Gauss-Newton Optimization

We study the accuracy of Gauss-Newton curvature in ridge-regularized nonlinear least squares. Under local regularity and persistence of level-set curvature magnitude along an exact-fit section, we prove uniform coexistence of two curvature regimes. Global minimizers exist, and every global minimizer has relative Hessian error below $(1+\sqrt2)/8$, while the same low-cost set contains a point with an indefinite Hessian and relative error at least $15/8$. One positive ridge cap works for all independent center and label perturbations in fixed neighborhoods and every positive ridge weight up to the cap. These neighborhoods do not shrink as the ridge weight tends to zero. A pointwise certificate based on the current prediction level set controls the normal, mixed, and tangent parts of the Hessian correction. We prove a sharp relative-error bound over the stated pointwise class when the prediction map and ridge vary. Analytic examples describe the roles of output alignment, curvature orientation, and persistence. A separate structural result gives full Jacobian row rank throughout low-cost sets and exact interpolation near a rank-deficient reference.

Physics-informed neural networks and finite element methods provide two different paradigms for the numerical approximation of partial differential equations: the former are commonly trained by minimizing pointwise strong residuals, whereas the latter are naturally built from weak variational formulations and the finite-dimensional systems obtained after discretization. In this work, we introduce a common framework based on the discretization of functional Gauss--Newton problems by finite families of linear measurements. We show that, through an appropriate duality pairing, the linear measurements can be represented by test functions. The resulting Gauss--Newton system is then precisely a Petrov--Galerkin discretization of the linearized functional problem. This perspective recovers pointwise collocation and natural-gradient constructions as particular cases, while making the choice of test functions an explicit algorithmic design choice. We specialize this framework to elliptic problems, where it naturally leads to weak residual formulations and to a hybrid finite element--neural construction acting on complementary approximation spaces. Numerical experiments support the proposed framework and demonstrate the effectiveness of weak Gauss--Newton formulations and hybrid finite element--neural approximations.

Optical motion capture delivers high-fidelity human motion, but its reliance on strict marker layouts and clean trajectories severely limits its real-world applicability. In practice, tracking systems frequently output unconstrained markers: sparse, noisy, and unordered point clouds with unknown or varying configurations. To bridge the gap between corrupted raw markers and parametric human models, we introduce DirtyMoCap, a robust, marker-layout-free framework. Our core insight is to map unordered marker observations to a fixed set of "proxy anchors" comprising skeletal joints and body surface points, which serve as a stable intermediate representation. We first initialize and track these anchors over long sequences using a recurrent sliding-window architecture. Then, a custom differentiable Gauss-Newton solver fits the SMPL-H model to the tracked anchors to recover full-body pose, translation, and shape. By explicitly deriving geometric residuals, our solver learns adaptive observation confidence, smoothness, and prior weights end-to-end, adapting dynamically to the reliability of the input data. Extensive experiments on diverse, noisy marker configurations demonstrate that DirtyMoCap successfully generalizes across arbitrary layouts using only a single trained model. It consistently outperforms state-of-the-art configuration-specific baselines in both joint and vertex reconstruction accuracy, while our custom CUDA solver achieves up to a 100x speedup over standard PyTorch implementations. We further apply DirtyMoCap to heterogeneous raw optical MoCap recordings of traditional Chinese martial arts, yielding a Kung Fu motion dataset of temporally coherent SMPL-H reconstructions. Code and data are available at https://wanglongzju.github.io/DirtyMoCap-Project-Page.

Muon's polar update is exact for an unweighted spectral geometry. We introduce GO-MUON, which uses a matched data-dependent geometry and reuses it across several optimization steps. Conditioned on any positive-definite left and right maps, its raw update exactly solves the corresponding weighted spectral oracle; this statement is independent of how the maps are estimated or how recently they were refreshed. For softmax cross-entropy, we quantify when the observed-label backward factor approaches the model Fisher and generalized Gauss--Newton factor. We also show that four-step refresh nearly preserves the tracking delay of slowly changing geometry while increasing stationary factor noise, making lazy geometry a compute--statistics tradeoff rather than a denoising mechanism.

