Writing an LLM from scratch, part 34b -- from bigrams to GPT-2, one component at a time (in JAX)

Giles Thomas completed building and training a GPT-2 small model from scratch using JAX, achieving a test loss of 3.418784, outperforming both his PyTorch model (3.538161) and the original GPT-2 small (3.499677). The training took 37 hours 15 minutes on an RTX 3090, and the model generated coherent text continuations.

This post is the capstone of the most long-running series on my blog https://www.gilesthomas.com/llm-from-scratch . In December 2024 , I started reading Sebastian Raschka https://sebastianraschka.com/ 's book " Build a Large Language Model from Scratch https://www.manning.com/books/build-a-large-language-model-from-scratch ", and worked through it carefully. Being who I am, despite trying to apply a strict "no side quests" policy, I found myself zooming off and digging into all kinds of things. It's time to wrap it up. I had decided that the endpoint would be to build and train an LLM from scratch just using my notes -- no reference to the book, no reference to the model code I'd written when following the book. After an X/Twitter https://x.com/gpjt/status/1985434030880293004 poll, I decided to use JAX for that, just to make sure that I really was building it from scratch and not regurgitating bits of PyTorch code like a bad coding LLM spitting out half-digested lumps of Stack Overflow. In my last post https://www.gilesthomas.com/2026/06/llm-from-scratch-34a-building-a-jax-training-loop-for-an-llm-training-run , I showed how I built a JAX training script that mirrored what I had built for the original PyTorch version of the model. To test it as I went along, I used it to train a really dumb "LLM", which instead of trying to predict the next token for every token in an input sequence, instead predicted the input -- that is, if you fed it The fat cat sat on the mat It would return the same thing. I called that an A-to-A model. In this post, I'll show you how I turned it into a GPT-2 model, and then trained it from scratch on my RTX 3090 using the parameter counts for the original paper's "small" size . What turned out really well with this is that I found a route that meant that almost every component I added made the model better That's not guaranteed -- sometimes different aspects of an AI model depend on each other, so adding A without also adding B makes things worse. But admittedly with a bit of backtracking in places I was able to find a route that shows a nice clear progression. The final training run took 37 hours 15 minutes -- compared to 40 hours, 38 minutes for an equivalent PyTorch model https://www.gilesthomas.com/2026/04/llm-from-scratch-32k-interventions-training-our-best-model-locally-gradient-accumulation . That is despite it being full-fat 32-bit -- the PyTorch one was using Automatic Mixed Precision AMP , which allowed it to use 16-bit calculations in places where it would be relatively harmless in terms of loss. When asked to continue "Every effort moves you", it came back with a decent response: Every effort moves you closer to your goals, but if you are unsure of what it takes, you don’t The model got 3.418784 loss on my held-back test dataset, as compared to my PyTorch model's 3.538161, and even more impressively, it was better than the original GPT-2 small's result of 3.499677 on the same dataset However, just as I found previously https://www.gilesthomas.com/2026/04/llm-from-scratch-32l-interventions-instruction-fine-tuning-tests , the OpenAI weights still beat mine consistently in instruction fine-tuning challenges. Let's get started. At the end of the last post, we had a solid training loop, using all of the tricks I'd picked up with my PyTorch code. The A-to-A model we were training with it looked like this: python from flax import nnx class GPTModel nnx.Module : def init self, vocab size, context length, d emb, n heads, n layers, qkv bias, drop rate, rngs, : self.token embedding = nnx.Embed num embeddings=vocab size, features=d emb, rngs=rngs, self.output head = nnx.Linear in features=d emb, out features=vocab size, use bias=False, rngs=rngs, def call self, xs : input embeddings = self.token embedding xs return self.output head input embeddings That was based on my preferred model of how LLMs work https://www.gilesthomas.com/2025/09/how-do-llms-work , where at the top level for a model, we feed in a sequence of token IDs, then: The A-to-A model basically skipped the second step completely: it would project to embedding space, then immediately project back to vocab space -- and after training, it was pretty good at mapping a sequence to itself. One interesting question is, if we train the same code, but this time try to get it to make next-token predictions, how good will it be at that? Obviously it can't be as good as a full LLM. But there are correlations between tokens; full stops will generally be followed by spaces, adjectives will normally be followed by other adjectives or nouns at least in English , and so on. It would be kind of like the predictive text systems on a phone, where at least until recently it would just use the last word you entered to generate a list of possible next words to select from. Old-school natural language processing has a name for this: bigrams. The idea is that you can work out statistically what the most common two-word pairs are, which allows you to make a guess at a next word from a single one. There are also trigrams, where you look at the last two words when predicting the next, then 4-grams, 5-grams, and so on. You'd build up a full probability table -- for every word in your vocab, you'd have the probability of every word coming next. So maybe even with that minimal model, we could get it to learn something similar to a set of token-level rather than word-level bigrams, which would then get the loss down. Obviously it wouldn't be as good as a full bigram table -- for our GPT-2 vocab size of 50,257, that would need parameters -- but perhaps it could approximate one. For comparison, the model we're using has just an embedding table and an output head, each mapping between 50,257 dimensions and 768, so that's parameters -- about 3% of the full table. An uninitialised model would hopefully have a loss of about 10.82, implying a perplexity https://www.gilesthomas.com/2025/10/llm-from-scratch-21-perplexed-by-perplexity equal to the vocab size. If we can train our dumb model to get better loss than that, then we'd have the beginnings of an LLM. That was a simple test to run. In my training code, I had a dataset class that looked like this: python class BigTrainDataset: def init self, all tokens, seq length, microbatch size : self.xs = all tokens :-1 .reshape -1, microbatch size, seq length self.ys = all tokens :-1 .reshape -1, microbatch size, seq length def getitem self, ix : return self.xs ix , self.ys ix def len self : return self.xs.shape 0 That is, the inputs, the xs , were the same as the targets, the ys . If we fed it The fat cat sat on the mat ...then we'd be training it to output exactly the same thing. The modified version for a real LLM would involve feeding it something like this: The fat cat sat on the ...and targeting this: fat cat sat on the mat That's a simple change -- that init method became this: self.xs = all tokens :-1 .reshape -1, microbatch size, seq length self.ys = all tokens 1: .reshape -1, microbatch size, seq length I did that, and kicked it off to train on the 92,209,152 tokens that I was somewhat arbitrarily using in the last post to test my training loop. The loss chart looked like this: That was pretty promising Loss came down from roughly 10.82 down to a fairly stable 6 or so by global step 768, and seemed to flatten out there. It's possible that further training could have got it down a bit more, but I decided again, somewhat arbitrarily to use the average train loss in the checkpoint period ending at step 937 as my starting point. If we could make changes that reduced that, then we'd be moving forward. For this model, that value was 5.909. So, what were the changes we needed to make to change our bigram-style model to a real, if small, LLM? Adapting from my how LLMs work https://www.gilesthomas.com/2025/09/how-do-llms-work post, a GPT-2-style LLM looks like this. We receive our sequence of token IDs, and then: Inside the Transformers blocks, we: So that gave me the checklist; looking at it, the most tempting next step was layer normalisation henceforth LayerNorm . It's used at the end of the core loop, and then twice in the Transformers blocks. What would happen if we coded it up, and then added it to the core only? The purpose of LayerNorm https://www.gilesthomas.com/2025/07/llm-from-scratch-16-layer-normalisation is to stabilise training. We constrain the values flowing through our model so that they have certain statistical properties that tend to make the whole thing more trainable. That would mean