search work well is to have an accurate idea of what’s hard, and what’s easy. Which constraints are “slack” - i.e. we can relax them, and probably be easily able to tweak the plan to handle them later? Which constraints are “taut” - i.e. we need to handle them up-front, because we probably won’t be able to easily tweak our plan to handle them later? If we mistakenly relax a taut constraint, then when we need to handle it later on, we’re likely to have to scrap a large chunk of the plan and redo it, or spend a lot of resources, or just fail altogether. It’s like showing up at the airport on December 22nd expecting to easily be able to get on a plane.

So, two kinds of useful information for plan-making:

  • Types of constraints which are reliably slack - e.g. using the trains or Uber/Lyft makes it reliably easy to get around the city.
  • Types of constraints which tend to be taut - e.g. getting good holiday plane tickets.

This information is especially valuable when there’s some strategy many people don’t already know which makes a broad class of problems easy, or when there’s some broad class of problems which are usually much harder than many people realize. 

With that in mind, this post is partly intended to fill a gap in my past writing. I have a whole sequence of posts with titles like:

… but I never quite properly explained why these posts are important, not just for modelling the world, but for problem solving using our existing world models.

Coordination as a Scarce Resource is a good example: the core idea is that coordination problems tend to be very taut in a wide variety of organizations (the post gives some qualitative evidence for this and an argument for why it might be). Imagine I start up a business on the assumption that my coordination problems will be easy - e.g. that I can easily identify and reach out to the people who my product would benefit, or that employees with different specializations will easily coordinate to share important information with each other. This business is likely doomed; I have ignored problems which are likely to be quite difficult/expensive to solve. Worse, this is often a non-obvious failure mode, especially for people who have not run a business before; many people do not realize the difficulty of coordination between employees with different specializations, or the difficulty of finding customers even when we have a product which would be great for them.

On the other hand, suppose I start up a business on the assumption that I can magically turn money into anything one might find at Walmart or Home Depot or a grocery store. This is likely to be fine; I can in fact do something pretty similar to that pretty easily. When making a business plan, I can largely ignore/relax constraints which require material goods, as long as I have a reasonable budget. Material goods are an abundant resource.

Hardest-First vs Easiest-First

One implication of the relaxation-based search view is that we should usually tackle problems hardest-part first. Advice to do the opposite is reasonably common, so it’s worth addressing why hardest-first is a good idea, and when/how easiest-first might be better.

The main argument is that, if we just solve an easy part of a problem, then we’ll likely have to throw out our solution and start over mostly-from-scratch when we get to the hard part. If I have a $1000 vacation budget, and flights cost $800, then planning my hotels without accounting for the flight budget will be basically useless. I’ll have to redo it all later. As a general rule, it’s the hard parts which most restrict the solution-space, so it’s the hard parts which most steer the overall form of solutions.

So why might one instead tackle an easy part first?

Well, tackling an easy part of a problem might provide some value other than a usable almost-solution for the main problem. It might help build confidence, or acquire generally-useful resources.

In particular, tackling an easy part of a problem can unearth information about the problem itself. If I’m entering a domain where I have little experience, where there’s lots of unknown unknowns, tackling an easy problem pushes me to interact with the system and “get my feet wet”. It gives my brain its first few data points about the problem. I may find out that I was completely wrong about which parts are hard vs easy!

Main takeaway: if we have a reasonable idea of what’s easy vs hard, then hardest-first makes sense; if we solve an easy part first, we’ll likely just have to throw away that effort later. But if we have no idea what’s hard vs easy, then the main priority is to figure that out, and tackling an easy-looking problem is one way to get our first few data points.

Duality: Mapping Tautness and Slackness in Unfamiliar Territory

More generally, how can we efficiently figure out which constraints are taut vs slack in a new domain? How do we map out the problem/solution space?

Here’s a concrete example: suppose we want to put something into orbit without using a big rocket. We don’t know anything at all about putting things into orbit, so to start out, we babble some dumb starting ideas to see where they fail:

  • Put the thing on a weather balloon to lift it to the edge of space, and then just use a small rocket for the last little push.
  • Shoot the thing out of a really big gun.
  • Build a really tall tower all the way to space, and throw the thing off the top.
  • Attach the thing to a long tether, spin it really fast, and release.

When we look at how these fail, two of them (the weather balloon and the tower) run into the same problem: the object can get to space that way, but then it just falls back down. Getting it into orbit requires accelerating the object to about 32 times the speed of sound, sideways.

