Toby Ord writes that “the required resources [for LLM training] grow polynomially with the desired level of accuracy [measured by log-loss].” He then concludes that this shows “very poor returns to scale,” and christens it the “Scaling Paradox.” (He continues to point out that this doesn’t imply it can’t create superintelligence, but I agree with him about that.)
But what would it look like if this were untrue? That is, what would be the conceptual alternative, where required resources grow more slowly?I think the answer is that it’s conceptually impossible.
To start, there is a fundamental bound on loss at zero, since the best possible model perfectly predicts everything—it exactly learns the distribution. This can happen when overfitting a model, but it can also happen when there is a learnable ground truth; models that are trained to learn a polynomial function can learn them exactly.
But there is strong reason to expect the bound to be significantly above zero loss. The training data for LLMs contains lots of aleatory randomness, things that are fundamentally conceptually unpredictable. I think it’s likely that things like RAND’s random number book are in the training data, and it’s fundamentally impossible to predict randomness. I think something similar is generally true for many other things—predicting world choice for semantically equivalent words, predicting where typos occur, etc.
Aside from being bound well above zero, there’s a strong reason to expect that scaling is required to reduce loss for some tasks. In fact, it’s mathematically guaranteed to require significant computation to get near that level for many tasks that are in the training data. Eliezer pointed out that GPTs are predictors, and gives the example of a list of numbers followed by their two prime factors. It’s easy to generate such a list by picking pairs of primes and multiplying them, the writing the answer first—but decreasing loss for generating the next token to predict the primes from the product is definitionally going to require exponentially more computation to perform better for larger primes.
And I don’t think this is the exception, I think it’s at least often the rule. The training data for LLMs contains lots of data where the order of the input doesn’t follow the computational order of building that input. When I write an essay, I sometimes arrive at conclusions and then edit the beginning to make sense. When I write code, the functions placed earlier often don’t make sense until you see how they get used later. Mathematical proofs are another example where this would often be true.
An obvious response is that we’ve been using exponentially more compute for better accomplishing tasks that aren’t impossible in this way—but I’m unsure if that is true. Benchmarks keep getting saturated, and there’s no natural scale for intelligence. So I’m left wondering whether there’s any actual content in the “Scaling Paradox.”
(Edit: now also posted to my substack.)
Strongly agree. I was making a narrower point, but the metric is clearly different than the goal—if anything it’s more surprising that we see so much correlation as we do, given how much it has been optimized.