that if it did help with this model -- placed in between the embedding layer at the start, and the output head at the end -- then we'd hope for loss to go down faster, and ideally finish at a lower level. Time to code it up NNX has its own LayerNorm implementation https://flax.readthedocs.io/en/stable/api reference/flax.nnx/nn/normalization.html flax.nnx.LayerNorm , of course as does PyTorch https://docs.pytorch.org/docs/2.12/generated/torch.nn.LayerNorm.html , but in the book, we implement it ourselves, and that felt like the correct path to take. Firstly, I implemented a dummy version: python class LayerNorm nnx.Module : def init self : ... def call self, xs : return xs ...and updated the core GPTModel to create and call one: python class GPTModel nnx.Module : def init ... : self.token embedding = nnx.Embed ... self.output norm = LayerNorm self.output head = nnx.Linear ... def call self, xs : input embeddings = self.token embedding xs normalised = self.output norm input embeddings return self.output head normalised And kicked off a training run for a few seconds just to make sure that it hadn't broken anything and that loss dropped -- being my first NNX module-inside-a-module, I worried that there might have been something non-intuitive that I had to do to get it to work. But everything seemed good -- loss was dropping, no errors. So, following the notes I made when I first learned about LayerNorm https://www.gilesthomas.com/2025/07/llm-from-scratch-16-layer-normalisation , I needed to make the values flowing through centred around zero by subtracting their mean, and then scale them to have a variance of one by dividing by the standard deviation details in those notes . The shape of the xs I had coming into my LayerNorm class's call was this: xs.shape= 6, 1024, 768 That was batch size, seq len, d emb . So we needed to do those operations strictly on the last axis, manipulating each embedding independently. JAX has a std function https://docs.jax.dev/en/latest/ autosummary/jax.numpy.std.html and a mean one axis parameter. The Array object repackaged those as methods, which was convenient, so I did a first cut test like this: python class LayerNorm nnx.Module : def init self : ... def call self, xs : jax.debug.print f"{xs.shape=}" means = xs.mean axis=-1 jax.debug.print f"{means.shape=}" stds = xs.std axis=-1 jax.debug.print f"{stds.shape=}" return xs That printed out these results: xs.shape= 6, 1024, 768 means.shape= 6, 1024 stds.shape= 6, 1024 ...which looked plausible; one number for each embedding vector. Could we broadcast them across the array? python class LayerNorm nnx.Module : def init self : ... def call self, xs : jax.debug.print f"{xs.shape=}" means = xs.mean axis=-1 jax.debug.print f"{means.shape=}" stds = xs.std axis=-1 jax.debug.print f"{stds.shape=}" normalized = xs - means / stds jax.debug.print f"{normalized.shape=}" return normalized This blew up: ValueError: Incompatible shapes for broadcasting: shapes= 6, 1024, 768 , 6, 1024 Fair enough. But mean and std have a keepdims kwarg that looked like it would help: python class LayerNorm nnx.Module : def init self : ... def call self, xs : jax.debug.print f"{xs.shape=}" means = xs.mean axis=-1, keepdims=True jax.debug.print f"{means.shape=}" stds = xs.std axis=-1, keepdims=True jax.debug.print f"{stds.shape=}" normalized = xs - means / stds jax.debug.print f"{normalized.shape=}" return normalized ...and it did xs.shape= 6, 1024, 768 means.shape= 6, 1024, 1 stds.shape= 6, 1024, 1 normalized.shape= 6, 1024, 768 Excellent. So the next step was to see if that would work even slightly. Interestingly loss started off a bit higher at 11.29 after the first global step -- so adding in the LayerNorm had actually made the model worse than it was -- but it seemed to be falling rapidly. Things weren't totally broken, at least. But there was more to LayerNorm than just zeroing the mean and scaling to the variance; we also needed to scale them up by a learnable amount, and then shift/bias them by adding on a different trainable amount. More precisely, both of those trainable amounts were different for each of the in this case 768 embedding dimensions. We needed two learnable vectors of length d emb . I hadn't noted it down at the time but I figured as it turned out, correctly that a sensible starting point for those values would be all-zero for the bias, and all-one for the scale. From this help page https://flax.readthedocs.io/en/stable/key concepts.html traced-vs-static-data , the way you create a trainable array associated with an NNX module is this: nnx.Param jax.random.normal rngs.param , dim, dim That code created a random vector, rather than the zeros/ones we needed, and we'd need to get the dimensions right. Because of the "Incompatible shapes for broadcasting" error I'd just had, I was feeling a bit paranoid about the latter, so I chose a shape of 1, 1, d emb , and wrote this: python class LayerNorm nnx.Module : def init self, d emb : self.scale = nnx.Param jnp.ones 1, 1, d emb self.bias = nnx.Param jnp.zeros 1, 1, d emb def call self, xs : jax.debug.print f"{xs.shape=}" means = xs.mean axis=-1, keepdims=True jax.debug.print f"{means.shape=}" stds = xs.std axis=-1, keepdims=True jax.debug.print f"{stds.shape=}" normalized = xs - means / stds jax.debug.print f"{normalized.shape=}" scaled and biased = normalized self.scale + self.bias return scaled and biased That looked pretty plausible, though in retrospect I think I was being overly cautious and didn't need the leading two axes for the scale and bias. The only thing I was unsure about was whether the nnx.Param wrappers I had put in were really making those arrays trainable. I put some code in to print them out and kicked off a run for a few minutes, and confirmed that they were changing in ways that seem plausible -- small non-zero bias, scale close to but not equal to one. That was all good Next, I spotted one issue. What if one of the standard deviations was zero? That would lead to a divide-by-zero error here: normalized = xs - means / stds Now, the standard deviation, if it's not zero, has to be positive -- so adding on a small value would fix that 1: normalized = xs - means / stds + 1e-5 With that in place, I felt that it was ready to go. Time to do a full training run I kicked that off, and it completed with this output: 2026-06-20 19:08:17.189721 Tokens seen: 92,209,152 2026-06-20 19:08:17.189724 Throughput: 95,383 tokens/second 2026-06-20 19:08:17.189734 Final train loss: 5.736 2026-06-20 19:08:17.189737 Done Loss looked like this: Let's look at the results for the previous run without LayerNorm for comparison: You can see that the new run, the first one, drops faster. It's harder to see from the chart, but it also finished up with a lower training loss at 937 my relatively arbitrary metric : 5.734 rather than 5.909. That was interesting The new model was basically doing the same thing -- predicting the next token based only on the "current" token, but loss was lower. My take is that if we had trained the non-LayerNorm model for longer, it might have managed to eventually grind out a better loss. But LayerNorm was doing its job -- it was stabilising training, and as a result we converged faster. That was a win I decided to run it through my old smoke test from the PyTorch training runs, and see how it completed "Every effort moves you": Every effort moves you can be a few years. -year-year-year-year-year-year- It was kind of impressive that it managed to finish the first line before it got stuck in a loop -- but it was understandable that we couldn't expect anything good yet. Each predicted token was based entirely on the token before it. What next? Back to our checklist: Inside the Transformers blocks, we: So, at this stage, for each input token we were predicting the next one based on the input token only -- like I said earlier, we were doing a somewhat roundabout way of building an approximation of a table of bigram probabilities. What would happen if we started paying attention to the tokens to the left? And what would be the simplest, dumbest way to do that? The real LLM has multiple layers of multi-head attention, each one also having a feed-forward network, some LayerNorms, and some shortcut connections. Single-head attention is easier to code, but even on its own, you'd expect it to be able to add some value. Each token would get at least some information from the ones to the left. And one layer, likewise, you'd expect might help a bit. I suspected that it wouldn't work on its own -- I expected I'd need shortcut connections too -- but decided to start with attention on its own. I modified the main class to have a single "Transformers" layer: python class GPTModel nnx.Module : def init ... : self.token embedding = nnx.Embed ... self.transformers layer = TransformersLayer d emb, qkv bias, rngs self.output norm = LayerNorm d emb self.output head = nnx.Linear ... def call self, xs : input