Nonlinear least-squares objectives form the foundation of scientific machine-learning tasks, yet even curvature-aware optimizers remain geometrically inconsistent at finite step sizes: Levenberg-Marquardt (LM) derives its direction from local Riemannian metrics but realizes it as a straight parameter update. Here we introduce RNC-LM, which carries the LM direction along a locally constructed curved trajectory in Riemann normal coordinates. A recursive reformulation of the geodesic equation generates arbitrary finite-order corrections while reusing the same damped Gauss-Newton matrix factorization, and curve length is controlled separately from damping. On a reaction-diffusion physics-informed neural network benchmark, RNC-LM reduces relative $L^2$ errors below $8\times10^{-3}$, whereas L-BFGS, LM and LM with geodesic acceleration remain near one. On a large-scale machine-learning potential fitting task with 985,160 configurations, fourth-order RNC-LM reaches a fixed training-error target with a $34\times$ wall-clock speedup over LM. These results establish finite-step geometric realization as a distinct optimizer design principle.

Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present $\textbf{DSGNAR}$: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. $\textbf{DSGNAR}$ couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative $\ell_2$ errors as low as $3\times10^{-16}$ in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to $\ell_2^{\text{rel}} = 4.75 \times 10^{-7}$ in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

Diffusion posterior sampling conditions diffusion priors on measurements, but data-consistency updates are typically scaled by hand-tuned guidance weights and can destabilize sampling under stiff, operator-dependent curvature. We replace scalar guidance with a per-noise-level damped Gauss--Newton correction computed in diffusion-state coordinates. The correction pulls likelihood gradients back through the denoiser, uses a one-sided curvature model that avoids forward denoiser Jacobians, and applies diffusion-calibrated rank-one damping aligned with the denoiser residual. Each correction is solved with matrix-free GMRES using automatic differentiation, and sampling proceeds with a variance-preserving Langevin transition with a closed-form drift/noise split. On FFHQ and ImageNet across inverse problems, it achieves competitive PSNR/SSIM/LPIPS while running markedly faster than most of the compared baselines; on accelerated MRI reconstruction, it achieves the best PSNR/SSIM among the compared baselines.

Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation through neural tangent kernel (NTK) analysis: for linearly coupled systems, the standard NTK's spectral radius grows as $Ω(γ^2)$ with coupling strength $γ$, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK $K_P = JH^{+}J^\top$ whose spectral radius is bounded by $S$ (number of networks), independent of $γ$. Adam's diagonal preconditioning destroys this projector structure -- inflating $λ_{\max}$ far above $S$ for any coupling type -- and its residual-dynamics kernel grows as $Θ(γ)$, placing its stable learning rate strictly between gradient descent and GN. For one-way coupling the limitation is class-wide: no diagonal preconditioner, fixed or adaptive, halves the driving residual in fewer than $Ω(γ)$ iterations ($Ω(γ^2)$ if fixed), whereas block-diagonal GN requires $O(1)$. We verify $Ω(γ^2)$ growth across linearly coupled benchmarks and confirm $λ_{\max}(K_P) = S$ in all three 1D systems, including nonlinearly coupled NP+P. Combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GradNorm) yields coupling-robust accuracy: across 222 experiments spanning three 1D systems and a 2D electroosmotic flow benchmark, SOAP+GradNorm maintains final-epoch $L_2$ accuracy across coupling strengths, with $\leq 2.3\times$ degradation in nonlinear NP+P while Adam+GradNorm fails ($L_2 > 0.1$). SOAP+GradNorm further scales to a 2D, 6-PDE electroosmotic flow at EDL-resolved conditions down to $\varepsilon = 0.01$ -- a regime all prior PINN electrokinetics studies have avoided -- where Adam+GradNorm fails entirely ($L_2 > 0.3$).

Authoring with a text-to-motion generator needs sparse anchors: chosen joints, at chosen frames, at given positions. Meeting them currently costs a conditioning branch trained for the task, or hundreds of per-clip optimisation steps in the architecture's native variables. In any generator that decodes a continuous state through a frozen differentiable decoder, the anchors ask for little and leave most of the state free: a few hundred numbers against a state of tens of thousands. Every control method is a choice among the states that satisfy them, and the choices differ along the directions the anchors cannot see and the motion can. SCAMP makes the choice that moves none of them: damped Gauss-Newton in the space of the anchors, through the frozen decoder alone, training-free, with one dimensionless damping constant. Every increment is a combination of the rows of the anchors' Jacobian, so the correction is orthogonal to everything the anchors never see, and the system solved is the size of the request rather than of the state. Applied unchanged to seven published generators spanning diffusion, token and latent designs, it matches or exceeds in anchor error every released control method it is measured against, and closes the anchors on hosts that ship none. Confined to those rows, a correction can only take the shapes the decoder admits, so what it costs belongs to the decoder, and holding the solver fixed makes that cost measurable: it divides by decoder family, windowed decoders staying within a small multiple of the unconstrained generator's foot skating where analytic recoveries multiply it several times over. Built to that criterion, our own generator reaches 0.083 m anchor error at FID 0.102 in 0.50 s per clip. The decoder's temporal support is a design criterion for controllable motion generation.

