🤖 AI Summary
This work addresses the limitations of conventional gradient-based optimization methods, which struggle to adapt to dynamic changes in length, curvature, and preconditioning implicitly induced by internal states under fixed geometric assumptions. The authors formulate optimization as a coupled system involving parameter trajectories, particle distributions, and a time-evolving Riemannian metric, explicitly distinguishing immutable obstacles from remediable geometric mismatches. They introduce the notion of “dynamic geometric complexity” and establish the first lower bound on geometric optimization difficulty based on affine-invariant distance. By leveraging gauge-invariant observables and Morse saddle-point flux analysis, they precisely characterize this complexity—in the setting of strongly convex quadratic objectives with a fully positive-definite metric oracle—as the affine-invariant distance from the relative logarithmic spectrum to the set of well-conditioned metrics.
📝 Abstract
Most gradient-based optimization methods move parameters through a fixed background geometry, even when their internal states implicitly define changing notions of length, curvature, and preconditioning. We introduce optimization geometrodynamics, a benchmark language in which optimization is a coupled evolution of a parameter trajectory, a transported distribution of particles, and a controlled time-varying Riemannian metric. The language separates invariant obstructions from improvable geometric mismatch: positive metrics preserve critical points and Morse indices, and cannot remove global geodesic-convexity obstructions, but can alter conditioning, distributional transport, and flux away from exact critical points. We introduce dynamic geometric complexity, the minimum geometric cost required to reduce an optimization difficulty observable. In the oracle benchmark model of strongly convex quadratic objectives with full positive-definite metric control, this complexity is exactly the affine-invariant distance from the relative log-spectrum to a low-condition-number set. We also analyze Hessian-matching flows, spectral Onsager relaxation, discrete exponential projection updates, gauge-invariant observables, and fixed-time local Morse-saddle flux. The paper is theory-only: its claims are formal statements with proofs, intended to provide invariants and benchmark costs against which implementable adaptive optimizers can be compared once their admissible metric families, curvature estimates, and discretization errors are specified.