embeddings = self.token embedding xs transformed = self.transformers layer input embeddings normalized = self.output norm transformed return self.output head normalized ...where that layer was actually just single-head attention: python class TransformersLayer nnx.Module : def init self, d emb, qkv bias, rngs : self.attention = Attention d emb, qkv bias, rngs def call self, xs : return self.attention xs Next, it was time for the Attention class. I'm not going to write yet another attention explainer -- I think my "How do LLMs work?" https://www.gilesthomas.com/2025/09/how-do-llms-work one does a decent job of that, and "The 'why' of attention, or: attention heads are dumb" https://www.gilesthomas.com/2025/05/llm-from-scratch-13-taking-stock-part-1-attention-heads-are-dumb works well too. So in the next bit I'll assume that you understand the basics. My first cut was basically just the maths up to the causal mask to get the attention scores: python class Attention nnx.Module : def init self, d emb, qkv bias, rngs : self.d emb = d emb self.W q = nnx.Linear d emb, d emb, use bias=qkv bias, rngs=rngs self.W k = nnx.Linear d emb, d emb, use bias=qkv bias, rngs=rngs self.W v = nnx.Linear d emb, d emb, use bias=qkv bias, rngs=rngs def call self, xs : Q = self.W q xs K = self.W k xs V = self.W v xs omega = Q @ K.T omega /= jnp.sqrt self.d emb causal omega = jnp.tril omega It did the projections into query, key and value space, worked out the attention scores with the array multiplication, normalised it by dividing by the square root of the number of dimensions in the Q-K embedding space, and then zeroed out the scores where a token was attending to tokens in its "future". There were a couple of problems, though. Firstly, that wouldn't work if we were working with batches, and secondly, zeroing out the non-causal scores wasn't quite correct. The batches first. Our incoming xs here would have the shape batch length, seq len, d emb . After the projections to the Q-K embedding space, both Q and K would also be shaped batch length, seq len, d emb . Now, the .T property on the JAX array class just reverses the axes, so the code above would give us K.T with the shape d emb, seq len, batch length . That would break Matrix multiplication in JAX expects all but the last two axes to represent batches, so we actually wanted K.T to have the shape batch length, d emb, seq len . That meant that what we actually wanted was to just transpose the last two axes. The JAX transpose function https://docs.jax.dev/en/latest/ autosummary/jax.numpy.transpose.html takes an axes parameter that allows you to specify the specific re-ordering of the input axes that you want. So I could rewrite the code like this: python def call self, xs : Q = self.W q xs K = self.W k xs V = self.W v xs omega = Q @ jnp.transpose K, axes= 0, 2, 1 omega /= jnp.sqrt self.d emb causal omega = jnp.tril omega As Q would have the shape batch length, seq len, d emb , and the transposed version of K would be batch length, d emb, seq len , they'd be compatible for matrix multiplication and give us a result that was batch length, seq len, seq len -- just what we wanted for attention scores. The next step was to fix the causal mask. The next step in this attention mechanism was going to be running the causal attention scores in causal omega through softmax over the last dimension, to convert them into attention weights. Now, our current code was zeroing out unwanted acausal scores, but a zero still contributes to softmax. If you want a particular value to come out of softmax guaranteed to be zero, you need to set it to minus infinity. I decided that the easiest way to do this was to create a causal mask -- a boolean array that matched the size of omega , but was full of True s: causal mask = jnp.ones like omega, dtype=bool Then I could zero out well, "false out" the cells in the mask related to unwanted future-facing scores, just like I was previously doing on the scores: causal mask = jnp.tril causal mask ...and then I could apply that mask to omega with jnp.where https://docs.jax.dev/en/latest/ autosummary/jax.numpy.where.html , telling it to create a new array, taking the value from omega where the mask had True , and -jnp.inf in places where it had False . causal omega = jnp.where causal mask, omega, -jnp.inf That seemed solid, so I just needed to run the result through jax.nn.softmax https://docs.jax.dev/en/latest/ autosummary/jax.nn.softmax.html , specifying that the last dimension was the one where it should apply the function, and that would give me the attention weights: attention weights = jax.nn.softmax causal omega, axis=-1 Finally, I just needed to use those attention weights to get the attention output by mixing in appropriate portions of the projection of the inputs into value space, V : return attention weights @ V As attention weights was shaped batch length, seq len, seq len , and V like Q and K was shaped batch length, seq len, d emb , the batch axes were at the start where they belonged, and the matrix multiplication would work and return something shaped batch length, seq len, d emb . With that, we were done The final single-head attention class looked like this: python class Attention nnx.Module : def init self, d emb, qkv bias, rngs : self.d emb = d emb self.W q = nnx.Linear d emb, d emb, use bias=qkv bias, rngs=rngs self.W k = nnx.Linear d emb, d emb, use bias=qkv bias, rngs=rngs self.W v = nnx.Linear d emb, d emb, use bias=qkv bias, rngs=rngs def call self, xs : Q = self.W q xs K = self.W k xs V = self.W v xs omega = Q @ jnp.transpose K, axes= 0, 2, 1 omega /= jnp.sqrt self.d emb causal mask = jnp.ones like omega, dtype=bool causal mask = jnp.tril causal mask causal omega = jnp.where causal mask, omega, -jnp.inf attention weights = jax.nn.softmax causal omega, axis=-1 return attention weights @ V I kicked off a training run with that, and it did work, in that loss went down over the course of the run -- but at the end of the run, the loss at step 937 was 5.934 -- significantly above the 5.734 I got on the previous run, with no attention. But that made sense As I'd said earlier, I suspected that this wouldn't help if we had no shortcut connection. Intuitively, if you want to work out what token should be at position , on average the most important other token you need to know about is probably whichever one is at position . Knowing about the tokens at , , and so on, could well be helpful -- maybe very helpful -- but not at the cost of not knowing about the one at . Now, single attention heads are just simple pattern-matchers. They can't learn complex rules, it's only by working together -- "horizontally", in multi-head attention or "vertically" across multiple layers -- that they can do complex things. What we were asking this head to do was to learn some way of gathering information about previous tokens, and also to keep the knowledge about the "current" one. That's a tall order for a dumb attention head In my mind, this is a large part of the benefit of shortcut connections. They are often presented as a way to make sure that during training, gradients flow smoothly from the output end of the model to the earlier layers. But I prefer to think of them as preserving the original embeddings, so that each layer doesn't completely replace what came into it, but instead does something closer to adding on its own notes -- like scholars adding commentary to a core text in the Talmud https://www.gilesthomas.com/2025/08/llm-from-scratch-18-residuals-shortcut-connections-and-the-talmud . In the training run above, the attention head was trying to learn how to preserve the meaning of the embedding it was working on, while also merging in information from earlier ones. If we added a shortcut connection, then it would only have to do the second of those two jobs. The code was simple: I updated the TransformersLayer module to do a shortcut connection: python class TransformersLayer nnx.Module : def init self, d emb, qkv bias, rngs : self.attention = Attention d emb, qkv bias, rngs def call self, xs : shortcut = xs att = self.attention xs return shortcut + att I kicked off a training run, and at the end it printed this: 2026-06-23 03:51:18.086097 Tokens seen: 92,209,152 2026-06-23 03:51:18.086099 Throughput: 90,442 tokens/second 2026-06-23 03:51:18.086108 Final train loss: 5.570 2026-06-23 03:51:18.086121 Done The loss chart looked like this: And, importantly, that training loss at step 937 which I was using as a metric was 5.553 -- a decent improvement over the previous best of 5.734. Even a dumb single attention head was able to do something useful, if it had a shortcut connection. I decided to run another qualitative smoke test: Every effort moves you can be able to get to get to get to get to get a lot of the way to get I mean, it was repetitive, but it was actually getting noticeably closer to making sense So that was excellent news. What next? Our checklist