Optimization Geometrodynamics: Variational Reduction and Interaction Curvature

• URL: http://arxiv.org/abs/2607.06723v2
• Date: Sun, 12 Jul 2026 05:07:25 GMT
• Title: Optimization Geometrodynamics: Variational Reduction and Interaction Curvature
• Abstract summary: We develop geometrodynamics as a variational theory of this hidden determinant pushforward.<n>For affine pre-reductions, the induced interaction curvature is the negative-semidefinite operator $-G*H-1G$.<n>We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.

Related papers

• Discretization and Statistical Consistency of Functional Flow Matching [0.0]
  We prove strong $L2$ convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions.<n>For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound.
  arXiv  Detail & Related papers  (2026-08-05T07:05:47Z)
• Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks [0.0]
  Positive quadratic networks admit the lowrank representation f_U(x)=xtop UUtop x, where UinmathbbRdtimes r is identifiable only up to right multiplication.<n>We study how this quotient structure governs training dynamics, curvature, recovery, and bias.
  arXiv  Detail & Related papers  (2026-07-28T12:04:46Z)
• Statistical inverse learning and $\ell^1$-regularization [0.27998963147546135]
  We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning.<n>We establish almost-sure exponents and derive non-asymptotic high-probability convergence rates in both prediction and $ell1$ reconstruction norms.
  arXiv  Detail & Related papers  (2026-07-08T14:32:08Z)
• Geometric--Nongeometric Optimizer Calculus: A Modular Language for Reachable Gradient Methods [0.0]
  We introduce geometric--nongeometric calculus, a modular language for auditing reachable gradient methods.<n>The main formal result is a direction-expressivity theorem: away from critical points, full positive-definite geometry expresses exactly the strict descent directions.
  arXiv  Detail & Related papers  (2026-07-08T09:40:44Z)
• Restricted Dynamic Geometric Complexity: Certificates for Structured Preconditioning [0.0]
  This paper records that benchmark in the present notation and develops restricted geometric complexity.<n>Main proved results are monotonicity and submanifold-distance principles, diagonal and block reachability as linear inequality feasibility problems.<n>The repository includes toy and synthetic matrix that check diagonal expression gaps, block primal/dual certificates, Kronecker spectral width, and Hessian-relative Kronecker candidate certificates on small quadratic instances.
  arXiv  Detail & Related papers  (2026-07-08T09:36:45Z)
• Is Spurious Correlation Removal Always Learnable? [56.28155520961125]
  Invariant learning can fail even when the structure is statistically identifiable.<n>We show a conditional computational barrier: under a black-box samplable supervised sparse recovery primitive, there exist emphamplable multi-dimensional environments.<n>Synthetic and real datasets illustrate the predicted gaps and transitions and motivate simple diversity diagnostics.
  arXiv  Detail & Related papers  (2026-06-11T05:49:43Z)
• Fitting Unknown Number of Hyperplanes with Manifold Optimization [57.48093263119306]
  Fitting unknown number of linearplanes to data is a fundamental yet challenging problem in machine learning.<n>Existing approaches often struggle with optimum optima or lack geometric consistency.
  arXiv  Detail & Related papers  (2026-05-27T14:02:20Z)
• Sample Complexity of Policy Gradient for Log-Growth Control [23.86784769848408]
  We study the sample complexity of policy gradient for log-growth control.<n>We prove that projected mini-batch policy gradient attains total sample complexity.
  arXiv  Detail & Related papers  (2026-05-26T07:24:14Z)
• BOOOM: Loss-Function-Agnostic Black-Box Optimization over Orthonormal Manifolds for Machine Learning and Statistical Inference [0.0]
  We present a systematic exploration of.<n>the set of $p.<n>orthonormal column-orthonormal escape through a.<n>novel applied to data data.<n>problems.<n>We show a diagonal optimization framework showing a.<n>strong, low-rank decomposition, independent decomposition analysis,.<n>and global convergence in under conditions.
  arXiv  Detail & Related papers  (2026-04-23T19:32:27Z)
• Variational Entropic Optimal Transport [67.76725267984578]
  We propose Variational Entropic Optimal Transport (VarEOT) for domain translation problems.<n>VarEOT is based on an exact variational reformulation of the log-partition $log mathbbE[exp(cdot)$ as a tractable generalization over an auxiliary positive normalizer.<n> Experiments on synthetic data and unpaired image-to-image translation demonstrate competitive or improved translation quality.
