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Blockwise Stabilized Adaptive Cubic Regularization with Subsolvers via Recurrence

Rodion Podorozhny

arXiv:2608.22129v2cs.LGmath.NA

TL;DR

Dense subproblem solves limit scalable cubic Newton methods, while many alternatives replace true block curvature or give up the exact cubic step. This paper introduces a blockwise optimizer with exact per-block cubic steps via matrix-free Krylov subsolvers, and reports strong FINER results against Adam and other optimizers.

  • Problem

    Dense cubic subproblem solves limit feasible block sizes, motivating scalable methods that retain true block curvature and exact cubic steps.

  • Method

    The method minimizes independent cubic models over true block Hessians using per-block adaptive constants, monotone full-loss guarding, and matrix-free Lanczos or recurrence-based Krylov subsolvers.

  • Results

    At 91.4M parameters, the introduced variants are the only evaluated cubic Newton methods with exact steps on every block; ARC-φ1 reaches 133.5 dB versus Adam’s converged 78.2 dB on FINER.

  • Takeaways & Limitations

    Per-block curvature adaptation makes the method practical at 91.4M parameters, while ARC-φ1 is especially effective on cross-coupled ill-conditioned or saddle-dominated landscapes.

  • Takeaways & Limitations

    The FINER full-convergence result is based on a single image and seed, while the broader comparison covers only two ESM images and has substantially higher cost than converged Adam.

Abstract

from arXiv · show

Cubic regularized Newton methods have the optimal $\mathcal{O}(ε^{-3/2})$ global rate, but a dense subproblem solve limits the feasible block size. Scalable Cubic Newton variants replace the true block curvature with a diagonal, low-rank, Kronecker-factored, or sketched surrogate and, most often, give up the exact cubic step. We introduce a blockwise optimizer that minimizes an independent cubic model per parameter tensor over the true block Hessian, under a per-block adaptive cubic constant and a monotone guard on the full loss. Arbitrarily large tensors are handled matrix-free in a Lanczos-built Krylov subspace, where we prove that the step minimizes the cubic model. The theory also supplies the $\mathcal{O}(ε^{-3/2})$ iteration complexity bound, a second-order guarantee, and monotone per-block descent. Four variants of this outer scheme are evaluated against the original adaptive regularization with cubics (ARC) optimizer, some other recent cubic Newton variants, Adam, SOAP, and L-BFGS. On a 91.4M-parameter implicit neural representation (INR), the variants introduced in this work are the only evaluated here cubic Newton methods whose steps stay exact on every block. Run to full convergence on FINER 2D image fitting, one of the ARC variants introduced here, ARC-$\varphi_1$, reaches 133.5 dB peak signal-to-noise ratio, while tuned Adam plateaus at 78.2 dB after about 70 minutes. In that time ARC-$\varphi_1$ reaches 95.6 dB.

1 Introduction

The paper targets spectral bias and scalability limits in second-order INR optimization by combining blockwise adaptive cubic regularization with exact true-curvature subsolvers. Its variants provide theoretical guarantees and strong empirical results, including exact cubic-model steps at 91.4M parameters and large FINER gains over Adam.

  • Motivation: First-order optimizers fit low-frequency INR components quickly while high-frequency components lag because kernel eigenmodes converge at rates proportional to their eigenvalues.This motivates a second-order optimizer for mitigating spectral bias in implicit neural representations.
  • Prior limitations: Existing scalable cubic Newton methods commonly use diagonal, low-rank, Kronecker-factored, or sketched curvature surrogates and often abandon the exact cubic step.Dense Hessian-based variants remain impractical beyond roughly 15k–20k parameters in the reported implementations.
  • Approach: The proposed optimizer minimizes an independent cubic model for each parameter tensor using the true block Hessian, adaptive per-block constants, and monotone full-loss acceptance tests.Large blocks use matrix-free Krylov methods, while small blocks can use explicitly formed lazy Hessians.
  • Scalability: At 91.4M parameters, ARC-Block and the per-block ARC control are the only evaluated cubic Newton methods whose steps remain exact in the cubic-model sense on every block.The model includes an 88.5M-parameter decoder tensor accessed through Hessian–vector products.
  • Empirical results: On FINER, CubicKrylov reaches 124.7 dB versus tuned Adam’s 65.2 dB, while ARC-φ1 reaches 133.5 dB versus Adam’s 78.2 dB after full convergence.The landscape fingerprint yields a dispatch rule distinguishing axis-aligned settings suited to first-order methods from cross-coupled or saddle-dominated settings favoring blockwise cubic steps.

2 Background

The background develops Chebyshev and cubic-regularization foundations for replacing dense solves with matrix-free polynomial or Krylov procedures. It emphasizes how adaptive cubic shifts control indefiniteness and the effective condition number while preserving access to exact model minimization in a subspace.

