Reiner Pope – The math behind how LLMs are trained and served

Summary

Dwarkesh Podcast Episode Summary

Episode Details

• Title: Reiner Pope – The math behind how LLMs are trained and served
• Host: Dwarkesh Patel
• Guest: Reiner Pope, CEO of Maddox (formerly of Google TPU architecture)
• Date: April 29, 2026
• Format: Inaugural blackboard lecture—graph and equation-driven, highly technical

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Episode Overview

This episode dives deep into the mathematical and infrastructure principles that underlie large language model (LLM) training and inference. Using graphs, equations, and visual explanations, Reiner Pope and Dwarkesh meticulously quantify resource trade-offs when running models on large GPU clusters. Topics range from batch size and latency to sparsity, mixture-of-experts (MoE) architectures, hardware limitations, pricing models, memory hierarchies, and parallels between cryptography and neural networks. The discussion offers a rare technical look at the economic, physical, and architectural forces shaping the evolution of AI.

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Key Topics and Insights

1. Batch Size, Latency, and Cost: The Fundamentals

• Batching:
  • Latency and cost trade-offs are heavily influenced by batching, which refers to serving multiple users’ requests together.
  • “If you do not batch together many users, the cost and the economics can be like a thousand times worse than if you do.” (Rainer Pope, 04:25)
• Compute vs. Memory:
  • Inference time is bounded either by the memory needed to load weights, or by compute throughput, depending on batch size and context length.
  • A “roofline” analysis is used to balance memory bandwidth and compute, revealing strong predictive power about bottlenecks.
• Lower Bound on Latency:
  • There is a lower bound on latency for a given hardware—"I need to read all my total parameters from memory into the chips, and that takes a certain amount of time. If I use all my memory bandwidth, I can’t do any better than that." (09:54)
• Batch Size Practicalities:
  • For frontier models, optimal batch sizes (based on hardware) often range around 2,000 unique sequences per inference (20:30).
  • The number of tokens processed per second is a function of batch size and time window, e.g., “around 128k tokens per second for a batch size of 2,000 and 20ms intervals.” (26:25)

Memorable Quote

“There's some lower bound on latency here ... for a given hardware configuration. There is a lower bound on latency, which is simply: I need to read all my total parameters from memory into the chips, and that takes a certain amount of time.”
— Reiner Pope (09:54)

• Latency from Batching:
  • Worst-case user request latency arises from missing the current batch; if a batch starts every 20ms, the worst-case wait is ~40ms. (22:21)
• Latency/calculation depends on hardware memory capacity and bandwidth (24:22)

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2. Memory and Compute Bottlenecks: Context Length and Sparsity

• Context Length Sensitivity:
  • Longer contexts shift the bottleneck from compute to memory. With dense attention, memory time scales linearly with context.
  • Sparse attention breaks this, scaling as the square root and thus allowing longer contexts at lower cost.

Memorable Quote

“I’m pretty excited about sparse attention. ... Some Deep SEQ papers that published sparse attention end up putting a square root in this term.”
— Reiner Pope (12:35)

• Sparsity Trade-offs:
  • Increasing sparsity in MoE architectures allows compute and memory to be amortized over larger batch sizes, but at a cost of increased total parameters.
  • Empirical studies show diminishing returns—increasing expert count gives only modest model quality gains for a huge increase in parameters (30:13).
  • “Keep running a larger batch size... from the view of the analysis we've done here, this is pure win, keep doing it, until you run out of users.” (30:51)

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3. Mixture of Experts (MoE) and Hardware Mapping

• MOE Layer Layout:
  • MOE layers consist of "router" layers and many "experts" (MLPs), with tokens routed to a subset of experts. (32:09)
  • Parallelism:
    • Experts are distributed across GPUs ("expert parallelism"), and full connectivity (“all-to-all”) within a rack is essential for efficiency.
    • When spreading MOE layers across multiple racks, limited inter-rack bandwidth becomes a bottleneck (36:50).
    • The largest expert layer that can be run efficiently is “bounded by one rack”—driving the need for larger interconnects. (36:50)

Memorable Quote

“One rack is actually the bounds for the size of an expert layer you can do. ... this has been part of what’s been driving towards larger and larger interconnect domains.”
— Reiner Pope (36:50)

• Rack and Interconnect Hierarchies:
  • Explained rack topology (NVidia vs. Google), difference between “scale up” (intra-rack, fast) and “scale out” (inter-rack, slower), and consequences for model parallelism (39:01).

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4. Parallelism Strategies and Pipelines

• Types of Parallelism:
  • Expert Parallelism is dominant for modern (sparser) models.
  • Pipeline Parallelism (dividing layers across racks) helps alleviate memory capacity limits but introduces complexity: pipeline bubbles, micro-batching, and memory-use tradeoffs (57:03).
  • Data Parallelism and legacy tensor parallelism are discussed but are less critical in current large-model setups.
• Trade-Offs in Pipeline Parallelism:
  • For inference, pipelining mainly helps with weight storage reduction (not latency or batch size), but is only beneficial for very large models; for most frontier deployment, inference stays within a single scale-up domain/rack (73:38).

Memorable Quote

“The physical and model architecture matches... we have experts, we're going to put them on different GPUs. Oh, we have different layers, we're going to put them on different racks.”
— Dwarkesh Patel (53:41)

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5. Memory Wall, Hierarchy, and Economic Constraints

• Memory Is Now a System Bottleneck:
  • Hyperscalers are spending up to 50% of CapEx on memory (63:59), particularly for high-bandwidth memory (HBM).
  • Simultaneously, “there is too much memory in some systems”—reflecting mismatches between capacity and actual requirements (64:18).
  • Memory Hierarchies:
    • HBM (fast/expensive), DDR, Flash, and even spinning disks; memory selection depends on how long KV caches need to be retained (119:47).
    • Engineered trade-offs between cost-per-token, storage cost, and retrieval latency.

Memorable Quotes

“If you have too many of these things sitting in HBM, if I fill up my HBM with just KV caches ... I can’t use that GPU.”
— Reiner Pope (115:09)

“I want the retrieval time to be equal to the hold time times the fraction of capacity ... that probably indicates that this is the two tiers of flash and spinning disk.”
— Reiner Pope (123:41)

• Pricing Correlates to Bottlenecks:
  • Commercial LLM APIs “leak” information about architecture and costs through price tiers: e.g., Gemini 3.1’s 50% premium above 200k context likely marks the compute/memory bottleneck crossover (95:38).
  • Decoding costs (output) are often 3–5x more expensive than prefill (input), reflecting that output is memory bandwidth limited (104:27).

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6. Scaling Laws, Overtraining, and Data Equilibrium

• Overtraining Relative to Chinchilla Scaling:
  • Optimal total compute splits training, RL-tuning, and inference into roughly equal shares.
  • Modern models are “100x overtrained” relative to Chinchilla scaling recommendations—i.e., models get trained on 200T tokens instead of the Chinchilla-optimal 2T (91:56).
  • Real-world deployment is determined by first-principles economic and hardware limits as much as by scaling curves.

Memorable Quote

“That is sort of wild that you can sort of first principles these kinds of numbers”
— Dwarkesh Patel (92:22)

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7. Cryptography and Neural Networks: Convergent Designs

• Shared Motifs:
  • Both cryptographic hashing and neural nets converge on architectures that “jumble” information, but with opposite goals (extracting versus hiding structure).
  • Feistel ciphers and reversible neural networks (like RevNets) use similar constructions for invertibility, enabling forward/backward computations in training while minimizing memory overhead (128:29–133:25).
  • Employing more compute to save on memory (RevNets) is the dual of how the KV cache spends more memory to save computation.

Memorable Quote

“Generally, spending more memory to save computation is profitable, given where hardware is at.”
— Reiner Pope (133:33)

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Key Timestamps

| Time | Segment / Discussion | |------------|-------------------------------------------------------| | 00:50-04:30 | Motivation for batching & blackboard analysis approach | | 06:00-11:30 | KB cache, memory/computation trade-offs, sparse vs dense attention | | 15:39-20:02 | Cost vs batch size, practical batch sizes, queuing, latency | | 26:21-30:51 | Tokens/sec, batch size as central system design constraint, model throughput | | 32:09-36:50 | Mixture of Experts (MOE) mapping to hardware, expert parallelism | | 40:48-43:40 | Physical rack constraints, cabling, Nvidia versus Google approaches | | 57:01-62:40 | Pipeline/microbatching diagrams for inference/training | | 63:57-67:27 | Memory as economic bottleneck, memory capacity calculation per GPU | | 69:31-73:02 | Pipelining, expert parallelism, limits in large-scale inference | | 91:56-94:00 | Overtraining relative to Chinchilla scaling law, compute budgeting | | 95:34-101:24| How API pricing reveals cost structures, context length inflection | | 115:09-124:01| Memory tier trade-offs (HBM, DDR, Flash, disk), cache write/read strategies | | 125:15-133:20| Cryptography vs neural network design convergence, differential cryptanalysis, RevNets |

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Notable Quotes

• “If you do not batch together many users, the cost ... can be like a thousand times worse than if you do.”
  — Rainer Pope (04:25)
• “Sparse attention ... end up putting a square root in this term.”
  — Rainer Pope (12:35)
• “The physical and model architecture matches. ... It’s interesting that the cutting matches the model architecture.”
  — Dwarkesh Patel (53:41)
• “Spending more memory to save computer is generally profitable, given where hardware is at.”
  — Rainer Pope (133:33)
• “That is sort of wild that you can sort of first-principles these kinds of numbers.”
  — Dwarkesh Patel (92:22)

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Concluding Thoughts

This episode provides a uniquely detailed, equation-heavy look at how the physical limitations, performance bottlenecks, and economic realities of GPU clusters shape not only how LLMs are trained and served but also how models’ capabilities and API prices evolve. Listeners gain a toolkit for understanding why latency, cost, sparsity, and model size scale the way they do, how batching and memory are the real levers in infra, and how these constraints influence the overall progress of AI. Pope’s hands-on experience designing both hardware and large-scale systems brings clarity to these intricate trade-offs—making this a must-listen (or must-watch) for anyone aspiring to understand the math and engineering behind modern AI.

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Further Reading/Resources

• Dwarkesh Podcast
• Rainer Pope’s blog post on scaling and blackboard book
• Deep SEQ papers on sparse attention and MoE architectures

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This summary omits advertisements and sponsor messages, focusing entirely on the technical and conceptual content of the lecture.

Transcript

[Lex Fridman]

Today I'm interviewing Rainer Pope, who is CEO of Maddox, which is a new chip startup. Previously he was doing TPU architecture and many other things at Google. This is a very different format from my usual interviews. This is going to be a blackboard lecture. We're going to get up in a second. We in fact built this whole new studio with specifically this format in mind and so it's a pleasure to get to inaugurate it with you. We're going to be talking about model architecture, ML, infra, many other things. And the reason I think it's an important topic is because once you actually understand how training and inference actually work in a cluster as we'll see a lot of things about why AI is the way it is, why AI architectures are the way they are, why API prices are the way they are, fundamentally also why AI progress is the way it is. Start making sense. And you need to understand the details to get there and you need a blackboard to understand the details. So Rainer, thank you so much for doing this.

[Rainer Pope]

Yeah, very happy to be here.

[Lex Fridman]

Just a heads up, this is a lecture with graphs and equations and all that stuff. So if you can, I would really recommend watching it on a video platform like YouTube. Okay, full disclosure, I am an angel investor in Maddox, but that's unrelated to this podcast. Reiner, maybe to kick us off, I'll ask this question. So we have a couple of companies like Claude and Codex and Cursor are offering something like Fast Mode where for 6x the price they'll give stream you tokens at 2.5x to speed mechanically. I'm curious what's going on here. Why is it the case that you can pay more to get faster latency? And two, could you keep going? Could you pay 100x more and somehow get even faster speeds or much, much faster speeds? And three, could you go the other way? Could you have something like quad code slow mode where if you are willing to wait for minutes on end, you could get even cheaper prices. So maybe this will help motivate the kind of analysis that you'll be doing through the leisure. Great.

[Rainer Pope]

I mean a little bit to jump to the conclusion. The big effect is batch size. But what we're going to do now is quantify exactly what that looks like and what its implications are on latency and cost. There's going to be another effect which is, you can call it speculative decoding or multi token prediction. We can maybe come back to that later. But I think the first thing that we'll talk through is batch size. So what I'd like to introduce is sort of the two principles of analysis. Firstly, we're going to look at a roofline analysis of how I run a transformer model on, on a cluster of chips. We'll take a sort of, let's say a Blackwell NVL 72 cluster, so a rack of 72 GPUs. And so the roofline analysis means we look at memory bandwidth and compute performance. And then the other side of that is that we're going to look at just two simple factors of the model, which are the time to operate on the weights and then the time to operate on the context, the KB cache. So let's jump in. What we're going to try and do is we're going to try and estimate the time that it takes to run an inference of a certain shape. Now, we're not perfect here. We can't exactly predict the time. And so instead we're going to approximate and so we're going to say that the time must be greater than or equal to a certain quantity. And so we're going to consider two different aspects. We're going to look at the time it takes to do the memory fetches and then the time it takes to do the compute. And it'll turn out that this actually gives us a very strong predictive power even with a simple model. So one by one, what is the time that it takes to do the compute? So there are really two things I need to do in the compute. I need to multiply by all of the active parameters and then I need to do some work on the attention. So multiplying by all the active parameters, I have a certain batch size that I'm running and then I've got a number of active parameters in my model. And then I'm just going to divide this by the compute throughput, which is the flops of the chip. So this is a hardware constant. So this actually accounts for all of the compute time for all of the weight matrix multiplies. There's a little caveat here. We've sort of ignored the time to do any of the attention computation, but that in general will be quite small in comparison to this. So we'll ignore this.

