Writing an LLM from scratch, part 34b -- from bigrams to GPT-2, one component at a time (in JAX)
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 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: 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. 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: 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: 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 , 257 2 = 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: That is, the inputs, the , were the same as the targets, the . If we fed it ...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: ...and targeting this: That's a simple change -- that method became this: I did that, and kicked it off to train on the 92,209,152 tokens that I was (somewhat arbitrarily) using in the last post to test my training loop. The loss chart looked like this: That was pretty promising! Loss came down from roughly 10.82 down to a fairly stable 6 or so by global step 768, and seemed to flatten out there. It's possible that further training could have got it down a bit more, but I decided (again, somewhat arbitrarily) to use the average train loss in the checkpoint period ending at step 937 as my starting point. If we could make changes that reduced that, then we'd be moving forward. For this model, that value was 5.909. So, what were the changes we needed to make to change our bigram-style model to a real, if small, LLM? Adapting from my how LLMs work post, a GPT-2-style LLM looks like this. We receive our sequence of token IDs, and then: Inside the Transformers blocks, we: So that gave me the checklist; looking at it, the most tempting next step was layer normalisation (henceforth LayerNorm). It's used at the end of the core loop, and then twice in the Transformers blocks. What would happen if we coded it up, and then added it to the core only? The purpose of LayerNorm 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: ...and updated the core to create and call one: 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 I had coming into my class's was this: That was . So we needed to do those operations strictly on the last axis, manipulating each embedding independently. JAX has a function and a one , both of which take an parameter. The object repackaged those as methods, which was convenient, so I did a first cut test like this: That printed out these results: ...which looked plausible; one number for each embedding vector. Could we broadcast them across the array? This blew up: Fair enough. But and have a kwarg that looked like it would help: ...and it did! 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 . 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: 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 , and wrote this: 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 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: Now, the standard deviation, if it's not zero, has to be positive -- so adding on a small value would fix that 1 : 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: 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": It was kind of impressive that it managed to finish the first line before it got stuck in a loop -- but it was understandable that we couldn't expect anything good yet. Each predicted token was based entirely on the token before it. What next? Back to our checklist: Inside the Transformers blocks, we: So, at this stage, for each input token we were predicting the next one based on the input token only -- like I said earlier, we were doing a somewhat roundabout way of building an approximation of a table of bigram probabilities. What would happen if we started paying attention to the tokens to the left? And what would be the simplest, dumbest way to do that? The real LLM has multiple layers of multi-head attention, each one also having a feed-forward network, some LayerNorms, and some shortcut connections. Single-head attention is easier to code, but even on its own, you'd expect it to be able to add some value. Each token would get at least some information from the ones to the left. And one layer, likewise, you'd expect might help a bit. I suspected that it wouldn't work on its own -- I expected I'd need shortcut connections too -- but decided to start with attention on its own. I modified the main class to have a single "Transformers" layer: ...where that layer was actually just single-head attention: Next, it was time for the 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: 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 here would have the shape . After the projections to the Q-K embedding space, both and would also be shaped . Now, the property on the JAX array class just reverses the axes, so the code above would give us with the shape . That would break! Matrix multiplication in JAX expects all but the last two axes to represent batches, so we actually wanted to have the shape ` . That meant that what we actually wanted was to just transpose the last two axes. The JAX function takes an 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: As would have the shape , and the transposed version of would be , they'd be compatible for matrix multiplication and give us a result that was -- 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 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 , but was full of s: 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: ...and then I could apply that mask to omega with , telling it to create a new array, taking the value from where the mask had , and in places where it had . That seemed solid, so I just needed to run the result through , specifying that the last dimension was the one where it should apply the function, and that would give me the attention weights: 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, : As was shaped , and (like and ) was shaped , the batch axes were at the start where they belonged, and the matrix multiplication would work and return something shaped . With that, we were done! The final single-head attention class looked like this: 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 module to do a shortcut connection: I kicked off a training run, and at the end it printed this: 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: I mean, it was repetitive, but it was actually getting noticeably closer to making sense! So that was excellent news. What next? Our checklist looked like this: Inside the Transformers blocks, we: Now, our single attention layer was lacking something. Without position embeddings, that layer has no idea what order the tokens before the one it's looking at come in. If it's considering the " cat" in ...it doesn't know if it's looking at "The fat cat" or "fat The cat". Position embeddings are simple, and might help, so that was the next step. These were trivial to add. We had this core code: So I just added a position encoding module in : ...and mixed it in with the token embeddings to create new, improved to be used in our "Transformers" layer: I kicked off a training run with that: 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: Pretty much indistinguishable to the previous one. But still, Loss Number Went Down, and that's what was important at this stage. It was time to try the next step. From the checklist: Inside the Transformers blocks, we: We had only one attention head right now. Individually, attention heads are dumb , so switching to multi-head attention seemed like a good thread to pull. At this point, my single-head attention code looked like this: 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 . Now, we could naively implement it just by spinning off 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: ...