looked like this: Inside the Transformers blocks, we: Now, our single attention layer was lacking something. Without position embeddings, that layer has no idea what order the tokens before the one it's looking at come in. If it's considering the " cat" in The fat cat ...it doesn't know if it's looking at "The fat cat" or "fat The cat". Position embeddings are simple, and might help, so that was the next step. These were trivial to add. We had this core code: python class GPTModel nnx.Module : def init ... : self.token embedding = nnx.Embed ... self.transformers layer = TransformersLayer d emb, qkv bias, rngs self.output norm = LayerNorm d emb self.output head = nnx.Linear ... def call self, xs : input embeddings = self.token embedding xs transformed = self.transformers layer input embeddings normalized = self.output norm transformed return self.output head normalized So I just added a position encoding module in init : self.position embedding = nnx.Embed num embeddings=context length, features=d emb, rngs=rngs, ...and mixed it in with the token embeddings to create new, improved input embeddings to be used in our "Transformers" layer: token embeddings = self.token embedding xs b, n = xs.shape position embeddings = self.position embedding jnp.arange n input embeddings = token embeddings + position embeddings I kicked off a training run with that: 2026-06-23 04:44:44.759768 Tokens seen: 92,209,152 2026-06-23 04:44:44.759771 Throughput: 88,618 tokens/second 2026-06-23 04:44:44.759779 Final train loss: 5.386 2026-06-23 04:44:44.759781 Done Pretty hard to distinguish from the previous one, but the metric I was tracking, that loss at step 937, had improved again We were down to 5.354 from 5.553 :- A quick qualitative smoke test didn't show that improvement, though: Every effort moves you can be able to get to get to get back to get back to get back to get back to Pretty much indistinguishable to the previous one. But still, Loss Number Went Down, and that's what was important at this stage. It was time to try the next step. From the checklist: Inside the Transformers blocks, we: We had only one attention head right now. Individually, attention heads are dumb https://www.gilesthomas.com/2025/05/llm-from-scratch-13-taking-stock-part-1-attention-heads-are-dumb , so switching to multi-head attention seemed like a good thread to pull. At this point, my single-head attention code looked like this: Q = self.W q xs K = self.W k xs V = self.W v xs omega = Q @ jnp.transpose K, axes= 0, 2, 1 omega /= jnp.sqrt self.d emb causal mask = jnp.ones like omega, dtype=bool causal mask = jnp.tril causal mask causal omega = jnp.where causal mask, omega, -jnp.inf attention weights = jax.nn.softmax causal omega, axis=-1 return attention weights @ V I decided to re-implement multi-head attention which I'll call MHA from here onwards from first principles rather than working strictly from my notes, and then to come back and check it. If you're looking at your browser's scrollbar with horror " stillonly 50%? " and really don't want to read a full derivation of MHA, you can skip straight to the first complete version of the code . The point of MHA is that we're running multiple copies of the calculation above in parallel -- let's pin down the name of the number of copies as n heads . Now, we could naively implement it just by spinning off n heads threads and running the existing code in each, but that wouldn't really take advantage of the GPU's inherent parallelism. I felt that we could rely on the fact that JAX's matrix multiplications treat all but the last two dimensions as "batches". For example, if you have two arrays with shapes: a, b, c, ..., l, m, n and a, b, c, ..., l, n, p ...then you can multiply them. A matrix multiplied by a one will be , so you'll get something that is a, b, c, ..., l, m, p The other dimensions so long as they match will essentially act as an batch. Now, right now we were just using a single batch dimension. Let's look at the core multiplication in the attention mechanism, which works out omega , the attention scores. I had this: omega = Q @ jnp.transpose K, axes= 0, 2, 1 Breaking that apart into two steps: K transpose = jnp.transpose K, axes= 0, 2, 1 omega = Q @ K transpose We got K from this line: K = self.W k xs Let's look at the shapes here. xs is our input embeddings for this layer; its shape is batch size, seq len, d emb . Projecting it through W k , which is shaped d emb, d emb gives us a shape for K of batch size, seq len, d emb again. Q , being a projection of xs through W q , which is the same shape as W k , will have the same shape as K . Now, that means that K transpose is batch size, d emb, seq len , and the calculation omega = Q @ K transpose ...is doing a batched matrix multiplication getting us the omega that we want, shaped batch size, seq len, seq len . But as I said above, there's no need to stop with just one batch dimension. Let's say that we have n heads heads, and that they each work with embeddings sized d head . Imagine that we've already somehow done multiple projections into the key and query spaces for each of our n heads heads, and that the results have somehow been put into arrays such that Q and K are shaped batch size, n heads, seq len, d head -- that is, we've gained an extra axis that keeps the projections for each head into its query-key space separate. We could use the fact that both of those two leading axes are basically just batch dimensions, and the existing single matrix multiplication will still work, with one tiny tweak: the current transpose is this: K transpose = jnp.transpose K, axes= 0, 2, 1 omega = Q @ K transpose ...to swap around the last two axes of a three-axis array. With one extra batch dimension, we'll need to take account of that and do this instead: K transpose = jnp.transpose K, axes= 0, 1, 3, 2 omega = Q @ K transpose That will be a multiplication of Q , shaped batch size, n heads, seq len, d head , with K transpose , shaped batch size, n heads, d head, seq len , which gives us an omega of the right shape, batch size, n heads, seq len, seq len . So, if we can start treating the heads as just another batch dimension, things seem simpler, at least for the attention score calculation. Let's continue down through the single-head code, and then come back later to how we might get the inputs into that double-batched shape. The next line after the omega calculation just scales the attention scores by a scalar: omega /= jnp.sqrt self.d emb That looked fine, just a broadcast division-by-float. We'd need to change that self.d emb to be d head in some manner, but that's all. Next: causal mask = jnp.ones like omega, dtype=bool The jnp.ones like will give us an array that's batch size, n heads, seq len, seq len full of True s. That seems reasonable. The next step: causal mask = jnp.tril causal mask What will that do? Well, per the tril documentation https://docs.jax.dev/en/latest/ autosummary/jax.numpy.tril.html jax.numpy.tril : When m.ndim 2 , jnp.tril operates batch-wise on the trailing axes. ...which sounded good. batch size and n heads would be treated as batch axes, which meant that the next line: causal omega = jnp.where causal mask, omega, -jnp.inf ...would work. Likewise, with the next line: attention weights = jax.nn.softmax causal omega, axis=-1 ...the axis to apply softmax to is explicitly stated as the last one, which is what we wanted. So at the end of all of those steps, we'd have attention weights shaped batch size, n heads, seq len, seq len , where the last axis had been softmaxed softmaxxed? . The next line looked a little trickier: return attention weights @ V In the single-head version we had attention weights of shape batch size, seq len, seq len , and V of shape batch size, seq len, d emb , so multiplying them gives us batch size, seq len, d emb In the new MHA code so far, we had our attention weights shaped batch size, n heads, seq len, seq len . So in order for the matrix multiplication to work, we'd need V to be shaped batch size, n heads, seq len, d head . That would give us a result shaped as batch size, n heads, seq len, d head . And conveniently, we'd already decided that the correct shape for Q and for K was batch size, n heads, seq len, d head . If we could use the same "magic" to do the projection into value space -- that is, to get V such that the heads formed a new batch-like axis like we had for Q and K -- then we'd be all set. So, at that point, I'd worked out the core of MHA. If we could get all of the inputs into the shape batch size, n heads, seq len, d head , and somehow handle an output of the shape batch size, n heads, seq len, d head , then we could use MHA code something like this: Q and K are batch size, n heads, len sequence, d head We need to convert K to batch size, n heads, d head, len sequence and then we get omega batch size, n heads, len sequence, len sequence omega = Q @ jnp.transpose K, axes= 0, 1, 3, 2 omega /= jnp.sqrt self.d head causal mask = jnp.ones like omega, dtype=bool tril treats all but the last two axes as batches so we're OK here. causal mask = jnp.tril causal mask causal omega = jnp.where causal mask, omega, -jnp.inf last axis is still OK. attention weights = jax.nn.softmax causal omega, axis=-1 attention weights is batch size, n heads, len sequence, len sequence V is batch size, n heads, len sequence, d head So this will come out as batch size, n heads, len sequence, d head weighted = attention weights @ V The next question was, how do we get our inputs into that shape? We could run them all through separate per-head weights -- that is, have an array with one per head, like W q 0 , W k 0 and W v 0 for the first one. But that, again, felt like it would be failing to take advantage of the GPU properly. The solution was to think of how matrix multiplications work. If you multiply two matrices, , the value in the result, in row , and column , is the dot product of row in and column in . So, imagine if you wanted to multiply by different versions of , let's call them , , and so on up to . If you imagine a new matrix, , which is basically all the s stacked side-by-side, then the dot-product understanding of multiplication makes it pretty clear that if you did , you would get the results of all of those separate multiplications, also stacked side-by-side. I'll call that kind of matrix a "striped" one, for want of a better word. Now, when we project our inputs into the embedding spaces used for attention, we have code like this: Q = self.W q xs We've initialised the weights, W k in this case, as an nnx.Linear , so what is happening under the hood here is basically: That is, it is just a matrix multiplication. 2 https://www.gilesthomas.com/feed/rss.xml fn-2 So if we imagine that W q is one of those "striped" matrices, holding all of the separate matrices to do the projections for all of the heads in a single one shaped d emb, n heads d head , then we could stick with the current code -- the Q = self.W q xs Our input xs would be shaped batch size, seq len, d emb , so the result would be batch size, seq len, n heads d head , and would have the projections for each head in the same vertical stripes as the separate heads' projection weights. Now, like PyTorch, JAX allows you to reshape https://docs.jax.dev/en/latest/ autosummary/jax.numpy.reshape.html arrays. You can take one axis of length say , and split it into two of lengths and respectively -- or, conversely, you can combine two axes of length and to one of . If our data had the shape batch size, seq len, n heads d head , we could reshape it like this: Q = self.W q xs .reshape batch size, seq len, n heads, d head ...and that would split things up. So we'd have Q shaped as batch size, seq len, n heads, d head . That's almost what we wanted We needed batch size, n heads, seq len, d head , and a simple transpose could sort that out: Q = jnp.transpose self.W q xs .reshape batch size, seq len, n heads, self.d head , 0, 2, 1, 3 Likewise for K and V , and that was our inputs sorted. Moving on to the output; it came from this: weighted = attention weights @ V ...and as we worked out above, it was shaped batch size, n heads, seq len, d head . I remembered that we wanted to run that through a single linear layer to combine all of the different heads' outputs into one. It felt like the best way to do that would be to get it back into a "striped" layout: batch size, seq len, n heads d head . This would be something like the inverse of the input-wrangling. That would need a reshape, but before I could do that, I'd need to get the axes that needed to be merged next to each other. If the input to the linear layer was going to be batch size, seq len, n heads d head , we'd need to convert it from batch size, n heads, seq len, d head to batch size, seq len, n heads, d head first: jnp.transpose weighted, 0, 2, 1, 3 ... and then we could just reshape it to batch size, len sequence, n heads d head : striped output = jnp.transpose weighted, 0, 2, 1, 3 .reshape batch size, len sequence, self.n heads self.d head Finally, we could run it through a linear layer, with in features set to n heads d head , and out features set to d emb . I put that all together, and decided to throw something extra into the mix. I remembered that Raschka's code had various checks to make sure that d head n heads == d emb , which seemed a little artificial -- I'd read that this was true of GPT-2, but wasn't a necessary restriction for GPT-style models, which makes sense. There's no obvious reason per se why the heads' embedding dimensions should sum up to the higher-level embedding dimensions. So I decided initially to just pass in d head and n heads to the constructor. In my training script I could force them to match the GPT-2 model, but if I wanted to use the code later for something different, I could vary them. Then I remembered that although the dimensionality of the embedding spaces for the query and the key vectors have to match because otherwise you can't multiply them to work out attention scores with , the value vector's dimensionality can in theory be different. So I decided to break d head into two separate d qk and d v parameters. python class MultiHeadAttention nnx.Module : def init self, d emb, n heads, d qk, d v, qkv bias, rngs : self.n heads = n heads self.d qk = d qk self.d v = d v self.W q = nnx.Linear d emb, self.d qk n heads, use bias=qkv bias, rngs=rngs self.W k = nnx.Linear d emb, self.d qk n heads, use bias=qkv bias, rngs=rngs self.W v = nnx.Linear d emb, self.d v n heads, use bias=qkv bias, rngs=rngs self.output projection = nnx.Linear self.d v n heads, d emb, use bias=False, rngs=rngs def call self, xs : batch size, len sequence, d emb = xs.shape For each of the below: The initial linear layer projects them to batch size, len sequence, d X n heads where X is qk or v as appropriate. The reshape makes them batch size, len sequence, n heads, d X The transpose makes them batch size, n heads, len sequence, d X Q = jnp.transpose self.W q xs .reshape batch size, len sequence, self.n heads, self.d qk , 0, 2, 1, 3 K = jnp.transpose self.W k xs .reshape batch size, len sequence, self.n heads, self.d qk , 0, 2, 1, 3 V = jnp.transpose self.W v xs .reshape batch size, len sequence, self.n heads, self.d v , 0, 2, 1, 3 Q and K are batch size, n heads, len sequence, d qk per above We need to convert K to batch size, n heads, d qk, len sequence and then we get omega batch size, n heads, len sequence, len sequence omega = Q @ jnp.transpose K, axes= 0, 1, 3, 2 omega /= jnp.sqrt self.d qk causal mask = jnp.ones like omega, dtype=bool tril treats all but the last two axes as batches so we're OK here. causal mask = jnp.tril causal mask causal omega = jnp.where causal mask, omega, -jnp.inf last axis is still OK. attention weights = jax.nn.softmax causal omega, axis=-1 attention weights is batch size, n heads, len sequence, len sequence V is batch size, n heads, len sequence, d v So this will come out as batch size, n heads, len sequence, d v weighted = attention weights @ V Transpose to batch size, len sequence, n heads, d v , then reshape to batch size, len sequence, n heads d v striped output = jnp.transpose weighted, 0, 2, 1, 3 .reshape batch size, len sequence, self.n heads self.d v Final linear layer to combine return self.output projection striped output Unusually for a case where I went off the reservation like this, the whole thing with the embedding space dimensionality didn't cause any problems at all But there was one small bug in this code, which I didn't discover until later -- we'll come to it by the end of the post. At this point, I did another of my short training runs, and: 2026-06-23 17:51:32.094308 Tokens seen: 92,209,152 2026-06-23 17:51:32.094311 Throughput: 85,682 tokens/second 2026-06-23 17:51:32.094321 Final train loss: 5.358 2026-06-23 17:51:32.094323 Done ...with a loss chart that looked like this: The training loss at the 937th global step was 5.336, only a tiny bit better than the 5.354 with single-head attention. That was quite possibly within the noise. Even though due to the d head n heads == d emb restriction I was enforcing in my training script the W q , W k , and W v arrays were the same size, I was creating that output projection , which would consume randomness and make things vary. If I were doing a proper scientific experiment to see if a single layer of MHA beat a single layer of single-head attention, I think I would have run both for more steps to see if the difference became more pronounced later. But for the purposes of this post, I decided to move on. My checklist now looked like this: Inside the Transformers blocks, we: Adding that simple neural network -- the FFN -- seemed like a good next step. The feed forward network https://www.gilesthomas.com/2025/08/llm-from-scratch-17-the-feed-forward-network is simple; you take the output of the MHA block, run it