Higher-order optimization algorithms such as Shampoo have been effectively applied in neural network training for at least a decade. These methods have achieved...

Higher-order optimization algorithms such as Shampoo have been effectively applied in neural network training for at least a decade. These methods have achieved significant success more recently when applied to leading LLMs. In particular, Muon (MomentUm Orthogonalized by Newton-Schulz) was used to train some of today’s best open source models, including Kimi K2 and GLM-5.

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As LLMs transition from simple text generation to complex reasoning, reinforcement learning (RL) plays a central role. Algorithms like Group Relative Policy...

As LLMs transition from simple text generation to complex reasoning, reinforcement learning (RL) plays a central role. Algorithms like Group Relative Policy Optimization (GRPO) power this transition, enabling reasoning-grade models to continuously improve through iterative feedback. Unlike standard supervised fine-tuning, RL training loops are bifurcated into two distinct, high-intensity phases: a…

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