MESH: Memory-Efficient Sinkhorn Optimization for Mixture-of-Experts Training

Optimizer state is the largest single line item in the memory budget of mixture-of-experts (MoE) training. On a 6.78B-parameter MoE language model AdamW keeps 50.6 GB of first and second moments to update 12.6 GB of bfloat16 weights. We study SkewAdam, an optimizer built on the observation that the three parameter populations of an MoE differ enough in size and gradient statistics that they should not receive the same state. Those populations are the dense backbone, the experts and the router. SkewAdam keeps float32 momentum plus a factored second moment for the backbone (5% of parameters), a factored second moment alone for the experts (95%) and an exact second moment for the router (<0.01%). The resulting state occupies 1.29 GB or 2.6% of AdamW's and peak training memory falls from 81.4 GB to 31.3 GB, within the budget of a 40 GB accelerator. In a controlled comparison from identical initializations over 82M tokens, SkewAdam reaches validation perplexity 108.4, ahead of AdamW (126.8), Muon (120.2) and Lion (393.7), and settles router load balance to within 1% of its uniform floor. The allocation is not what earns that perplexity. A tier ablation reaches the same value while carrying twenty times the state, so the tiers buy memory rather than accuracy. Same-platform runs separate what does earn it. Removing momentum costs 31 perplexity points (tuned Adafactor, 139.7) and replacing the factored second moment and its update clipping with a full second moment costs 10 (tuned AdamW, 118.5), so neither tuned baseline reaches the untuned tiered policy. Where optimizer state lives, these results suggest, matters at least as much as how much of it there is.

This paper showcases a memory-efficient training stack for Mixture-of-Experts (MoE) models. It is a training paradigm that combines and specializes various existing and novel parallelism techniques at different layers and stages of the Mixture-of-Experts (MoE) model training pipeline. It leverages these techniques to achieve maximal efficiency given the physical constraints of CPU, CPU memory, GPU HBM memory, and the CPU-GPU, GPU-GPU, and node-node communication bandwidth of the GPU cluster. It also contains a novel strategy for the optimizer step to achieve high throughput and memory efficiency, enabling practitioners to conduct lossless pre-training/fine-tuning of trillion-parameter scale models, at a million context length, with just under 12 8x H200 GPU nodes, with state-of-the-art throughput and memory efficiency. In our experiments, MoP delivers 4.7x--8.2x higher per-GPU throughput than a strongly-tuned FSDP2 baseline (with the gap widening at larger scale) and sustains training at context lengths up to 1M tokens, where the baseline runs out of memory beyond 64--128K.

Sparse Mixture-of-Experts (MoE) models offer a powerful way to scale model size without increasing compute, as per-token FLOPs depend only on k active experts rather than the total pool of E experts. Yet, this asymmetry creates an MoE efficiency paradox in practice: adding more experts balloons memory and communication costs, making actual training inefficient. We argue that this bottleneck arises in part because current MoE training allocates too many experts from the beginning, even though early-stage data may not fully utilize such capacity. Motivated by this, we propose EMO, a simple progressive training framework that treats MoE capacity as expandable memory and grows the expert pool over the course of training. EMO explicitly models sparsity in scaling law to derive stage-wise compute-optimal token budgets for progressive expansion. Empirical results show that EMO matches the performance of a fixed-expert setup in large-scale experiments while improving wall-clock efficiency. It offers a surprisingly simple yet effective path to scalable MoE training, preserving the benefits of large expert pools while reducing both training time and GPU cost.