But I think the paper fails to answer the most important question. It alleges that this isn't a statistical model: "it is not a statistical model that predicts the most likely next state based on all the examples it has been trained on.
We observe that it learns to use its attention mechanism to compute 3x3 convolutions — 3x3 convolutions are a common way to implement the Game of Life, since it can be used to count the neighbours of a cell, which is used to decide whether the cell lives or dies."
But it is never actually shown that this is the case. It later on isn't even alleged that this is true, rather the metric they use is that it gives the correct answers often enough, as a test for convergence and not that the net has converged to values which give the correct algorithm.
But there is no guarantee that it actually has learned the game. There are still learned parameters and the paper doesn't investigate if these parameters actually have converged to something where the Net is actually just a computation of the algorithm. The most interesting question is left unanswered.
(This can be shown by comparing that attention matrix to a "manually computed Neighbour Attention matrix", which is known to be equivalent to 3 by 3 conv.)
"We detected that the model had converged by looking for 1024 training batches with perfect predictions, and that it could perfectly run 100 Life games for 100 steps." This would be superfluous (and even a pretty bizarre methodology) if the shape of the attention matrix was proof that the Network performed the actual game of life algorithm.
Just to be clear, I am not saying that the NN isn't converging to performing some computation that would also be seen in other algorithms. I am saying that the paper does not investigate whether the resulting NN actually performs the game of life algorithm. The convolution part is certainly evidence, but I think it would have been worthwhile to look at the actual resulting Net and figure out if the trained weights together actually formed an algorithm. This is also the only way to determine the truth of the initial claim, that this isn't just a statistical model, but rather an actual algorithm.
How could it not be computing the game of life algorithm? Given that it gets 100% accuracy over multiple steps on a bunch random game boards it's never seen before.
And then based on the structure of the net, and by examining the attention layers and finding that it's doing 3 by 3 average pooling, we can see that the attention layer produces a set of tokens, where each token contains the information of the number of neighbours it had, and its previous state. This then goes through a classifier layer, which decides it's next state, given that information.
Further evidence for that: it was possible to use linear probes to confirm that the tokens that had been through the attention layer contained the information about the number of neighbours and the previous state.
From all of this, it's clear that the model is running the Game of Life properly.
> How could it not be computing the game of life algorithm? Given that it gets 100% accuracy over multiple steps on a bunch random game boards it's never seen before.
This is such an insane statement.
In what way? Maybe you mean something different when you say computing the game of Life algorithm.
Then afterwards you can check the neutral network against the exact algorithms.
When the attention grid is manually computed (to be equivalent to 3 by 3 conv), the model can be trained to be 100% perfect, verified by checking all 3 by 3 grid states. (And this manually computed attention matrix means that once the tokens reach the classifier layer, each token contains only the information of the relevant 3 by 3 grid, and the whole thing is deterministic as you say.)
However, when the model is computing the attention grid itself, just checking all 3 by 3 sub-grid states crop up is not enough, because the position of the sub-grids can impact the attention matrix, and also the state of other cells can impact the attention matrix. So as shown in the post, it does approximate 3 by 3 conv, but if it doesn't get the approximation quite right, there could be errors. But I would say that it's still computing the Game of Life algorithm in an interpretable way, it's just that maybe it has struggled to create a perfect 3 by 3 convolution via attention in that particular case. (To exhaustively check this, would require checking all 2 * (16x16) grids.)
It’s similar to comparing hardware radio and software-defined radio: Yes, we already know how to build a radio with hardware but a software-defined one offers greater flexibility.
Like for learning the English language we don’t fully understand the way LLMs work. We can’t fully characterize it. So we have debates on whether the LLM actually understands English or understands what it’s talking about. We simply don’t know.
The results of this show that the transformer understands the game of life. Or whatever the transformer does with the rules of the game of life it’s safe to say that it fits a definition of understanding as mankind knows it.
Like much of machine learning where we use the abstraction of curve fitting to understand higher dimensional learning we can do the same extrapolation here.
If the transformer understands the game of life then that understanding must translate over to the LLM. The LLM understands English and understands the contents of what it is talking about.
There was a clear gradient of understanding before understanding the game of life hit saturation. The transformer lived in a state where it didn’t get everything right but it understood the game of life to a degree.
We can extrapolate that gradient to LLMs as well. LLMs are likely on that gradient, not yet at saturation. Either way, I think it’s safe to say that LLMs understand what they are talking about. It’s just that they haven’t hit saturation yet. There’s clearly things that we as humans understand better than the LLM.
But let’s extrapolate this concept to an even higher level:
Have we as humans hit saturation yet?
There are only 512 training examples needed for that, and it would be a lot more interesting if a learning algorithm were able to fit that 3x3 convolution layer from those 512 examples. IIRC, and don't quote me on that, but that's not been done.
I firmly believe that differentiable logic CA is the winner, in particular because it extracts the logic directly, and thus leads to generalize-able programs as opposed to staying stuck in matrix multiplication land.
I shared it with a friend and he thought its wasn't that useful.
That made me happy since I knew my secret maybe safe.
The training method they're using is the same as used for quantized neural networks. Your 'neurons' being logic gates doesn't mean you're doing logic, it's still statistics.
https://patents.google.com/patent/WO2023143707A1/en?inventor...
Also, it may just be the nature of the task. Some tasks you might have more to learn, all the time, with each training sample potentially giving information that's different from all the others. But with this, once it gets close to the solution of Life, it's quick.
bonzini•1mo ago
montebicyclelo•1mo ago
If by small grid you are referring to the attention matrix plot shown, then that is not a correct interpretation. That diagonal-like pattern it learns, is 3x3 convolution, so it can compare the neighbours of a given cell.
Edit: and note that every grid it is trained on / runs inference on is randomly generated and completely unique, so it cannot just memorise examples
bonzini•1mo ago
yorwba•1mo ago
Using 2D RoPE instead would in principle allow scaling up as well, and maybe even period detection if you train it across a range of grids, but would eventually hit the same issues that plague long-context scaling in LLMs.
bernb•1mo ago
Would it be possible to train an LLM on the rules how we would teach them to a human?