[Lex Fridman]

Maybe I'll just ignore from time to time to ask some very naive questions or to clarify some basic points, but just for the audience, you're not serving one user at a time. The batch refers to the fact that you're serving many different users at the same time.

[Rainer Pope]

And that's a whole batch yeah, so I can motivate the batch at least a little bit. So I mean, we will see exactly why batch is such a favorable optimization. But what will turn out to be the case is that that if you do not batch together many users, the cost and the economics you get can be like a thousand times worse than if you do batch many two users together. And we'll be able to see that quite explicitly. And then number of active parameters. This is saying like if I look at, for example, a deep seq model, the deep seq v3 model has about 37 billion active parameters and then 700 billion total parameters. So we're focusing on just the ones that are active for a single. All right, that's all good. Okay, so we modeled compute performance. I'm going to keep writing equals, but in all of these cases you can think of this time as being at least this much. And maybe there'll be some terms we ignored on the memory side. What do we need to do with memory? We need to fetch all of the weights and so there is some time to fetch all of the total number of parameters, not just the active parameters. So there's waitfetchtime and then in addition there's a KVCACHE fetch time. So this actually depends on batch size. So for every element in the batch we have to fetch an entire context length worth of tokens and then there's a size per token. So like bytes for one token. And so this is a model parameter.

[Lex Fridman]

And maybe just back in. Let's just explain what the KBCACHE is real quick.

[Rainer Pope]

Yeah, so when I do a forward pass, let me draw actually how the autoregressive inference works. So this is during decode. So if I think I have a bunch of tokens of text, I'm growing a tensor because ultimately the tokens are represented as some tensor in some embedding dimension. And then in this direction I have the sequence length. The work of running a decode is I have to run each token through a, through a whole bunch of matrix multipliers over a bunch of different layers. And in general I'm going to have to do that work over all of these tokens. But then one step of decode is actually to produce just this one additional token out here. Yep. And so what I'm going to do there is I'm going to run a full forwards pass of multiplying by all of the weight matrices in the entire model. But then I've got this attention mechanism where this token sort of. It's like looking at all of the past tokens in this way. And what is it looking at specifically? It is looking at some internal representation that the model has produced of the tokens and we call that the kbcache. So this process of attending this single token, attending to all of the history of tokens, that's attention, it is mostly dominated by memory fetches rather than matrix multiplies. So we've got the amount of memory that we're fetching shown over here. And then this is of course just then divided by the memory bandwidth so the memory bytes per second. So in fact these equations here are actually enough for us to now draw some fit lines. And so the things that we'd like to look at are sensitivity to batch and then also which we'll draw separately to context links. So we said that the big effect you can get is like some trade off in latency versus cost in batch size. So let's draw them out. I think there's just really two graphs we want to draw. We'll first just draw batch size versus time here. So when we look at the shape of this, we've got a maximum of the sum and then another term. So let's look at these terms one by one and how they scale the time for compute and memory and how they show up. So let's first look at this compute time. This is just purely linear in batch size with no offset. So it is some curve like this, this is t compute. And then on the memory side we've got some portion here that is just this constant that is constant in some base offset here which is the weight. Fetch. Wait, fetch. And then finally we have this term here which is the kbfetch, which we're going to draw as the kbfetch, which is linear and batch size. So it looks like that. So the sum of this plus this maxed with this. So let's at least first draw the sum. So the two memory times in, in conjunction end up looking on this curved slope like this. And then we get a. The overall maximum is. I'll draw a little thicker. Here is the maximum of these two curves.

[Lex Fridman]

Makes sense.

[Rainer Pope]

Okay, so what does, what does this mean actually? So this is a latency plot. So if I grow my batch size, I get initially some not very strong dependence on batch size. And so there's some lower bound or latency here. Latency, Lower bound, lower bound. So this already partially answers the question for a given hardware configuration. And then we can talk about varying hardware configuration. But for a given hardware configuration, there is a lower bound on latency, which is simply. Simply I need to read all of my total parameters from memory into the chips, and that takes a certain amount of time. If I use all of my memory bandwidth, I can't do any better than that.

[Lex Fridman]

It seems like the way you've drawn the slopes for compute time and how the KB grows and what implication the KB has on memory time.

[Rainer Pope]

What if this were above or below?

[Lex Fridman]

Yeah. Or is that necessarily the case? Because if this is always true, then as batch size grows, compute always dominates kv, which suggests that if you have big enough batch size, maybe memory is never an issue.

[Rainer Pope]

Yeah, this is really sensitive to the context length, so I think we should come back and explore this. As you vary the context length, the KBFetchtime will go up and up, and so that'll cause a transition from compute limited to memory limited.

[Lex Fridman]

And is there something especially significant about the slope being exactly the slope of the compute time?

[Rainer Pope]

Yeah, whenever we have balance points, it kind of says that you're getting it exactly right. And so for the particular context length where the slopes match, that says I am equally memory bound and compute bound, which is a really desirable place to go.

[Lex Fridman]

But suppose it's like this is a very simple algebra problem. But suppose the optimal is 100k context length and you go to 200k context length, does your MFU go down to 50%? Does it have a humongous impact on MFU to be slightly outside of context length? Optimal range, Goldilocks zone.

[Rainer Pope]

That's right. So that is true as modeled here. There is a key point here that I'm modeling this context length as well. I'm modeling the memory fetch as linear in context length. That actually depends on model architecture. It is true for many of the all of the model architectures with dense attention. Sparse attention actually scales much better than that. Got it.

[Lex Fridman]

And is sparse attention that everybody uses in practice?

[Rainer Pope]

I'm pretty excited about sparse attention. It's hard to know what the labs are using. Deep SEQ has published a sparse attention mechanism. I'll just put a plug in that sparse attention. Some of the Deep SEQ papers that have published sparse attention end up putting a square root in this term. Okay, so so far we've looked at the latency. It's kind of hard to read off cost from this. So if I think what does cost mean to run this inference, I'm going to use the GPU for a certain number of seconds, like 1 millisecond or 20 milliseconds or something like that. And I have to pay the Rental time for that time. So like it's $2 an hour per GPU or something like that. So, so that's the cost of this inference. But how many tokens have I processed during that inference? That is the batch size. And so what we actually want to plot is going to be the cost versus batch size, which is like T over B versus batch size. This is the cost per token. So we have to imagine dividing each of these three curves by. By B. So multiplying by this reciprocal. And so what we end up with there is the compute curve is going to. It was linear. We divide by B. That makes it a constant here. This is tcompute. The kvfetch was linear. Now it becomes a constant as well, kbfetch. And then the weight fetch was constant and now we're divided by B. And so it becomes this hyperbole. And so again we're going to compute the max of the sum. So the sum of these two terms shifts the parabola up sum of the KBFetch and the weight fetch gives us a sort of a higher parabola that's like this. And then we're going to take the max with the compute here. So we end up with this being the overall shape that we care about. So again we see some limiting behavior. The cost initially starts very high at batch size of one. Actually, it almost goes to infinity. It's because we've got so many weight fetches which are not amortized over a large batch size. But then as we increase the batch size, the weight fetches become amortized over so many different batch elements that their cost grows very small and eventually the compute time ends up driving the cost. So there is a limiting, like lower bound on cost, Which is this one here.

[Lex Fridman]

Yeah, so Claude code slow or codex slow or whatever would just live on this line. And it wouldn't help much because you're not able to amortize the KV values over a much bigger batch.

[Rainer Pope]

Yeah, they're unique per batch. The compute is also unique per batch. And so what is the minimum work you can do per batch after amortizing it? Everything else away.

[Lex Fridman]

So this point where you are no longer memory bandwidth bound, what practically, how big a batch do you need to like, how big are the batches practically

[Rainer Pope]

for frontier models, you can just solve for that actually, and it's not even particularly sensitive to model architecture. So let's go ahead and do that. So what we're talking about is we're going to say when the memory time is equal to the compute time, that's what that question is for now, I'm going to discard the. Because we're focused on what the batch size is. And really there's a question of when the weights are amortized over the multiplies. I'm going to focus on comparing the weight fetch time to the wait, multiply time. I'm going to disregard the kbfetch term just to simplify the analysis so we can get a kind of a clean answer out. So we're going to equate this portion with this with these two terms.

[Lex Fridman]

Yep.

[Rainer Pope]

So writing that out, we get N. Number of total parameters over memory bandwidth is equal to batch size times number of active parameters divided by the compute performance. So looking over here, everything on the top, these are model parameters. Everything on the bottom, these are hardware parameters. It turns out to be nice to rearrange them such that we have the hardware parameters on one side. So this is equivalent to. Oops over memory bandwidth being equal to batch size times number of active parameters divided by the number of total parameters. So this is a hardware parameter. Actually. This actually ends up being a dimensionless constant. If you look in terms of flops, what are the dimensions of this? This is multiplies per second. This is bytes per second. So that's not quite dimensionless. But what you do is you say like multiplies per second times. Let's say I'm doing FP4. So I do like how many FP4 multiplies per second times the fact that each FP4 is half a byte. And so I can actually make this ending up being dimensionless. And this ends up being on most GPUs around 300, somewhere around 300.

[Lex Fridman]

And sorry, has that ratio changed over time as we've gone from model generation to model generation where the flops keeps increasing.

[Rainer Pope]

So this is a hardware parameter. To what extent has the hardware changed? So from like A100 to H100 to B100, the flops has increased substantially. The memory banner has also increased substantially and it has remained reasonably stable. And we can express this one as well. This is a sparsity parameter and I might even phrase it slightly different. Let's solve for batch size in total. We end up with. So we're just moving this back over to the other side. We end up with batch size needs to be bigger than approximately 300 times sparsity. So for example, if I have 100, like I activate in deep seq, I activate 32 out of 256 experts. So this would be like eight for deep ticking. Got it. Okay, so this actually gives you a ballpark, which is remarkably accurate to practice. Generally, people will go a little bit larger than this. They don't really want to be exactly at the balance point because real world efficiencies aren't as good as a roofline analysis would say. But take this and maybe double it or triple it.

[Lex Fridman]

Okay, so Basically it's like 2 to 3,000 tokens per batch. But then if you included the KB cache.

[Rainer Pope]

Yes.

[Lex Fridman]

The implication would be that the optimal batch size should grow larger.

[Rainer Pope]

So this is like we solved for the equivalence between when compute time is equal to memory time. If I add in more memory bandwidth, like something that consumes more memory bandwidth, then I have less available for the weight loads. And so I need to grow the memory bandwidth more and therefore the batch size more.

[Lex Fridman]

This seems incredibly small. This would be like less than one sequence, right?

[Rainer Pope]

Yeah. Okay, so I guess this is. Keep in mind that I'm talking about the number of tokens that I'm generating one more token for. So it's actually 2000 unique sequences.

[Lex Fridman]

Got it.

[Rainer Pope]

Okay.

[Lex Fridman]

We're just talking about a single forward

[Rainer Pope]

pass on these sequences.

[Lex Fridman]

This is like the. Do you think of it like the bash as the number of sequences rather than like.

[Rainer Pope]

That's right. Okay, cool.

[Lex Fridman]

Yeah. When I'm prepping for interviews, I often talk to experts in the field. So for Reiner, I chatted with two of Jane Street's engineers, Clark and Axel. Clark, who works on low latency trading systems, walked me through why Jane street uses FPGAs to make sure that they have predictable nanosecond latencies.

[Rainer Pope]

You can just build these giant grids of compute very easily that do exactly what you need to touch 100 megabytes of SRAM and then get your response back in tens of nanoseconds, very easily.

[Lex Fridman]

And that's basically impossible on Cinnamon. He then went on to explain why CPUs just wouldn't work for this kind of thing. And so if you have a clock that's going every 3Ns, you actually have

[Rainer Pope]

several bytes of information at a time to make your decision.

[Lex Fridman]

That's as opposed to a CPU where

[Rainer Pope]

you'll just collect up a whole packet.

[Lex Fridman]

You know, let's say a 1500 byte

[Rainer Pope]

packet, and then you say, okay, this packet's ready. Here you go, CPU. You can start thinking about it now.

[Lex Fridman]

FPGAs allow you to react to the earliest part of the packet as it arrives, rather than having to wait for the full thing. We also talked about liquid Cooling, network design and many other things. If you're interested in this stuff, Jane street is hiring, you can check out their open [e-mail] dwarkash and if you want to watch the full prep conversation, we posted it there too. If you've got a Frontier model and you are actually doing inference, surely they must have more than 2,000 concurrent users.

