Matrix-Preconditioned Optimizers

• Matrix-preconditioned optimizers are a class of methods that use matrix-valued preconditioners to adjust gradient updates based on curvature and higher-order statistics.
• They employ strategies such as Kronecker-factored, diagonal, and low-rank representations, balancing computational load with effective curvature adaptation.
• These techniques improve convergence speed and stability across applications by reweighting update directions to better handle anisotropies in parameter space.

Matrix-preconditioned optimizers are optimization algorithms that modify the standard gradient update by multiplying the gradient by a matrix–valued preconditioner, rather than a scalar or coordinatewise factor. This preconditioning transforms the geometry of the underlying optimization problem, often equalizing curvature or reweighting update directions based on higher-order statistics such as the covariance or Fisher information matrix. The resulting updates adapt more flexibly to anisotropies and correlations in parameter space than methods relying on vector-based (i.e., coordinatewise) rescaling. Matrix-preconditioned methods include structure-aware stochastic optimizers for deep learning, diagonal scaling algorithms for linear systems, block or Kronecker-factored methods for large neural networks, and both theoretical and empirical frameworks for learned and adaptive preconditioning.

1. Foundational Principles of Matrix Preconditioning

The core principle of matrix preconditioning is to transform the optimization landscape so that the condition number of the effective Hessian or gradient covariance is reduced, thereby accelerating and stabilizing convergence. In its generic form, the preconditioned gradient step is:

$x_{t+1} = x_t - \eta P_t \nabla f(x_t)$

where $P_t$ is a positive-definite matrix that may be data-dependent, iteratively accumulated, or learned on the fly. Preconditioners can be full (dense), block-diagonal, Kronecker-factored, or diagonal, with choices determined by statistical power and computational constraints.

An early instantiation in stochastic gradient optimization is Shampoo (Gupta et al., 2018), which preserves the tensor or matrix structure of neural network weights by maintaining per-dimension preconditioners (such as left and right preconditioning matrices for a weight matrix), so that expensive full-matrix updates are avoided yet rich second moment structure is retained. The update rule for a weight matrix $W_t \in \mathbb{R}^{m \times n}$ with gradient $G_t$ is:

$$L_t = L_{t-1} + G_t G_t^T, \quad R_t = R_{t-1} + G_t^T G_t, \quad W_{t+1} = W_t - \eta L_t^{-1/4} G_t R_t^{-1/4}$$

which yields a Kronecker product preconditioner $L_t \otimes R_t$ acting on the vectorized gradient.

In deep learning, such structured approaches combine the curvature handling efficacy of full-matrix AdaGrad or (inverse) Fisher preconditioning with scalable memory and runtime complexity. Extensions include blockwise, Kronecker factorizations, or even polar decomposition-based steps for matrix weights (Lau et al., 27 May 2025).

2. Algebraic and Algorithmic Strategies

There exist both explicit and implicit strategies for building matrix preconditioners:

• Structure-Aware Preconditioning: Shampoo and Kron-based optimizers update per-dimension or block Kronecker factors, reducing both storage and computation (O(m^2 + n^2) per layer for Shampoo, O(rn) for dynamic low-rank AdaGram (Matveeva et al., 28 Aug 2025)).
• Learning-Based Approaches: LODO (Liao et al., 2022) and Optimus (Gärtner et al., 2022) employ neural networks (e.g., deep, sparse block-shaped, or transformer-based architectures) to meta-learn preconditioners, representing highly expressive positive-definite matrices via parameterized networks and optimizing these via hypergradient descent.
• Probabilistic Inference and Low-Rank Adaptation: Probabilistic inference over matrix–variate Gaussian models actively constructs low-rank Hessian approximations to form preconditioners with only a subset of eigen-directions "explained" at each step (Roos et al., 2019, Matveeva et al., 28 Aug 2025). Fast symmetric factorization (e.g., Cholesky updates, Sherman-Morrison-Woodbury identity) underlies efficient recursive or dynamic low-rank maintenance.
• Lie Group and Manifold-Based Updates: Preconditioners may be restricted to certain matrix Lie groups (diagonal, upper-triangular, Kronecker–factored, or general linear group with positive determinant) to enforce invariance, avoid singularity, and enable stable multiplicative parameterization (Li, 2018, Li, 2022).
• Polynomial and Splitting Preconditioning: In structured linear systems (e.g., block tridiagonal KKT conditions in control), polynomial multi–splitting preconditioners can be constructed, with optimal parametrizations designed to minimize eigenvalue spread and maximize iterative solver convergence (Yang et al., 19 Mar 2025).

The table below summarizes prominent scheme types and key features:

Preconditioner Type	Structure/Representation	Example Algorithms
Kronecker-factored	Per-dimension or block Kronecker	Shampoo, KrADagrad, KFAC
Diagonal	Coordinatewise scaling	AdaGrad, Adam, Jacobi, Optimal Diag
Polar decomposition	Orthogonal + nuclear-norm scaling	PolarGrad, Muon
Low-rank/dynamic	Truncated basis updates/SVD	AdaGram, Probabilistic PGD
Matrix-free/Lie group	Lie algebra/connected group	Black Box PSGD, Matrix Lie SGD

3. Theoretical Guarantees and Convergence Analysis

Matrix-preconditioned optimizers have been rigorously analyzed with respect to regret bounds, convergence rates, and spectrum control:

