Matrix-Based Optimizers

Updated 5 September 2025

Matrix-based optimizers are algorithms that operate directly on matrices, leveraging spectral properties and low-rank structures for enhanced efficiency.
They apply advanced methods like spectral analysis and monotonicity theory to enable robust optimization in fields such as deep learning, signal processing, and quantum computing.
Efficient matrix preconditioning and structured reformulations allow for faster convergence and improved numerical stability in large-scale problems.

Matrix-based optimizers are algorithms and computational frameworks in which the principal optimization steps operate directly on matrices or matrix structures, rather than treating parameters or design variables as flat vectors. These optimizers leverage the algebraic, spectral, or structural properties of matrices, enabling efficient exploitation of problem-specific features such as low-rank structure, symmetry, or monotonicity. Widely adopted in disciplines ranging from signal processing and control to deep learning, scientific computing, and quantum algorithms, matrix-based optimizers encapsulate a diverse set of approaches—from explicit matrix-valued updates in learning algorithms, to structured semidefinite programming, to compiler-level routines that optimize low-level matrix-matrix operations. Theoretical developments in spectral operator analysis, monotonicity, and differentiability underpin advanced methods, while practical implementations exploit matrix properties for computational acceleration, stability, and hardware-level efficiency.

1. Foundational Principles and Spectral Operators

The mathematical foundation of matrix-based optimizers is rooted in the notion that many optimization problems can be most naturally and efficiently formulated with matrices as primary objects. Central to this is the formalization of spectral operators—matrix-valued functions generated by applying a scalar function to each singular value or eigenvalue of an input matrix (Ding et al., 2014). This generalizes the classical Löwner operator and is crucial in regularized or non-smooth settings, as encountered in low-rank matrix recovery, compressed sensing, and structured convex optimization.

Key properties established for spectral operators include:

Well-definedness and continuity: Ensuring functional mappings behave robustly across the relevant matrix domains.
Directional differentiability and $\rho$ -order semismoothness: Addressing optimization in the presence of non-smoothness, providing formulas for the B(ouligand)-differentiability and G-semismoothness.
Characterization of Clarke's generalized Jacobian: Precise analysis of subdifferential sets (B-subdifferential) for spectral operators, captured via limits of derivatives from approximating sequences.

A systematic treatment, for example, proceeds by representing the derivative-term from the nonsmooth part via decompositions leveraging the SVD (or EVD) and partitioning indices according to the active (nonzero) and inactive (zero) blocks. The limiting structure is codified in identities such as

$\lim_k M_k^T @'(W_k) D(H) N_k = \lim_k [M^T @'(W_k) D(H) N],$

where $M_k, N_k$ are orthogonal factors from the SVD, and $D(H)$ encodes divided differences. This formalism is foundational for robust subdifferential calculus and for developing efficiently convergent iterative solvers.

2. Matrix-Monotonic Optimization and Structured Solution Forms

Matrix-monotonic optimization exploits the order structure of the cone of positive semidefinite matrices to decouple high-dimensional or multi-variable optimization problems into tractable subproblems amenable to closed-form or efficiently solvable forms (Xing et al., 2018, Xing et al., 2020). The core insight is as follows:

For optimization objectives that are monotonic in the eigenvalues (or singular values) of a PSD matrix argument—for example, trace or determinant-based metrics—it is possible to diagonalize the problem via unitary transformations.
Variables are decomposed as $X = F Q_x$ , where $F$ is optimally structured (often diagonal or block-diagonal), and $Q_x$ is an auxiliary unitary that can often be optimized separately or set to align with problem symmetries.

Typical application domains include MIMO transceiver design, robust sensor networks, and multihop relaying, with constraints encompassing sum-power, per-antenna power, shaping, and joint constraints. The matrix-monotonic framework enables:

Derivation of Pareto-optimal (often diagonalizable) structures for matrix variables, e.g.,

$F_{\mathrm{opt}} = U_{(\Pi)} \Lambda_F U_{(\mathrm{Arb})}^H,$

with $U_{(\Pi)}$ from an EVD of the effective channel matrix and $\Lambda_F$ determined by a water-filling or similar scheme.