[Rainer Pope]

Yeah.

[Lex Fridman]

Is there any added latency from the fact that you need to have the whole batch fill up, or is it. If you have a reasonable amount of users, it's so unlikely that you wouldn't. It would not take you 100 milliseconds to fill up the next 2,000 slots.

[Rainer Pope]

Yeah. The way to think about this, I guess we think of it as, like, when does the train depart as a model? So let's say I've picked a batch size that I'm going to run at. Maybe I pick this batch size. And by the way, this intersection point is the same intersection point here. So I pick this batch size. I know that it's going to take, for example, maybe it's something like 20 milliseconds is a common place to send up landing. What I'm going to produce is. So this is a timeline of what is running on the GPU. It's going to start a new batch every 20 milliseconds, regardless. This is 20, this is 40. You can think of this as a schedule for the train. A new train departs over 20 milliseconds. Any passengers who are ready board the train. If the train is full, then they wait to the next train. If the train is not full, the train's going to go anyway. And so in terms of what that means for queuing latency, it means that the worst case is that a request arrives just after the train departed. It has to wait for the next train, so that's up to 20 milliseconds. And then it has to wait for that train to complete. And so the worst case latency is 40 milliseconds.

[Lex Fridman]

How is the 20 milliseconds derived?

[Rainer Pope]

I mean, rule of thumb. But where it comes from is not fully explained yet. But so far we've focused on memory bandwidth and compute time. When we look at memory, the other consideration is that we want to use all of the memory capacity we have. And so generally we're going to use all of that memory capacity to store the weights or the kbs. And so we just want to read like in the time of doing a forward pass, maybe we want to read all of the memory capacity into the chip. And so that is capacity divided by bandwidth that tends to be 20 milliseconds. On many different generations of HBM, the units make sense.

[Lex Fridman]

You would have a byte divided by bytes per second.

[Rainer Pope]

Yeah. So for example, I mean on, on I think the Rubin generation, it is something like 288 gigabytes divided by 20 terabytes per second. And this looks like it comes out to about 15 milliseconds.

[Lex Fridman]

Let me just make sure I understand what it's saying. I mean I understand why the units can't the sort of unit analysis, but what is it saying is. We can evacuate and replace the HBM in this amount of time. And so we don't want to be in a situation where the HBM is not big enough that we're not actually able to keep write everything we want to it or take everything out of it, or we don't want to be in a situation where our ability to write back and forth is so big or sorry, so small compared.

[Rainer Pope]

Yeah, there's sort of two scenarios. Why don't we pick a latency that is bigger than 15 milliseconds and if I think what that means, it means I actually have time to read the HBM like twice. By the way, most of HBM accesses is reads, not writes. It's like almost all reads because the weight matrices are read only and then almost all of the KB cache accesses are reads. So in like let's say I run 30 milliseconds I can read all of HBM twice. But what's the point of that? I don't want to read the weight matrices twice. I don't want to read the KVs twice.

[Lex Fridman]

Yeah, makes sense. Makes a ton of sense. Okay, so a couple of actually quick questions. One, if it is the case that the optimal batch size is something like 2000 and that actually true, it's totally dependent on sparsity. It's not dependent on the model size or anything.

[Rainer Pope]

I mean sparsity shows up in model size, but beyond that it only depends on sparsity, not on scale.

[Lex Fridman]

But that's a very interesting result. And that seems to imply that you can. One question is how much of a push towards centralization is it that you would have these economies of scale from inference from batching? Yeah, but it seems like it's not

[Rainer Pope]

that big a deal.

[Lex Fridman]

I don't know. Is 2000 users at the same time a lot? It doesn't seem like a lot.

[Rainer Pope]

We can do a bit of analysis on this which would be actually it's like you can think of it in terms of number of users, but maybe a more productive way to think of it is in terms of number of tokens per second. So what does this batch size mean in terms of tokens per second of the system? So tokens per second. Tokens per second is going to be equal to the batch size. We run a batch, many tokens, and then we do that every T. So every time intervals, which is, let's say this thing is equal to the 15 milliseconds or 20 milliseconds number. So this ends up being batch size itself times about 60. So like 64 times B. And so this ends up being around 2000 times 64. So like 128k tokens per second. So this is sort of in more digestible units. It's hard to reason about concurrent users. But what is the global traffic for a system? When you look at some of the announcements, sometimes the API providers will brag about how much traffic they have. The numbers that I've remembered from some announcements of Gemini last year were in the hundreds of millions of tokens per second worldwide. So about a thousand. Like this is one thousandth of that we're interested.

[Lex Fridman]

But I mean, Gemini is big, so that's actually one thousandth of Gemini is a lot. To be competitive at scale, you need

[Rainer Pope]

to be able to serve at least 1000th of Gemini.

[Lex Fridman]

Yeah, that's interesting.

[Rainer Pope]

Cool.

[Lex Fridman]

Okay, so the more sparsity you have, the less compute you need. And it does seem that as batch sizes get bigger, compute ends up being the bottleneck according to this analysis. So then the question is, how far can you take sparsity? That is to say, as the sparsity ratio increases, as you have fewer and fewer active parameters relative to total parameters, how much is performance of the model degrading? And is it degrading faster than you're saving compute by increasing the sparsity factor.

[Rainer Pope]

Yeah, so quality of the model rather than speed of the model. So unfortunately we're not able to answer that analytically. That is an empirical question of model quality. Best I can do is pull up a paper and answer that empirically.

[Lex Fridman]

Should we follow the paper now or

[Rainer Pope]

does it make sense? So this paper, this is unified laws for routed language models. It's a somewhat old paper by this stage, but one of the things that they did is looked at. If I keep increasing sparsity, what is the model quality impact? This answer is very sensitive to the actual choice of mixture of experts. Mixture of experts has been around for a really long time. I think it was maybe even back in 2017. But the techniques have changed a lot. DeepSeq mixture of experts was a big change in how it worked. There have been older papers which are GSHAR switch transformer. So the actual empirical results are going to depend on all of that. But on one of the older techniques that is shown here, you can see if I hold constant the number of active parameters at a certain size and then I increase the sparsity, which they call expert count here, the quality keeps increasing and then if you imagine drawing a horizontal line from 1.3b dense across, you end up seeing that for example, in this case the 64 expert 370 million activated parameters model is as good as a dense 1.3 billion model.

[Lex Fridman]

So in some sense it's actually not amazing returns where you need to increase total parameters 100 fold to get the equivalent of 10x as many active parameters.

[Rainer Pope]

Yeah, I mean actually even more so. Yeah, it's a huge increase in parameter count for a modest increase in.

[Lex Fridman]

Yeah, so in this case actually it's.

[Rainer Pope]

What is it, 4x64x for 4x.

[Lex Fridman]

Yeah. So while it is true, I guess that you get this benefit of being able to economize on your compute time if you increase sparsity, naively it would seem like oh, that's a trade off worth making. But if you're decreasing this by 2x and then having this go up by 8x every time you double sparsity.

[Rainer Pope]

So is that good or bad? Actually, even from a memory point of view, keep in mind you are doubling this portion of the memory fetches which is amortized by batch and so just keep running a larger batch size. From the point of view of the analysis we've done here, this is pure win. Keep doing it, keep doing it until you run out of available users, basically. So there's actually this equivalence between if I want to go sparse or if I have a lot of users, I can go to a much sparser model. So from that point of view it's a reasonable trade off. The other trade off that shows up here is that it also consumes memory capacity, which we've only reasoned about memory bound with it, but it also consumes memory capacity.

[Lex Fridman]

So let me just make sure I understood. You're saying we want bigger, we want to spend less time computing, therefore we do more sparsity. To make that work, we need bigger batch sizes, which means we need more memory capacity to have more sparsity.

[Rainer Pope]

Yeah. So I mean, maybe this would be a good point to actually talk about how a mixture of experts layer is typically laid out on a rack of GPUs or something like that.

[Lex Fridman]

Yeah, yeah, that makes sense.

[Rainer Pope]

Yeah.

[Lex Fridman]

Where were we?

[Rainer Pope]

Sparse mixture of experts. Maybe how we lay that out on a gpu. So let's zoom in on the mixture of experts layer first and sort of draw what that looks like. So we typically will have some kind of a router layer which is making the decision of where we route the experts the tokens to. So we get tokens coming in here, they go through a router layer and then we have a bunch of different experts. I'll draw a few more to line some up and then the router will make a decision. Which experts am I going to route to? And it'll be a small fraction of them, maybe 1 in 32. So maybe it'll make a decision to route to this one, maybe this one and maybe this one. These experts. So each expert itself is a normal mlp. It has an up projection and then a down projection with a non linearity in between. And then finally we sort of do the inverse operation. So where we were broadcasting things out here, we're going to bring them back in and sum them up. So bringing them in like this. And then finally we have our residual connection. The token is also passed through here and it gets added to the result of the MOE layer. So this is a normal MOE layer. What I want to talk through is how this is mapped to a GPU rack and what this means for communication. Because I think this will start to show some of the limits of how fast we can go. Yeah, so the standard practice here, and it is the best solution, is to use expert parallelism. So that means different experts go on different GPUs. So if we take something like a deep SEQ model, they have 256 experts. Let's say we want to run that on a Blackwell rack. So there are 72 GPUs. We have a divisibility problem. This is not a power of two. So we'll just like simplify and say we're only going to use 64 of them. Just ignore the other eight, it's not a big deal. And so we have four experts per GPU. Very simple. For the sake of the diagram, I'll actually just say let's say we have two experts per gpu. So we end up just putting these are the GPU boundaries. Every pair of experts is on its own gpu. And then we can look at the communication cost. We had some experts stored some Tokens stored centrally here, they get routed to all of these experts. And so there is some communication cost paid here, there's the same communication cost paid on the output. And then the hope is that this does not become communication limited. Now, what is the traffic pattern here? The traffic pattern here is that any GPU in fact will be talking to any other GPU depending on the decisions made by their model. So this is an all to all traffic pattern.

[Lex Fridman]

So when you say any GPU in the pretense the router is more than one gpu.

[Rainer Pope]

Yeah, So I drew this as one router. In reality, you would actually have many copies of the router. And so you would have as many

[Lex Fridman]

routers as GPUs, in fact, as the incoming traffic.

[Rainer Pope]

Yeah. So these are 64 GPUs. These are 64 GPUs. It's actually the same GPUs. We just draw them as separate because they're serving different purposes. So at this point, any GPU can be sending to any other gpu. So this all to all pattern of communication that shows up and how the Blackwell racks are configured is a perfect fit for the the communication pattern that the MOE actually wants to do. However, if you think maybe I want to do like maybe one rack is too slow and I want to do two racks, then I have this challenge that maybe I've got some sort of rack boundary drawn outside here like this, and I no longer in fact have all to all communication between all the GPUs in two racks. And so the rack to rack communication ends up being a substantial bottleneck. So the fundamental thing here is that one rack is actually the bounds, the size of an expert layer you can do. And so this has been part of what's been driving towards larger and larger interconnect domains.

[Lex Fridman]

Yeah, it may be worth you explaining what exactly a rack is. The differences in bandwidth between a rack and within a rack, and the all to all versus not all to all nature of communication within versus outside.

[Rainer Pope]

Yeah, and this is a place where it starts to be very different, in fact, between Nvidia for example, and Google and then others, including us. So generally a rack is a physical structure. It's a few meters tall, meter or 2 wide, depends on configuration, and it stores some number of GPUs or XPUs, which is typically about 64. What constrains it being a certain size, is power delivery, weight and cooling ability. It ends up being about this size in many cases because of these physical constraints. So then when I deploy a data center, a Data center may have thousands of these racks. So I've got one of these tall racks, it's got a bunch of GPUs in it and so on. And then I put another rack next up.

[Lex Fridman]

You make it sound so easy.

[Rainer Pope]

Yeah, I just drop them in. In Nvidia's case, the communication topology is actually they put the GPUs on the outside of the rack and then they put these switches on the inside of the rack. So what this ends up being is that there's a set of switches in here, these are the NV switches. And then they run a bunch of cables. Every single GPU has cables going to the switches in the middle. So every GPU goes to the switches in the middle and then the switches have connections to all the GPUs. So all of the GPUs can talk to all the other GPUs in just like two hops, going to the switch, going to the other GPU. Now when I want to leave the rack, I end up going via a different path. The GPUs have also a much slower connectivity, which is typically about eight times slower, which is. So the green that I drew here in GPU cases is the NVLink. More generally, it's called the scale up network. This is the scale up network. You will typically also have a scale out network which allows you to connect to some data center switch. So data center switch and then all of the GPUs will have some connectivity up to some data center switch somewhere. But this is about times like this is the scale out and it tends to be about eight times slower in bad width. So the challenge, if you want to, for example, lay out a mixture of expert layer across two racks is that half of the GPUs here are going to be wanting to talk to the GPUs of GPUs here. And so just on average, when I look at where the tokens on these GPUs want to go, half of the tokens want to go inside the rack. That's great. They can use the fast scale up network, but half the tokens are going to want to leave the rack and go to the other rack. And that's not as good. They're going to need to use a much slower network. And so that becomes the bottleneck on the all to all pattern. A different choice would be, well, why don't I have a big switch here and sort of like, and connect everything to some big switching, like much bigger switch that actually combines the two racks together. There are many ideas in this direction, but in general it becomes the reason you have this sort of hierarchy of switches rather than one big switch is to manage the cabling congestion. You just need to run a large number of cables.

