I hadn't uploaded the models that I trained using JAX to the Hugging Face Hub because Transformers has been PyTorch-only since version 5 (though they say they're working to add interoperability with JAX in the future), so it would have been tough to get them working natively with AutoModelForCausalLM and the like. But then it dawned on me that I'd already written a conversion script that could take my JAX safetensors files and convert them into ones compatible with my PyTorch code. It's actually those converted models that I use for my evals -- so I could use my existing PyTorch script to upload them. So, I've now uploaded PyTorch-compatible versions of all of my JAX-trained models: "Writing an LLM from scratch, part 34b -- from bigrams to GPT-2, one component at a time (in JAX)" gpjt/jax-no-mha-bias-no-dropout -- the first full LLM trained in the post, in the "Adding LayerNorm" section. gpjt/jax-no-mha-bias-with-dropout -- the second full LLM trained in the post, in the "Dropout" section. gpjt/jax-with-mha-bias-no-dropout -- the third full LLM trained in the post, in the "Adding bias to the MHA output projections" section. "Why do OpenAI's GPT-2 weights beat mine? Part three: testing overtraining" gpjt/jax-with-mha-bias-no-dropout-extended -- the single-epoch, double-Chinchilla-tokens model. gpjt/jax-with-mha-bias-no-dropout-2-epoch -- the model trained on two epochs over the Chinchilla-optimal number of tokens. "A quick(ish) Chinchilla check" gpjt/jax-with-mha-bias-larger-chinchilla-1 -- the slightly-larger model. gpjt/jax-with-mha-bias-larger-chinchilla-2 -- the slightly-smaller model. I've also added links to the posts in question.

I recently overtrained a couple of GPT-2 style models, training them both on 40 tokens per parameter rather than the 20 per parameter that is generally regarded as "Chinchilla-optimal". The normal heuristic is that instead of doing that, you should scale up the number of tokens and the number of parameters equally -- so I would have been better off scaling up the model by 2 and the token count by the same amount. By doing that, I should expect to get a better model in terms of loss on my held-back test set than I did with my 40-tokens-per-parameter models. My training machine poppy wasn't doing anything, so I decided to give that a go. Would the Chinchilla rule-of-thumb hold up? As you might expect, it did. But it was a surprisingly close-run thing, and could conceivably have been in the noise. Let's take a look. The Chinchilla heuristic If you already know all about the Chinchilla paper -- regular readers in particular must be sick and tired of it by now :-) -- then click here to skip this section. In "Training Compute-Optimal Large Language Models", which is always called the Chinchilla paper after the name of the model they trained at the end, the authors tried to work out the optimal number of tokens to train an LLM on based on its number of parameters. In particular, they were pushing back on a trend they were seeing at the time, where people were making models ever-larger, but not increasing the amount of data they were training on. The authors were all at Google DeepMind, and this was the kind of project that only a large lab could do: they trained "over 400 language models ranging from 70 million to over 16 billion parameters on 5 to 500 billion tokens". Their conclusion was "for compute-optimal training, the model size and the number of training tokens should be scaled equally: for every doubling of model size the number of training tokens should also be doubled". They don't actually state an overall optimal number of tokens to train on in the paper, but in table 3 they provide an estimate of the optimal training FLOPs and tokens for models of various sizes, and it's approximately 20 tokens per parameter. That number has become a heuristic, and people talk about a model as being trained for the Chinchilla-optimal number of tokens. Models that were trained on fewer tokens per parameter are referred to as "undertrained", and models that were trained on more as "overtrained". It's worth noting that overtraining a model is not, in itself, a bad thing. If you have a model of a particular size and you continue training it past the Chinchilla-optimal number of tokens, it will -- in general -- get better. The point of the heuristic is that doing that is not the best way to spend whatever budget you have in terms of compute time. You'll get better results, as they say, by scaling the number of tokens and the number of parameters equally. But let's say you're creating a model for specific target hardware -- say, a mobile device. You have a hard restriction on how large the model can be -- the device has only so much RAM to hold it. So it might make sense to overtrain to get a better model. 