Rounding in Preconditioner Space: Redesigning 4-bit AdamW Optimizer-State Quantization

Optimizer-state quantization is commonly designed for Adam's dense, parameter-aligned first- and second-moment arrays. This abstraction breaks for memory-efficient optimizers, whose states may be factored, confidence-modulated, or maintained in a projected space, so similar reconstruction error can produce different update error. We formulate optimizer-state quantization as a joint problem over representation, topology, and update semantics. We then introduce Adaptive Log-Space (AL) quantization for non-negative states. AL fits each block's observed nonzero logarithmic interval and reserves a separate code for exact zero, enforcing $q = 0 \Leftrightarrow x = 0$; signed momentum and state precision remain independently selectable. Controlled probes show that adaptive ranges reduce update error and temporal drift, exact-zero reservation preserves dormant states, and state topology constrains useful block granularity. End-to-end language-model training evaluates the resulting policy across dense, factored, confidence, and projected optimizer states. On TinyLlama-1.1B, AL8 with uniform 8-bit momentum reaches 72.90 perplexity versus 73.54 for bitsandbytes 8-bit AdamW, with comparable optimizer-state storage and higher throughput. CAME matches reference-level final perplexity across three seeds when its non-negative states use AL16, while a semantic grouping-and-protection policy closes most of quantized Adafactor's 100K-step late-loss gap. These results make state topology and update semantics first-class design constraints for optimizer quantization.

The Muon optimizer has emerged as a compelling alternative to Adam for training large language models, achieving remarkable computational savings through gradient orthogonalization. However, Muon's optimizer state is more sensitive to quantization errors: because the orthogonalization discards the magnitudes of singular values and retains only directional information, even small quantization errors in singular vector directions are amplified in the update. In this work, we propose MuonQ, a low-bit Muon training framework built on the principle of directional fidelity optimization. First, we apply a pre-quantization normalization so that each step introduces quantization errors of the same magnitude, preventing the accumulated error from developing a preferred direction. Second, we introduce a structural decomposition that separately quantizes the dominant singular components via power iteration, ensuring that quantization errors perturb only singular value magnitudes rather than rotating singular vector directions. Third, we adopt $μ$-law companding quantization to allocate higher resolution to densely packed momentum values, shifting the quantization objective from outlier preservation to dense-region distinguishability. Together, these techniques enable stable 4-bit quantization of Muon's optimizer states. Pre-training experiments on GPT-style and LLaMA-style models demonstrate that MuonQ at 4-bit precision closely matches full-precision Muon in both training loss and downstream task accuracy, while reducing optimizer state memory by up to 7.3 $\times$. Our code is available at https://github.com/YupengSu/MuonQ.

AdamW is a default optimizer for deep learning, but its moment states add two parameter-sized buffers to training memory, increasing the cost of large-scale pretraining. We propose Gefen, a memory-efficient optimizer that automatically shares second-moment estimates across parameter blocks and quantizes the first moment using a learned codebook. Gefen reduces AdamW's optimizer memory footprint by up to 8x while maintaining performance, saving 6.5 GiB per billion parameters. Prior work shares second moments across parameters grouped along the Hessian's block-diagonal structure, but relies on hand-specified architectural rules and leaves unexplained why such grouping works. We prove that large mixed Hessian entries constrain the ratio of squared gradients toward one, explaining why shared second moments are accurate when the squared gradients they pool are similar. The Hessian need not be computed: its block structure is inherited by squared gradients, allowing blocks to be found directly. Gefen therefore infers block structure from initial squared gradients, requiring no architecture-specific metadata or user-tuned hyperparameters beyond AdamW defaults. Gefen learns an exact histogram-based dynamic-programming quantization codebook and reuses the blocks for first-moment scaling. Across diverse pretraining experiments, Gefen achieves the lowest peak optimizer memory among compared methods that maintain AdamW-level performance. In single-machine and distributed training, the reduced footprint enables larger microbatches and substantially improves throughput over AdamW, making Gefen a drop-in replacement that can train larger models or use larger global batch sizes. We provide the complete Python implementation, including fused CUDA kernels at https://github.com/ndvbd/Gefen