[Lex Fridman]

So is that question you just asked, basically, why isn't it a bigger scale up?

[Rainer Pope]

Yeah, exactly. Why not just have a million chips and scale up?

[Lex Fridman]

Or what has changed that is allowed in really to go from Hopper was 8 then Blackwell is 72, and now Ruben will be. Is it 500 something?

[Rainer Pope]

Yeah, 500 and something. Yeah.

[Lex Fridman]

What has allowed that to happen from

[Rainer Pope]

Hopper to Blackwell is mostly just the decision to switch from trays as the form factor. One of these is a tray to switching to racks as the form factor. That's a product decision. There wasn't a substantial technical barrier there. Switching from, from the like 64 to 500 or so. There's a bit of chance in math there, but there is at least a genuine 4x increase which is coming from a much more complicated and difficult rack design. And so that is actually like new physical design to run more cables.

[Lex Fridman]

And the cable complication is just the cost of figuring out which cable hops to which cable or like, what's your signal.

[Rainer Pope]

Let's sort of zoom in on this and look at the wire density. I'll draw this diagram just once more. So we have a bit of a cleaner version to work with and a larger version. Let's say I have some switches in the middle and let's say I'm going to have. Initially I'm going to start with just two GPUs on each side or two trays of GPUs on each side. And let's say maybe each tray wants to have two cables coming out of it. So I get some kind of. I physically run vertical cables that look like this running to the switches. Now if I want to double the number of GPUs in Iraq, I need to run like literally twice the density of cables. So I need to run these as well.

[Lex Fridman]

Extremely naive question, but if you look at a physical data center, it seems like there's a lot of space within a rack. I don't know. Just like the cables are like really big.

[Rainer Pope]

Yeah. So there is space outside the rack, inside the rack. Like these racks are like. I mean, as they become more optimized, these racks are very tight. So there's connector density going from. From the tray into the rack and the rack's backplane and then the backplane itself has a really high density. There are other physical constraints including bend radius of cables, you don't want to snap them, and so on.

[Lex Fridman]

So it's literally the physical space to put a cable that's constraining it. I had no idea. Interesting. That seems surprising that the rack is so big and we can't just stuff more cables in there.

[Rainer Pope]

Yeah, so, I mean, rack design is not my expertise, but when I talk to folks and what are the constraints they're up against, it's a combination of so what are the big physical things you're optimizing for? Space weight of the rack, it's actually really heavy and so you need enough metal to not sag and fall. But then you add more metal and it's heavier and then power and cooling and so all of those are competing for modern racks are pushing all of those to very extreme physical limits.

[Lex Fridman]

Deep work is by its nature quite aversive, so even things which seem like work like Slack and email can be easy ways to distract yourself. So I often wish that I could just turn the Internet off. But if I'm prepping for an interview, even if I have the papers and books on hand, it's still super useful to be able to do a back and forth on an LLM so I can break down concepts and research follow ups. Google's new Gemaphore is the first open model that allows me to have this kind of fully disconnected focus machine. It's small enough to run on my laptop, but good enough to actually be useful. So to prep for this episode, I downloaded Reiner's scaling block book and shut off the Internet. I was able to have Gemma help me understand the material and answer my questions. If you want an LLM that you can run locally on your laptop or even your phone, you should check out Gemma 4. When was GPT 4 released again? It was 2022 or 202333 and it was rumored to be over 1 trillion parameters. And it seems like only now and within the last six months have models been getting released that are significantly more parameters than the model released three years ago, when supposedly there should have been. This scaling in the meantime is the reason that we were just waiting for racks with enough memory to pull the 5 trillion parameter model along with its cache for enough users for a lot of sequences or RL if you're doing RL. Kind of a similar consideration of actually holding the KBCache for all the batch of problems you're trying to solve. So if you look at hopper, you had eight hoppers and I think that's 640 gigabytes as of 2022 with Blackwell finally which was deployed what 2020 very

[Rainer Pope]

recently, maybe last year.

[Lex Fridman]

Last year you finally have a scale up with on the order of like 1020 terabytes which is enough for a 5T model plus KBCache.

[Rainer Pope]

Yeah, deploying in larger scale up domains is a huge unlock. I've drawn here the sort of Nvidia Blackwell deployment. The Google deployment has actually had very large scale domains for a long time

[Lex Fridman]

and that also explains why Gemini seemed to be ahead. Like Gemini 2.5 was a successful or it just seems like Gemini has that successful pre train for longer than some

[Rainer Pope]

of the other labs. Not having been there at the time, I'm not sure how much is coming from successfully deploying higher sparsity ratios which could be. It could also be. I mean there's a whole bunch of actual modeling things of like specifically how do you do the mixture of experts? We've seen the deep SEQ like the deep SEQ mixture of expert has said actually activate more experts, but finer grained experts was a big innovation. I'm sure that there are many other innovations on the model architecture as well as on the training data. It's kind of hard to disentangle all of them. But what shows up in terms of the limits of what you can do, the active parameters as we saw, is limited by the compute cost and then the total parameters is limited by the scale up size. Yep.

[Lex Fridman]

When you're operating within a single scale of domain, is that a consideration specifically for either forward or backward or specifically for pre fill versus decode or is it preferred to always be within a scale up whatever kind of workload you have, whether you're doing a pre training run or whether you're doing RLL generation or whether you're doing inference for users.

[Rainer Pope]

Yeah, really interesting. Okay, so to answer that question, we're going to need to talk about the communication patterns that so we've talked about the mixture of expert communication pattern that is this all to all? This all to all. All to all. All to all very strongly favors full connectivity, which is what we've kind of just shown here and it favors being within one rack. There are other kinds of parallelism besides expert parallelism which we just showed here in the literature is tensor parallelism. With the trend towards smaller experts, this has become much less relevant. So we can ignore that. But the other two things that we have available are data parallelism and pipeline parallelism and they can be a much better fit for using multiple racks. So let's focus on pipeline parallelism specifically, this is one layer of moe. I'm going to have like 100 more layers up above. I could decide at this point, for example, to move to a different rack, change rack. Now, is that going to become a communication bottleneck? So we can actually just solve for when this becomes a communication bottleneck. But before we do that algebraically, let's just sort of visualize it out and sketch the path. So we're going to have a bunch. This is another MOE layer and we're going to have another mo. Your layer here and so on. So let's say I change rack here and then some number of layers later, I change rack here as well. So our methodology that we're going to use to determine whether we have a communication bottleneck in this point where we change rack is we're going to compare. This is the scale out scale out bandwidth requirements to the scale up bandwidth requirements. So let's write this. I mean the hint is going to be that there's a lot more sends here. Like we're sending many things here, whereas we're only sending one thing here. And then we're also maybe doing it many times. So that's going to be what makes the difference.

[Lex Fridman]

Can I try to guess, just out of curiosity, to see if I'm actually understanding it seems like you're sending batch size into the rack in here. Yes, but the communication within a rack is sort of batch size times number of GPUs.

[Rainer Pope]

Yeah. So number of activated GPUs. Right. So I don't send to this GPU at all. Right. So there's an explosion from one to it's three times larger here in this diagram. The key thing is that I didn't even need to send to this GPU at all. And so that's a big saving.

[Lex Fridman]

I see.

[Rainer Pope]

Yeah. Okay, so we're going to talk through sort of how much more what is the slowdown of to what extent is scale up a bottleneck over scale out? So we will directly jump to the ratio of the time spent on scale up time on scale up over, the time spent on scale out. So this is the quantity we're talking about. And the first consideration is that the scale up is like scale up is eight times faster than scale out generally. And so at a baseline, if the bandwidths were the same, we would have this one over eight, which is coming from bandwidth bandwidth. But then we have some amount of expansion in how much data we're sending. So if one token comes in here, then this one token gets routed to in the deep sea case, it'll get routed to maybe 32 experts or 16 experts. Gets routed to some number of experts. So this is the number of activated experts. Number of activated experts. And then it also, the same thing applies on multiple different layers. So maybe I'm going to run two layers. So there's also multiple times number of layers per stage.

[Lex Fridman]

And don't you need to multiply the whole thing by two for the.

[Rainer Pope]

Yes, yes. And there's a factor of two. Thank you. So what we would like is for the scale up time to be greater than the scale out time because the scale up time is the more important and precious resource. And so we want this one, we would like this number to be greater than or equal to one. And this really doesn't seem hard. There's just a factor of H that we need to overcome. So we need the product of these three things to be bigger than eight. Typically we have a fairly large number of activated experts. It could be eight by itself. And then we can increase the number of layers per stage a lot until we satisfy this.

[Lex Fridman]

I see.

[Rainer Pope]

So what this ends up looking like is that I can in fact have an entire pipeline of racks where one rack does one layer and then I move on to the next rack and I do another layer and then I move on to the next rack and I can do another layer.

[Lex Fridman]

It's interesting to me that the best parallelism strategy in practice ends up being one which physically resembles the actual architecture. It's not some galaxy brain thing. It's like, oh, we have experts, we're going to put them on different GPUs. Oh, we have different layers, we're going to put them on different racks. Isn't that. I feel that's interesting that the physical

[Rainer Pope]

and the model architecture matches. Like the cutting matches the model architecture.

[Lex Fridman]

Yeah, exactly.

[Rainer Pope]

Yeah.

[Lex Fridman]

I mean it could have been something wackier with tensor parallelism and whatever.

[Rainer Pope]

Yeah. So I mean, I think a way to think of it is, okay, the galaxy brain way to think of it is what are all the different dimensions in which a model is scaled up? And so it is scaled up by layers. It is scaled up by the Demodel dimension, it is scaled up by the DFF dimension, it is scaled up by the number of experts. Every single one of those numbers you can choose to cut along. And if those numbers are big enough, it eventually becomes profitable to get along there. And we have selected two of them. The other two in the way models are typically sized are not profitable.

[Lex Fridman]

So there's a talk by ILYA where he says, today we know not to do pipeline parallelism. And Horacey gave my friends and me. I hate that it sounds like a Dr. Seuss quote, but he gave us a lecture on these different kinds of parallelisms. And you said, the problem with pipeline parallelism is that other than the bubbles, it creates these architectural constraints on. Like Kimi, for example, has these residuals where attention attends to a.

[Rainer Pope]

Fewer back or something. Yeah.

[Lex Fridman]

Layers a few back. And so that becomes hard to implement in this way.

[Rainer Pope]

Yeah. And I guess we didn't really fully articulate even what is the benefit that we're getting from pipelining. And so these complexities are real. Pipelining is a massive hassle, but it does give you some benefits and then you can then decide whether those benefits are worth the costs. The biggest benefit that shows up. So it has some benefits in inference, maybe bigger benefits in training in inference. What are we saving on? Are we saving on memory time or compute time? Not really. We're just moving the memory time from one chip to another chip or one rack to a different rack. There's no actual benefit in runtime. However, what we are saving on is that the memory capacity is the amount of memory used per rack. If we think that the memory in a rack is a bottleneck, then there's a constraint on how sparse we can go. Pipelining allows us to massively reduce that bottleneck, I guess.

[Lex Fridman]

But the opposite connotation to this, which. Actually, before this interview, I was chatting with Axel, who's a GPU performance engineer at Jane Street. He was explaining, well, to do pipelining you had to do micro batches rather than full batches. And if you do micro batches, then you're by definition not able to amortize loading the weights across all the users or all the sequences. And so the positive connotation on that is you don't have to use as much memory. The negative connotation of that is that you can't amortize loading the weights across all those users. Maybe it's worth explaining why you had to do micro batches, because you can't.

[Rainer Pope]

So we draw the pipeline bubble. Yeah. Okay, so why do we do. What is this micro batching that shows up in pipeline parallelism? So

[Lex Fridman]

the.

[Rainer Pope]

I'll focus on inference first. It's a slightly simpler problem. So. And I'm going to draw. So this is time and then this is which rack rack we're on. And so the idea is that maybe I'll have like four racks. So I've got an inference that is Going to step through these four racks in some time like this. So great. This is inference number zero. It runs at a certain batch size and it steps through all the pipeline stages like this. Now if we were to say, well, we're going to run inference number one here, this is clearly a massive waste. Right. Like three quarters of the time each of the racks is doing nothing. So we don't actually run inference one here. We run it as soon as we can, which is immediately after inference 0 finishes like this and we keep going. So if we hadn't filled this in, we would call this the pipeline bubble. When I've drawn it in this inference context where we're only going in a forwards pass, it's like obvious, why would you do the stupid thing? But in a training context it's maybe less obvious. But in the inference context it's sort of really natural to make this change.

[Lex Fridman]

Oh, interesting. So this is sort of obvious, but the difference between micro bash and bash doesn't matter at all in inference because you can just call whatever you want. Whatever.

[Rainer Pope]

Yeah.