1 But if you're not so limited in how many parameters you can use, then you should indeed scale the model up, and that's what I wanted to try. How would that work? Scaling the model A week or two back, I was investigating whether I could make my GPT-2 style models better at a specific instruction-following task by overtraining them. The details of that experiment aren't important here, but what it meant was that I had three GPT-2-style models, each of exactly the same size, roughly 163M parameters A Chinchilla-optimal one, which I'll call jax-gpt2-chinchilla here. One trained on twice the Chinchilla-optimal tokens, jax-gpt2-2x-chinchilla. One trained on the Chinchilla-optimal tokens, with two epochs (so that it was trained for as long as #2): jax-gpt2-2-epoch-chinchilla When I tested them against a held-back test set of sequences -- stuff that they'd never seen before -- they got results rather like you might expect: Test loss jax-gpt2-2x-chinchilla 3.324953 jax-gpt2-2-epoch-chinchilla 3.326482 jax-gpt2-chinchilla 3.418784 A lower loss is better, and you can see that the longer-trained models were noticeably better than the Chinchilla-optimal one. The difference between them was tiny; they were trained starting with the same initial weights, and the training runs themselves were deterministic, but a difference of 0.05% in loss doesn't seem like it could be meaningful -- an extra batch for one or one fewer for the other could easily swap them around, you'd think. Now, these models each had 163,009,536 parameters -- they were the small-size model from the GPT-2 paper, modified to not have QKV bias or weight-tying. jax-gpt2-chinchilla had been trained on 3,260,190,720 tokens (rounded up to fit into a round number of full batches), and the other two on 6,520,381,440 tokens each -- double the amount (rounded up too). What I needed to do for my Chinchilla check was to try training a model that used the same amount of compute, scaling the parameters and the number of training tokens equally. Because training compute increases roughly linearly with both parameters and tokens, that would mean scaling both up by 2, giving us: 163,009,536*2≈230,530,296parameters ...and thus 4,610,605,920 tokens. How to scale the model up? In the GPT-2 paper, they train four models: Name Parameters 2 Layers d_emb MHA heads 3 small 124M 12 768 12 medium 345M 24 1024 16 large 762M 36 1280 20 xl 1542M 48 1600 25 I wanted to scale my own model up from 163M parameters to about 231M. Which of those numbers would I want to increase, and by how much? The first thing that stands out is that the number of heads is always 1/64th of the number of embedding dimensions. So that sorted that one out. I just needed to adjust the number of layers, and the number of embedding dimensions, but ensure that the latter was a multiple of 64. I decided to see if I could fit some kind of curve to the relationship between the number of parameters and the GPT-2 authors' choices. This was made a bit more complicated by one thing: they were using weight-tying, and I was not. That meant that they re-used the embedding matrix at the start of the LLM as an output head at the end -- which is why they had 38M fewer parameters. Embeddings and the output head make up a surprisingly large percentage of the parameters for small models like this -- about 47% without weight-tying, 23% with. I couldn't work out a solid way to scale things up and wound up doing some rather messy hacking around in a spreadsheet. I came up with two proposed model sizes that were within a couple of percentage points of the right size: Name Layers d_emb MHA heads Parameters % diff slightly-larger 15 896 14 235,621,120 +2.21% slightly-smaller 14 896 14 225,978,368 -1.97% Interestingly, I found that because d_emb could only change in increments/decrements of 64, it was a pretty coarse control -- my first attempt at making a slightly-smaller model changed it to the next step down, 832, but that led to a model that was 9.25% too small. That was an interesting first lesson. I'd previously been thinking of the Chinchilla rule as being something like "don't double the tokens, just scale the model and the tokens equally". But that "just" was wrong. Scaling a model is hard -- even with just two dials to fiddle with, like in this case, it was tricky to get something right -- and I can't say for sure that my choices were the right ones. Anyway, the next step was to double-check that these models would use the right amount of compute to train. Training FLOPs As I said earlier, the compute time scales roughly linearly with the number of parameters. Let's dig into that "roughly". Different kinds of parameters take different amounts of FLOPs to train, and scale differently with things like the embedding dimensions, sequence length, and so on. Now, for very large models, a lot of that comes out in the wash, but with tiny models like these where the embeddings make up such a large proportion of the parameters, it might matter. Conveniently, in appendix F of the Chinchilla paper, they provide a set of formulae for estimating the number of training FLOPs for a normal dense LLM like these ones. I coded that up into a script that, given the JSON configuration files I was using for my models and training runs, would work out the number of FLOPs for a single epoch of training. It didn't take account of the fact that my real training runs round the number of tokens up so that we do a round number of full batches, but I felt that so long as the results weren't very close that wouldn't matter. I got these results (multiplying the two-epoch numbers by two): Est. training FLOPs jax-gpt2-chinchilla 3,544,967,596,946,227,200 jax-gpt2-2x-chinchilla 7,089,935,193,892,454,400 jax-gpt2-2-epoch-chinchilla 7,089,935,193,892,454,400 slightly-larger 7,419,664,885,127,577,600 slightly-smaller 6,804,429,215,367,168,000 The numbers were indeed different enough that I wasn't worried