Lila Soto: Hugo, long week — but I came across something that made it feel shorter, honestly.
Hugo Vance: Mm. What was it.
Lila Soto: I was reading the 2020 Kaplan et al. paper — the foundational OpenAI scaling laws work, Dario Amodei and Tom Brown also on it — and there's this moment where they just... graph the relationship between compute and loss, across six orders of magnitude, and it's a straight line. On a log-log plot, which means it's a power law. And I thought: this is when prediction replaced surprise.
Hugo Vance: That is a striking way to put it.
Lila Soto: I mean — before that paper, AI breakthroughs felt like weather. They arrived. After it, at least in principle, you can schedule them. That cultural shift is what I keep sitting with, not just the math.
Hugo Vance: Yes. And that is precisely what I find worth examining carefully — because the field went from 'here is a stable empirical relationship' to 'here is the blueprint for the future' very quickly. What the Kaplan et al. result actually shows is that cross-entropy loss follows approximate power laws across model size, dataset size, and compute. That is specific and bounded. What got built on top of it is considerably more ambitious.
Lila Soto: So Neural Scaling Laws — that's what we're in today. And the thing I actually want to figure out is whether the line keeps being straight, or whether at some point the map just stops matching the territory.
Hugo Vance: Now that is the right question. And I think to answer it we have to first understand why a power law appears at all — because nothing about language obviously suggests it should compress that cleanly.
Lila Soto: Wait — it really shouldn't, should it. Language is messy.
Hugo Vance: Extraordinarily. Which makes the straight line on that log-log plot either a profound discovery about information, or a very elegant artifact of the specific conditions under which we train these models. And the field has not fully reckoned with which one it is.
Lila Soto: But that's the thing I can't quite land — if language is so messy, why would the math cooperate at all? Like who told neural networks to follow a formula?
Hugo Vance: We don't know. That's the honest answer. There is no mechanistic theory underneath the regularity — just the regularity itself, replicated.
Lila Soto: Huh. That's — I mean, that's either exciting or unsettling.
Hugo Vance: Both, I'd say. Now — before the 'why,' let me give you the 'what,' because I think the intuition is actually quite clean once you have the right picture. Think of a musician practicing scales. Every time they double their total hours at the piano, they get a predictable percentage better — not in random bursts, not in sudden explosions of ability. A steady fractional gain, each doubling buying the same fraction as the last. That is a power law. Loss falls as a power of compute, written as loss proportional to N raised to some exponent alpha. On a log-log plot — that's both axes logarithmic — that relationship becomes a straight line. And the line is what Jared Kaplan and his colleagues at OpenAI were seeing in 2020, across more than six orders of magnitude.
Lila Soto: Six orders of magnitude — that's a factor of a million in scale?
Hugo Vance: More than a million, yes. Model size, dataset size, compute — all of it. The line held.
Lila Soto: Okay but — mm — why is cross-entropy loss the thing they're measuring? Why not just, benchmark scores? Can the model do this task, yes or no?
Hugo Vance: Because benchmark scores are lumpy. A model either gets the question right or it doesn't — there's no signal in between. Cross-entropy loss is, I suppose you could say, more fine-grained. It measures how confident the model is about the next token at every single step. So if the model assigns 40% probability to the correct word versus 60%, that registers, even if a benchmark would just mark both as wrong. You can see the improvement accumulating continuously rather than waiting for it to cross some threshold.
Lila Soto: It's like — instead of asking 'did the student pass the test,' you're watching how they're breathing during the test.
Hugo Vance: That is actually — yes, that is a reasonable way to put it. And the important consequence is that the power-law relationship only becomes visible because you're using that continuous signal. Which, incidentally, is why this relationship has now been found not just in language models but in vision models, in diffusion models — multiple architectures, multiple domains. Kaplan and the OpenAI team saw it first in language, but the fact that it appears structurally across all of these suggests it's not an artifact of how transformers happen to be built.
Lila Soto: So it's — it might be something about how neural networks absorb information, period. Not a quirk of one model family.
Hugo Vance: That is the leading interpretation. And here's where I'd be cautious — we are pattern-matching across architectures and calling it a structural property, but we do not have the deeper explanation that would let us predict *when* it breaks. The line has been straight. Whether it stays straight is a different claim entirely.
Lila Soto: But that's exactly where it — I mean, the line not staying straight is kind of what already happened, right? Because Hoffmann et al. at DeepMind, 2022 — they basically said the recipe was wrong.
Hugo Vance: Yes. The Chinchilla paper. And that is worth sitting with precisely because it came only two years after Kaplan et al.
Hugo Vance: Two years. What Hoffmann and DeepMind showed was that the field had been systematically overbuilding on parameters and undertraining on data — that for a fixed compute budget, the optimal strategy is to balance model size and training tokens. Many of the large models already deployed had too many parameters and not nearly enough data to justify them. A smaller model, trained on substantially more data, could outperform a much larger undertrained model at identical compute cost.
Lila Soto: It's like — the map was accurate, but everyone had been holding it upside down.
Hugo Vance: That is a fair description. Now, one could argue — and I have heard this argument — that a self-correcting framework is a sign of health. Science revises. That is what it is supposed to do.
Lila Soto: Yeah, I'd grant that, but — okay, the correction arrived within a single research cycle. Which means anyone who used Kaplan's ratio to plan a major training run in 2021 was working from a table that had already expired. That's not a philosophical problem, that's a practical one.
Hugo Vance: Precisely. And this is where I'd be cautious about the confidence being placed in any single formulation right now.
