4 ms·

Actually understanding a function means that it can be computed to arbitrary precision. Memorizing inputs and outputs only allows you to interpolate between val

Actually understanding a function means that it can be computed to arbitrary precision. Memorizing inputs and outputs only allows you to interpolate between values you’ve already seen. That breaks down quickly for anything somewhat chaotic. You can try the following function in ChatGPT yourself: f(x) = x * (sin(sin(x)))^2 Ask it to give you the values of f for integers of x between -10 and 10. I tried it 5 times and it was never close. I chose this for several reasons. One, it’s very unlikely that it’s memorized the answers somewhere on the internet. Two, it’s pretty chaotic if you look at a graph of it, so interpolation won’t work. (It is bounded by x and 0 for all values but for large absolute values of x it varies wildly.) And three, memorizing values won’t get you anywhere since it becomes much more chaotic as x increases. I also disagree that approximations are simpler. As you can see, the actual function is only sine and multiplication. To approximate this function would be far harder.

int_19h 4y ago
It's not exactly a secret that GPT is bad at processing math in inputs, which seems to have something to do with the token representation being poorly suited for it, for starters. But I thought we were talking more generally about functions mapping inputs to outputs - the "perfect understanding" function is the one that captures all the relationships between the entities in the input and thus can always give a perfect answer to the extent that training data contains it, while an approximated "partial understanding" function internally has a much simpler model of those entities and their relationships (which can still turn out to be still good enough). Mapping inputs to outputs is an especially simple model, but it's still a model and an approximation. And yes, in your example, of course it's simpler to just store the function itself - provided that you know in advance what it is, which, to remind, GPT does not. But when we're dealing with f(question)=answer of chatbots, the function that GPT ends up approximating is decidedly not simple.

int_19h 4y ago

It's not exactly a secret that GPT is bad at processing math in inputs, which seems to have something to do with the token representation being poorly suited for it, for starters. But I thought we were talking more generally about functions mapping inputs to outputs - the "perfect understanding" function is the one that captures all the relationships between the entities in the input and thus can always give a perfect answer to the extent that training data contains it, while an approximated "partial understanding" function internally has a much simpler model of those entities and their relationships (which can still turn out to be still good enough). Mapping inputs to outputs is an especially simple model, but it's still a model and an approximation. And yes, in your example, of course it's simpler to just store the function itself - provided that you know in advance what it is, which, to remind, GPT does not. But when we're dealing with f(question)=answer of chatbots, the function that GPT ends up approximating is decidedly not simple.