[Lex Fridman]

It only matters in training because there is an optimal batch size.

[Rainer Pope]

Yes.

[Lex Fridman]

And before you do the backward step, you want to have accumulated before you do a full backward step, you want to have accumulated all the sequences in that bash. And if you want to do pipeline and training, in order to avoid that bubble, you need to.

[Rainer Pope]

Should we draw the training diagram? Yeah, let's do that. So this is the inference diagram and I'll call this 4. Just so we don't have the wrong thing showing up there. So let's do the same thing for training. Now we've got a forwards pass, but at some stage we're going to have to transition to a backwards pass. So we'll do some number of batches in the forwards pass. And then we're going to transition to the backwards pass for everyone all in one go. So the inference part is the same here, but then we do a hard stop at this point and then transition everyone to backwards pass.

[Lex Fridman]

Similar numbering like this, it may be worth clarifying. The reason there is that hard stop is because you want to do a whole batch at once for the backward step and then there is an optimal size for how big that batch should be.

[Rainer Pope]

Yeah, I mean smaller is always better, actually is a way to put it. But it's like from a ML convergence rate perspective, smaller is always better because basically you're getting the freshest information from the gradient descent. But total training time perspective, total training time perspective, smaller is worse from a systems perspective. And so the optimum is the trade off between those two. So you pick a batch size and then for that batch size you do some amount forwards and then some amount backwards. You asked why is there even a hard stop the pipeline parallelism? Because of the fact that you've got this idle time here, which is the bubble. There are so many techniques in the literature for how to lay this out differently. And avoid that there are more complicated schemes called like zero bubble or one forward, one backward, which sort of interleave the forwards and the backwards in complicated ways. But you can mine Bitcoin in that, right? Right. More usefully, you can do the weight gradient step, but you can also mine Bitcoin.

[Lex Fridman]

Yeah.

[Rainer Pope]

So in inference, actually the effect of pipelining on anything you care about like batch size or latency actually is neutral. It doesn't improve it, it doesn't make it worse. So if you look at the latency of this inference running it if it were pipelined versus if it were all on one rack, if it were all on one rack, we would just slide all of the boxes down and still put them in a row and the latency would be the same. So pipelining is neither better nor worse for latency, but it does mean that you just use less memory per rack, like memory capacity. Because now instead of needing the whole model, you only need a quarter of the model.

[Lex Fridman]

It makes tons of sense. So basically no brainer to use pipelining during inference. But there's this harder trade off during training.

[Rainer Pope]

So even in inference, in fact it is not used a ton. It reduces your memory capacity requirements. There's actually a huge surplus. Like I think you're saying that a rack of Blackwell has many terabytes, maybe tens of terabytes of. That's much bigger than a trillion parameter model. A trillion parameter model only needs 1 TB and so it already fits, in fact. And so there's not a huge benefit from pipelining because you're reducing a number that's already pretty small. But it does say that theoretically maybe you had too much memory there and maybe you could have done a different, like built a different hardware that has less memory. In fact, if you were designing your hardware and you said I actually didn't need that much memory because I don't need the weights to fit in one rack, I can fit the weights in eight racks, then I could have maybe built a hardware that didn't have so much HBM per gpu.

[Lex Fridman]

Last week, Pora C was kind enough to give me and my friends a great lecture on large Scale pretraining systems. And there were some concepts that I wanted to animate for a write up on my blog, like how weight shard and gradients flow depending on the parallelism that you're using. So I gave Cursor my lecture notes and a sketch that I'd made during the lecture, and I asked it to visualize a specific hierarchical collective that Horace had explained. The first version was already pretty good, and then I was able to use design mode to select and tweak any specific components. From there, I was able to do all of this without a clear end state in mind. Cursor's Composer 2 fast model was quick enough that I was able to iterate almost instantaneously. I could try an idea, test the results in the built in browser and immediately make any changes. I went through 10 different versions in under 20 minutes. If you want to check out this animation, I published it along with the lecture notes in a blog post. The link is in the description. And if you want to try out this kind of iterative design flow for yourself, go to cursor.comloracache to get started. So, macro question. Everybody's talking about the memory wall right now. Memory's getting super expensive. There's not enough memory. Smartphone volume will go down 30% because there's not enough memory. Hyperscalers are spending. This is shocking. Dylan said they're spending 50% of their capex this year on memory. On memory.

[Rainer Pope]

Oh, that's believable.

[Lex Fridman]

What is hyperscaler? Capex. That's like high hundreds of billions, maybe a trillion. And they're spending half of that on memory. Okay, so that is this huge constraint. That's why we're not going to get new laptops and phones this year. But at the same time, we have too much memory. Like people are willing to put too much memory into these systems.

[Rainer Pope]

Right. So this is.

[Lex Fridman]

Why is Jetsense shoving all this memory into these racks if you don't need it?

[Rainer Pope]

Yeah. So in the equations we had here before we erased them, we were doing memory time. So memory bandwidth and compute bandwidth. Let's now start looking at memory capacity. So we'll start off with just like memory capacity without even thinking about parallelism scheme. And so the capacity of memory or the demand on memory is the number of total parameters plus so this is what we need to fit the weights in some system that we are using. And then we need to fit the KVs as well. So KVs go as batch size times the length of the context times the bytes per token. Okay. So What I was arguing about in this context and the case I was making for pipelining is that we will actually there are some techniques that allow us to solve this. Are there techniques that allow us to solve this? So let's consider. So we're going to run this on some number of GPUs and we're going to say we're going to have one extent, which is E is going to be the expert parallelism. So how many, when we had this charting of expert layer across many GPUs, how much of that, to what extent do we do that? How many GPUs? So we're going to say that this is, for example, 64. And then P is going to be the extent of pipeline pipelining. And so this is the number of racks, which, who knows, maybe we'll pick four or something like that. What we want to calculate. So this is like the total memory requirement across the system. But now I'm going to calculate a, a memory requirement per gpu. So per GPU memory requirement we're going to have. I guess I'll use a lowercase C mem and. Well, obviously we just take all of these numbers and divide it by enp really easy. So it's this N total plus the batch times, length of context times, bytes per toke. All of this is divided by E times P. Okay, so this is like why is this correct to divide it this way? Well, we're saying we knew that the parameters were perfectly divided Amongst all the GPUs in a rack. The layers are perfectly divided amongst the different racks. So that works here. And somehow we're going to arrange. I'll hand wave exactly how somehow we can arrange the same perfect sharding of the contexts across GPUs in a rack and based on layer across racks.

[Lex Fridman]

And sorry, four is the number of racks.

[Rainer Pope]

Yeah, for example. So this is the place where we actually need to go back and analyze this batch size B. And you were making this comment that there's micro batching versus global batching. So let's come back to this pipelining diagram here. We've got one batch going forward here and then as I drew it, it kind of just like disappeared. That's not really correct if you think about how decode is working. I have a bunch of tokens that I have generated already. I do one forwards pass where I generate a new token and then I push like, then I write that to my KB cache and then I do another forwards pass that generates the next token. So I'm actually Going to be running this Batch 0 in a loop. So in fact I go forwards. Once I finish, I can start the next iteration of the loop up here. Yeah. So we'll just fill this in. We'll have a. Oh, nice.

[Lex Fridman]

Yes.

[Rainer Pope]

Yeah. So we've got the 2 or 3. 2 and 3. 2, 3. So let's split this batch. This batch will be the global batch size. So B is going to be the number of micro batches times the batch of like the batch size per micro batch. So how many micro batches do we need? So the number of micro batches in this diagram is 40123 and then the batch size per like the micro batch size. This is still this 2000ish number. This is the one that is like. This is the like 2,000 times sparsity. Sorry. No, this is the 300 times sparsity. 300 times sparsity.

[Lex Fridman]

This is how big the train that takes up every 20 milliseconds is, right?

[Rainer Pope]

Yes. This is going to be the 20 milliseconds train. So the global batch size is the number of micro batches times the local batch size. Local batch size is set by this hardware parameter. The number of micro batches. Well, the number of microbatches is as small as possible, such that we can wrap around and not leave any idle time when we wrap around. So if we had fewer, we would have this idle time when we wrap around. And so you can sort of just visually see that it is equal to the number of pipeline stages. I mean sort of proof by visual here it is four and it's four this way as well. But you can sort of look and see that it goes along here and then it wraps around number of pipeline stages. Yeah.

[Lex Fridman]

And sorry, very basic question. This is what is actually done.

[Rainer Pope]

Okay.

[Lex Fridman]

As in a frontier model today we'll actually have during inference have pipeline for sure.

[Rainer Pope]

During massive scale training. This is done. It can be done for inference. I'm actually going to make the case for why it is less attractive. It is useful for weights but not so useful for kg. Yeah. The big challenge is. So let's fill this in. The micro batch size here ends up being equal to the number of pipeline stages. When we go back and substitute all of that into here, We get a number of pipeline stages times this little B showing up in here. And then when we factor this out, I'm going to split this into like this plus into two terms. We get the full division by E times P over here. We still have division by E times P over here. But The P's cancel, this P and this P, they cancel. And so what we find, if you increase the number of pipeline stages, the memory footprint for the number of weights keeps going down and down and down, but the memory footprint for the number of activations stays constant. So it doesn't actually work. Most of your memory ends up once you do enough pipelining and it's really not much, even two is often enough. This term becomes very small. This becomes the dominant term. The KB cache becomes the dominant term.

[Lex Fridman]

Yeah, I know this is wrong. I'm trying to think about why my train of logic here is wrong. If you have many different. If you're pipelining through many different stages, the KV values are not shared between layers. So why would it not help to be pipelining across multiple layers? Because then you don't have to store.

[Rainer Pope]

Yeah, you only need to store like one layer rather than two layers of KVs. Right? Yeah. So it helps from that perspective. You're right. What's competing with that though is that you need to be keeping all of the racks usefully busy at a time. And so the number of sequences that are in flight simultaneously has gone up.

[Lex Fridman]

Yeah, makes sense. Makes sense, Makes sense.

[Rainer Pope]

So those exactly cancel and you end up not getting a saving per gpu. Right.

[Lex Fridman]

This is going back fundamentally to the point of you're not able to amortize across KVCaches.

[Rainer Pope]

Well, so first we did you can't amortize KV caches across batch size and now we're saying you also can't shard it across pipeline stages. It sucks from both of those points of view.

[Lex Fridman]

Yeah, interesting. Okay, so then what is done during inference?

[Rainer Pope]

So I mean, the Deep SEQ paper reports what they do, which is like they just do a lot of expert parallelism. In effect, you should increase your expert parallelism up to your scale up domain size and then do very little pipelining. Maybe none at all, Maybe two. Just enough to make the weight storage not too big of an issue. Those are the only two parallelisms that really make sense. In the past there was tensor parallelism, which was cutting up within an expert, but the experts are so small now. That is not a profitable optimization.

[Lex Fridman]

So this goes back to the question, does that mean that Frontier Labs when they're doing inference are just basically within a single scale up?

[Rainer Pope]

Yes. I mean, you can look at how it depends on model size. You could have a very large model like one that exceeds the memory of a rack, and there you should be doing a bit of pipelining. Maybe it's extremely sparse for example and that would be a reason to do it.

[Lex Fridman]

So I guess this goes back to the question about or this goes back to the promise at the beginning of the lecture, which was this will actually tell you about AI progress as well. To the extent it is the case that model size scaling has been slow until recently because let me make sure I understand the claim. The claim would not be you could have trained across more. More racks. It was just that it would not have made sense before. We didn't have the ability to do inference for a bigger model easily.

[Rainer Pope]

Actually I make the. So pipelining doesn't help with context length. It totally helps with model size. And so because of the ability to do pipelining, at least a rack should not be a constraint on your ability to fit the model parameters. I guess the other consideration you're asking like why hasn't it scaled up more and why did bigger scale up domains help? So we talked through one aspect of that, which is we kind of said it's not because of memory capacity. We have a solution to the memory capacity at least with respect to model size. Not with respect to KVCACHE size, but at least with respect to model size, we have a solution to memory capacity. The other issue that shows up is latency.

[Lex Fridman]

I was just about to ask, so what is the going from rack to rack, what is the latency cost per hop?

[Rainer Pope]

This is very much dependent on the hardware. I can't say with a lot of authority. I think it's probably on the order of a few milliseconds. But it could be off by an

[Lex Fridman]

order is for a realistic number of how many pipelining stages you might have.

[Rainer Pope]

Yeah. Okay, so that's not on a small number of pipelining stages. This is not a huge latency impact.

[Lex Fridman]

I guess it's 10 milliseconds per token.

[Rainer Pope]

That's right.

[Lex Fridman]

2 times 4ish. Or I don't know how many said,

[Rainer Pope]

but yeah, yeah, 10 milliseconds per token

[Lex Fridman]

is actually a lot.

[Rainer Pope]

Yeah. If it goes from 20 to 30. Right. Or something like that. Yeah. So just to chart the path that it goes through here you're going from your GPU or TPU or whatever to a network card which then goes to like a top of rack switch and then hops over to the other rack and does the same thing in reverse. So you sort of have to sum up the latencies of these different things.