through a biased linear layer to expand it from d emb to 4 d emb , then run it through the GELU activation function, then shrink it back down to d emb with another linear layer. I didn't really see any value in writing my own implementation of GELU, given that even in the book we were just given code for an approximation to type in. So, using jax.nn.gelu https://docs.jax.dev/en/latest/ autosummary/jax.nn.gelu.html , I wrote this: python class TransformersLayer nnx.Module : def init self, d emb, n heads, d qk, d v, qkv bias, rngs : self.attention = MultiHeadAttention d emb, n heads, d qk, d v, qkv bias, rngs self.ffn = nnx.Sequential nnx.Linear in features=d emb, out features=d emb 4, use bias=True, rngs=rngs , jax.nn.gelu, nnx.Linear in features=d emb 4, out features=d emb, use bias=True, rngs=rngs , def call self, xs : shortcut = xs att = self.attention xs post attention = shortcut + att fed forward = self.ffn post attention return fed forward + post attention Note that I added in a shortcut connection around the FFN as well, so that it didn't overwrite what was there, but only "added on its notes". I kicked that off, and it ran for ten minutes or so, but then OOMed: 2026-06-23 18:20:01.376377 Saving checkpoint 51%|██████████████████████████████████████████████████████▊ | 481/938 10:20<11:47, 1.55s/it, loss=5.758, tps=76,185 W0623 18:20:12.602631 2860192 bfc allocator.cc:514 Allocator GPU 0 bfc ran out of memory trying to allocate 2.93GiB rounded to 3149744640 requested by op If the cause is memory fragmentation maybe the environment variable 'TF GPU ALLOCATOR=cuda malloc async' will improve the situation. Adding TF GPU ALLOCATOR=cuda malloc async didn't help. I spent some time trying to dig into what might be causing it, but eventually noticed something interesting: in nvtop , the VRAM usage was consistently 75% throughout. Now I knew that JAX pre-allocates 75% of VRAM when it starts up, but I'd been assuming that it would try to grab more if it needed it. It turned out I was wrong with that assumption -- it grabs 75%, but that's all you ever get The solution turned out to be the XLA PYTHON CLIENT MEM FRACTION https://docs.jax.dev/en/latest/gpu memory allocation.html environment variable. If you set that to, say, 0.90 , then JAX will pre-allocate 90% of the VRAM, and you can use all of that. You can also make it allocate as-needed with XLA PYTHON CLIENT PREALLOCATE=false , and there are various other settings you can control with other environment variables on that linked page .Anyway, setting it to 0.90 to grab 90% of VRAM worked, and I was able to get a successful run: 2026-06-24 00:29:34.864880 Tokens seen: 92,209,152 2026-06-24 00:29:34.864882 Throughput: 77,596 tokens/second 2026-06-24 00:29:34.864900 Final train loss: 5.341 2026-06-24 00:29:34.864902 Done The loss chart was this: ...and the training loss at global step 937 was 5.295, compared to the 5.336 from MHA alone. Another tiny improvement, another one that could have been in the noise. Again, if I were doing a proper experiment, I'd do a longer run, but for now, I decided to move on. The checklist looked like this: Inside the Transformers blocks, we: Now, my gut instinct was that the layer normalisation inside the Transformers blocks was of most value as a way of stabilising training over deep networks. And with one layer, it didn't seem like the right time to add it. Instead, I decided to add on multiple layers. For GPT-2 small, you have 12 layers. That was already being passed in to my GPTModel 's init method as n layers , so I just replaced this: self.transformers layer = TransformersLayer d emb, n heads, d qk, d v, qkv bias, rngs ...with this: self.transformers layers = nnx.Sequential TransformersLayer d emb, n heads, d qk, d v, qkv bias, rngs for in range n layers ...and then just renamed it where it was called; this: transformed = self.transformers layer input embeddings ...became this: transformed = self.transformers layers input embeddings I kicked it off, and it completed However, the loss chart was telling: Ouch. Loss started dropping quite nicely, but then things got out of control and it settled down at a loss that was essentially that of a random model. At step 937, we were at 10.75, so just a hair less than the 10.82 that randomly guessing next tokens would give. Well, LayerNorm is specifically meant to stabilise training, and the checklist looked like this: Inside the Transformers blocks, we: ...and the only remaining step was that LayerNorm in the Transformers blocks, so it was time to add it in As per the checklist, we do the LayerNorm after we've taken our copy for the shortcut connection, just before MHA, and then likewise after the second shortcut copy, before the FFN. As I understand it, this was a GPT-2 innovation -- previously, people had done normalisation after those steps, but this pre-norm setup turned out to work better. The code changes were simple. I added two LayerNorm modules to the TransformersLayer class, and then called them in the appropriate places taking the opportunity to tidy up the variable naming in the forward pass while I was there : python class TransformersLayer nnx.Module : def init self, d emb, n heads, d qk, d v, qkv bias, rngs : self.attention norm = LayerNorm d emb self.attention = MultiHeadAttention d emb, n heads, d qk, d v, qkv bias, rngs self.ffn norm = LayerNorm d emb self.ffn = nnx.Sequential ... def call self, xs : shortcut = xs xs = self.attention norm xs xs = self.attention xs xs = xs + shortcut shortcut = xs xs = self.ffn norm xs xs = self.ffn xs return xs + shortcut I kicked it off and ran it, and got these results: 2026-06-24 03:07:51.128966 Tokens seen: 92,209,152 2026-06-24 03:07:51.128969 Throughput: 23,399 tokens/second 2026-06-24 03:07:51.128979 Final train loss: 5.359 2026-06-24 03:07:51.128981 Done That certainly looked much healthier However, when I looked at the loss at step 937, it was 5.311 -- a tiny bit higher than the single-layer MHA example, which got 5.295. I'd been willing to play a bit fast and loose with this loss number and allow myself to accept a win when the loss went down a tiny bit, even if it was such a small amount that it could have been within the noise. But increasing loss -- even if it could also be within the noise -- was a step too far. I decided that in this specific case, I'd be strict and test the hypothesis that longer training runs would demonstrate an improvement between one single layer without pre-norm, and multiple layers with pre-norm. I had to remember that these training runs would not be comparable with the earlier ones. In the training script, I had a learning rate schedule like this https://www.gilesthomas.com/2026/06/llm-from-scratch-34a-building-a-jax-training-loop-for-an-llm-training-run learning-rate-scheduling : That straight-line warmup period and the following cosine decay were 5% and 95% of the training run respectively, which meant that for example global step 937 of the short runs we had been doing would be at a completely different point in the schedule than the same step would in these longer runs. However, they would be comparable to each other, and that was what mattered. After some humming and hawing, I decided that a full Chinchilla-optimal for the full model training run over 3,260,190,720 tokens, rounded up to fit into a round number of global steps, would be a nice experiment. I expected it to run comfortably overnight for the single-layer run, and take a bit less than two days for the multi-layer one. So I kicked off the first. Just over 11 hours later: 2026-06-24 16:39:52.178045 Tokens seen: 3,260,252,160 2026-06-24 16:39:52.178049 Throughput: 81,733 tokens/second 2026-06-24 16:39:52.178058 Final train loss: 4.324 2026-06-24 16:39:52.178061 Done Here's the loss chart: The last checkpointing period in that run ended at global step 33,164, and the training loss then was 4.165 -- indeed, it had been at around 4.17 for quite some time, though the trend still seemed to be a tiny bit downward. So then I kicked off a run of the full version -- multiple layers, with pre-norm in the Transformers blocks. Just over 37 hours later: 2026-06-26 06:55:39.956609 Tokens seen: 3,260,252,160 2026-06-26 06:55:39.956614 Throughput: 24,151 tokens/second 2026-06-26 06:55:39.956625 Final train loss: 3.637 2026-06-26 06:55:39.956629 Done The "Final train loss" line at the end said it all, really But here's the loss chart: ...and the loss at step 33,164 was 3.399. Definitely quite an improvement over the 4.165 that a single layer got. Again, at some point I might do the equivalent tests for the earlier results where improvements appear to be pretty much in the noise. It would be good to be sure that the changes really did have the impact I think they did. But for now: our checklist was looking like this: Inside the Transformers blocks, we: Everything was checked off. So was this journey over? Well, there was one thing that the original PyTorch