Lila Soto: Okay — so this is where the model ladder work comes in, and this one I did not expect. Akshita Bhagia led the study. The idea is you train small models at roughly one percent of the target compute budget, fit the scaling law parameters from those, and then predict what the big model will do before you spend anything on it. And the accuracy — within approximately two percentage points of absolute error.
Hugo Vance: That is — well. Two percentage points.
Lila Soto: Think about what that means concretely. There's a research team, say, at a lab — they have a small cluster, a few GPUs, a Friday afternoon. They run the ladder experiment. And by the end of the day they know, with two-point accuracy, what a thirty-million-dollar training run is going to produce before a single dollar of it is committed. That is not forecasting in the abstract. That is schedulable knowledge.
Hugo Vance: It is almost eerie. And I mean that carefully — the forecasting power is genuine. What gives me pause is that it depends entirely on the power-law relationship remaining stable. Which is exactly what the Chinchilla episode called into question. If the underlying recipe can shift in two years, the ladder's predictions are only as reliable as the regime you're currently in.
Lila Soto: Right — and that's the core issue. The ladder works beautifully inside the power law. But there are places where the power law may not be the whole story — emergent ability artifacts, domain generalization failures, what happens when you scale inference-time compute instead of training compute — and that's the part I want to get into next.
Hugo Vance: Yes. And those are not minor caveats at the margin — they are the cases where the straight line may simply stop being straight, and we have no ladder for that.
Lila Soto: And Schaeffer is the first place I want to pull that thread — because Benjamin Schaeffer's 2023 paper is called 'Are Emergent Abilities of Large Language Models a Mirage?' and that title alone is doing a lot of work.
Hugo Vance: Yes. The metric artifact hypothesis. What Schaeffer argues is that those dramatic benchmark jumps — the ones that look like sudden capability appearing at scale — largely disappear when you switch to continuous evaluation metrics. The underlying cross-entropy loss was improving smoothly the whole time. The discontinuity was in the measurement, not the model.
Lila Soto: But — okay, some capability jumps are genuinely real in practice, aren't they? Like a model suddenly being usable for a task it wasn't before?
Hugo Vance: I'd grant that. Crossing a usability threshold is real — it matters enormously to practitioners. The point Schaeffer makes isn't that nothing changed. It's that the underlying dynamics were smooth all along, and what looked like a phase transition was actually the metric hitting a step function. The model didn't leap. The ruler changed.
Lila Soto: Oh. That's — that's a meaningful distinction.
Hugo Vance: It is. And it matters for forecasting because if emergence is an artifact, then the power law's smooth predictions are actually vindicated. But — well, here's the crack — it also means the capability that practitioners care about can still jump discontinuously even inside a smooth curve. So you can't simply read the loss curve and know when something becomes deployable.
Lila Soto: Which is the whole point of the ladder model. But then you've got Sorscher — Ben Sorscher — and the data pruning finding, which is I think the most genuinely strange one to me.
Hugo Vance: Strange is fair. What Sorscher and collaborators showed is that if you rank training examples by quality and prune the low-quality ones, the improvement in performance doesn't follow a power law anymore. It can go exponential. Which means — and I want to be precise here — the power law may be a ceiling that reflects the quality of data we've been using, not a ceiling that reflects something fundamental about neural networks.
Lila Soto: Wait. So the power law isn't the ceiling — it's just the ceiling for this particular kind of data?
Hugo Vance: That is exactly the implication. If you can beat the power law with data curation, then Kaplan's straight line isn't a law of nature. It's the line you get when you train on the internet, unsorted.
Lila Soto: I mean — that reframes everything about what OpenAI and DeepMind have been optimizing for.
Hugo Vance: It does. And inference-time scaling compounds it further — because that's a regime where you're not adding training compute, you're adding compute at test time, and the dynamics may follow entirely different curves. We don't have a Kaplan paper for that yet.
Lila Soto: And then domain generalization is kind of the quieter version of the same problem, right? A model scales beautifully on its training distribution and then you hand it something genuinely new —
Hugo Vance: And it degrades. Sharply. Raw scaling doesn't fix that. The power law predicts pre-training loss, not what happens when the distribution shifts. The field has been conflating those two things — loss reduction and real-world capability — and I think that is the structural limit that no ladder model currently addresses.
Lila Soto: So scaling laws are reliable — for what they actually measure. The problem is we've been assuming that's the same thing as capability.
Lila Soto: I keep wanting to say 'scaling laws are wrong' and that's — that's not it, is it. The power law is real. Kaplan's line across six orders of magnitude, replicated in vision, in diffusion, in language — that held. It's just that what the line is measuring and what we want to know are... not the same thing.
Hugo Vance: We found the map. We're still figuring out what territory it covers.
Lila Soto: Yeah. And I think the part that lands for me — the quiet part — is that the ladder model, the Bhagia result, the two-percentage-point accuracy at one percent of training cost — that works. It genuinely works. Inside the regime where cross-entropy loss is the thing you care about. The question is whether pre-training loss is still the thing you care about once you're asking about domain generalization, or inference-time scaling, or what happens when Sorscher's data pruning actually breaks the power-law ceiling.
Hugo Vance: And if those regimes don't follow the same curve, then — well, practitioners are back to running experiments. Which is, I should note, not a catastrophe. It's just not the schedulable future the field was beginning to assume.
Lila Soto: It might be the last era where you could do that. Forecast capability without running the experiment, I mean. That window may already be closing.
Hugo Vance: That is an honest place to leave it. I think we both buy that much — the forecasting was real, and narrow, and extraordinary while it lasted.
