What “Cleverer” Means
There is much talk of general intelligence and of spectacular abilities said to appear all at once, as if by magic, at a certain size of model. But beyond the enthusiasm and the controversy, both ideas can be tested and measured rigorously. In this lesson we train seven models of increasing capacity on exactly the same task, to see what the passage from helplessness to mastery really looks like.
Terms: generality breadth and depth a leap (emergence) the measuring ruler general intelligence
The word “general”
Throughout this course we have measured many concrete things: percentages of accuracy on specific tasks, digits hit, signs retained, points classified on a spiral. But we have never measured “intelligence” as an abstract essence. We have already learned that any score in a ranking is a property of the the pairing between the model and the test chosen, not an isolated virtue of the model. Starting from this, the word “general” takes on a far more precise, useful and modest meaning.
We say that a model is *general* in so far as it successfully solves a wide range of different tasks, including problems it never met during training. To describe this behaviour it is useful to look at two complementary axes: breadth (how many kinds of task it can tackle) and depth (how competent and reliable it is in each of them). A multiplication table is extremely deep but narrow: it does one thing perfectly and nothing else. A five-year-old child is very “broad” and moderately “deep”: it can speak, run, draw and recognise emotions, even if it cannot do advanced arithmetic. In current debates, the phrase “artificial general intelligence” (AGI) describes a system that would be both broad and deep, comparable to a capable person across the great majority of intellectual and practical tasks.
Still, this generality is no fixed number that can be measured with a single instrument. It always depends on the list of problems chosen for the evaluation. And whoever settles the list inevitably influences the answer as well. That is why, to the question whether a system has become “general” or not, a single test can never give an absolute conclusion.
The leaps
It is often claimed that certain cognitive capacities of neural networks “appear” all at once: a model of a given size seems entirely helpless before a request, while a model only a little larger suddenly solves it brilliantly. This idea — often called an *emergent ability* — can be tested and checked directly here, on a small scale, using the concepts built up so far.
We shall take seven models similar to the one in earlier lessons, with hidden layers from 4 units up to 32 units, and train them on the same arithmetical task: adding two two-digit numbers, writing the result digit by digit. Then we measure each model on the same set of new questions, unmet in training, by two criteria in parallel: what percentage of the individual digits are correct, and in how many cases the whole answer (all three digits) is perfectly correct.
Looked at together, the two curves tell remarkably different stories about the same seven models. The percentage of digits hit rises smoothly and gradually as the network grows in size, with no surprise and no magical leap. The curve of the complete answer, by contrast, stays near zero for a long time in the small models and then shoots suddenly upward. Looking only at total success, the leap looks like a lightning discovery; but looking at competence per digit, the progress was continuous and steady from one end to the other.
The same figure, another ruler
Where, then, does the feeling come from that the model made an enormous leap, as if it had suddenly seen the light? The answer is surprisingly simple: not from any abrupt change inside the network, but from the way we chose to mark it.
Think of opening a safe with a three-wheel combination. To open the safe you must hit all three digits in a row. If you get even one wrong, the door stays locked and you score zero. If a model begins to learn and hits each individual digit in 7 cases out of 10, that looks like reasonable performance. But to open the safe it needs three successes linked together, in the same attempt. Because of this strict requirement, it will hit the whole code only in about one attempt out of three.
If the model improves only a little — going gradually from 7 to 9 digits hit out of 10 — something spectacular happens: the rate of opening the safe jumps abruptly from a third to nearly three quarters of the attempts! Seen exclusively through the safe test (“Did the door open? Yes or No”), it looks like an overnight miracle, a sudden awakening. In reality the machine made only modest, uniform progress on each wheel separately. The “leap” is merely an illusion created by a rigid ruler that accepts no half measures.
The table below shows this effect clearly. We evaluate the same seven models using three rulers of different severity and note the size at which the ability seems to switch on (that is, passes the threshold of 50% success on new examples).
Look how the apparent moment of the “leap” moves across the screen with the ruler chosen: on a gentle ruler (which asks for only one correct digit), the ability seems to appear very early, even in small models; on the severe ruler (where we demand everything perfect), it seems to settle in far later. The models are exactly the same between the two rows — only the demands of the observer changed.
This does not mean that real thresholds do not exist in nature or in technology. It means only that when we hear that an artificial-intelligence model has acquired a spectacular ability “all at once”, the first thing to check is the ruler by which it was measured.
Breadth, seen close up
Even within an apparently single task such as addition, very different degrees of difficulty lie hidden: additions with no carry (where the digits add directly), additions with one carry, and more complex additions with two successive carries. Looked at closely, these are three distinct sub-tasks, and the breadth of a model can be read from the number of levels it truly masters.
Small models master the simplest case first, without a carry. As the network is given more parameters, it learns in turn to handle the first carry and, only at large capacities, the double carry. The breadth of the model therefore grows in discrete steps, conquering one category at a time, while the overall average often masks these internal thresholds. That is why, when it is claimed that a system has reached general performance, the key question to ask is: which specific categories were included in the test, and which borderline cases were left out?
What cannot be predicted
In the lesson on scaling we saw that mathematical laws predict precisely the fall of the loss function (the error of prediction). But what the scaling laws cannot predict is which concrete practical abilities will switch on as the error falls. The graphs above show us clearly why: the same continuous fall in the loss can translate, through the ruler of a compound test, into an abrupt step, and where that step settles depends on the internal structure of the requirement.
From this follow all the uncertainties in current debates about the future of artificial intelligence. We do not yet know whether the beaten road — more data, more parameters, more computation — will suffice to reach the breadth and depth of human thought in all its nuances, or whether we shall stop at an unseen ceiling that will demand new fundamental principles. There are remarkable researchers on both sides of this dilemma, and this course does not set out to settle an open debate. What we can do is to put the questions properly: with transparent benchmarks, multiple rulers and a clear-eyed analysis of the evidence.
The concept that follows, “superintelligence”, moves attention from mere performance to a still deeper question: what happens to a goal when it is pursued with unbounded efficiency by an ever more powerful system of optimisation? In the next lesson we shall explore this challenge with the figures and the mathematics on the table.