Daily Papers

Scaling laws for large language models depend critically on the optimizer and parameterization. Existing hyperparameter transfer laws are mainly developed for first-order optimizers, and they do not structurally prevent training instability at scale. Recent hypersphere optimization methods constrain weight matrices to a fixed-norm hypersphere, offering a promising alternative for more stable scaling. We introduce HyperP (Hypersphere Parameterization), the first framework for transferring optimal learning rates across model width, depth, training tokens, and Mixture-of-Experts (MoE) granularity under the Frobenius-sphere constraint with the Muon optimizer. We prove that weight decay is a first-order no-op on the Frobenius sphere, show that Depth-μP remains necessary, and find that the optimal learning rate follows the same data-scaling power law with the "magic exponent" 0.32 previously observed for AdamW. A single base learning rate tuned at the smallest scale transfers across all compute budgets under HyperP, yielding 1.58times compute efficiency over a strong Muon baseline at 6times10^{21} FLOPs. Moreover, HyperP delivers transferable stability: all monitored instability indicators, including Z-values, output RMS, and activation outliers, remain bounded and non-increasing under training FLOPs scaling. We also propose SqrtGate, an MoE gating mechanism derived from the hypersphere constraint that preserves output RMS across MoE granularities for improved granularity scaling, and show that hypersphere optimization enables substantially larger auxiliary load-balancing weights, yielding both strong performance and good expert balance. We release our training codebase at https://github.com/microsoft/ArchScale.

Model merging offers a training-free way to combine multiple post-trained expert models, but merging experts obtained through reinforcement learning (RL) remains challenging. Existing spectral merging methods often assume that leading singular directions contain the main task signal, while lower-energy residual components can be compressed, selected, or attenuated to reduce interference. We find that this assumption does not hold for RL task vectors: after decomposing each task vector into a leading spectral head and a residual component, both parts can independently recover substantial behavior knowledge, while exhibiting different merging properties. The head is highly concentrated and informative but more prone to sharp cross-expert conflicts, whereas the residual component is more dispersed and provides a more stable basis for aggregation. Based on this observation, we propose ResMerge, a residual-based spectral merging framework for RL experts. ResMerge first constructs a stable residual backbone with Spherical Residual Consensus Adaptation, which estimates a reliability-weighted consensus direction on the Frobenius sphere. It then reintroduces leading-head information through a Lightweight Head Correction module gated by positive cross-expert agreement. Experiments across multiple RL expert groups and capability domains show that ResMerge better preserves expert capabilities than representative task-vector and spectral merging baselines.

For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric mathcal D-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic mathcal D-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. Over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric D-module defines an F-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.