Unification of diverse performance metrics within a single formalism, simplifying both analysis and implementation.
Physical interpretability, such as the mapping of $F^H \Pi F$ to "matrix-SNR" allocations over channel eigenmodes, and insights into trade-offs induced by fairness or robustness requirements.

For multi-variable matrix problems, key theoretical results show that right-unitarily invariant constraints permit blockwise decoupling and iterative optimization—each variable admits an optimal local structure that reduces search dimensionality (Xing et al., 2020).

3. Matrix Optimization in Applications: Numerical Methods, Deep Learning, and Scientific Computing

Practical matrix-based optimizers encompass a range of algorithmic strategies, including:

Compiler-level optimization of matrix multiplication: Advanced search (e.g., G-BFS, N-A2C) for GEMM configurations yields major computational savings, such as 24–40% improvements in deep learning training and inference, by efficient cache tiling and hardware adaptation (Zhang et al., 2019).
Structured problem reformulation: For instance, bilinearization and mixed-integer programming (MIQCQP, MILP) are employed to recast matrix product constraints in material science and genomic treatment planning, yielding orders-of-magnitude speedups over heuristics or enumeration (Kocuk, 2020).
Memory-efficient optimization via factorizations: Innovations such as SMMF enable the optimization of momentum tensors for arbitrary-rank parameter structures in deep models, reducing memory by up to 96% while matching convergence rates of dense methods (Park et al., 12 Dec 2024).
Sparse matrix operations in autodiff frameworks: High-performance, automatic differentiation-enabled implementations (e.g., PyTorch CSR kernels) facilitate training and deployment of optimizers for very large-scale sparse systems, including preconditioned iterative methods and GCNs (Nytko et al., 2022).
Quantum algorithms for matrix-based optimization: Mirror descent methods generalized to the space of density matrices and the use of matrix multiplicative weights update enable scalable quantum and classical semidefinite programming (Nannicini, 8 Aug 2024).

Notably, deep learning applications have seen significant adoption of matrix-based preconditioners, including methods that exploit Kronecker, block-diagonal, or low-rank approximations of the Fisher Information Matrix (FIM) for robust adaptivity and efficient handling of curvature (Gomes, 26 Apr 2025, Gong et al., 11 Feb 2025).

4. Advanced Preconditioning and Matrix-aware Gradient Methods

State-of-the-art matrix-based optimizers rely critically on matrix-aware preconditioning, designed to exploit the anisotropy or inherent directional biases in gradient statistics—particularly in large neural network training:

Preconditioner Diagonalization: Transforming the gradient or its covariance into a basis where the preconditioner is nearly diagonal (e.g., via SVD of the gradient) facilitates effective scaling with reduced computational burden (Nguyen et al., 11 Feb 2025).
Low-rank and hierarchical FIM approximations: Structured approximations, such as RACS and Alice, systematically balance memory usage and curvature fidelity via tensor-product or subspace methods and compensatory updates, achieving ≥2× convergence rate improvements over traditional Adam in LLMs while maintaining SGD-like or slightly higher memory cost (Gong et al., 11 Feb 2025).
Adaptive block-diagonal Kronecker methods: Approximating the FIM by diagonals of Kronecker factors, as in AdaFisher, allows for second-order preconditioning with per-iteration computation and storage comparable to first-order methods, yet achieves faster convergence and improved generalization (Gomes, 26 Apr 2025).
Polar decomposition and nuclear norm scaling: PolarGrad and related methods precondition updates by extracting the orthogonal "direction" and scaling by the nuclear norm, yielding null-gradient consistency and more robust handling of ill-conditioning compared to matrix sign or vectorized methods. This leads to more stable and accelerated convergence, especially for "fat" matrix layers in deep models (Lau et al., 27 May 2025).

The distinction between curvature-anisotropy (as addressed by vector-based preconditioners such as Adam) and gradient-anisotropy (addressed by matrix-aware normalization such as orthogonalization or polar decomposition) is shown to have direct consequences for training stability, speed, and scaling properties.