about the batch-rounding. And the good news was that slightly-larger and slightly-smaller would indeed use slightly more and slightly less compute to train than the overtrained models -- about 4.6% more and 4% less respectively. A true Chinchilla-equivalent model would lie somewhere between them. It was time to train some models! Training I kicked off the run for the slightly-larger model first. Because it was bigger than the 163M models I'd been training, I couldn't fit such large batches into my VRAM; previously I'd been running with a batch size of 6, and now I could only fit in a batch of 4. Luckily, though, I was using gradient accumulation, so by bumping that up from 16 steps to 24 steps I could keep the same overall batch size and keep the training runs comparable. Even despite that, the training run ran out of VRAM about 60 hours in -- I'm guessing due to VRAM fragmentation, as I did not have TF_GPU_ALLOCATOR set to cuda_malloc_async -- but I was able to restart from the most recent checkpoint and complete the run. After just less than four days total training time, it completed. When it was done, I copied the last checkpoint 4 over to my dev box, perry, and ran my standard smoke test against it, asking it to complete "Every effort moves you" with 20 tokens, using greedy sampling. I got something reasonably coherent: Every effort moves you. I’m not sure what you’re thinking. I’m Next, I converted the safetensors file -- which had been saved by my JAX code -- into a format compatible with my PyTorch code, because that's what I use for evals. I ran another smoke test (this one with temperature 1): Every effort moves you through the motions for your life, your soul, your body, and your soul’s happiness Very spiritual. Next, it was time to work out the loss on my held-back test set: giles@perry:~/Dev/ddp-base-model-from-scratch (main)$ uv run test_loss.py datasets/ ../jax-gpt2-from-scratch/runs/full-llm-full-train-with-mha-output-bias-larger-chinchilla/model.json ../jax-gpt2-from-scratch/runs/full-llm-full-train-with-mha-output-bias-larger-chinchilla/checkpoints/latest/pytorch-model.safetensors Fetching 4 files: 100%|███████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 4/4 [00:00<00:00, 3485.09it/s] 100%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 3200/3200 [07:11<00:00, 7.42it/s] Loss against our test dataset: 3.280028 Well, it was certainly better than the 3.324953 that the best of the overtrained models got -- but only by a bit over 1% better. Interesting! I decided to train the second model, slightly-smaller. This one crashed mid-way through with an error that I've seen before: jax.errors.JaxRuntimeError: INTERNAL: CUDA error: Failed to end stream capture: CUDA_ERROR_STREAM_CAPTURE_INVALIDATED: operation failed due to a previous error during capture [executable_name='jit_train_step'] I'm going to have to investigate that more in future, but for now, I just restarted from the checkpoint, and again after a bit less than four days, I had a model. The JAX smoke test was solid: Every effort moves you forward. The best way to get started is to start with a free trial. You can ...and so was the PyTorch one: Every effort moves you forward in love with our products. I love the way it’s easy to use. Both quite commercial this time! It was time for the proper test loss eval: giles@perry:~/Dev/ddp-base-model-from-scratch (main)$ uv run test_loss.py datasets/ ~/Dev/jax-gpt2-from-scratch/runs/full-llm-full-train-with-mha-output-bias-larger-chinchilla-2/model.json ~/Dev/jax-gpt2-from-scratch/runs/full-llm-full-train-with-mha-output-bias-larger-chinchilla-2/checkpoints/latest/pytorch-model.safetensors Fetching 4 files: 100%|███████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 4/4 [00:00<00:00, 3151.24it/s] 100%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 3200/3200 [06:45<00:00, 7.90it/s] Loss against our test dataset: 3.292937 So, slightly worse than the 3.280028 from the larger model, better than the 3.324953 from the best overtrained one. Time to put this all together. Results Here's an updated version of the table from the start of this post; I've added in the two new models, and the improvement they each had over jax-gpt2-2x-chinchilla in both absolute terms and as a percentage rounded to 3sf. Test loss Improvement Improvement % slightly-larger 3.280028 0.044925 1.35% slightly-smaller 3.292937 0.032016 0.962% jax-gpt2-2x-chinchilla 3.324953 - - jax-gpt2-2-epoch-chinchilla 3.326482 - - jax-gpt2-chinchilla 3.418784 - - Now, unlike the overtrained models, prior to training these two new ones started with different initial weights to the jax-gpt2-chinchilla one -- after all, they had to, because they had more of them! A while back, I did a bit of analysis of how random variation in weight initialisation can change the resulting test loss. It wasn't anything in-depth, but I trained three models with different explicit seeds set prior to the model initialisation, but with the same seed set before the training run started 5. Those three models wound up with test losses of 3.681356, 3.673943, and 3.664345. Doing statistics with three data points is a bit flaky, but the cost of training models is so high that I'll leave the Proper Science to the likes of Google DeepMind and wing it :-) Mean: ~3.673215 Sample variance: ~0.000073 Standard deviation (SD): ~0.008529 Now, piling