Adam Improves Muon: Adaptive Moment Estimation with Orthogonalized Momentum

TL;DR

This paper introduces NAMO and NAMO-D, new optimizers that enhance Adam by integrating orthogonalized momentum with noise adaptation, leading to improved training performance on large language models.

Contribution

The paper presents NAMO and NAMO-D, the first principled methods combining orthogonalized momentum with norm-based noise adaptation in Adam-type optimizers.

Findings

NAMO and NAMO-D outperform AdamW and Muon on GPT-2 pretraining.

NAMO-D achieves further gains with an additional hyperparameter.

Both algorithms have proven optimal convergence rates under standard assumptions.

Abstract

Efficient stochastic optimization typically integrates an update direction that performs well in the deterministic regime with a mechanism adapting to stochastic perturbations. While Adam uses adaptive moment estimates to promote stability, Muon utilizes the weight layers' matrix structure via orthogonalized momentum, showing superior performance in large language model training. We propose a new optimizer and a diagonal extension, NAMO and NAMO-D, providing the first principled integration of orthogonalized momentum with norm-based Adam-type noise adaptation. NAMO scales orthogonalized momentum using a single adaptive stepsize, preserving orthogonality while improving upon Muon at negligible additional cost. NAMO-D instead right-multiplies orthogonalized momentum by a diagonal matrix with clamped entries. This design enables neuron-wise noise adaptation and aligns with the common near block-diagonal Hessian structure. Under standard assumptions, we establish optimal convergence rates for both algorithms in the deterministic setting and show that, in the stochastic setting, their convergence guarantees adapt to the noise level of stochastic gradients. Experiments on pretraining GPT-2 models demonstrate improved performance of both NAMO and NAMO-D compared to the AdamW and Muon baselines, with NAMO-D achieving further gains over NAMO via an additional clamping hyperparameter that balances the competing goals of maintaining a well-conditioned update direction and leveraging fine-grained noise adaptation.