5. Benchmarks, Hyperparameter Sensitivity, and Scaling Behavior

Extensive benchmarking reveals nuanced efficacy and limitations for matrix-based optimizers:

On smaller-scale models (e.g., $\sim$ 130M parameters), matrix-based methods often deliver $\sim$ 1.3–1.4× speedup—in terms of steps or tokens to target loss—compared to optimally tuned AdamW. However, this edge diminishes to $\sim$ 1.1× for models in the $\gtrsim$ 1B parameter regime (Wen et al., 2 Sep 2025). The improvement is inversely proportional to model scale.
Large-scale benchmarks emphasize the critical importance of rigorous, regime-aware hyperparameter tuning. Small deviations or blind hyperparameter transfer can easily distort comparative rankings and exaggerate purported speedups.
Intermediate learning curves can be misleading; optimizer rankings may flip due to learning rate decay or warm-up scheduling, underscoring the necessity of final checkpoint evaluation for realistic assessment.
Detailed implementation of preconditioners (e.g., number of Newton–Schulz or polar decomposition steps, periodicity of SVD rotations, low-rank compensation) directly affects both speed and stability; practical deployment must balance infrequent, efficient updates against the risks of stale statistics or introduced numerical instability.

The table below summarizes representative empirical results traced to the data:

Optimizer	Model Scale	Speedup vs AdamW	Notes
Muon, SOAP, Kron	0.1B–0.13B params	1.3–1.4×	Matrix-based, well-tuned regimes
Muon, SOAP, Kron	1.2B params	1.1×	Edge decreases with scale
AdaDiag, AdaDiag++	LLaMA (60M–350M)	$\sim$ 2×	Validation perplexity halved in steps
Alice (low-rank FIM)	LLaMA up to 1B	≥2×	Low-memory, faster than Adam
SMMF (memory-optimized)	CNNs, Transformers	≤4% of baseline	Up to 96% memory saving

A plausible implication is that for future LLM scales, architectural and hardware advances must synergize with preconditioner innovation to sustain further improvement.

6. Matrix-Based Optimization Beyond Deep Learning

Matrix-based optimizers have broad impact in diverse fields:

In materials science, MIQCQP reformulations enable rigorous control of multilayer film reflectance, outperforming classical heuristics and enumeration, especially in small or odd-layer cases (Kocuk, 2020).
In control and systems engineering, zero-order matrix optimization frameworks yield improved complexity bounds and controller tuning efficacy for symmetric and positive-definite matrix gain selection, with reduced experimental burden (Maass et al., 2021).
In quantum computing and semidefinite programming, matrix multiplicative weights update (a matrix analogue of mirror descent with von Neumann entropy regularization) provides foundational iterative methods for both classical and quantum SDPs (Nannicini, 8 Aug 2024).
Data-centric scientific computing benefits from array algebra and mathematics-of-arrays approaches that directly formulate entire algorithms and hardware mappings via dimension-lifting, optimizing both performance and energy on modern accelerators (Mullin, 2023).

7. Future Directions and Open Challenges

Despite significant advances, several open research directions remain:

Extending low-rank and block-structured matrix preconditioners to regimes that require adaptive updating and non-stationary data, with guarantees on numerical robustness and minimal hyperparameter tuning.
Development of matrix-based optimizer kernels and algebraic primitives optimized for next-generation hardware, including custom sparse or low-rank formats, and seamless integration with distributed and asynchronous systems.
Closing the gap between theoretical convergence rates and practical stability, particularly in highly non-convex, large-scale problems, and quantifying the trade-offs induced by structured approximations or hybrid diagonalization schemes.
Systematic benchmarking protocols for optimizer comparison across architectures, dataset regimes, and final-task metrics, including the adoption of best practices for hyperparameter sweep and evaluation protocol.
Exploration of quantum and hybrid quantum-classical algorithms for large-scale matrix optimization tasks, leveraging block-encodings and efficient subroutines from quantum algorithmics.

Matrix-based optimizers constitute a foundational and rapidly evolving field, synthesizing deep mathematical analysis (spectral and monotonicity theory), advanced algorithmic design (preconditioning, factorization, structured approximation), and large-scale empirical validation (benchmarks on vision, language, and scientific datasets). Their utility is evident both in core optimization methodologies and as a substrate enabling efficient deployment and new learning paradigms in modern computational science and engineering.