[Lex Fridman]

So this is the same thing as the DC data search.

[Rainer Pope]

It may in fact go up to a datacenter switch and back. Depends on your deployment configuration and because

[Lex Fridman]

it's decode and sequential, it's also not the like they stack up across the stage. You can't do them at the same time.

[Rainer Pope]

That's right, yeah.

[Lex Fridman]

Okay, so I guess this brings us back to the question then. Is the size of the scale up at all relevant to why AI model sizes or whatever have been what they have been over the last few years, whether through training or through inference?

[Rainer Pope]

Yeah, so I mean we talked about latency of the hop, of this hop. There is also just the, the same TMEM latency. The memory time latency is actually substantially, massively improved by larger scale up domains. So I'll recall tmem down here. TMEM for the weights, TMEM of weights. This was equal to the number of total parameters divided by the memory bandwidth. Which memory bandwidth are we talking about here? Is it just one GPU or. It's. In fact it is the number of GPUs that I can use in parallel to load these weights. So I can't use different pipeline stages in parallel because they're not running at the same time, but I can use all the GPUs in my scale up domain in parallel to load the weights. And so this is actually extremely effective. So basically I end up with a term here. This memory bandwidth term itself is equal to like scale up size times memory bandwidth per gpu. Yeah. Times GPU bandwidth. And so this term doesn't increase a lot. It maybe increases 1.5 or 2x per generation, but this one increased by like a factor of eight from Hubble.

[Lex Fridman]

So the reason the bigger scale of matter is not the memory capacity of the whole scale scale up, but really the memory bandwidth.

[Rainer Pope]

Yeah, pipelining totally solves the capacity problem, but scale up size helps solve the

[Lex Fridman]

bandwidth problem and the bandwidth problem helps you do longer context lengths, which is more and more relevant as these models get more enchantic.

[Rainer Pope]

Yeah, it lets you just run the model at lower latency. As a first thing, if I just do a very fast model and it's On a little H100 box, the latency will be really high.

[Lex Fridman]

Yeah. Okay. A super tangential question. There's chinchilla scaling, which tells you how big should a model be relative to the amount of data you're going to train it on. But now obviously you're not just trying to optimize for the highest quality model you can get with training. Compute. You want the best results a user can get as a mixture of training and inference computer. So then there's a question of how Much should you overtrain a model such that that compute amortized over training and inference is minimized to get a certain performance. But now with RL inference or rl, there's another consideration which is you're going to do some minor pre training. That pre training will be used both for RL generation and then for inference for the final user. And by overtraining here, I mean while it would have been more efficient just from a training computer perspective, to have a bigger model that you train for less time because it can learn faster, maybe you get a smaller model, you spend more compute training it than you otherwise would have, but now it's cheaper to give it to users, basically. Okay, let me do the question more concrete. How much more than chinchilla optimal are models overtrained and has that changed as a result of RL generation?

[Rainer Pope]

This is a place where we have to do a bit of guesswork because the updated scaling laws and the model traffics are not reported and have to guess there. But one way to look at it, Let me first just make a general heuristic claim. If I have some cost and I've got a total cost, which is a sum of cost A and cost B, maybe this is the training cost and this is the inference cost. And so I want to minimize this sum. For many curves that end up being the case, the minimum tends to be where the costs are equalized. That's something of a heuristic claim. But there are many examples where it's true where one is one over X and the other one is X, for example, they tend to be minimized at the point where they equal each other. It's also true for E to the X and E to the minus X and all kinds of other things. So basically I've got some curve that's going down, some other curve that's going up, and they tend to be minimized at this equal point. Heuristically I will conjecture that that is true for the setup you described as well. Actually showing that that would be true would require looking at the scaling laws and and fitting these weird exponents. But things that do follow power laws tend to have this property. So I'll just make that claim and move on. So we're going to say that the cost of training plus the cost of inference, we want to equalize these. We'll do pre training only first because it's a little. Well, actually we can do all of it in general. So actually we'll cost of it as cost of pre training.

[Lex Fridman]

So

[Rainer Pope]

number of active Params times the data on pre training. So that's the cost of pre training. There's a factor of six out here, which is the number of flops. There's the famous 6nd formula. And then in RL we have approximately the same thing. We've got like same number of active parameters. But now the amount of data is the RL data. There is this extra efficiency multiplier or inefficiency. Like the inefficiency

[Lex Fridman]

which is the fact that you're not trading on all your rollouts.

[Rainer Pope]

Well, yeah, there's that. And then the other perhaps even bigger inefficiency is that this involves a substantial amount of decode and often decode runs at less MFU than training.

[Lex Fridman]

Okay. So if you're doing a backward pass on every single generation in RL, it would be 6nd.

[Rainer Pope]

Yeah. So this could be a smaller number. Right. Like this could be somewhere. So it would at least be two. Yeah. Somewhere in the range of two to six. So we'll just like, we'll say somewhere in the range of 2 to 6 and leave it at that.

[Lex Fridman]

Yeah.

[Rainer Pope]

And then we can add in the inference cost. The inference cost is 2 Number of active times the data in inference.

[Lex Fridman]

I think the way I said it was super garbled just for the audience. Maybe forward plus backwards per parameter is six, forward alone is two. That's why rl, where you're definitely going to generate all the trajectories, but you might or might not train on all the trajectories is two to six.

[Rainer Pope]

Yes. Yeah. Thank you. And then inferences is just two. So we're going to solve for essentially maybe equality of all three of these terms, I.e. ballpark, where people are going to be labs have more information on what is productive in doing more rl, for example, than versus doing more pre training. I don't have that information, but I think a good ballpark is 33% split between each of them.

[Lex Fridman]

Actually, I'm not sure there's an intuition for that. Another naive model could have been that RL plus pre training would be 50% and inference would be 50%.

[Rainer Pope]

Yeah, that's also a valid answer as well. Because this is heuristic. I can't really argue for one versus the other. They don't differ by that much. Like 33 versus 25 is only a smaller factor of. So let's pick one of them all equal. Seems simple enough. And so we're just going to solve for equality of them. It's pretty straightforward. We can immediately see that the number of activated parameters totally disappears. And so let's factor that out and we're going to just say that data in pre training. I decided to do it your way. It's a little bit nicer actually. So data in pre training plus this. Oh, I didn't have the inefficiency over here either. Inefficiency data in pre training plus some multiple of alpha times the data in RL is just going to end up equal to some beta times the data in inference. And then let's just roughly size the alpha this alpha. It's going to be. This is like the, it's maybe somewhere in the range of 2 to 6, 2 to 6 over 6 from this term compared to this term. And then we've got an inefficiency term which I would say is maybe in the range of like 30%, something like that. So this alpha is going to be something like one on 10, one over 10, let's say. And this beta here is actually the same. It's a third, it's 1/3 times 30%. So it's also equals 1 in 10, something like that.

[Lex Fridman]

If both of them are 1 in 10, that kind of implies that there's never a backward pass on RL. Yeah.

[Rainer Pope]

Okay, we can make this like 2 in 10. Make it a bit bigger.

[Lex Fridman]

Yeah.

[Rainer Pope]

So yeah, like just write it out once more. Like this is 2 over 10, this is 1 over 10. So the number of inference tokens you have, and this is just a function of like I've got hundreds of millions of tokens per second times my model is deployed for, I don't know, two months before I shift to the next version. That should determine the number of tokens in RL pre training. And then I guess we didn't do the equivalence between pre training and idle rl. So we'll do that here. Data pre training should be equal to like 2 over 10 times data in RL for them to be cost equivalent. So sorry, this one over, I got it backwards. We pay more cost when it's inefficient. So this needs to be one over. So this tracing this back forward, this thing ends up actually being as written here. It's like. Yeah, so this is like 1.5 and this is 1

[Lex Fridman]

billions of dollars of the compute just flowed the other direction.

[Rainer Pope]

Yeah, right. I think if you do it with a spreadsheet and actually you might notice when the money's going down the drain. Yeah. So I think this, all of these end up being close as modeled here. This 30% may have been a little bit too generous. So let's say something like 1.5 here and leave this as a 1 here. So I think at this point you can almost read it off like the number of inference tokens should be about the same as the number of pre training tokens should be about the same as the number of RL tokens within factors that we're not able to reason about.

[Lex Fridman]

Sorry, I'm making a basic algebra mistake. It seems like there should be less RL tokens than pre training tokens.

[Rainer Pope]

Yeah, that's in general. Right. Because RL is less efficient in terms of machine time. And so if you're trying to equalize the RL and pre training time, then you should have fewer tokens in order to have the same wall time.

[Lex Fridman]

This is quite interesting that I never thought about it in terms of how much equalizing in terms of data.

[Rainer Pope]

I think starting with equalizing in cost is right. But depending on how you model the cost, this comes close to equalizing data

[Lex Fridman]

that if every single user who uses Basically for GPT to be trained optimally, every single user who uses GPT5, the total amount of tokens that they stream should equal the total amount that have gone into pre training. And the total amount of tokens that have gone into pre training is the sum of all human knowledge. So each model should generate the sum of human knowledge on the output that it gets on the input.

[Rainer Pope]

Yeah. So I mean, which way are people going to err? Like if you think that people's power of prediction is not perfect and also you run the risk that you make a model that is not a frontier model and then you just throw it away, then that kind of changes the cost trade off because there's some probability that applies to the inference. And you should derate the inference tokens by some amount. Right.

[Lex Fridman]

And then can we back out how much more compute than chinchilla optimal for a given sized model?

[Rainer Pope]

So I think we just have to make some real world assumptions here in order to do that. So the inference tokens we should totally be able to count. Right. So let's say a few hundred million. I don't know, Maybe it's like 500 million tokens a second now. I don't really know. 500 million tokens a second times a model is deployed for two months before it becomes obsolete. I don't really know. I can't do this in my head. Can you type it into a computer?

[Lex Fridman]

2.6 times 10 to the 15th.

[Rainer Pope]

Okay. 2.6 times 10 to the fifteenth. Okay, this number is probably too large because this is going to be multiple models in a family. So let's make it like 5 times smaller or 10 times smaller or something like that. Okay, so we're estimating maybe 50 million tokens per second per specific model. The model is live for two months and so this comes out to around 200 trillion tokens. And then we want to compare that to active parameters on a frontier model. I don't actually know the latest rumors, but some do.

[Lex Fridman]

You know, somebody taught me 150 trillion active prems. Sorry, I meant tokens trained on 150 trillion tokens.

[Rainer Pope]

Interesting.

[Lex Fridman]

Which is similar.

[Rainer Pope]

Yeah, that's actually similar. So data on pre training, this is

[Lex Fridman]

not well cited, but you want me

[Rainer Pope]

to not remove that? And I think often active prems number of active prems could be in the range of like 100 billion, something like that, maybe a bit larger. So I'm assuming active prems of about 100 billion. And so multiply by 20 to get the chinchilla token count. So chinchilla D. Chinchilla would be around 2 trillion. And we see like we're about 100 times larger than that actually.

[Lex Fridman]

What does decentrala actually mean?

[Rainer Pope]

Like the token count for pre training that the chinchilla scaling law would recommend, I guess.

[Lex Fridman]

Oh, I see. So how much is it overtrained? Got it.

[Rainer Pope]

So yeah, like the ratio of this 200 trillion or 100 trillion parameters over the chinchilla optimal of 2 trillion, that's the amount it's overtrained, which is like a factor of 100 overtrained, perhaps.

[Lex Fridman]

That's what. Okay, so if you consider this right here, to the extent this isn't the right ballpark, just by thinking about, okay, you kind of want everything to be equal in terms of compute. If that OpenAI also realizes that and they're serving a certain amount of tokens per second, that tells you how much data went into the pre training of GPT5, even if it's like 50% off or something. That is sort of wild that you can sort of first principles, these kinds of numbers.

[Rainer Pope]

This is why you should just approximate everywhere because there's so big error bars on this. But yeah, it's kind of empowering to just set A equal to B and figure it out.

[Lex Fridman]

Yeah, yeah, that's super cool. Okay, so in the spirit of trying to deduce things, we can publicly look up the prices of the APIs of these models and maybe you can learn something from that. So first, with longer context, Gemini 3.1 is 50% more expensive if you go over 200k tokens than if you're below 200k tokens. I mean, at a high level, I understand why that might that be, but why specifically 50%?

[Rainer Pope]

Yeah, so I mean, why specifically 50%? Let's sort of. So the high level, even in the first place is there is some amount of increasing cost with context length. And, and we can bring that back up. That was the memory time versus the compute time. Okay. So we've put up these same equations from before of the time for memory fetches, which is the weights and the KB cache, and then the time for the compute, which is just the matrix multiplications for the weights. I will also draw the cost curve, But this time I'll do it as a function of context length instead of as a function of batch size. So this is time over. Yeah, just time. And so this is the cost curve as a function of context length. We'll draw the compute. The cost of the compute is actually constant as a function of context length. There's no dependence here on context length. In reality there is some dependence, but it is very mild dependence, so we'll ignore it. So this is the time for the compute, This one. And then we'll also draw the dependence of the memory fetch on context length. And this starts at a large number for the weights and then grows gradually with the context length. So maybe here, and then grow gradually with context length. And so you take the maximum and you see there is this inflection point here. So this is the costs that for example, Gemini might be paying. And then you think, how might you put a pricing structure on top of that? You would like to ensure that no matter what the context length is, you are still profitable.