statistical flakiness on statistical flakiness, we'll compare these. You'd normally expect about two thirds of results to be within one SD of the mean, 95.4% to be within two SDs, and 99.7% to be within three. Three SDs on that (yes, different, I know) distribution is 0.025587. That's smaller than both of the improvements that our Chinchilla-optimal runs had over the overtrained ones. So what does that tell us? Well, perhaps not much given the statistical flakiness. But I think it is useful directionally. It suggests that we might be able to take these results seriously as an improvement, and that Chinchilla held: scaling up the model and the number of tokens evenly did give us a better model than just scaling up the number of tokens. In particular, the fact that the loss for slightly-smaller was lower -- even though it had 4% less compute spent on it than the overtrained models -- was encouraging. But it's certainly far from a slam-dunk. A larger test, training lots of overtrained models and lots of Chinchilla-optimal ones, all with different random seeds, would give actual real serious data. Not worth it for me, and perhaps not for anyone. Conclusion I wanted to do a quick sanity check of the Chinchilla heuristic of 20 tokens per parameter. I came up with results that were certainly in line with it -- perfectly so in terms of the ordering of the models I trained. But the effect was small enough that I could imagine that it was in the noise, especially given the small numbers of models I'm able to train. I'll chalk it up as a tentative success. In addition, I learned one useful thing: when talking about scaling up a model to more parameters, you actually have to think quite hard about where you want to put those parameters. I wound up doing a rough curve-fit to the models in the GPT-2 paper, but I have no idea if that was optimal. At some point I should try to dig up some research into optimising embedding dimensions, numbers of layers, and so on. But not now, as I've a bunch of other stuff I want to investigate first. Anyway, I hope you found this experiment interesting, and as ever, comments and questions welcome below. Thanks for reading! I'm less familiar with arguments for under-training -- that is, for fewer than 20 tokens per parameter. I've heard that these days, modern LLMs get a lot more reinforcement learning than they do pre-training, and perhaps that might mean that some very big ones are undertrained prior to RL? I'm uncertain. It's unlikely to be raw lack of data; even for those of us outside the big labs, FineWeb has 18.5T tokens. On its own, that would be enough to train a 0.925T-parameter model, and given that you can apparently do four epochs over the same data before you start getting diminishing returns, that takes us up to 3.7T. That's frontier-lab size, and I'm sure they have better datasets than FineWeb. ↩ Parameter counts are from the paper, apart from the "small" model, which is known to be wrong -- I used my own calculation, and the result is in line with what I've seen elsewhere. ↩ The paper doesn't mention the number of heads; these numbers are from "Build a Large Language Model (from Scratch)", and match up with the ones on this Hugging Face page. ↩ Regular readers might have noticed that I'm ignoring what I've been calling the "best" checkpoint. I've come to the conclusion that because for my training script, "best" means best in terms of training loss, and the training loss changes based on what training data the model has seen recently, it's actually not a very useful metric and just confuses things. At some point I'll probably re-introduce pre-checkpoint evals and use that for "best", which would be the right way to do it. ↩ At the time I was using dropout, so training runs were not deterministic without a known seed. ↩

This post is the capstone of the most long-running series on my blog. In December 2024 (!), I started reading Sebastian Raschka's book "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 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, 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. 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, the OpenAI weights still beat mine consistently in instruction fine-tuning challenges. Let's get started. The starting point -- A-to-A 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: 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, where at the top level for a model, we feed in a sequence of token IDs, then: Firstly, we convert them into embeddings, so we get a sequence of vectors, one for each token. We do this by a lookup into a table, but we can see it conceptually as a projection via a matrix, from vocab space (where a particular token ID is a one-hot vector) to embedding space. Next, we do the magic with our Transformers layers, getting embeddings for the next token. After these layers, the embedding at position n in the output sequence is for the predicted token to come after the token at position n in the input sequence, considering that input token and all other tokens to its left. Finally, we project those back from embedding space to logits, this time actually using a real matrix (in the form of a linear layer), the output head. The logits (after being run through softmax) represent the probabilities for each token of it being the next one. 