2026-05-02 · Research

The Math Behind How LLMs Are Trained and Served

Axiom — Physical AI Research

What this is about: Reiner Pope, CEO of MatX and former Google TPU architect, gave a 2-hour blackboard lecture on Dwarkesh Patel's podcast that deduced the internal operations of AI labs from nothing more than public API prices, a handful of equations, and a roofline model. This post is a structured synthesis — for the researcher who wants the mechanism, not just the takeaway.

1. Who Is Reiner Pope and Why His Voice Matters

Reiner Pope is CEO of MatX, a semiconductor startup that raised $500 million in February 2026 to build processors specifically optimized for large language model operations. He was previously at Google, where he led AI software development for the Tensor Processing Unit (TPU) — Google's custom AI chip, now in its fifth generation.

He co-authored the JAX scaling book, one of the few rigorous treatments of ML training infrastructure at scale. He co-authored papers on TPU architecture and compiler efficiency. Mike Gunter, his co-founder at MatX, also came from Google's TPU team.

The combination is rare: Pope understands chip design, compiler theory, model architecture, and the economics of inference. On the podcast he walks through a blackboard derivation that lets you predict API prices, understand why MoE models are structured the way they are, and reason about future hardware trends — all from first principles.

As Dwarkesh says in the introduction: "There are less than a handful of people in the world who understand the full stack of AI, from chip design to model architecture, as well as Reiner."

2. The Core Framework: Roofline Analysis

The foundation of everything is a roofline model of how a transformer runs on a GPU cluster. It's simple but powerful. For any inference run, the time to produce a token is:

T = max(tcompute, tmemory)

Two constraints bind simultaneously. You can't go faster than your compute throughput, and you can't go faster than your memory bandwidth. The actual time is whichever is larger.

Each of those terms has a specific form. Compute time is:

tcompute = (B · Nactive) / FLOPs

Where B is batch size (how many parallel user requests you're serving), Nactive is the number of parameters that actually fire for each token, and FLOPs is the raw compute throughput of the hardware. For a dense model like GPT-4, Nactive equals the full parameter count. For a Mixture-of-Experts (MoE) model like DeepSeek V3, only a fraction of parameters are active per token.

Memory time is:

tmemory = (Ntotal + B · lenctx · KVbytes/token) / mem_bw

Two components here. First, you always have to load all the model weights from HBM (High Bandwidth Memory) into the compute units — Ntotal parameters, not just the active ones. This cost doesn't scale with batch size; it's a fixed tax per forward pass. Second, you have to load the KV cache for every element in the batch. The KV cache grows with context length and batch size, and this term does scale with B.

3. Why Batch Size Is the Fundamental Lever

Understanding those two equations gets you surprisingly far. Draw a graph of latency (time to produce one new token) vs. batch size:

The sum of these gives you the total memory time curve, and the maximum against compute time gives you the actual latency. The key insight:

There is a minimum latency floor that you cannot beat by reducing batch size. Even if you serve one user at a time (B=1), you still have to load all Ntotal parameters from HBM. That takes a fixed amount of time that no batching trick can reduce.

This is exactly why "Fast Mode" on Claude or Codex works. These services let you pay 6x the price for 2.5x the token generation speed. What you're actually buying is a larger batch allocation — the system schedules your request alongside fewer others, so your tokens don't wait in the queue behind a massive batch. The lower bound on latency is set by hardware physics, not by pricing policy.

Could you pay 100x more for proportionally faster speeds? Not indefinitely. Once your batch is small enough that compute is no longer the bottleneck — you're dominated by memory fetching the model weights — adding more money doesn't help. You've hit the memory wall. To go faster, you need different hardware.

4. The 300 FLOPs/Byte Ratio and Optimal Batching

Modern AI hardware has a characteristic ratio:

FLOPs / memory_bandwidth ≈ 300

On Blackwell NVL72 (Nvidia's current rack-scale system), you get roughly 20 terabytes/second of memory bandwidth and tens of petaflops of compute. The ratio is about 300. This number is load-bearing for everything that follows.

Setting compute time equal to memory time (the point where both resources are saturated) gives you the optimal batch size:

Boptimal = (FLOPs / mem_bw) · (Ntotal / Nactive) ≈ 300 · (1 / sparsity)

For a dense model where all parameters are active, optimal batch size is ~300. For a MoE model with 8-way sparsity (like DeepSeek V3, which has 37B active out of 700B total), you need at least 300 × 8 = 2,400 tokens in flight to saturate the compute. Sparsity lets you fit a massive model in memory while keeping hardware utilization high — you just need bigger batches.

This is why the API economics work the way they do. If you're serving a MoE model at low batch, you're leaving 99% of your FLOPs idle. You need to amortize the weight fetch cost over enough concurrent users to make the economics work. And that "enough" is measured in thousands.

5. Why MoE Lives Within One Rack

Mixture-of-Experts models route each token to a subset of experts (specialized feed-forward layers) via an all-to-all communication pattern. Any GPU's tokens may need to route to any other GPU's experts. On a single rack, NVLink connects every GPU to every other GPU at full bandwidth — 900 GB/s bidirectional. This is a perfect fit for all-to-all communication.

