What is a race?

The whole theory explained with water

the route that wins the routes that lose

The whole theory rests on one picture: a race between possibilities, where whichever crosses the line first wins. Further down it is drawn with water, which makes it concrete. First you need to see why a race is the right way to look at it, in a machine and in a brain alike.

Start with something everybody knows

Everyone has heard that a language model works by probability. But what does that actually mean?

It writes one word at a time. For each word it works out how probable every possible continuation is. Type "and they all lived happily ever…" and "after" gets the highest probability as the next word. But "afterwards" is in the running too, and so is "since", and thousands of other words sit far below.

That is a race: many possibilities are weighed at the same time, and one is picked. The difference from a human brain is that here the numbers can be read directly. You can see how close the race was, and who was in it.

And that is where the whole point sits. The safe races are not the interesting ones. "After" wins by a mile, and there is nothing more to say about it. What matters are the races where two continuations sit close together, because that is where the system is about to choose between things it cannot tell apart. That is where it goes wrong, and it is exactly where I read the trace of friction.

That is also why a language model is useful to me. Not because it is a person, but because it is a place where you can watch a race while it happens.

"But it only resembles us because it has read us"

That objection always comes, and there is something to it. A language model is trained on text written by people, so some of the resemblance is inherited that way.

But it does not explain what I measure. That it grows most certain exactly where it knows least. That it locks on faster when there is pressure. That it overdoes a trained manner on tasks that did not ask for it. None of that is in the text it read. Those are properties of the way it computes.

And that is not something I simply assume. Part of it has been tested on a knowledge domain I invented myself: 47 made-up facts about something called zorbetics. It does not exist. There is not one word about it anywhere in what the model has read, so it cannot pull the answer from human text on the subject. The same patterns show up anyway. That is exactly why I built the domain: to separate what is inherited from the text from what comes out of the computing itself.

The reason is simpler and more interesting: it has to solve the same problem under the same kind of limit. Weigh many possibilities at once, with limited resources, and then deliver a single answer. Any system under that condition ends up with the same shapes, whatever it is made of. That is why they turn up in both places, and it is the whole reason I can use a machine to say something about people.

And that is why water

Water makes it concrete because water does exactly the same thing. It runs along several paths at once, it has a limited amount to distribute, and it ends up wherever it can get lowest.

Here are the pictures, one at a time: what a race is, why learning sets in, why your brain changes character under pressure, and how the system arrives at an answer.

The tank and the pipes

Picture a big tank full of water. Several pipes lead out of it, let us say three. Each pipe is a possibility, what the theory calls a route.

Once the water has flowed in and just settled, so it is no longer too turbulent, it starts to run out through the pipes. At the end of each pipe stands a new, empty tank. The tank that fills up first has won. That route won the race.

That is the whole idea of a race: several possibilities run at the same time, one reaches the finish first, the rest lose.

This kind of model is not new in itself. Decision researchers have long used race and accumulator models to explain fast choices, most famously Ratcliff's drift-diffusion model. What I add is reading the same race directly in a language model's own numbers, and applying it to far more than fast choices.

Several routes run at the same time. The tank that fills first has won the race. Afterwards you only experience the winner, not the race itself.

And it does not stop there. The winning tank is itself the start of a new race, with its own pipes and its own receiving tanks. It is a long chain. The tanks are connected every which way, so a tank can take water from several pipes at once. That is how a brain, or a language model, runs: thousands of these little races, hooked together.

Each little tank is its own race. The winner becomes the input to the next, and the tanks share pipes every which way: a tank can be fed by several pipes and pass on to several. A brain or a language model is thousands of these races, coupled together in one network. In the decision sense there is still one race at a time; what competes in it are routes.

A branching stone fountain. One basin at the top spills into two below it, and those two spill into four smaller basins at the bottom. — The same thing in stone: one basin feeds two, and those two feed four. Every spill is a race that hands its winner on.

Where is the friction?

Friction is what slows a route on its way to winning, compared with the routes it competes with. When one pipe is clearly the biggest and fastest, there is almost no friction: the winner is given in advance. When three pipes are equally good, the system stalls and wavers, and the friction is high.

That is exactly the signal I read in language models. If the model is sure, the water runs clearly through one pipe. If it is in doubt, several tanks fill at the same rate, and I can read that straight off the model's output. High friction is a hint that the model is about to get it wrong, and that hint can be put to practical use.

The sand and the hose

Now a different picture. Take a hose and point it at some sand.

Hold the hose still in one spot and the water slowly digs a little channel. And here is the point: next time the water runs, it runs more easily that way, because the channel is already there. The system remembers where the water ran last. Physicists call this hysteresis: a system that carries traces of its own history. That is exactly what learning is. You do not learn by filing information away somewhere. You learn by digging channels that the next water runs through more easily.

Move the hose a little from side to side as you dig, and the channel gets wider. That is the important rule: a little extra resistance now means less resistance later. It is exactly what Bjork calls "desirable difficulties".