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 50,2572=2,525,766,049 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 2×50,257×768≈77 million parameters -- about 3% of the full table.) An uninitialised model would (hopefully) have a loss of about 10.82, implying a 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: 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? Building GPT-2: a checklist Adapting from my how LLMs work post, a GPT-2-style LLM looks like this. We receive our sequence of token IDs, and then: Convert them into embeddings. ✔ ️done Add on position embeddings. Run these embeddings through multiple successive Transformers blocks. Layer normalisation Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings Layer normalisation Run multi-head attention Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original Take a second copy of that one Layer normalisation again Run it through a simple neural network Add the results of that back in. 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? LayerNorm The purpose of LayerNorm 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, of course (as does PyTorch), but in the book, we implement it ourselves, and that felt like the correct path to take. Firstly, I implemented a dummy version: class LayerNorm(nnx.Module): def __init__(self): ... def __call__(self, xs): return xs ...and updated the core GPTModel to create and call one: 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, 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 and a mean one, both of which take an axis parameter. The Array object repackaged those as methods, which was convenient, so I did a first cut test like this: 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? 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: 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, 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: 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: Convert token IDs into embeddings. ✔ ️done Add on position embeddings. Run these embeddings through multiple successive Transformers blocks. Layer normalisation ✔ ️done Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings Layer normalisation Run multi-head attention Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original Take a second copy of that one Layer normalisation again Run it through a simple neural network Add the results of that back in. 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? A single layer of single-head attention 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: 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: 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?" one does a decent job of that, and "The 'why' of attention, or: 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: 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 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: 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 Trues: 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, 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, 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: 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 n+1, on average the most important other token you need to know about is probably whichever one is at position n. Knowing about the tokens at n−1, n−2, and so on, could well be helpful -- maybe very helpful -- but not at the cost of not knowing about the one at n. 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. 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: 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: Convert token IDs into embeddings. ✔ ️done Add on position embeddings. Run these embeddings through multiple successive Transformers blocks. part-done -- one layer only Layer normalisation ✔ ️done Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings ✔ ️done Layer normalisation Run multi-head attention part-done -- single-head attention only Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original ✔ ️done Take a second copy of that one Layer normalisation again Run it through a simple neural network Add the results of that back in. 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. Position embeddings These were trivial to add. We had this core code: 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: Convert token IDs into embeddings. ✔ ️done Add on position embeddings. ✔ ️done Run these embeddings through multiple successive Transformers blocks. part-done -- one layer only Layer normalisation ✔ ️done Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings ✔ ️done Layer normalisation Run multi-head attention part-done -- single-head attention only Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original ✔ ️done Take a second copy of that one Layer normalisation again Run it through a simple neural network Add the results of that back in. We had only one attention head right now. Individually, attention heads are dumb, so switching to multi-head attention seemed like a good thread to pull. Multi-head attention 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 ("still only 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 m×n matrix multiplied by a n×p one will be m×p, 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 a×b×c×...