Preconditioned Matrix Norms in Optimization

• Preconditioned matrix norms are defined by altering the underlying metric with positive-definite preconditioners to modify the geometric and spectral properties of matrices.
• They provide a unified framework linking classical conditioning theory with modern optimization techniques such as quasi-Newton methods, adaptive optimizers, and steepest descent using various base norms.
• This concept is pivotal in applications ranging from deep learning to signal processing, where it improves convergence rates, stability, and invariance under reparameterizations.

A preconditioned matrix norm is a matrix norm defined with respect to an explicit preconditioning transformation—usually via left and/or right multiplication or via a change in the underlying metric—so that the geometric or spectral properties of the matrix are modified to improve computational, stability, or convergence qualities. This abstraction provides a unified perspective in numerical linear algebra, convex optimization, signal processing, machine learning, and large-scale data science, connecting classical conditioning theory, iterative solvers, and the design of geometry-aware optimization algorithms.

1. Definition and Formalization of Preconditioned Matrix Norms

A preconditioned matrix norm arises when the underlying geometry or metric in which the matrix acts is altered by a positive-definite operator (the preconditioner). Specifically, for a matrix $G \in \mathbb{R}^{m \times n}$, let $L \in \mathbb{R}^{m \times m}$ and $R \in \mathbb{R}^{n \times n}$ be symmetric positive-definite matrices. The $(L, R)$-preconditioned matrix norm associated with a base norm $\|\cdot\|$ (such as Frobenius, spectral, or entrywise $\ell_p$) is defined as

$\|G\|_{(L, R), \cdot} = \| L \, G \, R \| .$

Alternatively, with a diagonal positive-definite matrix $D$, the $D$-preconditioned norm (entrywise scaling) is

$\|G\|_{(D), \cdot} = \| D \odot G \| ,$

where $\odot$ denotes the Hadamard (elementwise) product.

This generalization subsumes “classical” norms ($L = I$, $R = I$), preconditioned vector or operator norms, and norm choices implicit in quasi-Newton, natural-gradient, and adaptive optimization methods.

2. Unifying Framework for Optimization Algorithms

The preconditioned matrix norm concept provides a unified framework connecting:

• Steepest descent in general norms: Each geometry is encoded by the choice of base norm and preconditioner; for example, using the spectral norm versus Frobenius, or left/right scaling versus entrywise scaling.
• Quasi-Newton methods: Here, curvature information is encoded in Kronecker-factored preconditioners $L = (H^L)^{-1/2},\ R = (H^R)^{-1/2}$, yielding update steps of the form $\Delta W_t = (H^L)^{-1} G_t (H^R)^{-1}$.
• Adaptive (diagonal) optimizers: Preconditioners $D$ are computed from running averages of gradient statistics, as in Adam, AdaGrad, or SANIA, producing normed updates $\Delta W_t = D^{-1} \odot G_t$.

This framework shows that state-of-the-art optimizers, such as Muon, KL-Shampoo, SOAP, SPlus, as well as SGD and Adam, fit into a single geometric principle in which all are steepest descent methods in appropriately preconditioned norm geometries (Veprikov et al., 12 Oct 2025).

3. Mathematical Properties: Invariance and Geometry

A central theoretical advance is the systematic treatment of invariances in the preconditioned norm setting.

• Affine invariance is ensured if, for a reparameterization $W \mapsto A_L W A_R$, the preconditioners transform as $$L^{(\mathcal{L}_{\text{new}})} = L^{(\mathcal{L})} A_L$$ and $$R^{(\mathcal{L}_{\text{new}})} = A_R R^{(\mathcal{L})}$$, making the optimizer’s trajectory coordinate-system independent.
• Scale invariance is established if a reparameterization $W \mapsto A \odot W$ yields preconditioner transformation $$D^{(\mathcal{L}_{\text{new}})} = A \odot D^{(\mathcal{L})}$$.
• LMO characterization: The linear minimization oracle (LMO) in the preconditioned norm yields explicit update rules:

$$\begin{aligned} \mathrm{lmo}_{(L,R),\cdot}(G) &= L^{-1} \, \mathrm{lmo}_{\cdot}(L^{-T} G R^{-T}) \, R^{-1} \ \mathrm{lmo}_{(D),\cdot}(G) &= D^{-1} \odot \mathrm{lmo}_{\cdot}(D^{-1} \odot G) \end{aligned}$$

where $\mathrm{lmo}_{\cdot}$ is the LMO for the base norm (e.g., the spectral or entrywise norm).