We have thus identified one slack constraint, and one taut constraint:

  • Getting to space is relatively easy (since even our initial list of dumb ideas had two decent ways to do that).
  • Going sideways at mach 32 is harder.

Now that we know getting to space is relatively easy, we can relax that part of the problem when searching for further solutions - for instance, we can ignore air resistance, and then account for that later by adding a 100 km tall tower to our solution or something. On the other hand, going fast enough sideways is relatively hard, so we should probably now focus on ways to accelerate an object to mach 32 and see how those ideas fail.

More generally, the strategy looks like this:

  • Try out some random or simple solution, and see where it fails.
  • Try to extend the failures to more general constraints, and see where the loopholes are.
  • Try to extend the loopholes to full solutions, and see where the failures are.
  • Try to extend the failures to more general constraints, and see where the loopholes are.

In our orbit example, we first list a bunch of dumb ideas. Two of them fail in a similar way, so we extend that failure to a more general constraint: we need to go sideways at mach 32. Then, we look for “loopholes in the constraint” - i.e. efficient ways to accelerate an object to mach 32.

This strategy shows up in many places. In math, for instance:

  • Try a proof, and see where the loopholes are.
  • Try to extend the loopholes into counterexamples, and see where they fail.
  • Try to extend the counterexample-failures into proofs, and see where the loopholes are.

In the process of iterating like this, our brains typically build up estimates of tautness and slackness. In a math problem, we might notice that one class of counterexamples is easily circumvented by a few different proof-strategies; those counterexamples are “slack constraints” on our proof, and we can probably ignore them during our search. Other counterexamples come up again and again, blocking a wide variety of proof-strategies; those counterexamples are “taut constraints”, and we should focus on them when searching for a proof.

One neat thing about this general strategy is that partial solutions serve as “constraints” in our search for constraints. This is duality: just as proofs are counterexamples of counterexamples, solutions are constraints on constraints. Just like our brains keep track of the tautness and slackness of constraints, they can also keep track of the tautness and slackness of solution-strategies - e.g. if it’s really easy to catch an Uber/Lyft, then that’s a very taut constraint-on-constraints for trip-planning. We can then use relaxation-based search to find constraints, just like we use it to find solutions - e.g. look for parts of the trip where we can’t just take an Uber/Lyft, since those parts are more likely to be taut constraints on the trip-plan.

For a more visual example of that very meta idea, check out this post, which has a neat visual example applying the technique to mazes.

To start off, just draw some dumb starting solution, and then see where it violates the constraints…


Humans naturally use relaxation-based search in our day-to-day lives: we ignore the “easy” parts of a problem (i.e. relax slack constraints), and just solve the hard parts (taut constraints), then go back and account for the easy parts later. This works well in our high-dimensional world because we can usually express problems using constraints which only depend on a few dimensions each, and some are much more taut than others. When booking plane tickets, I usually only worry about three or four variables.

In order for relaxation-based search to work well, we need to know what things are taut/slack. Thus the importance not only of identifying classes of things which are easy, but also classes of things which are hard. Try to start a business on the implicit assumption that coordination between employees with different specialties is easy, and we’re in for a rough ride. Thus the value in noticing broad classes of taut/slack constraints, especially constraints people often don’t notice are taut/slack.

For solving problems, it’s usually best to tackle the hardest parts first, if we know which parts are the hardest. Otherwise we’re likely to need to throw out large chunks of our solution when it comes time to handle the hard parts. That said, if we don’t know which parts are hard - if we’re tackling a problem with lots of unknown unknowns - then starting with some easy-looking parts can provide useful data.

More generally, we can map out the slackness and tautness of constraints on a problem by trying out a bunch of solutions, seeing where they fail, generalizing the failures to constraints, seeing where the loopholes are, and iterating. Constraints which block broad classes of solutions are more taut; constraints which are usually easily circumvented are more slack. This process also lets us switch back and forth, building up an idea of tautness and slackness for both constraints and solutions, much like switching between searching-for-proof and searching-for-counterexample in math.


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3 comments, sorted by Click to highlight new comments since: Today at 6:39 AM

This seems surprisingly related to a thought I've had about the inefficiency of markets in consumer goods. When buying things, people usually only have room to optimize on a small number of taut constraints. One of them is going to be price; you get one, maybe two more if you're lucky. If some product or service has more axes of variation than "what does it cost" and "does it work at all", they will typically get almost negligible optimization pressure applied to them.