×l 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 Trues. That seems reasonable. The next step: causal_mask = jnp.tril(causal_mask) What will that do? Well, per the tril documentation: 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, X·Y, the value in the result, in row r, and column c, is the dot product of row r in X and column c in Y. So, imagine if you wanted to multiply X by n different versions of Y, let's call them Y0, Y1, and so on up to Yn. If you imagine a new matrix, Yall, which is basically all the Yxs stacked side-by-side, then the dot-product understanding of multiplication makes it pretty clear that if you did X·Yall, 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: Q=xs·Wq That is, it is just a matrix multiplication. 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 arrays. You can take one axis of length (say) m×n, and split it into two of lengths m and n respectively -- or, conversely, you can combine two axes of length m and n to one of m×n. 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 Ω=QKT), 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. The result was this: 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: Convert token IDs into embeddings. ✔ ️done Add on position embeddings. ✔ ️done Run these embeddings through multiple successive Transformers blocks. part-done -- one layer only Layer normalisation ✔ ️done Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings ✔ ️done Layer normalisation Run multi-head attention ✔ ️done Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original ✔ ️done Take a second copy of that one Layer normalisation again Run it through a simple neural network Add the results of that back in. Adding that simple neural network -- the FFN -- seemed like a good next step. Adding the FFN to the Transformers block 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, I wrote this: 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 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: Convert token IDs into embeddings. ✔ ️done Add on position embeddings. ✔ ️done Run these embeddings through multiple successive Transformers blocks. part-done -- one layer only Layer normalisation ✔ ️done Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings ✔ ️done Layer normalisation Run multi-head attention ✔ ️done Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original ✔ ️done Take a second copy of that one ✔ ️done Layer normalisation again Run it through a simple neural network ✔ ️done Add the results of that back in. ✔ ️done 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. 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: Convert token IDs into embeddings. ✔ ️done Add on position embeddings. ✔ ️done Run these embeddings through multiple successive Transformers blocks. ✔ ️done Layer normalisation ✔ ️done Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings ✔ ️done Layer normalisation Run multi-head attention ✔ ️done Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original ✔ ️done Take a second copy of that one ✔ ️done Layer normalisation again Run it through a simple neural network ✔ ️done Add the results of that back in. ✔ ️done ...and the only remaining step was that LayerNorm in the Transformers blocks, so it was time to add it in! Adding LayerNorm 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): 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: 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: Convert token IDs into embeddings. ✔ ️done Add on position embeddings. ✔ ️done Run these embeddings through multiple successive Transformers blocks. ✔ ️done Layer normalisation ✔ ️done Project them back from embedding space to vocab space. ✔ ️done Inside the Transformers blocks, we: Take a copy of the input sequence of embeddings ✔ ️done Layer normalisation ✔ ️done Run multi-head attention ✔ ️done Add the copy back in so that the version that came out of MHA is something more like an "annotation" of the original ✔ ️done Take a second copy of that one ✔ ️done Layer normalisation again ✔ ️done Run it through a simple neural network ✔ ️done Add the results of that back in. ✔ ️done 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. Dropout I'd found in my lengthy interventions experiments 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.) 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: Once in the main body, just after we've worked out the embeddings. Twice in the transformers block: once after attention (but before the shortcut is mixed back in), and once after the FFN (ditto) Inside multi-head attention, on the attention weights (which surprised me). 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 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 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!), 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! Evals, first try -- and fixing an MHA bug I wanted to evaluate these models against the ones I got using the old PyTorch code: specifically, the last local training run 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, 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 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. Adding bias to the MHA output projections 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. Evals, take two I updated my conversion script 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. 