This affine and scale invariance is critical because it ensures that optimization dynamics are robust to reparameterizations and feature rescalings often encountered in deep learning applications (Veprikov et al., 12 Oct 2025).

4. Algorithmic Instantiations and Specialized Methods

Within the preconditioned norm framework, many classical and modern optimization algorithms can be interpreted as special cases or combinations:

Method	Preconditioner Structure	Base Norm
SGD	$L=I$, $R=I$ (none)	Frobenius
AdaGrad/Adam	$D$ diagonal (entrywise)	Entrywise $l_2$ or $l_\infty$
Shampoo, KLShampoo	$L, R$ Kronecker-factored from Hessian	Frobenius
Muon	$L=I, R=I$, but spectral norm	Spectral
MuAdam/SANIA	$D$ diagonal (adaptive) + spectral	Spectral

Recent hybrid methods such as MuAdam (spectral + adaptive elementwise preconditioning) and MuAdam-SANIA (spectral + SANIA-like update for scale invariance) directly emerge from this abstraction, combining complementary strengths of spectral and diagonal geometries. In these methods, preconditioning steps adapt to curvature both globally (via SVD or approximate Newton–Schulz for spectral norm) and locally (via elementwise statistics), enabling robust and efficient optimization under varying data and model scaling (Veprikov et al., 12 Oct 2025).

5. Theoretical Foundations and Implications

Preconditioned matrix norms allow a precise characterization of how update steps utilize information about the local geometry of the loss landscape:

• Curvature adaptation: The preconditioner encodes (approximate) second-order information, thereby accelerating convergence in ill-conditioned or anisotropic problems.
• Geometric robustness: Correct invariance properties guard against the optimizer trajectory being unduly influenced by parameterization artifacts.
• Norm-induced constraints: The LMO approach allows (potentially non-Euclidean) norm constraints that can reflect problem priors or robustness criteria.

This perspective also clarifies the trade-offs between different classes of methods in terms of their ability to exploit curvature, adapt to geometry, and retain invariance, and enables the systematic design of new optimizers by mixing and matching preconditioners and norms.

6. Empirical Performance and Applications

The paper’s experiments highlight several applications and practical implications:

• Scale invariance is verified empirically; optimizers with proper preconditioned norm structure (SANIA, MuAdam-SANIA) yield invariant training dynamics regardless of input scaling, whereas standard AdamW and Muon can degrade when features are rescaled.
• LLM and GLUE task benchmarks: The proposed MuAdam and MuAdam-SANIA perform on par or surpass AdamW and Muon in distillation and LoRA fine-tuning of transformer models and in character-level language modeling tasks.
• Generalizability: The preconditioned norm framework encapsulates popular methods across diverse architectures, demonstrating adaptability from point-wise to layer-wise update rules and from Euclidean to strictly non-Euclidean geometries.

A plausible implication is that the preconditioned norm abstraction can guide the principled development of new optimization algorithms for deep models, especially as model size and heterogeneity in architecture increase.

7. Broader Significance and Directions

Establishing preconditioned matrix norms as a foundational concept bridges numerical linear algebra, optimization theory, and scalable algorithm design for modern machine learning. This provides:

• A rigorous language for describing and analyzing a wide class of optimization algorithms, connecting root concepts from matrix analysis (norms, spectral theory, invariance) with practical algorithm design.
• A framework for studying generalization, robustness, and model scaling properties as induced by the optimizer’s underlying geometry.
• Tools for further research—by varying the base norm or preconditioning structure, more specialized or application-tailored methods can be developed.

Given this, the systematic abstraction and implementation of preconditioned matrix norms will likely continue to inform both theoretical advances and practical tools in large-scale optimization and learning (Veprikov et al., 12 Oct 2025).