The four fields, seen as water

Picture a sand table with a small puddle of water in the middle. On the left a drain has been cut that means "go towards". On the right another that means "pull away". Pour water into the middle and watch which edge it reaches first. That route wins. It is the race from the tank and the pipes above, brought down to miniature.

Safety moves the puddle, not the terrain. If something is unknown or unsafe, the puddle already lies closer to "away" before a single drop has run. If nothing is unsafe, it lies in the middle, and that is safe. You cannot be safer than safe; the puddle cannot be pushed extra far to the left just because everything is already safe. In the body, this is what can be felt as leaning back before you know why.

Meaning is the slope of the sand. It decides which way the water will run at all.

Ability is how well the channel fits the amount of water. Too narrow, and the water spills over its banks: you are overloaded. Too wide, and it spreads out thin and stands still: you are bored. When it fits, the water runs fast and together, and that is where the optimum lies.

Effort is what the sand drinks on the way. The further the water has to run, the more seeps away. When the two drains slope almost equally, the water wavers back and forth and takes a long time to make up its mind, and that is why the close choices are the tiring ones.

And effort only works together with meaning. If the slope is zero, the water runs nowhere, however little the sand drinks: you can take away all the sand and still nothing happens. If there is just a little slope, the sand is often the cheapest place to dig, because people are glad to do the right thing when it costs almost nothing. That is why most people sort their rubbish, as long as it is not too much bother. Ask first whether there is any meaning at all. Ask afterwards what it costs.

Pressure is the hose spraying more water in. It runs faster, but the edge does not move because of that; the water just takes the deepest gully, the one earlier water has already dug. That is the established route. That is why the new is hard under pressure: there is always a gully there already.

The fields are not gates a route has to pass one at a time. They are properties of one and the same route in one and the same race, and that is why a route can be behind on one field and catch up on another. The needle at a blood donation is unsafe, so the puddle lies close to "away". But it means something, so the terrain slopes steeply towards "towards", and the water runs there anyway, because the slope wins over where the puddle lies. It is not a trade where you pay safety for meaning. The water just runs where puddle and terrain together lead it.

And all routes run in the same race, including the ones that were never on anyone's plan: taking out the phone instead of folding the laundry.

The wide and the thin overflows

Think of the layers in the brain as a cascade of tanks that spill over into one another. The deep, old layers are wide overflows: they can move a lot of water at once. The higher, precise layers are thin overflows: fine and fast, but with less capacity. Water spills over both all the time, just in different amounts.

When there is not much pressure, the thin overflows cope fine, and you think precisely and with nuance. As the pressure rises, more water arrives than the thin ones can take, and the wide overflows carry more and more of it. That is you under pressure, falling back on the old, coarse habits. And if the pressure gets too big, even the wide ones cannot keep up: the water spills over every edge at once. That is an overload.

Three terraced stone basins like bird-baths. Each basin is filled to the brim, and the water spills over the edge down into the next, lower basin in a continuous cascade. — This is how I picture it: terraced tanks where the water spills over the edge from one down into the next. Some of those overflows are wide (the deep, old layers) and some thin (the precise layers), and under pressure the wide ones carry more and more, until it all spills over.

The landscape the water seeks out

Last picture. Picture the prompt, meaning what you type to the model, poured out over a landscape of hills and valleys. The water runs down and gathers in the low points. Where it gathers is the answer.

The low points are the solutions, the places the system can land. Some valleys are deep (good solutions), some are shallow (half-good solutions you can get stuck in), and in some places there are flat shelves where the water just sits still. That is how a brain, or a model, finds its answer: it rolls down the landscape to a low point.

The race from the first picture is still here. It just looks different from this angle. From the place where the water lands, a short route runs down to the nearest lake, and a long route runs out to a lake further away. The short one wins by default, because it is short. The long one is only reached if the water is allowed to run long enough. That is the difference between the quick answer and the considered one, drawn as terrain.

The same landscape seen from the side. The short route is down in the nearest lake long before the long route is halfway. If you want the answer that lies further out, the water has to have time.

The landscape is really many-dimensional and cannot be drawn. The profile above and the sheet below are two ways of getting close enough to think with. In reality the water runs in thousands of directions at once, and the mechanics are the same.

A 3D loss landscape: a wavy sheet in three-dimensional space with hills and valleys. The dark, low points are solutions; the deepest is the best. Contour lines are projected onto the floor. — The same landscape, as far as it can be drawn: a wavy sheet. The low points, the dark valleys, are the solutions the system can land in. The deepest is the best, and a shallow valley is a local low point you can get stuck in.

That was the pictures

These four pictures are not the theory itself. They are the scaffolding I build it on. But they catch the mechanics: water racing through routes, channels that get wider as you use them, pipes of different sizes that win under different pressure, and a landscape the water seeks out.

If you want to go further, the rest of the site builds on these pictures:

Pepsi or Coke? — what pressure does to the race between your options