[Lex Fridman]

Interesting.

[Rainer Pope]

And so we've got a two tier pricing structure. Maybe we've got something that looks like this up to some max context.

[Lex Fridman]

Fascinating.

[Rainer Pope]

So I think it says something about, given that the bump is at 200k, it probably means that this is somewhat aligned with this crossover point. Maybe not exactly aligned with.

[Lex Fridman]

Fascinating.

[Rainer Pope]

So we can actually probably even complete that calculation just to see where it lands out. We can solve for the number of bytes per token if we sort of make some assumptions about the number of active parameters. So solving for the number of bytes per token, we're going to assume that the point where we equalize the time of memory and the time of compute is at, let's say 200k tokens. So we equalize these two. We're also going to just assume that the batch Size is large enough that the the memory time spent on weights is negligible. So we'll forget about this and we'll focus on the actual memory time spent on KBCache. So that ends up saying copying this term over batch times lencontext times bytes per token over MEM bandwidth is going to be equal to number of activated params over flux and then we're going to solve for bytes per token. Batch size was missing here, shows up here and then it cancels out by the time we get to here. And I dropped the LEN context so we can plug in numbers. This number, this is. This, well is the reciprocal of the number that we saw before. Yeah, this is like 1/800 which is reasonably stable across many different hardware platforms. We conjecturally said that maybe number of activated tokens is like 100 billion. And like the context we said was 200k, something is wrong here. The length of the context should be on the denominator, not the numerator.

[Lex Fridman]

1667 like about almost two kilobytes.

[Rainer Pope]

That is plausible actually. So you said around two kilobytes. So let's just do a sanity check for this for what this could be. There are two mechanisms that people do attention with a small number of bytes per token. One is dense attention with a lot of reuse across layers. So character AI has a blog post talking about that alternating long and short context. And in the character AI kind of model, which also showed up in the GEMMA models, the global context, which is really what we're talking about here, global context was shared across all the layers. And so to get this two kilobytes you could get that for example, as a d head of 128 is typical and then the number of bytes is typically number of attention layers times

[Lex Fridman]

2

[Rainer Pope]

times d head times number of Q heads. So this is the number of unique contexts per layer. Do you share the context across many layers or do you use it only once? So in character AI like models, this number is one we said this is 128 and this is a choice which typically ranges from one. Sorry, this is KV heads I meant.

[Lex Fridman]

So there is written a head and a KV head is that the KV

[Rainer Pope]

heads are the heads that are stored in memory, like store the contents of the previous tokens. The queue heads are the, the retrieval heads there they're only used temporarily and they're used by the attending token. So in this autoregressive context I've got KV heads associated with all of the context and then Q heads associated with this new token here.

[Lex Fridman]

But this head, the 128.

[Rainer Pope]

Oh, this number is actually the same for. Oh, sorry. This D head is the dimension of the vector. And number of KV heads is typically in the range of 1 to 8. So like it is totally plausible to get this by for example having 8kv heads and a D head of 128. That gives you exactly this number. Or you could have like fewer kv heads but more layers. Yeah, so this is one way to get there via dense attention. There's also a way to get there via sparse attention where you increase all of these numbers but then you have like a line over sparsity term. So yeah, I mean I think this number is plausible if maybe a little bit small.

[Lex Fridman]

It's funny that they would leak so much information through their API pricing.

[Rainer Pope]

I mean you are incentivized to price close to your costs because otherwise someone could scoop you.

[Lex Fridman]

Maybe we can learn something about the difference in input versus output prices and what that tells us about decode versus pre fill in these models. And I think last I checked it's like 50% more expensive or something like that.

[Rainer Pope]

I don't remember. What I've seen in the past is like three or five times more expensive.

[Lex Fridman]

Okay, that makes more sense. Let's say it's five times more extensive. Okay, this is the compute to process the next token in decode. Suppose you're doing pre fill where you're not just processing the most recent token, you're processing all the tokens in parallel. So I want to say that it would be this times len length, prefill

[Rainer Pope]

length of the pass. In general. Yeah, if we say like if we can think of decode as being a pass with one and then prefill being a pass with many.

[Lex Fridman]

Okay, yeah, yeah. So maybe like prefix.

[Rainer Pope]

Sure, whatever.

[Lex Fridman]

Okay. Memory. So you're not storing the kvcache for the tokens that are the pre fill tokens.

[Rainer Pope]

I think maybe, maybe sort of. Let's draw actually how pre fill shows up here, if I may clarify. So we do a bit of decode like this, we may actually come back and do more pre fill. If you think this is a chat session, the user says something, the AI generates response and then the user says something else and we pre fill this. So maybe this is the more common, this is the general case rather than this.

[Lex Fridman]

In fact, this is like you read a file or something.

[Rainer Pope]

Read a file or just like the AI is responding to user input or tool call or anything that's not hyper generated. Yep.

[Lex Fridman]

Okay. Okay, suppose we're here. So you will need to load basically the. You will have calculated all of this previously. So just the KV of everything that came before. But what is the memory cost of this? Well, memory bandwidth cost of this. If you're doing flash attention, it would.

[Rainer Pope]

Yeah, it's basically temporary. It doesn't even go to main memory. Just ignore it.

[Lex Fridman]

Okay, so then it would just be everything that came before. So is it not just that then?

[Rainer Pope]

Yeah, there's actually no adjustment at all to the memory time. Great.

[Lex Fridman]

Oh, so it's a very trivial change to accommodate. So this term is making it 5x more expensive. Now why would that be? Or what does that tell us about. What are we trying to learn here? What does that actually tell us? What variable does it help us clamp? Well, the compute has presumably gotten five like the only thing that could have changed is the compute is 5x more expensive as a result.

[Rainer Pope]

So yeah, this is the time for one pass, but actually the amount of tokens is that much larger. So I guess we want the cost per token in fact. Or the time per token.

[Lex Fridman]

Sorry, I'm not sure I understood. This is for processing the next token in prefix.

[Rainer Pope]

Well, actually for processing the entire batch. So at this cost we have processed this many tokens like len. Lennerprefill. Yeah, I guess of the paths, not this prefix, but it's this cost.

[Lex Fridman]

Okay, look what's just changed as a passion. So this is 5x more expensive Input is 5x more expensive.

[Rainer Pope]

No, output is more expensive.

[Lex Fridman]

Output is 5x more expensive.

[Rainer Pope]

So the result we want to work towards is that pre fill is compute limited and decode is memory bandwidth limited.

[Lex Fridman]

Why don't we do this? Why don't we just chart it with like lenpass on the x axis.

[Rainer Pope]

Yep.

[Lex Fridman]

T on T on the Y axis.

[Rainer Pope]

T. We want the cost per token so it'll be T over some stuff. T over length of the pass. Yeah, that'll be great.

[Lex Fridman]

Okay, so. Okay, it gets me confused about this Len pass is the. It seems like this should be higher when you're doing pre fill.

[Rainer Pope]

Prefill has a bigger length pass. Yeah, right.

[Lex Fridman]

But then why is it cheaper?

[Rainer Pope]

Why is it cost higher? Yeah, yeah. So I mean we're gonna. It's this division by length pass that actually makes it all. So okay, this is gonna divide out. This is gonna divide out, but then we're gonna get all of this is gonna divide by length of the pass and it's gonna make the memory cost cheaper.

[Lex Fridman]

Okay. Yeah, Let me think about this. Then okay, so let's do one line for. Basically we'll have four different lines. Let's. So let's do prefill first. And so actually let's do decode first.

[Rainer Pope]

Oh, so actually length of the pass when it's one, that is decode when it is bigger, that is prefill.

[Lex Fridman]

Okay, I see, I see, I see. That makes sense. Okay, getting back to it. So T compute, if you have basically just this divided by length pass is just this amount. So this actually does not vary based on T. So it'll just be some flat value like this. And this is T compute. And then this is like this is.

[Rainer Pope]

That's decode. Decode.

[Lex Fridman]

Right. Now tmem, if you have this whole thing divided by lempass. Well, it doesn't really matter what's up there. It'll just be something that looks like this.

[Rainer Pope]

Right? Yeah.

[Lex Fridman]

Say this is T mem. This is decode again. So as the length of the prefix goes up or pass, your memory bandwidth time declines. And that means that to the extent that you were bottlenecked on memory bandwidth before, you can avoid being bottlenecked on memory bandwidth. The fact that they are charging 5x less for pre fill than decode does suggest that they are bottlenecked on memory bandwidth to quite a degree. Such that for them at least. Because T is equivalent to cost. Right. It's the cost of renting a compute. This is actually like this. This would be at 1 and this would be at 5.

[Rainer Pope]

That's right. That's right. Yeah.

[Lex Fridman]

So it is in fact tremendously memory bandwidth bottleneck. The real graph looks something like. The real graph looks something like. Like that.

[Rainer Pope]

Yeah. I mean it still crosses, but yeah, yeah.

[Lex Fridman]

So yeah, let me do it this way.

[Rainer Pope]

Yeah, that's right.

[Lex Fridman]

And then this is the gap on decode between the memory and the compute time.

[Rainer Pope]

Yeah, yeah.

[Lex Fridman]

Okay. Interesting. Another interesting one would be why cache hits are so much cheaper. Yeah, okay, so if I remember correctly, cache hits are like 10x. It's more expensive to write to cache according to the pricing on all these models. But if you do hit a cache, it's 10x cheaper. So what is going on with. Presumably this is the cost of keeping something in HBM rather than just evacuating it. But if you do keep it in hbm, then it's cheaper to load again.

[Rainer Pope]

Right. So there's two ways you can produce tokens or the kvcache for a token. You can just produce it from scratch by computing it from the underlying like token IDs, which are tiny, or you can previously have produced it and stored it in a memory somewhere. So the cost ratio is really talking about the ratio between those two mechanisms of producing it. A cache miss means you've deleted it from all your memories and you have to recompute it from the tokens directly. In fact, you can maybe even take that a step further and think about which memory tier do you store it in. So you could store it in hbm. There are other slower and cheaper memories than HBM like DDR on your host or Flash as well. And so one of the things you can do is a calculation of where it makes sense to be in each memory tier. And this is related to how long you're going to store for. So we want to look at the cost of storage in a few different memory tiers and also the cost of rematerialization. So remat means the cost to rebuild all of the KBCache from scratch, having it after you deleted it. So we rematerialize it. And so basically this is going to cost the length of the context. Actually we'll look at cost per token so that we don't need to carry around this length of context everywhere. So to rematerialize one token of kvcache, I just need to run. I need to run a forward pass on the whole model. And then. So this is going to be the compute time. I have to rerun the compute at whatever speed my GPU does it and then I multiply it by my GPU dollars per second.

[Lex Fridman]

Sorry, extremely naive question. Why is there not a quadratic term?

[Rainer Pope]

Yeah, so there is a quadratic term in. It shows up in the compute. As an approximation. I chose to remove it. I'll just show you sort of quickly what that looks like. It's because. So you have the. If you look at the cost per token or the number of flops per token, there is the flops that are coming from doing the weight matrix multipliers as a function of context length. And then there is the number of multiplies that comes from doing the kvcache, which goes up linearly with the amount of stuff you attend to. The slope on this is so low that when you draw it like this, it's very well approximated by a flat line. So you start to notice the effect of the quadratic or the linear term up in the millions of tokens or so. So just not super relevant.

[Lex Fridman]

So what is the reason that there's no company which has over a million token context length if this is true?

[Rainer Pope]

Yeah, so there are two costs of long context. One is the memory bandwidth cost, which we've spent a lot of time analyzing. That's this thing. And then the other one is the compute cost. The compute cost is almost always and sort of actually forced by fundamental principles to be a much smaller slope than the memory bandwidth cost. And so the primary thing that limits you to have really large contexts are memory bandwidth, memory capacity, which is exactly this effect.

[Lex Fridman]

And so there's this idea that Dario said on the podcast and others have said which is we don't need continual learning for AGI in context learning is enough. And if you believe that, then you have to think that we had to get to 100 million billion context length to have an employee that is the equivalent of working with you for a month. Now maybe that's no longer true as far as attention or something. But yeah, if you think that then some ML infra thing would have to change to allow for 100 million like the memory bandwidth to allow for 100 million token context lengths.

[Rainer Pope]

I mean sparse attention gives you a get out for sure because you get this square root gives you a big improvement. But I think it's like if you look at the history of context lengths of models from earlier models like GPT3 maybe to GPT4, I don't remember when the transition happened exactly. They shot up from about 8k to 100k to 100k and then for the last year or two they've all been hovering around there. I think that actually indicates that is sort of the reasonably balanced cost point. And going massively beyond that would be cost prohibitive.

[Lex Fridman]

Not because of the compute cost, because

[Rainer Pope]

of the memory bandwidth cost. Yeah. So I actually don't see a very good path to solving that. Like the HBM is where it's at where it is. It's not getting hugely better.

