Optimization Geometrodynamics: Variational Reduction and Interaction Curvature

Jul 16, 2026

Uploaded by Hazel

Adaptive optimizers rely on hidden positive operators that change how gradients translate into parameter updates. The work develops optimization geometrodynamics as a variational theory of this hidden geometry, using infimal pushforward to eliminate equivalent hidden realizations across optimizer hierarchies. Smooth nondegeneracy yields hidden susceptibility and an effective Schur-complement curvature. A determinant-one affine-invariant SPD action map is central, leading to a globally convergent, linearly convergent solver with residual, distance, and posterior stopping certificates, plus finite identification via active-subspace spectral reduction.

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Introduction

Related Work

Reduction and Interaction

Action Geometry and Solver

Finite Recovery

arXiv :2607 .06723v2 [math .OC] 12 Jul 2026 Optimization Geometrodynamics: Variational Reduction and Interaction Curvature Zavier Li [zavierli888@gmail. com](zavierli888@gmail. com) Xidian University Xi’an, China 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. Smooth nondegeneracy gives the hidden susceptibility and the Schur-complement curvature seen after relaxation. In particular, affine pre-reduction perturbations induce the negative-semidefinite interaction curvature −G∗ H −1G, whose mixed entries integrate to finite mechanism contrasts. The main realization is the determinant-one affine-invariant SPD action map P →7 PA. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. More importantly, a general strongly-convex fiber theorem and its explicit action residual yield a solver that converges globally and linearly from every feasible initializer, with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. The solver remains globally convergent conditional on supplied rigorous residual-error and radius majorants. Its dense spectral kernel is restricted to an active subspace of dimension r ≤ 2m, giving an explicit spectral-arithmetic operation bound after one-time input reduction. For nested shape-normalized quadratic actions, canonical multi-secant projections obey an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold d − 1; this conclusion requires a known scalar gauge. Together these results turn the action bundle from a geometric characterization into an exact iteration with posterior certificates and a finite-identification theory. 1 Introduction Adaptive optimizers use hidden positive operators—moments, curvature surrogates, Kronecker factors, or tensor preconditioners—to turn visible gradients into motion (Duchi et al., 2011; Kingma & Ba, 2015; Martens & Grosse, 2015; Gupta et al., 2018; Morwani et al., 2024; Jordan, 2024; Essential AI et al., 2025; Crawshawet al., 2025) . Writing ˙θ = − gradgt f does not select which hidden metric should represent an observed action when many metrics realize it. We study that ambiguity as a variational reduction problem. For hidden state x, visible action y = π (x), and deformation energy E, the infimal pushforward (π! E)(y) = inf E (x) π (x)=y eliminates hidden realizations and composes across nested maps. Convex-fiber regularity selects a canonical lift; smooth nondegeneracy gives Dσ⋆ = −E1σEσy , D2 (π! E) = Eyy − Eyσ E1σEσy . When mechanism amplitudes enter affinely before reduction, the corresponding interaction curvature is D2uu (π! E) = −G∗ H −1G ⪯ 0 , and its mixed entries integrate to finite contrasts for the specified mechanism protocol. Our main realization is the determinant-one SPD action map πA (P ) = PA. For full-column-rank A, every admissible fiber has a global analytic parameterization by a smaller determinantone SPD factor and is closed and totally geodesic under AIRM. Projection of the identity therefore gives a unique analytic canonical controller. A general strongly-convex fiber engine connects this selection to computation, while the action bundle makes its logarithmic residual explicit. A sublevel first-return argument yields the sharp radius majorant L0 = ψ (D0 / √2) , nonasymptotic iteration bounds obtained by exact inversion of the linear rate, and posterior controller and objective certificates. Relative matrix-function error remains globally convergent when a numerical routine supplies rigorous error and radius majorants. The solver has an exact active realization on s

FAQs

What hidden information do adaptive optimizers introduce, and why does it matter?

Adaptive optimizers introduce hidden positive operators (e.g., moments, curvature surrogates, Kronecker factors) that determine how visible gradients become parameter motion. The paper treats the resulting ambiguity in the choice of hidden metric as a variational reduction problem.

How does infimal pushforward help in the proposed framework?

Infimal pushforward eliminates hidden realizations that produce the same visible action and composes this elimination across nested maps. It also enables a canonical selection via convex-fiber regularity and yields explicit derivatives for the reduced action.

What guarantees does the main solver provide?

The solver is globally convergent and globally linearly convergent from every feasible initializer. It comes with nonasymptotic bounds on value, controller-distance, and residual, and includes posterior-certified stopping certificates, with conditional convergence under rigorous inexact residual and radius majorants.

Jul 16, 2026

Uploaded by Hazel