Improve implementations of Dion2

[kwangjunahn]

In this pull request, a more efficient Dion2 implementation is provided.

Previous implementation had a somewhat complicated logic for sharding the Newton-Schulz computation across distributed_mesh. I won't get into the details of it too much, but roughly speaking, it kept the momentum states "batch"-sharded across different iterates (i.e., the full momentum matrix is taken care of by its owner device). This had some benefits of being able to do single matrix math efficiently. However, this led to "asymmetric" optimizer states across devices, which led to some checkpointing issues (as pointed out by #18)

Hence, in this new implementation, we take a different approach. Let me describe the main difference for the FSDP case below.

1. Similar to how muon.py is handling the optimizer states, we just simply keep the optimizer states sharded in the same way parameters are.
2. We then choose select_dim carefully according to the matrices are sharded. In the case of row-sharding, we select along rows, and in the case of column-sharding, we select along columns. (This way, we don't have to communicate over devices to compute the row/column norms.)
3. After we choose the $\alpha$-fraction submatrix on each shard, we only communicate submatrix via all-to-all, which leads to communication saving! (only $\alpha$-fraction of the matrix needs to be communicated.)
4. We then do Newton-Schulz on the chosen submatrix and then communicate the result back to the respective devices via another all-to-all.
5. We then use appropriate foreach ops + list ops to update the corresponding parameter matrix shards with the orthonormalized shards.

This leads to not only compute savings (because only submatrix gets orthonormalized), but also communication savings (all-to-all's are on the submatrices.)

Moreover, now the optimizer states do not cause any issues with checkpointing.

Other than the new Dion2 implementation, I have made some changes to README.md, as well as train.py. For train.py, I just made things a bit easier for users to configure distributed mesh, and also naming convention for the wandb runs.

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[lalo]

maybe its worth mentioning here or on readme that it is top-k based on L1 norm, if I'm understanding the impl correctly

[kwangjunahn]

Yes, you are right. I have added more detailed comments.

[lalo]

if param is bfloat16 momentum and variance will be bfloat16 right?, is that ok or you want fp32 here?

[kwangjunahn]

I see, I think usually master weights are kept in fp32. This _get_or_initialize_state mirrors muon.py, so I would just keep it as it is for unity.

[lalo]

comment: logging the norms would be interesting for tuning the fraction/k hyperparam, to see if the distribution is heavy tailed or flat.

[kwangjunahn]

I agree that it would be nice to log that stat. I think users can easily add that feature, so I would just leave as is.