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 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 , the attention scores. I had this: Breaking that apart into two steps: We got from this line: Let's look at the shapes here. is our input embeddings for this layer; its shape is . Projecting it through , which is shaped gives us a shape for of again. , being a projection of through , which is the same shape as , will have the same shape as . Now, that means that is , and the calculation ...is doing a batched matrix multiplication getting us the that we want, shaped . But as I said above, there's no need to stop with just one batch dimension. Let's say that we have heads, and that they each work with embeddings sized . Imagine that we've already somehow done multiple projections into the key and query spaces for each of our heads, and that the results have somehow been put into arrays such that and are shaped -- 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: ...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: That will be a multiplication of , shaped , with , shaped , which gives us an of the right shape, . 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 calculation just scales the attention scores by a scalar: That looked fine, just a broadcast division-by-float. We'd need to change that to be in some manner, but that's all. The will give us an array that's full of s. That seems reasonable. The next step: What will that do? Well, per the documentation : When , operates batch-wise on the trailing axes. ...which sounded good. and would be treated as batch axes, which meant that the next line: ...would work. Likewise, with the next line: ...the axis to apply 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 shaped , where the last axis had been softmaxed (softmaxxed?). The next line looked a little trickier: In the single-head version we had of shape , and V of shape , so multiplying them gives us In the new MHA code so far, we had our shaped . So in order for the matrix multiplication to work, we'd need to be shaped . That would give us a result shaped as . And conveniently, we'd already decided that the correct shape for and for was . If we could use the same "magic" to do the projection into value space -- that is, to get such that the heads formed a new batch-like axis like we had for and -- 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 , and somehow handle an output of the shape , then we could use MHA code something like this: 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 , and 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 Y 0 , Y 1 , and so on up to Y n . If you imagine a new matrix, Y all , which is basically all the Y x s stacked side-by-side, then the dot-product understanding of multiplication makes it pretty clear that if you did X · Y all , 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: We've initialised the weights, in this case, as an , so what is happening under the hood here is basically: That is, it is just a matrix multiplication. 2 So if we imagine that 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 , then we could stick with the current code -- the Our input would be shaped , so the result would be , 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 , we could reshape it like this: ...and that would split things up. So we'd have Q shaped as . That's almost what we wanted! We needed , and a simple transpose could sort that out: Likewise for and , and that was our inputs sorted. Moving on to the output; it came from this: ...and as we worked out above, it was shaped . 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: . 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 , we'd need to convert it from to first: ... and then we could just reshape it to : Finally, we could run it through a linear layer, with set to , and set to . 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 , 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 and 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 = Q K T ), the value vector's dimensionality can in theory be different. So I decided to break into two separate and parameters. The result was this: 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: ...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 restriction I was enforcing in my training script) the , , and arrays were the same size, I was creating that , which would consume randomness and make things vary. If I were doing a proper scientific experiment to see if a single layer of MHA beat a single layer of single-head attention, I think I would have run both for more steps to see if the difference became more pronounced later. But for the purposes of this post, I decided to move on. My checklist now looked like this: Inside the Transformers blocks, we: Adding that simple neural network -- the FFN -- seemed like a good next step. The feed forward network is simple; you take the output of the MHA block, run it through a biased linear layer to expand it from to , then run it through the GELU activation function, then shrink it back down to 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 , I wrote this: 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: Adding didn't help. I spent some time trying to dig into what might be causing it, but eventually noticed something interesting: in , 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 environment variable. If you set that to, say, , 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 , and there are various other settings you can control with other environment variables on that linked page). Anyway, setting it to to grab 90% of VRAM worked, and I was able to get a successful run: The loss chart was this: ...and the training loss at global step 937 was 5.295, compared to the 5.336 from MHA alone. Another tiny improvement, another one that could have been in the noise. Again, if I were doing a proper experiment, I'd do a longer run, but for now, I decided to move on. The checklist looked like this: Inside the Transformers blocks, we: Now, my gut instinct was that the layer normalisation inside the Transformers blocks