The Gauss-Newton matrix is widely viewed as a positive semidefinite approximation of the Hessian, yet mounting empirical evidence shows that Gauss-Newton descent outperforms Newton's method. We adopt a function space perspective to analyze this phenomenon. We show that the generalized Gauss-Newton (GGN) matrix projects the Newton direction in function space onto the model's tangent space, while a Jacobian-only variant obtained by applying the least squares Gauss-Newton matrix to non-least squares losses projects the function space loss gradient onto this same tangent space. Both projections eliminate distortions from the model's parameterization. Specifically, the evolution of the prediction-target mismatch depends on the model's parameterization through the matrix $JJ^\top$ where $J$ is the Jacobian of the model with respect to its parameters. The projections effectively replace $JJ^\top$ with the identity. We call this effect error whitening. Once the parameterization is removed, the prediction-target mismatch evolves according to dynamics dictated by the structure of the loss and the projection produced by the optimizer. Error whitening is a special property of Gauss-Newton descent that rigorously distinguishes it from Newton's method. We empirically demonstrate that Gauss-Newton optimizers follow the theoretically predicted function space dynamics and outperforms Newton's method, Adam, and Muon across case studies spanning supervised learning, physics-informed deep learning, and approximate dynamic programming.

In multiclass softmax cross-entropy, the full generalized Gauss-Newton (GGN) curvature couples all output logits through the softmax covariance, making curvature-vector products harder to scale as the number of classes grows. We show that the standard multiclass GGN can be decomposed exactly into a true-vs-rest term and a positive semidefinite within-competitor covariance term. Fast Gauss-Newton (FGN) retains the first term and drops the second, yielding a positive semidefinite under-approximation of the multiclass GGN that is exact for binary classification. The derivation uses an exact true-vs-rest scalar-margin representation of softmax cross-entropy: the loss and gradient are unchanged, and the approximation enters only at the curvature level. Exploiting the FGN curvature structure, the damped update can be written as an equivalent whitened row-space system with one row per mini-batch example. We solve this system matrix-free by conjugate gradient using Jacobian-vector and vector-Jacobian products of the scalar margin map. Targeted mechanism experiments and an evaluation on a fixed-feature multiclass head support the predictions from the decomposition: FGN stays closest to the full softmax GGN when competitor mass is concentrated or damping is large, and deviates as the dropped within-competitor covariance grows.

We study hyperparameter optimization (HPO) from a numerical-optimization perspective and propose a multi-objective damped Newton--Gauss--Newton search method. Rather than perturbing each hyperparameter separately or treating model evaluations as independent trials, the method uses performance changes between successive full hyperparameter vectors to construct an iterative secant approximation of the local sensitivity matrix. Each iteration therefore requires only one new full-vector model evaluation while jointly updating all optimized hyperparameters. A Tikhonov-regularized Gauss--Newton system stabilizes the update when the number of hyperparameters exceeds the number of performance objectives. The search is initialized from readily available empirical/default settings of the underlying learner, without requiring a separate initialization search. We evaluate four-dimensional XGBoost HPO on three public classification datasets against exhaustive grid search, random search, and tree-structured Parzen estimator (TPE) optimization. On Breast Cancer Wisconsin, the proposed method matches the best validation accuracy of a 320-configuration grid search with slightly better log loss and ROC--AUC. Across three datasets and three seeds, predictive performance remains competitive with random search and TPE while using substantially fewer search iterations. A supplementary proprietary eight-dimensional threshold-optimization case study further demonstrates joint multi-parameter optimization under competing performance targets and reveals non-monotonic, oscillatory trajectories that motivate damping and best-iterate selection. Overall, the results establish iterative secant sensitivity as an evaluation-efficient local alternative to exhaustive HPO.