[Lex Fridman]

And why doesn't solve it.

[Rainer Pope]

Sparse attention is a big improvement. Maybe that is priced in already. Perhaps it's not an infinite improvement because if you go too sparse you lose too much quality. But yeah, I mean the empirical result is that the complex lengths haven't been increasing that much. And I think it's because there is no solution to the memory wall.

[Lex Fridman]

Interesting.

[Rainer Pope]

So going too sparse just means you're attending to a very small subset of the tokens and the quality will get worse. Makes sense. So what is the cost of these different ways of producing resynthesizing the kvcache? Computing it from scratch is based on my GPU time. I have to do a certain amount of multiplies in order to. Of GPU time that I spend in order to produce it. Storing hbm. This really goes as my. I think I had a number here which was the bytes per token. So I need to. I need to just have some number of bytes per token and then I need to store this in the hbm. So it's going to use up some of my HBM capacity. So a way to think of this is that like if I have too many of these things sitting in my hbm, if I fill up my HBM with just KV caches that I'm not using, I can't use that gpu. And so how do I price that? Maybe I say that the cost of it is proportional to the fraction of the HBM I'm using. So there's also times GPU dollars. And then let's just do one more memory tier and say something like DDR store in DDR instead. The same kind of thing goes up for Flash and for DDR. I put these in the wrong columns. Actually, I meant to make two columns. The distinction I want to make is that there is the cost to retrieve and then there's a cost to store. Costs to hold on. And so this is like. This is a cost per second, whereas this is like an instantaneous cost. So rematerialization has a cost to retrieve and has zero cost to store it because we've deleted it. This is the one that I put in the wrong location. This is actually the cost to hold on. So I will rewrite it. Okay, so we have. This is the like if we're just storing it in hbm, it has this sort of cost profile. And then if we store in DDR, it's actually going to take some time. So it's like we get the same thing here. Bytes. Bytes per token over DDR capacity times DDR cost per second. But now this has a cost to retrieve that is higher than the HBM because we need to copy it into the hbm. And so this is bytes per token over DDR bandwidth. Bandwidth. And then this consumes some amount of the DDR as well.

[Lex Fridman]

Then every scale up has DDR and Flash.

[Rainer Pope]

This is really a deployment question. And so you can choose that Nvidia does deploy in this form. It has both.

[Lex Fridman]

Why isn't the cost to retrieve HBM the memory bandwidth or the bytes divided by memory bandwidth?

[Rainer Pope]

Yeah, I mean, it depends what you define a retrieve to be. Here I'm defining retrieve to be move it into HBM so that you can start actually doing inference on it. And so sort of by definition, because

[Lex Fridman]

if it's already in hbm, you can be doing comp while you're getting it

[Rainer Pope]

from HBM to svr, for example. So these are three things and I guess I ordered them wrong. In general, if you're balancing two costs and you've got different tiers in the memory hierarchy, you should expect as this cost goes up, this cost should go down. So you can kind of see where the zeros are. And I should have ordered them this one first, this one second, and this one third. So if you're going to hold onto it for a very short amount of time, then all of this is multiplied by the hold time. This one is, and so is this one.

[Lex Fridman]

And interestingly, they have different prices to write for. And you specify this in the API for five minutes versus an hour.

[Rainer Pope]

Yeah.

[Lex Fridman]

Which suggests that the five minutes is HBM and the hour is DDR.

[Rainer Pope]

I think that's like. I think that's a pretty good assumption. It could, if you look at the numbers, it might also turn out that it's one tier down and it's DDR versus flash. Is there?

[Lex Fridman]

Yeah. Okay, interesting. And the price difference I think was. I'll look it up. Okay, so the base, base input tokens is 5 per million tokens, which means rebound. Yeah, that's 5.

[Rainer Pope]

This is 5.

[Lex Fridman]

To retrieve, quote unquote and then to write to. Presumably HBM write for 5 minutes is 6.25.

[Rainer Pope]

So actually we might actually be able to determine the which memory tier it is by the durations. Actually, the durations probably tells it to actually.

[Lex Fridman]

Five minutes versus one hour.

[Rainer Pope]

Yeah, exactly. I think this will probably end up being. It's going to be the drain time of the memory tier that you're in. And so what that means is, given that I know I'm going to be holding something for five minutes, I would like to have pick a memory that I can read every five minutes. Like I can read the whole memory once per five minutes, ballpark. So that is the drain time of the memory. So if I take the storage capacity over storage bandwidth, bandwidth, I would like this to be like equal to five minutes or something like that. And so actually we did this calculation for HBM for HBM. We know that this number is 20 milliseconds. So HBM is much too short, like much too small. DDR could be about an order of magnitude or two off from this. And so this is probably in the order of like. Actually, I think it might even be in the seconds, like 1 to 10 seconds. And then this is really. I don't have these numbers memorized, but generally, as you go just lower tiers, flash is plausibly in the order of one minute. And then like spinning disk, which is massively different, I think is on the order of one hour. So this might actually identify that the tiers are probably flash and spinning disk.

[Lex Fridman]

Sorry, why is this the calculations? The storage cap divided by the bandwidth.

[Rainer Pope]

So you've got a bunch of different memory tiers. Like we've listed four of them. Your choice of which memory tier is like, you want to minimize the cost. And so what fraction of the device are you using? You're using some fraction of the device for the holding onto it, and then you're using some fraction of the device to retrieve it. And so let's say I'm using 10% of the device and I want to equalize those two fractions. That's a sign that I've hit the right thing. So let's say I've got some runtime here. Like I'm going to hold on for all of this time. So this is the time hold. And then there's going to be some amount of time here, which is time retrieve. And I want, I mean, basically to equalize the costs, these two costs. I want the retrieval time to be equal to the hold time times the fraction of capacity. Because this is the retrieval time. Yeah, I mean, this is how many other things I can hold simultaneously.

[Lex Fridman]

Basically, just like, hey, you want to store things in there for so long such that the amount of time it's in there is kind of the time to get all your things in there and out.

[Rainer Pope]

Yeah, basically, I think that probably indicates that this is the two tiers of flash and spinning disk. I'm kind of shocked to see spinning disk being used at all because it's such an old technology.

[Lex Fridman]

But yeah, I mean, it's also crazy that it's so slow that it takes an hour to load its full capacity into it.

[Rainer Pope]

It's a really unattractive technology, but it's useful in some places.

[Lex Fridman]

So we're sitting down because I want to ask you some questions that I guess don't need a blackboard. You have this extremely interesting blog post where you talk about how at a high level, the architecture of different cryptographic protocols looks a lot like neural networks. And there's this conversion evolution where they both need to jumble information across all their inputs. For cryptographic protocols, it's to make sure that there's like each new input into a hash function will totally scramble what happens for neural networks, of course, they need to consider how this piece of information changes what you should make of this other piece of information. That has an extremely interesting point. I guess at a high level, the difference in what they're trying to do, in some sense they're trying to do the inverse thing, which is cryptographic protocols are trying to take information which has structure and make it look indistinguishable from randomness. And neural networks are trying to take things which are look like random protein sequences, DNA garbled text, and extract higher level structure from it. So they have similar high level mechanisms, but they're actually kind of trying to do the opposite things. I wonder what you make of that.

[Rainer Pope]

Yeah, so I mean, the mixing, I tried to look for other examples where mixing, like scrambling mixing shows up as well. There's actually almost even like a physical example where like you're stirring something, you're making a cake and you want to stir the batter. And literally the idea first stir it this way and then stir it this way is actually not too bad of an approach. But beyond that, back to the digital world, there are some differences. And the one you call out is a pretty strong difference the way it shows up. What makes neural nets, if you just randomly initialize a neural network, actually maybe it's a reasonable cryptography cipher as well, because the random initialization is going to jumble stuff in a complicated way. It may even do what you want, who knows. The thing that makes it interpretable is the gradient descent. So you can differentiate a neural network and get a meaningful derivative. And we do a lot of work to not overcomplicate the derivative. So the residual connection keeps it contained and simple and the and so does like the layer norm stuff that we do. One of the biggest attacks against cryptographic ciphers is also to differentiate the cipher. Ciphers run in a different number field. They run in the field of two elements, so just binary. Whereas neural nets run like in theory in the field of real numbers. And so you have to differentiate with respect to binary numbers, but you can absolutely differentiate a cipher. And this is called differential cryptanalysis. And basically what it says is that if you take a small difference of the input, it's quite difficult to make the difference of the output be small. The whole job of a well designed cipher is to make the difference in output very large. So I guess the distinction is that the optimization goals at that point are about complexifying. They don't have the same residual connections or layer norms.

[Lex Fridman]

Yeah, I mean, I guess a place where the two merge is backdoors. Okay. So with a backdoor you're trying to hide, what do you consider an input? It's not an input into the forward pass, but it's an input into the backward pass, but you're trying to hide an input into the backward pass.

[Rainer Pope]

This is like an adversarial context. Yeah. So, yeah. I mean, in fact, this is actually a place where you get exactly the sort of avalanche property that ciphers have as well. Adversarial attacks on typically like image classification models. Right. Are. Can I find a perturbation of the image that. A very, very small perturbation of the image that totally changes the classification, totally changes the output. That is the common case in ciphers, whereas that's the undesired case in neural nets. For sure. Yeah.

[Lex Fridman]

Okay. So I was asking you, have neural networks actually been used for cryptography? And we realized it might be better to just do this on a blackboard. Yeah. So I'm curious, are they actually being used for cryptography?

[Rainer Pope]

Yeah. So using neural nets for cryptography. Well, in general cryptography, like creating a new cipher is a very, very dangerous proposition. Almost all of them are broken, like 99% of them are broken. So probably a bad place to start. But the other direction has been very, like, in at least one very clear case, quite productive. So there is this construction in. Sorry, a construction that exists in ciphers and then was imported into neural nets called a Feistel cipher. Feistel network. So the idea is that you may have some function F which is not invertible, but you like the function because it does interesting things, like it does an mlp, for example, or it mixes it in an interesting way. You'd like to build something out of this that is invertible. So the construction we're going to make is going to actually be a two input function rather than a one input function. And we're going to apply F of X. We need to actually remember what X was. So we're going to stick X over here so that we can work backwards. And then we also count drop Y. So we're going to remember Y and we're going to add them together and so we form this tuple. So the way to invert this, like if you think I have this output and I want to recover X and Y, well, I can easily recover X. It's right there. I just read it off. And then to recover Y, if this thing was called Z, I can recover Y by Z minus F of X, because I've already recovered X. So that means that this construction is invertible. This was used in ciphers. A ton still is used. It's one of the main mechanisms of constructing ciphers. Often you want ciphers to be invertible, especially the layers of ciphers. You want to be invertible because that has been no cryptographic properties. This has actually been ported over into neural nets. There's a 2017-18 paper called Revnets Reversible networks. And what it does is it actually makes the entire like you can apply it to any network. Like a transformer network. I do a forwards pass, but then I can actually run the entire pass backwards as well. So the whole neural network is invertible with exactly this construction. And so this paper reversible networks applied to some layer, like a transformer layer. For example, we've got this function F which is our transformer layer. Now normally we would have just an input and then a residual connection coming out and it gets added like this over here. But now the variation of this is going to be we've got two inputs, X and Y. So we've got X and Y inputs. X goes through the function, gets added to Y. And then this becomes the new X, the output X. And then this X becomes the output Y. So really what this is doing, this is actually sort of doing. If you think of two layers back, this is actually the thing you mentioned before. It's actually doing the residual connection from two layers back like this Y came from the previous layer and was the residual connection there. But because of this construction, the whole thing is invertible. Why do I care? What does invertible matter for? The big thing that it can be interesting for is for training. If I think of a forward passive training. So I will. Let's say I have four layers. I run them in 0, 1, 2, 3 order. I have to write all of the activations to HBM. And so I get an HBM footprint here that is kind of like linear in number of layers. So this actually can be the largest memory footprint during training. And so this is normal training. And then I run the backwards pass and I read it kind of in reverse. Like I run them sort of forward pass goes forward, backward pass goes backwards and I have to read them back out. The idea of this Revenants paper is that because it's invertible, I don't need to store this at all. I can completely rematerialize it when I'm running my backwards pass. So I run my forwards pass and then when I'm running my backwards pass, I'm simultaneously in lockstep undoing all of the forwards pass steps that I did in order to have the activations that I need here. So this ends up being a memory saving, which is a nice idea.

[Lex Fridman]

Interesting. And in some sense, you're spending more compute to save memory.

[Rainer Pope]

That's right.

[Lex Fridman]

Yeah. Interesting, huh? Actually, it's kind of the opposite of what you're doing with the kvcache. The kvcache? You're spending more memory to save computer.

[Rainer Pope]

Yeah, spending more memory to save computers. Generally profitable, given where hardware is at. Yeah.

[Lex Fridman]

Interesting.

[Rainer Pope]

Cool.

[Lex Fridman]

That was super fun, right? Thank you so much for doing it. I feel like it really vindicated the vision behind the studio and the blackboard.

[Rainer Pope]

Cool.

[Lex Fridman]

Thanks so much for doing it.

[Rainer Pope]

Thanks.