Muon in Associative Memory Learning: Training Dynamics and Scaling Laws

Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory. We study Muon in a linear associative memory model with softmax retrieval and a hierarchical frequency spectrum over query-answer pairs, with and without label noise. In this setting, we show that Gradient Descent (GD) learns frequency components at highly imbalanced rates, leading to slow convergence bottlenecked by low-frequency components. In contrast, the Muon optimizer mitigates this imbalance, leading to faster and more uniform progress. Specifically, in the noiseless case, Muon achieves an exponential speedup over GD; in the noisy case with a power-decay frequency spectrum, we derive Muon's optimization scaling law and demonstrate its superior scaling efficiency over GD. Furthermore, we show that Muon can be interpreted as an implicit matrix preconditioner arising from adaptive task alignment and block-symmetric gradient structure. In contrast, the preconditioner with coordinate-wise sign operator could match Muon under oracle access to unknown task representations, which is infeasible for SignGD in practice. Experiments on synthetic long-tail classification and LLaMA-style pre-training corroborate the theory.

Reasoning in Diffusion Large Language Models is Concentrated in Dynamic Confusion Zones

Diffusion Large Language Models (dLLMs) are rapidly emerging alongside autoregressive models as a powerful paradigm for complex reasoning, with reinforcement learning increasingly used for downstream alignment. Existing trajectory-based RL methods uniformly allocate policy gradients across denoising steps, implicitly treating all steps as equally important. We challenge this assumption by analyzing trajectories with several step-level metrics: entropy-based uncertainty, Confidence-Margin (CM) uncertainty, and Rate of Entropy Change (RoEC). These reveal structured "zones of confusion": transient spikes in uncertainty and instability that strongly predict final success or failure, while most steps remain stable. We propose Adaptive Trajectory Policy Optimization (ATPO), a lightweight step-selection strategy that dynamically reallocates gradient updates to these high-leverage steps without changing the RL objective, rewards, or compute budget. Using a hybrid RoEC+CM rule, ATPO delivers substantial gains in reasoning accuracy and training stability across benchmarks, showing that exploiting trajectory dynamics is key to advancing dLLM RL.