code had that my new code didn't: dropout. I'd found in my lengthy interventions experiments https://www.gilesthomas.com/2026/02/llm-from-scratch-32a-interventions-baseline-model that dropout seemed to make models worse. It was, I felt, a smart idea back in the days when people had little data and did multiple epochs, each sweeping over everything, but it made less sense nowadays with single-epoch training runs over very large datasets. Though I do have some intuitive ideas about why it could still help https://www.gilesthomas.com/2025/03/dropout-and-mandatory-vacation . Still, it would be good to show that it harmed loss for this model as well. Checking my notes, I found that there were four places where dropout was applied: The changes are tiny and rather dotted around the code, so rather than showing you isolated bits of code, if you'd like to see it you can take a look at the code at this point https://github.com/gpjt/jax-gpt2-from-scratch/blob/5fe6e745b55a33a3b55d3d7ce39f480e29ed0d77/gpt.py and search for "dropout". When I started running that, I got an error when saving the first checkpoint: TypeError: JAX array with PRNGKey dtype cannot be converted to a NumPy array. Use jax.random.key data arr if you wish to extract the underlying integer array. This was happening deep inside the bowels of Safetensors, but it made a lot of sense. The nnx.Dropout object needs to keep track of the state of the random number generator, and that meant that the to flat state function that I was using https://www.gilesthomas.com/2026/06/flax-and-safetensors might return a structure that had something that contained that state, and was not compatible with Safetensors. I decided that I'd cheat a little bit here. If I skipped the dropout layers when I saved my checkpoints, like this: for tuple key, array in flat state: key = ".".join str key for key in tuple key if "dropout" not in key: simple dict key = array ...then I'd be able to save them. This would have a problem -- if I restarted from a checkpoint, the dropout pattern after the restart would mirror the dropout pattern from the start of the training run, because the random seed it started with would not have come from the checkpoint, but just the initialisation code. I felt that this would not have a serious impact, though, and given that I'd not had to restart from checkpoints so far, I wrongly, as it turned out decided it wouldn't matter. I kicked off the run, and... after four hours, it OOMed. I cursed, decided that I'd nurse this run through anyway despite my dropout checkpointing concerns , and kicked it off again. Three hours later, it OOMed again. I happened to be away from home at the time, logging in to my machine remotely thanks, Tailscale https://tailscale.com/ , and on looking at nvtop , I realised that the X window system on my machine was using a gig or so of VRAM. I was running the training run in a tmux session, which meant that I could kill X and not lose state, so I did that, and adjusted the XLA PYTHON CLIENT MEM FRACTION environment variable I was using -- it had been 0.90, so I bumped it up to 0.95. I kicked it off again, and... 2026-06-28 22:06:47.669676 Tokens seen: 2,640,052,224 2026-06-28 22:06:47.669683 Throughput: 23,019 tokens/second 2026-06-28 22:06:47.669691 Final train loss: 3.776 2026-06-28 22:06:47.669694 Done Note that the tokens seen only relates to the period since the restart, which is why it was lower. One more loss chart: ...and the training loss at step 33,164 was 3.524, higher enough than the 3.399 I got without dropout that I was comfortable that it wasn't in the noise. That was very reassuring. Once again, if this was a proper scientific experiment I'd fix the issue with saving dropout, and run it completely from scratch -- or, at least, run it all the way through from scratch without restarts, even if I had to try several times to get it done. But I don't think that "replaying" dropout would make the loss any worse. And for this experiment, I felt this was enough. So: checklist complete. GPT-2 model coded up. It was time for some evals I wanted to evaluate these models against the ones I got using the old PyTorch code: specifically, the last local training run https://www.gilesthomas.com/2026/04/llm-from-scratch-32k-interventions-training-our-best-model-locally-gradient-accumulation that used exactly the same training hyperparameters, and only differed in that it was trained using AMP -- 32-bit floats in general, but using 16-bit where the framework thought it would not be harmful. In order to do exactly the same evals, I decided it would be easiest to write a conversion script to take the Safetensors files written to my JAX checkpoints, and write out new files that were compatible with the PyTorch model code -- then I'd be able to use the original PyTorch eval code. I put something together https://github.com/gpjt/jax-gpt2-from-scratch/blob/fd170272e54844229642619b9c69993f0f18778f/convert model to pytorch.py , converted my last two models -- the full runs with and without dropout -- and tried to load them up. Unfortunately there was an error: RuntimeError: Error s in loading state dict for GPTModel: Missing key s in state dict: "trf blocks.0.att.out proj.bias", "trf blocks.1.att.out proj.bias", "trf blocks.10.att.out proj.bias", "trf blocks.11.att.out proj.bias", "trf blocks.2.att.out proj.bias", "trf blocks.3.att.out proj.bias", "trf blocks.4.att.out proj.bias", "trf blocks.5.att.out proj.bias", "trf blocks.6.att.out proj.bias", "trf blocks.7.att.out proj.bias", "trf blocks.8.att.out proj.bias", "trf blocks.9.att.out proj.bias" You might remember that back when I went through multi-head attention, I mentioned that I'd made a mistake. Somehow, I'd misremembered, and thought that the output projection -- the one that mixes together all of the different heads' outputs -- was a linear layer without bias, despite my original notes https://www.gilesthomas.com/2025/04/llm-from-scratch-12-multi-head-attention being perfectly clear that it did have bias. The good news was that if I disabled bias in the PyTorch code, I could load the safetensors files that I had. So the two models I'd trained so far were not useless, and could actually work as a kind of natural experiment into the benefits of having that bias there. But anyway, in order to do things properly, I was going to need to fix the bug and train yet another model. The fix was simple, I just replaced this in MultiHeadAttention : self.output projection = nnx.Linear self.d v n heads, d emb, use bias=False, rngs=rngs ...with this: self.output projection = nnx.Linear self.d v n heads, d emb, use bias=True, rngs=rngs Then it was time to kick off yet another training run. After another 37 hours: 2026-07-05 10:09:22.147819 Tokens seen: 3,260,252,160 2026-07-05 10:09:22.147823 Throughput: 24,072 tokens/second 2026-07-05 10:09:22.147832 Final train loss: 3.650 2026-07-05 10:09:22.147834 Done ...with this loss chart: ...and the training loss at step 33,164 was 3.398 -- almost exactly the same as the 3.399 that I got in the no-dropout training run without MHA bias above Well, now it really was time for the evals. I updated my conversion script https://github.com/gpjt/jax-gpt2-from-scratch/blob/9ab47bc589973b3d8d18a5248b405ef3f17f09fe/convert model to pytorch.py to handle the bias on the MHA output projections, and used it to convert the three models -- the un-biased ones, with and without dropout, and the biased one, without -- to the PyTorch format, then ran the loss test that I had been using to compare the old models on each. Here are the results, compared to the previous models, and OpenAI's: | Test loss | | |---|---| | OpenAI weights: medium | 3.231442 | JAX, with MHA bias, no dropout | 3.418784 | JAX, no MHA bias, no dropout | 3.420089 | JAX, no MHA bias, with dropout | 3.476802 | | OpenAI weights: small | 3.499677 | 1xrtx3090-stacked-interventions | 3.538161 | 8xa100m40-stacked-interventions-1 | 3.577761 | | Cloud FineWeb, 8x A100 40 GiB | 3.673623 | 1xrtx3090-baseline | 3.683835 | 8xa100m40-baseline | 3.691526 | | Cloud FineWeb, 8x H100 80 GiB | 3.724507 | | Cloud FineWeb, 8x A100 80 GiB | 3.729900 | | Cloud FineWeb, 8x B200 160 GiB | 3.771478 | | Local FineWeb train | 3.943522 | | Local FineWeb-Edu extended train | 4.134991 | | Local FineWeb-Edu train | 4.166892 | That was a pretty amazing result -- I'd clearly proven that JAX trains much better models than PyTorch 3.5% better in the best case. Well, OK, no. My guess is that the difference was probably something like better luck with the initial weights on the JAX side, plus the improvement from not using AMP https://www.gilesthomas.com/2026/04/llm-from-scratch-32h-interventions-full-fat-float32 . Anyway, the important thing was that the JAX models were in the same kind of loss range as the PyTorch ones -- and while a 3.5% improvement in loss was more variation than I'd been expecting, it was definitely the right ballpark. Now, one