  arXiv  Detail & Related papers  (2026-02-02T15:48:44Z)
• Latent Object Permanence: Topological Phase Transitions, Free-Energy Principles, and Renormalization Group Flows in Deep Transformer Manifolds [0.5729426778193398]
  We study the emergence of multi-step reasoning in deep Transformer language models through a geometric and statistical-physics lens.<n>We formalize the forward pass as a discrete coarse-graining map and relate the appearance of stable "concept basins" to fixed points of this renormalization-like dynamics.<n>The resulting low-entropy regime is characterized by a spectral tail collapse and by the formation of transient, reusable object-like structures in representation space.
  arXiv  Detail & Related papers  (2026-01-16T23:11:02Z)
• Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations [57.179679246370114]
  We identify the distribution of random perturbations that minimizes the estimator's variance as the perturbation stepsize tends to zero.<n>Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length.
  arXiv  Detail & Related papers  (2025-10-22T19:06:39Z)
• On the Optimal Construction of Unbiased Gradient Estimators for Zeroth-Order Optimization [57.179679246370114]
  A potential limitation of existing methods is the bias inherent in most perturbation estimators unless a stepsize is proposed.<n>We propose a novel family of unbiased gradient scaling estimators that eliminate bias while maintaining favorable construction.
  arXiv  Detail & Related papers  (2025-10-22T18:25:43Z)
• Beyond Coordinates: Meta-Equivariance in Statistical Inference [0.0]
  Optimal statistical decisions should transcend the language used to describe them.<n>How do we guarantee that the choice of coordinates does not subtly dictate the solution?<n>We first analyse the optimal combination of twoally normal estimators under a strictly convex trace-AMSE risk.
  arXiv  Detail & Related papers  (2025-04-14T19:40:39Z)
• Rao-Blackwell Gradient Estimators for Equivariant Denoising Diffusion [55.95767828747407]
  In domains such as molecular and protein generation, physical systems exhibit inherent symmetries that are critical to model.<n>We present a framework that reduces training variance and provides a provably lower-variance gradient estimator.<n>We also present a practical implementation of this estimator incorporating the loss and sampling procedure through a method we call Orbit Diffusion.
  arXiv  Detail & Related papers  (2025-02-14T03:26:57Z)
• SPARE: Symmetrized Point-to-Plane Distance for Robust Non-Rigid 3D Registration [77.13381026159111]
  We propose SPARE, a novel formulation that utilizes a symmetrized point-to-plane distance for robust non-rigid registration.<n>The proposed method greatly improves the accuracy of non-rigid registration problems and maintains relatively high solution efficiency.
  arXiv  Detail & Related papers  (2024-05-30T15:55:04Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Smoothing the Edges: Smooth Optimization for Sparse Regularization using Hadamard Overparametrization [10.009748368458409]
  We present a framework for smooth optimization of explicitly regularized objectives for (structured) sparsity. Our method enables fully differentiable approximation-free optimization and is thus compatible with the ubiquitous gradient descent paradigm in deep learning.
  arXiv  Detail & Related papers  (2023-07-07T13:06:12Z)
• Convergence of Adam Under Relaxed Assumptions [72.24779199744954]
  We show that Adam converges to $epsilon$-stationary points with $O(epsilon-4)$ gradient complexity under far more realistic conditions. We also propose a variance-reduced version of Adam with an accelerated gradient complexity of $O(epsilon-3)$.
  arXiv  Detail & Related papers  (2023-04-27T06:27:37Z)
• Optimal Extragradient-Based Bilinearly-Coupled Saddle-Point Optimization [116.89941263390769]
  We consider the smooth convex-concave bilinearly-coupled saddle-point problem, $min_mathbfxmax_mathbfyF(mathbfx) + H(mathbfx,mathbfy)$, where one has access to first-order oracles for $F$, $G$ as well as the bilinear coupling function $H$. We present a emphaccelerated gradient-extragradient (AG-EG) descent-ascent algorithm that combines extragrad
  arXiv  Detail & Related papers  (2022-06-17T06:10:20Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
  We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
  arXiv  Detail & Related papers  (2020-04-09T17:54:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.