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 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 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. Conclusion It's been a long trek. I started reading "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, 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 (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. (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. 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, 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, 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 -- 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?) 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. ↩ In PyTorch, linear layers are stored as the transpose of the matrix that would allow you to do that, so it would be: Q=xs×WqT Also, note that for simplicity (heh) I'm disregarding bias in this discussion. ↩

I'm porting my PyTorch LLM code to JAX, using Flax as the neural network layer. For various reasons I wanted to use Safetensors to store checkpoints of the model. It took a little while to get it working; here's the trick I learned. If you look at the Safetensors docs, you'll see that it doesn't mention a JAX implementation -- indeed, searching for "safetensors jax" at the time I'm writing this gives you a link to this GitHub repo by Alvaro Bartolome -- which was last updated in 2023. However, if you look more closely at the docs, they do have a link to the Flax API. I feel this is somewhat misnamed, as it is actually a JAX API. There's no reference (again, as of the time of writing) to Flax in the source -- it's all just JAX code. And in fact Bartolome's library uses it under the hood. There is one problem, though. The API works with simple single-level dictionaries, with strings mapping directly to JAX arrays. For example, the save_file function has this signature: def save_file( tensors: Dict[str, Array], filename: Union[str, os.PathLike], metadata: Optional[Dict[str, str]] = None, ) -> None This can cause problems if you're not careful. If you look at the Flax documentation on checkpointing, it suggests that you use Orbax 1, which has its own API and file format, but then goes on to say: When interacting with checkpoint libraries (like Orbax), you may prefer to work with Python built-in container types. In this case, you can use the nnx.State.to_pure_dict and nnx.State.replace_by_pure_dict API to convert an nnx.State to and from pure nested dictionaries. I initially put two and two together -- that and the dictionary-based API for Safetensors -- and got five, and tried feeding one of those "pure" dicts into Safetensors. I got a very confusing error: SafetensorError: dtype object is not covered It's worth digging in to why that happens. The problem is that although Safetensors is expecting a dict of strings mapping to tensors, it doesn't check that that is what it actually gets. And while the dictionaries from nnx.State.to_pure_dict are "pure", they are also nested (as the docs say!). Even for the simple model I was working with, I got a structure like this: { 'output_head': { 'kernel': Array([...], dtype=float32) }, 'token_embedding': { 'embedding': Array([...], dtype=float32) } } So, we had strings mapping to dicts, and those dicts mapped from strings to the JAX arrays. More complex models would have had deeper dict structures. Now, internally inside Safetensors, the Flax/JAX API is a simple wrapper. It iterates over the keys in the dictionary it's been provided with, and tries to convert their respective values into NumPy arrays. It does that by passing them into NumPy's asarray function, which accepts things like lists, tuples, and NumPy arrays, and converts them into arrays. JAX's own Array class exposes an interface that it recognises, so they're converted without trouble. Once it's done that, it passes the result to a lower-level Rust implementation that actually converts everything to Safetensors format. But because Safetensors didn't check types, in my case it was iterating over the top level of the dict, trying to convert the values to NumPy arrays, and got something like this: { 'output_head': numpy.array({'kernel': Array([...], dtype=float32)}, dtype=object), 'token_embedding': numpy.array({'embedding': Array([...], dtype=float32)}, dtype=object) } That is -- because it assumed that the values in the top-level dict were JAX Arrays, it blindly tried to convert them to NumPy arrays. But they were dicts (that happened to map from strings to arrays) -- and if you ask asarray to create an array based on a random object, it happily does so and wraps that object in a NumPy array, with a dtype of object. When that is then fed into the lower-level Rust code that is trying to write the file, it encounters NumPy arrays that have a dtype it can't handle, object -- hence that error: SafetensorError: dtype object is not covered It all makes sense when you read through the code, but I was a bit perplexed for a while! I think all this might be the reason why Bartolome created his GitHub repo. In the README, he says that: There are no plans from HuggingFace to extend safetensors to support anything more than tensors e.g. FrozenDicts, see their response at huggingface/safetensors/discussions/138. So the motivation to create safejax is to easily provide a way to serialize FrozenDicts using safetensors as the tensor storage format However, you don't