thing I had found in the past was that the OpenAI weights -- and some of my own models, like the Fineweb-Edu ones -- were consistently better https://www.gilesthomas.com/2026/04/llm-from-scratch-32l-interventions-instruction-fine-tuning-tests at an instruction fine-tuning test than their test loss scores would indicate. Would that hold here? The IFT eval code fine-tuned each model on the Alpaca https://crfm.stanford.edu/2023/03/13/alpaca.html dataset until validation loss started rising, then used the model prior to the start of the rise to generate responses for a test set. These were saved, and then run past an OpenAI model so that they could be compared with each other: You are judging the comparative capabilities of a number of different LLM models. They have been trained to follow instructions. The input was this: {input} An example correct output is this: {correct output} Please produce a score of between 0 and 100 for each model, and respond with a JSON structure like this note that the number of models may differ from this example : { "Model 1": {"score": XXX, "comments": "optional comments"}, "Model 2": {"score": YYY, "comments": "optional comments"}, "Model 3": {"score": ZZZ, "comments": "optional comments"} } ...where the XXX, YYY and ZZZ are the scores for the respective models. You can optionally add the "comments" field if you want to explain your reasoning. Here are the models' responses: Model 1 {model 1 response} Model 2 {model 2 response} Model 3 {model 3 response} ...with the model order randomly changed for each query to avoid any position bias. The methodology seemed solid, but I was uncertain about the "train until loss starts rising", as it meant that different models had wildly different amounts of fine-tuning -- between two and seven epochs. On the one hand it felt "unfair" to certain models that they'd get less training than others. On the other hand, if the less-trained models had been trained past the point where their validation loss started rising, then assuming that loss would continue to rise, further training would actually be a disadvantage rather than an advantage. I decided to stick with the original plan, and train until validation loss started rising. I did, however, switch the judge model from the GPT 5.4 that I used in my last IFT test to GPT 5.5. Here are the results: | Test loss | IFT epochs | IFT score | IFT rank | | |---|---|---|---|---| | OpenAI weights: medium | 3.231442 | 2 | 41.62 | 1 | JAX, with MHA bias, no dropout | 3.418784 | 4 | 19.25 | 4 | JAX, no MHA bias, no dropout | 3.420089 | 3 | 14.66 | 11 | JAX, no MHA bias, with dropout | 3.476802 | 4 | 12.94 | 15 | | OpenAI weights: small | 3.499677 | 2 | 26.73 | 2 | 1xrtx3090-stacked-interventions | 3.538161 | 4 | 17.79 | 6 | 8xa100m40-stacked-interventions-1 | 3.577761 | 4 | 10.29 | 16 | | Cloud FineWeb, 8x A100 40 GiB | 3.673623 | 7 | 20.71 | 3 | 1xrtx3090-baseline | 3.683835 | 6 | 15.11 | 9 | 8xa100m40-baseline | 3.691526 | 4 | 14.74 | 10 | | Cloud FineWeb, 8x H100 80 GiB | 3.724507 | 4 | 13.25 | 14 | | Cloud FineWeb, 8x A100 80 GiB | 3.729900 | 4 | 14.50 | 12 | | Cloud FineWeb, 8x B200 160 GiB | 3.771478 | 4 | 16.03 | 8 | | Local FineWeb train | 3.943522 | 7 | 13.73 | 13 | | Local FineWeb-Edu extended train | 4.134991 | 7 | 16.70 | 7 | | Local FineWeb-Edu train | 4.166892 | 7 | 18.68 | 5 | More interesting datapoints As before, you can see that low loss is not particularly well-correlated with a high score on this instruction fine-tuning test. The OpenAI weights continue to lead the pack, and while one of our new JAX models did quite well, it's still beaten by the Cloud FineWeb, 8x A100 40 GiB model. But what was important here, just as with the loss, was that the new JAX models landed in the same ballpark as the PyTorch ones. They did, and so I could be confident that they were doing essentially the same thing. And that meant that, after 18 months, I had reached the end of my LLM from scratch journey. It's been a long trek https://www.gilesthomas.com/llm-from-scratch . I started reading https://www.gilesthomas.com/2024/12/llm-from-scratch-1 "Build a Large Language Model from Scratch " on 22 December 2024. I was planning to breeze through over the Christmas break, but somehow it morphed into being a curriculum onto which I could hang projects to learn the fundamentals of LLMs, beyond what was in the book. In May 2025, I had my first real conceptual breakthrough when I realised that attention heads are individually dumb https://www.gilesthomas.com/2025/05/llm-from-scratch-13-taking-stock-part-1-attention-heads-are-dumb , and as I continued, the second big one came later on in the same month, when the concept of embeddings as being projections between vocab space and embedding space https://www.gilesthomas.com/2025/05/llm-from-scratch-15-from-context-vectors-to-logits and the converse projection in the other direction that happens in the LLM's output head became clear. In August I had the first moment where I felt that the standard teaching approach to LLMs might not be the full story; shortcut connections are normally explained as a way to fix vanishing gradients, while I felt that a better way to see them was a way to allow attention and the FFN to "annotate" the existing information, similarly to how Jewish scholars have annotated the original text of the Talmud https://www.gilesthomas.com/2025/08/llm-from-scratch-18-residuals-shortcut-connections-and-the-talmud . The results in this post seem to point in that direction, given how even a single layer of attention was massively helped by adding them. By early December, I had essentially finished the book, and felt I wanted to try to train my first base model from scratch on my RTX 3090 https://www.gilesthomas.com/2025/12/llm-from-scratch-28-training-a-base-model-from-scratch . It worked, and wasn't far off the quality of the original GPT-2 small. I was really surprised that I could do that with consumer hardware, and became interested perhaps obsessively so with whether I could match OpenAI's weights. In January 2026, I trained a model using DDP on Lambda Labs https://www.gilesthomas.com/2026/01/llm-from-scratch-29-ddp-training-a-base-model-in-the-cloud , and then spent the following months training model after model, trying to work out which interventions -- learning rate scheduling, gradient clipping, etc -- would improve the loss. I wrapped that up in late April https://www.gilesthomas.com/2026/04/llm-from-scratch-32m-interventions-conclusion , with the interesting finding that although I'd been able to get the test loss pretty low, that didn't seem to map cleanly to performance in my instruction fine-tuning tests. In other words, Loss Number Goes Down is an interesting technical game to play, but doesn't cleanly map to real-world performance. The final step was this post, and the previous one https://www.gilesthomas.com/2026/06/llm-from-scratch-34a-building-a-jax-training-loop-for-an-llm-training-run -- could I, using my notes, implement GPT-2 completely from scratch in JAX without referencing the book? And as you've read, the answer was a definite yes Of course, as with any long-running project, there are some loose ends -- from this post alone, there's the interesting fact that JAX trained faster than PyTorch perhaps torch.compile could close the gap? and had a larger possible batch size for full-fat 32-bit. And the fact that fixing the multi-head attention bias bug didn't seem to help with the loss much was interesting too. But those are really details, and there's so much beyond them to learn. Longer-context LLMs: position embedding improvements like RoPE, efficiency tricks like flash attention and attention variants like DSA. Mixture of experts models. How do optimisers really work? Do they work? https://x.com/mgostIH/status/2024087619756318866 And plenty more. So it's time to draw a line under this series, and start thinking about what comes next. It's been a blast; if you've been reading along, I hope it's been as useful and fun to read as it was to write. And as always, comments, questions and corrections very welcome below. On looking back at Raschka's code, after having worked through all of this, there's a slight difference. I do this: normalized = xs - means / stds + 1e-5 ...whereas he does this: norm x = x - mean / torch.sqrt var + self.eps Now, the standard deviation is the square root of the variance, so if you ignore the small numbers -- 1e-5 in my case, and self.eps in his -- the calculations are the same. But there is a difference once those are taken account of. I don't think it's large enough to have any serious effect in these runs, though. ↩ https://www.gilesthomas.com/feed/rss.xml fnref-1 In PyTorch, linear layers are stored as the transpose of the matrix that would allow you to do that, so it would be: Also, note that for simplicity heh I'm disregarding bias in this discussion. ↩ https://www.gilesthomas.com/feed/rss.xml fnref-2