Adding Layerwise GN (https://arxiv.org/pdf/2510.09378)

Is your feature request related to a problem? Please describe.

Large-batch LLM pretraining currently relies on first-order or approximate second-order optimizers (e.g., AdamW, SOAP, Muon, Shampoo).
However, even the strongest of these still under-utilize curvature information — requiring significantly more iterations to reach the same loss compared to a full Gauss-Newton (GN) preconditioner.

For example, the paper The Potential of Second-Order Optimization for LLMs: A Study with Full Gauss-Newton (arXiv:2510.09378) reports that:

• GN reaches a loss of 3.25 in 54 steps
• SOAP requires 292 steps to reach the same loss
  → roughly 5.4× fewer iterations, and GN also extends the critical batch size.

Describe the solution you’d like

Add a Layerwise Gauss-Newton (GN) preconditioning mode, with an optional “Full GN (oracle)” flag for research comparison.

1. Optimizer Core

• Integrate GN updates via JVP-based preconditioning, avoiding explicit Hessian materialization.
• GN acts as a plug-in preconditioner that wraps existing optimizers (SOAP / Muon / Shampoo).
• Use inner-loop Muon or AdamW to minimize the quadratic objective under GN preconditioning, with optional line search for stability.

2. Layerwise GN Variant (Default)

• Compute per-layer GN updates (ignore cross-layer curvature).
• Nearly matches Full GN on medium-scale LLMs and large batches — requiring only ~1.4× more steps than Full GN but ~3.4× fewer steps than SOAP.

3. Why GN?

• GN captures curvature from the loss surface only (positive semi-definite),
  avoiding negative-curvature instability of full Newton updates while significantly improving iteration efficiency at scale.

Describe alternatives you’ve considered

• Existing SOAP / Muon / Shampoo implementations already provide approximate second-order preconditioning,
  but they lack full curvature fidelity and plateau earlier in large-batch regimes.
• Extending these optimizers with GN-based preconditioning could preserve backward compatibility
  while improving step efficiency.
• The GN-prox-linear variant was analyzed but offered little gain, suggesting the loss curvature alone captures most of the benefit.

Additional context

• Treat Full GN as a research-only configuration (≈ 4–5× slower wall-clock).
• Layerwise GN is the practical, scalable variant to evaluate for improved step efficiency and batch scaling.
• Recommended evaluation setup: 45M and 150M-parameter models under large-batch regimes.
• GN update formula:
  [
  \theta_{t+1} = \theta_t - G^{-1} g,\quad G = J^\top \nabla_z^2 L, J
  ]
  implemented efficiently via JVPs without explicit Hessian storage.

Start by reviewing the existing SOAP, Muon, and Shampoo implementations and the repository's optimizer entry points. Research the JVP-based Gauss-Newton design, layerwise approximation, inner-loop optimizer, and optional full-GN flag before defining the integration. Done means a scalable layerwise mode, a research-only full-GN configuration, and evaluation on the specified 45M and 150M models.