Parallelizing the Muon Optimizer: A Deep Dive into Sharding and Replication Strategies

In the world of large-scale machine learning, optimizing performance is a constant challenge. One recent discussion on social media about the Kimi Moonlight paper led to an interesting exploration of parallelization strategies for the Muon optimizer. This article delves into the technical details of how we can effectively parallelize the Muon optimizer using different sharding and replication techniques.

Premise

Let's start with a model that is already partitioned using PP (Pipeline Parallelism), FSDP (Fully Sharded Data Parallelism), and TP (Tensor Parallelism). For our purposes, we can ignore pipeline parallelism (PP) since it doesn't affect the optimizer directly. Instead, we focus on how Fully Sharded Data Parallelism (FSDP) and Tensor Parallelism (TP) shard parameters.

Each parameter ( W ) is sharded into multiple shards ( w_1, w_2, \ldots, w_n ), which are distributed evenly across the ranks. Each parameter shard has a corresponding gradient shard ( g_i ). For simplicity, we'll also ignore small parameters like norms or modulation, as their correct analysis is more latency-driven than bandwidth or compute-driven.

Variants

Here are the key variants of parallelizing the Muon optimizer:

1. Replication

The simplest baseline approach is to duplicate the momentum and compute it on all GPUs. This involves the following steps:

• Gather Gradients: ( G = \text{allgather}(g_i) )
• Update Momentum: ( M_t = \mu M_{t-1} + G )
• Compute Update: ( X_t = \mu M_t + G )
• Apply Normalization: ( O_t = \text{NS}(X_t) )
• Update Weights: ( W_t = W_{t-1} + \eta_t O_t )

However, this approach is inefficient because it requires:

• Extra communication (as much as FSDP backwards does)
• Extra computation for both scalar momentum/parameter updates and matrix multiplications in the normalization step

2. NS Replication

A more efficient strategy involves sharding the update for ( M_t ) and ( W_t ), while keeping the normalization step ( O_t = \ldots ) replicated. This approach is free if the data type of ( g_i ) matches that of ( M_t ).

• For PyTorch FSDP, this condition is typically met if the underlying base model parallelized has fp32 master weights and the momentum is also in fp32.

Let ( m_i ) be the local shard of ( M ). The steps are as follows:

1. Gather Gradients: ( G = \text{allgather}(g_i) )
2. Update Local Momentum: ( m_t = \mu m_{t-1} + g_i )
3. Compute Local Update: ( x_t = \mu m_t + g_i )
4. Apply Normalization (Replicated): ( O_t = \text{NS}(x_t) )
5. Update Weights: ( W_t = W_{t-1} + \eta_t O_t )

Implementation Details

To implement the NS Replication strategy, you need to ensure that:

• The gradients are gathered across all ranks.
• The local momentum and update steps are performed on each shard independently.
• The normalization step is replicated across all ranks to maintain consistency.

Here’s a high-level pseudocode for the NS Replication approach:

# Pseudocode for NS Replication in PyTorch FSDP

def muon_optimizer_step(grads, momentum, weights, learning_rate, mu):
    # Gather gradients from all ranks
    G = allgather(grads)
    
    # Update local momentum
    m_t = mu * momentum + G
    
    # Compute local update
    x_t = mu * m_t + G
    
    # Apply normalization (replicated step)
    O_t = normalize(x_t)
    
    # Update weights
    W_t = weights - learning_rate * O_t
    
    return W_t, m_t

# Example usage in a distributed training loop
for epoch