• Shampoo achieves regret $O(\sqrt{T})$ in the online convex optimization framework, matching the optimal rate for worst-case adversarial settings (Gupta et al., 2018).
• Convergence proofs for structured preconditioners rely on trace and geometric mean inequalities (e.g., Ando's inequality), and show that Kronecker-factored alternatives do not substantially degrade performance relative to full-matrix AdaGrad.
• Polynomial preconditioners for block tridiagonal systems are parametrized to cluster spectrum, with optimal reduction (up to 76%) in condition number and corresponding lower conjugate gradient (CG) iterations for KKT systems (Yang et al., 19 Mar 2025).
• SVD-based diagonalization of full-matrix preconditioners (e.g., AdaDiag++) enables diagonal adaptation in a rotated basis where the covariance is made near-diagonal, offering sample efficiency improvements and up to 2x speedup in foundation model training (Nguyen et al., 11 Feb 2025).
• Subgradient methods for diagonal preconditioning—using affine pseudoconvex reformulations—yield provable convergence and scalability to large matrices (Ghadimi et al., 27 Sep 2025), with O(1/𝜖²) iteration complexity for the worst-case condition number and closed-form solutions for criteria such as the ω-condition number (trace/determinant average).

4. Practical Implementation and Scaling Considerations

Matrix preconditioners draw a balance between computation, memory, and spectral effectiveness. Several implementation advances have broadened their applicability:

• Tensor and block-wise strategies enable scaling to networks with millions of parameters, as in Shampoo's incorporation into TensorFlow and deployment on image classification networks (ResNet, Inception) and LLMs (Gupta et al., 2018).
• Quantized preconditioners, including 4-bit quantization with Cholesky decomposition and error feedback, minimize memory overhead with negligible loss of convergence and model performance, halving the footprint per layer (Li et al., 14 Dec 2024).
• Approximate preconditioner computation—e.g., Jorge's binomial expansion for inverse roots—removes bottlenecks on GPU by replacing inversion with multiplication/addition, permitting wall-clock parity with first-order methods (Singh et al., 2023).
• Integration with adaptive optimizers is possible through diagonalization in a rotated space (via SVD), which is compatible with Adafactor and Adagrad memories while retaining fast convergence (Nguyen et al., 11 Feb 2025).
• AI-driven parameter search (Bayesian optimization with graph neural surrogates) can optimize hyperparameters of MCMC-based matrix inversion preconditioners, reducing the search cost and improving Krylov iteration convergence in large-scale sparse systems (Lebedev et al., 22 Sep 2025).

5. Comparative Empirical Performance

Matrix-preconditioned optimizers consistently yield superior or at least competitive convergence rates compared to first-order or diagonal methods, especially in regimes where curvature information or parameter correlations are significant.

• Shampoo yields faster loss reduction on image classification and LLMing tasks, with training throughput per step on par with SGD and Adam (Gupta et al., 2018).
• KrADagrad demonstrates robustness to limited floating-point precision, outperforming Shampoo on ill-conditioned synthetic problems under 32-bit precision, while matching generalization performance on practical deep learning datasets (Mei et al., 2023).
• Learnable preconditioners (LODO, Optimus) and transformer-based rank-one update methods can match or surpass BFGS and Adam in classical optimization as well as complex high-dimensional real tasks, with O(n log n) per-step cost (Liao et al., 2022, Gärtner et al., 2022).
• Structure-aware updates—using polar decomposition and nuclear norm scaling (PolarGrad)—address gradient anisotropy in matrix weights and outperform both Muon (orthogonalization only) and Adam (diagonal curvature) in matrix regression and LLM pretraining (Lau et al., 27 May 2025).
• Empirical scaling studies in LLM pretraining reveal that matrix-based optimizers (Muon, SOAP, Kron) offer up to 1.3–1.4x speedup over AdamW in small-to-medium model scales (≈130M parameters), but the relative gain diminishes to ≈1.1x in the 1B+ parameter regime (Wen et al., 2 Sep 2025). The decreasing marginal benefit with size suggests a nuanced trade-off between computational overhead and effective utilization of parameter structure.

6. Domain-Specific Applications and Future Directions

Matrix-preconditioned optimization underpins efficient algorithms in both deep learning and scientific computing:

• In large-scale PDE-constrained optimization (e.g., control, inverse problems), block-preconditioners for saddle-point systems cluster spectrum near unity, yielding mesh-independent GMRES convergence and substantial reduction in iterations and CPU time (Mirchi et al., 2019).
• In robotics, polynomial and parallelizable preconditioners enable real-time optimal control on GPU by consolidating eigenvalue spectrum and reducing CG iterations in block tridiagonal systems (Yang et al., 19 Mar 2025).
• Optimal diagonal (or block-diagonal) preconditioning via convex or pseudoconvex optimization yields near-theoretically minimal condition numbers, benefiting both deep stochastic optimization and scientific linear algebra (Qu et al., 2022, Ghadimi et al., 27 Sep 2025).
• Memory-efficient variants, such as quantized Cholesky approaches, address the bottleneck of preconditioner storage for state-of-the-art neural networks without loss of accuracy (Li et al., 14 Dec 2024).

Future work includes: adaptive and online updating of structure-aware preconditioners in streaming or distributed settings (Qu et al., 2022); improved theoretical characterizations for loss surfaces where high-order structure or curvature adaptation plays a dominant role; and scalable learning-based frameworks that can generalize across model architectures and training regimes (Liao et al., 2022). The main open challenge remains achieving an optimal blend of expressiveness, computational tractability, and end-to-end benefits in both statistical efficiency and wall-clock runtime, especially as model dimensionality and batch sizes continue to grow.