was of most value as a way of stabilising training over deep networks. And with one layer, it didn't seem like the right time to add it. Instead, I decided to add on multiple layers. For GPT-2 small, you have 12 layers. That was already being passed in to my 's method as , so I just replaced this: ...with this: ...and then just renamed it where it was called; this: ...became this: I kicked it off, and it completed! However, the loss chart was telling: Ouch. Loss started dropping quite nicely, but then things got out of control and it settled down at a loss that was essentially that of a random model. At step 937, we were at 10.75, so just a hair less than the 10.82 that randomly guessing next tokens would give. Well, LayerNorm is specifically meant to stabilise training, and the checklist looked like this: Inside the Transformers blocks, we: ...and the only remaining step was that LayerNorm in the Transformers blocks, so it was time to add it in! As per the checklist, we do the LayerNorm after we've taken our copy for the shortcut connection, just before MHA, and then likewise after the second shortcut copy, before the FFN. As I understand it, this was a GPT-2 innovation -- previously, people had done normalisation after those steps, but this pre-norm setup turned out to work better. The code changes were simple. I added two modules to the 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): I kicked it off and ran it, and got these results: 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: 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: The "Final train loss" line at the end said it all, really! But here's the loss chart: ...and the loss at step 33,164 was 3.399. Definitely quite an improvement over the 4.165 that a single layer got. Again, at some point I might do the equivalent tests for the earlier results where improvements appear to be pretty much in the noise. It would be good to be sure that the changes really did have the impact I think they did. But for now: our checklist was looking like this: Inside the Transformers blocks, we: Everything was checked off. So was this journey over? Well, there was one thing that the original PyTorch code had that my new code didn't: dropout. I'd found in my lengthy interventions experiments 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: 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: This was happening deep inside the bowels of Safetensors, but it made a lot of sense. The object needs to keep track of the state of the random number generator, and that meant that the 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: ...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 , 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 session, which meant that I could kill X and not lose state, so I did that, and adjusted the environment variable I was using -- it had been 0.90, so I bumped it up to 0.95. I kicked it off again, and... Note that the tokens seen only relates to the period since the restart, which is why it was lower. One more loss chart: ...and the training loss at step 33,164 was 3.524, higher enough than the 3.399 I got without dropout that I was comfortable that it wasn't in the noise. That was very reassuring. Once again, if this was a proper scientific experiment I'd fix the issue with saving dropout, and run it completely from scratch -- or, at least, run it all the way through from scratch without restarts, even if I had to try several times to get it done. But I don't think that "replaying" dropout would make the loss any worse. And for this experiment, I felt this was enough. So: checklist complete. GPT-2 model coded up. It was time for some evals! I wanted to evaluate these models against the ones I got using the old PyTorch code: specifically, the last local training run 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: 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. The fix was simple, I just replaced this (in ): ...with this: Then it was time to kick off yet another training run. After another 37 hours: ...with this loss chart: ...and the training loss at step 33,164 was 3.398 -- almost exactly the same as the 3.399 that I got in the no-dropout training run without MHA bias above! Well, now it really was time for the evals. I updated my conversion script 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: 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: ...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: More interesting datapoints! As before, you can see that low loss is not particularly well-correlated with a high score on this instruction fine-tuning test. The OpenAI weights continue to lead the pack, and while one of our new JAX models did quite well, it's still beaten by the Cloud FineWeb, 8x A100 40 GiB model. But what was important here, just as with the loss, was that the new JAX models landed in the same ballpark as the PyTorch ones. They did, and so I could be confident that they were doing essentially the same thing. And that meant that, after 18 months, I had reached the end of my LLM from scratch journey. It's been a long trek . 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 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: ...whereas he does this: Now, the standard deviation is the square root of the variance, so if you ignore the small numbers -- in my case, and 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: Also, note that for simplicity (heh) I'm disregarding bias in this discussion. ↩ 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. 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 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. 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 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. 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 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. 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 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. 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 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. 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 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 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 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 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 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 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 ). On looking back at Raschka's code, after having worked through all of this, there's a slight difference. I do this: ...whereas he does this: Now, the standard deviation is the square root of the variance, so if you ignore the small numbers -- in my case, and 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 × W q T Also, note that for simplicity (heh) I'm disregarding bias in this discussion. ↩
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