Cross-rack communication (scale-out) is roughly 8x slower. If your MoE experts span multiple racks, the all-to-all becomes a bottleneck. This is why MoE layer boundaries are drawn at the rack level — it's not a software convention, it's a hardware constraint baked into the physics of NVLink.

This also explains why large MoE models (DeepSeek V3, Mistral's MoE variants) are architected the way they are. The routing is designed to keep communication within a NUMA node or a single rack. Cross-rack expert communication would destroy the latency budget that the attention mechanism depends on.

6. Pipeline Bubbles and Why Ilya Killed Pipelining

Training a large model across multiple GPUs using pipeline parallelism (splitting model layers across stages) creates "bubble" inefficiencies. At the start of each batch, the GPUs hosting the last layers sit idle waiting for the first layers to finish. At the end of the batch, the GPUs hosting the first layers sit idle waiting for the last layers. In a d-layer pipeline with N devices, you can have up to (d-1)/d fraction of the compute idle at any moment.

Why not overlap multiple batches to fill the bubbles? Because you need to consolidate gradients and update the model weights before processing the next batch. That synchronization barrier is unavoidable. The bubbles are structural.

Reiner cites Ilya Sutskever: "As we now know, pipelining is not wise." The deeper reason is that pipelining forces architectural constraints on the model. When your residuals, attention patterns, and layer connections are split across pipeline stages, you can't easily iterate on architecture. Research velocity suffers — and in frontier AI labs, that's the greatest sin. You can always add more chips; you can't easily undo an architecture that slows down your research cycle.

7. The 100x Over-Training Hypothesis

Here's where it gets genuinely surprising. The Chinchilla scaling law (Hoffmann et al., 2022) said: train a model on roughly 20 tokens per parameter for compute-optimal results. A 70B model should need about 1.4T tokens of training data. That was the consensus.

Reiner argues that with RL (Reinforcement Learning) in the post-training phase, the picture changes. RL requires its own compute budget — you're generating rollouts, evaluating them, and updating the model. If you believe the optimal allocation has equal compute going to pre-training, RL, and inference serving, then with RL inefficiency factored in (decode-phase MFU is ~3x lower than prefill-phase), you can derive:

Dpretrain = DRL = Dinference

And since RL and inference each cost roughly as much as pre-training (when you include the inefficiency multipliers), the total training compute for an RL-trained model may be 6x the Chinchilla-optimal pre-training budget — or more, depending on how much RL you do.

Bessemer's robotics report and Capgemini's Physical AI report both flag that the key bottleneck is data. If RL indeed requires 100x more compute than Chinchilla-optimal pre-training, then the data required to train that compute is the actual constraint. This connects directly to the fleet learning problem — you need physical interaction data at scale to train RL policies for robots, and you're going to need enormous amounts of it.

8. Deduced from API Pricing: The Memory Cost of Long Context

One of the most striking demonstrations in the talk: you can reverse-engineer the KV cache cost from public API pricing. If a provider charges a premium for 128K context vs. 32K context, and you know the memory bandwidth cost, you can estimate the bytes-per-token overhead of the KV cache.

This works because the marginal cost of long context is dominated by KV cache memory bandwidth — each token you generate has to attend to the entire history, which means fetching the entire KV cache. At 128K tokens, that's a significant memory operation that doesn't scale with compute, only with memory bandwidth. Providers that don't price this premium into long-context requests are subsidizing long-context users with short-context users' hardware.

9. The Convergent Evolution of Neural Nets and Cryptography

The final section of the lecture draws a striking parallel. Cryptography evolved from ad-hoc confusion functions to rigorous mathematical structures (lattices, LWE) because side-channel attacks kept revealing weaknesses in informal systems. Neural networks are on a similar trajectory.

Early neural networks were hand-wavey constructions. The field is converging toward rigorous understanding: you can write down exact equations for training cost, inference latency, and hardware utilization. The math isn't approximate — it's exact enough to deduce what a lab is doing from public prices. This is the discipline maturing.

10. Why This Matters for Physical AI

This is the math layer that sits beneath VLA (Vision-Language-Action) training costs. When you reason about how much compute it takes to train a robot policy, you're ultimately doing the same accounting: FLOPs per token, memory bandwidth, optimal batch size, KV cache scaling with context length.

For robotics specifically, a few direct connections:

What this means for the experiment loop: Before running any physical experiment, you can and should estimate the compute required. The equations above are the right tool. If you're running policy gradient RL on Shrike Lite, you're not hitting the memory wall yet (the model is small). But as you scale to larger models, the batch size economics and KV cache costs will matter. And the broader point: the entire AI infrastructure stack is legible if you're willing to do the math. The labs' internal operations, the pricing on API menus, the chip roadmaps — it's all derivable from first principles. That's a useful skill for someone running physical experiments.

References and Further Reading

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