Pro-KLShampoo: Projected KL-Shampoo with Whitening Recovered by Orthogonalization
Authors: Ruotong Sun, Ermin Wei
Organizations: †Department of Industrial Engineering & Management Sciences, Northwestern University · ∗Department of Electrical & Computer Engineering, Northwestern University
Abstract
Optimizers that exploit the matrix structure of gradients are central to modern LLM pre-training, with two distinct frontiers: explicit Kronecker-factored preconditioning -- most recently KL-Shampoo, which estimates the preconditioner via KL divergence minimization -- and orthogonalization of the gradient momentum, exemplified by Muon and analyzed as steepest descent under the spectral norm. The two routes are typically developed in isolation. We make a structural observation about KL-Shampoo's Kronecker preconditioners: their eigenvalue spectra exhibit a \emph{spike-and-flat} shape -- a few dominant eigenvalues followed by an approximately uniform tail -- across layers and training stages, holding exactly under a rank-ρ signal-plus-noise gradient model. We exploit this structure by restricting one of KL-Shampoo's Kronecker factors to a parametric family aligned with the spike-and-flat shape: full spectral structure on a tracked r-dimensional subspace, single shared eigenvalue across the remaining n−r directions. On these directions, we apply orthogonalization. An identity shows that this orthogonalization recovers the algebraic form of full KL-Shampoo's preconditioner. On four pre-training scales (GPT-2 124M / 350M, LLaMA 134M / 450M), Pro-KLShampoo consistently outperforms KL-Shampoo at every subspace rank we test in validation loss, peak per-GPU memory, and wallclock time to reach each loss level.
Shampoo and its efficient variant, SOAP, employ structured second-moment estimations and have shown strong performance for training neural networks (NNs). In practice, however, Shampoo typically requires step-size grafting with Adam to be competitive, and SOAP mitigates this by applying Adam in Shampoo's eigenbasis -- at the cost of additional memory overhead from Adam in both methods. Prior analyses have largely relied on the Frobenius norm to motivate these estimation schemes. We instead recast their estimation procedures as covariance estimation under Kullback-Leibler (KL) divergence minimization, revealing a previously overlooked theoretical limitation and motivating principled redesigns. Building on this perspective, we develop KL-Shampoo and KL-SOAP, practical schemes that match or exceed the performance of Shampoo and SOAP in NN pre-training while achieving SOAP-level per-iteration runtime. Notably, KL-Shampoo does not rely on Adam to attain competitive performance, eliminating the memory overhead introduced by Adam. Across our experiments, KL-Shampoo consistently outperforms SOAP, Shampoo, and even KL-SOAP, establishing the KL-based approach as a promising foundation for designing structured methods in NN optimization. An implementation of KL-Shampoo/KL-SOAP is available at https://github.com/yorkerlin/KL-Methods
Shampoo is one of the leading approximate second-order optimizers: a variant of it has won the MLCommons AlgoPerf competition, and it has been shown to produce models with lower activation outliers that are easier to compress. Yet, applying Shampoo currently comes at the cost of significant computational slowdown, due to its expensive internal operations. In this paper, we take a significant step to address this shortcoming by proposing \method (for \textbf{D}istributed \textbf{A}ccelerated \textbf{SH}ampoo), a faster implementation of Distributed Shampoo based on two main new techniques: First, we show that preconditioner blocks can be stacked into 3D tensors to significantly improve GPU utilization; second, we introduce the Newton-DB iteration and the Chebyshev polynomial approximations as novel and faster approaches for computing the inverse matrix roots required by Shampoo. Along with these algorithmic contributions, we provide a first in-depth analysis of how matrix scaling critically affects Shampoo convergence. On the practical side, our GPU-aware implementation achieves up to 5.6× faster optimizer steps compared to the well-optimized Distributed Shampoo, while Newton-DB attains the lowest validation perplexity per iteration among all tested methods. Our code is available at https://github.com/IST-DASLab/DASH.
Ionut-Vlad Modoranu, Philip Zmushko, Erik Schultheis +2
The massive scaling of Large Language Models (LLMs) has made pretraining increasingly cost-prohibitive. While low-rank representation and orthonormal weight matrices could in principle reduce parameter counts and computational overhead, most existing methods rely on static rank selection and do not enforce weight orthonormality due to high computational cost. This paper introduces TSVD, a framework that maintains low rank and strict orthonormality throughout the training process. It utilizes a spectral energy-based heuristic for adaptive rank selection, and a caching mechanisms to maintain orthonormality. Theoretical analysis justifies the advantage of the approach in pretraining dynamics and experiments across various model scales demonstrate that it is effective empirically. TSVD matches or exceeds the performance of full-parameter baselines while significantly reducing compute requirements. The approach thus offers a well-founded, practical, and scalable path toward efficient high-performance LLM pretraining.