need to use that library to serialise simple Flax models. Consider how PyTorch models get serialised to Safetensors; my LLMs have keys with names like out_head.weight, pos_emb.weight, and trf_blocks.0.att.out_proj.weight. They're "flat" dictionaries mapping strings to PyTorch Tensors, similar to what Safetensors wants for these Flax ones, but they use dots to separate different levels, with integers for list items and strings for field names. Looking at the pure-dict structure I had for my model: { 'output_head': { 'kernel': Array([...], dtype=float32) }, 'token_embedding': { 'embedding': Array([...], dtype=float32) } } ...you can see that you could walk the dictionary structure to generate keys like output_head.kernel and token_embedding.embedding. That would be easy enough to code up. But -- as Adithya Dsilva points out on GitHub -- you can get there even faster by using nnx.to_flat_state. That returns a (non-dict) structure like this: FlatState([ (('output_head', 'kernel'), Param( # 786,432 (3.1 MB) value=Array([[ 2.3581974e-02, 3.0957451e-02, -3.5088759e-02, ..., -4.5880198e-02, 5.3717274e-02, -2.6590331e-02], ..., [-9.6302675e-03, -3.3276502e-02, 5.7173111e-02, ..., -7.9063717e-03, 2.0532632e-02, 5.4753982e-02]], dtype=float32) )), (('token_embedding', 'embedding'), Param( # 786,432 (3.1 MB) value=Array([[ 0.00273973, -0.01754938, 0.04656043, ..., -0.04276522, -0.03986642, -0.00781331], ..., [ 0.01421758, -0.0219186 , -0.01701825, ..., -0.00793659, 0.00500103, 0.03839901]], dtype=float32) )) ]) If you iterate over that FlatState, you get tuples where the first element is that tuple of strings, like ('output_head', 'kernel'), and the second is a Param object wrapping the JAX Array. The tuples mirror the dot-separated string format in the PyTorch-style Safetensors files. Param objects also implement an interface that asarray can understand, so you can quickly and easily convert the FlatState to a regular dict for Safetensors: from safetensors.flax import save_file ... model_state = nnx.state(model) flat_state = nnx.to_flat_state(model_state) simple_dict = {} for tuple_key, param in flat_state: key = ".".join(str(key) for key in tuple_key) simple_dict[key] = param save_file(simple_dict, "model.safetensors") (You need to wrap key in a str because if you have a nnx.Sequential in your model, the item in the tuple will get an integer index rather than a string). You can go the other way pretty easily too; given a model, you can load the saved checkpoint into it like this (because from_flat_state accepts raw JAX Arrays in place of explicit Params): from safetensors.flax import load_file ... simple_dict = load_file("model.safetensors") dict_flat_state = {} for key, array in simple_dict.items(): elements = key.split(".") list_key = [] for element in elements: try: list_key.append(int(element)) except ValueError: list_key.append(element) dict_flat_state[tuple(list_key)] = array new_flat_state = nnx.from_flat_state(dict_flat_state) nnx.update(model, new_flat_state) A little more work than I'd ideally like, but given that it can be tucked away in general save_checkpoint/load_checkpoint functions, not too big a deal. Hope that's of use for other people coming across this problem! I'm beginning to feel a bit swamped with all of these libraries with names ending in -ax. It reminds me of the names of the characters in Asterix's village... ↩

In my last post I showed the somewhat-scary temperatures I was getting on the MikroTik 10GBASE-T SFP+ module I have plugged into nigel, the 10Gb/s switch I have in my study. As I mentioned then, the plan was to try using some of the mini-heatsinks that people use on Raspberry Pis, to see if that would help. Here's how it went. I bought a 40-piece set of heatsinks made by the improbably-named VooGenzek on Amazon for €8, and attached two of them like this -- see the bottom module, with the yellow cable: That was 24 hours ago, and here's a chart of temperatures from that module showing the 24 hours before and after: You can see the big drop-off in the middle of the chart; it even overshot a bit (I'm guessing because the heatsinks absorbed a bunch of heat initially when I put them on). The difference looks more dramatic than it is! See where the Y-axis starts. But given that the weather has been pretty much the same today as it was yesterday, that looks like a 3.5°C improvement. Not great, but not nothing either. In the copious discussion about the last post on Hacker News, one of the most popular comments -- from xxpor -- was that there are two generations of SFP+ modules for this kind of thing; an older one, using a Marvell chip, and the newer one using one from Broadcom. blunden on the ServeTheHome forums made the same point. They both mentioned that a good indicator of which type a module is using is that the older ones tend to be rated up to 30 metres, while the newer ones are rated up to 100. This one is a MikroTik S+RJ10, which definitely is one of the older ones -- the specific chip is mentioned in the docs. I'm not sure which chip the Protectli modules in my router reggie are -- they're these modules -- but they say they're rated up to 30 metres, so I guess they're probably the older type too. Looking into switching those out might be a good next step! I probably won't do that in the short term, though, unless I start getting issues as we move into summer.