For example, the problem of buying an airplane ticket already has a bunch of taut constraints -- the right place, the right time, the right price. The wifi on planes is generally garbage; whether or not there is wifi has risen up to the point where it gets a bit of optimization pressure, but how well the wifi works is way below the required bar. There is no incentive to improve it. Similarly the food on planes.

(To be honest, this applies to employment relationships, too, which might explain some things about how people generally find jobs to suck.)

This concept is often discussed in the subfield of AI called planning. There are a few notes you hit on that were of particular interest to me / relevance to the field:

The key is that we can usually express the problem-space using constraints which each depend on only a few dimensions.

In Reinforcement Learning and Planning, domains which obey this property are often modeled as Factored Markov Decision Processes (MDPs), where there are known dependency relationships between different portions of the state space that can be represented compactly using a Dynamic Bayes Net (DBN). The dynamics of Factored MDPs are easier to learn from an RL perspective, and knowing that an MDP's state space is factored has other desirable properties from a planning perspective.

I expect getting to the airport to be easy. There are many ways to get there (train, Uber/Lyft, drive & park) all of which I’ve used before and any of which would be fine.
I want to arrive at the airport an hour before the plane takes off, that constraint only involves two dimensions: my arrival time at the airport, and the takeoff time of the flight. It does not directly depend on what time I wake up, whether I pack a toothbrush, my parents’ plans, cost of the plane tickets, etc, etc.

You are actually touching on what seems to be three kinds of independence relationships. The first is temporal, and has something to do with options having identical goal states. The second is regarding the underlying independence relationships of the MDP. The third isn't technically an independence relationship, and is instead in regards to the utility of abstraction. In detail:

  1. It doesn't matter which option you take (train, Uber/Lyft, drive & park) because they all have the same termination state (at the airport). This shows that we plan primarily using subgoals.
  2. Certain factors of the state space (your parents' plans, whether you pack a toothbrush, cost of the place tickets) are actually independent of each other, i.e. your parents' plans have no real physical consequences in your plan at any time, e.g. you can walk and chew gum at the same time. This shows that we plan with a factored understanding of the state-action space.
  3. The time you wake up does indeed matter in your plan, but the exact time does not. For your planning purposes, waking up any time before you must leave your house (including factoring in packing, etc.) is permissible and functionally equivalent in your plan. All possible states of being awake before your out-the-door time collapse to the same abstract state of being awake on-time. This shows that we plan using abstract states (a similar, but subtly different point than point 1).

More generally, how can we efficiently figure out which constraints are taut vs slack in a new domain? How do we map out the problem/solution space?

We can use the three kinds of independence relationships I mentioned above to answer these questions in the RL/Planning setting:

  1. So long as you can learn to consistently reach a specific state, you can use that state as a subgoal for planning and exploration. This principle is used in some existing RL literature (I'm a student in this lab).
  2. If you can figure out the underlying representation of the world and discern independence relationships between state variables, you can focus on making plans for subsets of the state space. This idea is used in some planning literature.
  3. If you discover a consistent way to get from any set of states A to a single state b, you can treat all states in A as a single abstract state a, so long as b is relevant to the rest of your plan. This abstraction principle allows one to derive a smaller, discrete MDP (much easier to solve) from a bigger, continuous one. This is actually the theme of the literature in point 1, and here is the source text (to be more specific, I am an undergrad working in George's lab).

I read about half this post before realizing that this concept is intuitively familiar to me from the process of translating poems/songs: very often a poem or song will have certain specific bits that are going to be extra important to get exactly right (e.g. the title, or something conceptually loadbearing, or a particularly clever or emotionally impactful line) or unusually hard for some reason (e.g. it's trying to get across a very specific or finicky or culturally specific concept, or using clever wordplay, or it's self-referential like "a fourth, a fifth, a minor fall, a major lift"), and when considering a new translation it usually makes sense to start brainstorming from those bits, because every line you write will affect what can go near it, so it makes sense to start with the lines where I'll be lucky if I can think of one good thing that scans, and then from there try to fill in the other lines (which hopefully have more degrees of freedom) with things that not only scan but also rhyme with the hard parts. (Also because sometimes you won't think of anything for the hard parts, and then it might not make sense to invest a bunch of time in working on the rest of the thing.)