  • Chebyshev preconditioning: Chebyshev second-kind preconditioning applies a polynomial approximation to inverse curvature through a Hessian–vector-product recurrence.Its normalized-spectrum relaxation contracts the worst-case residual by at most approximately 0.22 per application once the degree is sufficient.
  • Chebyshev limitations: Standalone Chebyshev preconditioning requires degree L ≈ 1.28√κ for fixed contraction, so ill-conditioning causes degree growth.Negative eigenvalues additionally induce hyperbolic growth and require clamping safeguards.
  • Cubic Newton: Cubic regularized Newton minimizes a cubic model whose global solution is characterized by a shifted linear system and a one-dimensional secular equation.The shift is selected so the shifted Hessian is positive semidefinite or positive definite in the non-degenerate case.
  • Positioning: This paper differs from prior Krylov cubic Newton work by treating non-convex objectives, building and regularizing subspaces per parameter block, and adapting polynomial degree to the cubic shift.The original single-subspace ARC implementation stalls at 4.0 dB on the 91.4M-parameter ViSIR landscape.

3 The Per-Block Cubic Step and its Krylov Subsolver

The method solves each block’s cubic model in a Krylov subspace after shifting the Hessian to handle negative curvature, with adaptive degree control and acceptance safeguards. The theory establishes exact subspace minimization, optimal stationarity complexity, second-order guarantees, and monotone blockwise descent.

  • Spectral shift: The cubic model’s global minimizer is obtained from a secular equation whose admissible shift exceeds both zero and the magnitude of the most negative Hessian eigenvalue.This shift makes the operator processed by the polynomial solver positive definite in the non-degenerate case.
  • Adaptive Krylov dimension: The shifted operator bounds the effective condition number and determines a self-adaptive Chebyshev or Lanczos degree for the Krylov subspace.When the gradient-driven shift dominates negative curvature, the required degree is constant independent of the original condition number; near strict second-order points, the method recovers the √κ regime.
  • Krylov subsolver: The Lanczos representation builds the same Krylov subspace as the Chebyshev recurrence, and Rayleigh–Ritz extracts the best step contained in that subspace.The stored-basis implementation uses O(n_bL) memory, whereas the fixed-coefficient recurrence uses O(1) auxiliary vectors.
  • Guarantees: The Krylov step minimizes the cubic model over the true block Hessian within its constructed subspace, and ARC-relative accuracy preserves the O(ε^-3/2) outer rate.The theory also supplies an approximate second-order stationarity guarantee.
  • Scope: The guarantees establish approximate second-order stationarity rather than global minimality for general smooth non-convex objectives.Global-minimum guarantees require additional structure such as gradient dominance, which sinusoidal INR landscapes do not satisfy.
  • Blockwise stabilization: Per-block acceptance against a monotone full-loss guard gives f(x_k+1) ≤ f(x_k) and individual decrease bounds when each block constant satisfies M_b ≥ L_H,b.Per-block adaptive constants avoid the joint-acceptance failure that stalls large multi-block models.

4 ARC-Block Algorithm

ARC-Block partitions parameters into small and large tensors, applying exact cubic steps to small blocks and matrix-free CubicKrylov steps to large blocks. Per-block acceptance, adaptive regularization, and fresh gradients stabilize monotone progress while keeping large-block oracle cost independent of block size.

  • CubicKrylov step: CubicKrylov computes large-block cubic Newton steps in a Lanczos-built Krylov subspace whose degree is bounded by Chebyshev analysis, while the Chebyshev recurrence itself is not used.The method uses HVPs to build the Lanczos basis and reduced tridiagonal model.
  • Two-phase update: Small blocks receive lazy exact cubic steps from cached eigendecompositions, whereas large blocks use per-block CubicKrylov trials with accept-or-reject tests.The algorithm partitions blocks at a size threshold and processes small and large blocks in two phases.
  • Per-block stabilization: Per-block acceptance and individual Mb values prevent a good large-block update from being rejected by another block and restore monotone progress.The guard compares each candidate against the loss after Phase A and preceding accepted large-block updates.
  • Adaptive regularization: Two-sided adaptation decreases Mb by ×2 after acceptance and increases it by ×4 after rejection, allowing each block to find its own curvature scale down to Mmin = 10^-6.The floor is retained to keep the step radius finite, while asymmetric factors bound rejected-step overhead.
  • Fresh gradients: 31.7 vs. 23.1 dB PSNR results when fresh Phase-B gradients are used instead of reusing a stale step-start gradient on nano-ViSIR.The stale-gradient variant lowers final PSNR by approximately 8.5 dB.
  • Cost accounting: For L ≈ 10, the Krylov oracle charge is independent of block size, although HVP wall-clock time and stored-basis memory still scale linearly with nb.This makes cubic regularized second-order steps feasible for large tensors where only HVPs are affordable.

5 The Exponential-Relaxation (φ1) Step Subsolver

The φ1 subsolver replaces the cubic step rule with a finite-horizon integration of quadratic-model gradient flow, while retaining the blockwise outer scheme and stabilized acceptance control. It approaches Newton behavior on stiff modes, but has weaker first-order complexity and requires explicit control of negative-mode amplification.

  • Step rule: The φ1 step is the exact finite-horizon solution of gradient flow for the local quadratic model, represented as s = −H(G,h)g.It is also the continuous-time limit of gradient descent with fixed total horizon h.
  • Matrix function: H(G,h) = hφ1(−hG) remains well-defined for singular G through the entire-function series, so the step requires no invertibility assumption.The matrix function is applied to the block Hessian or its Lanczos projection.
  • Adaptive control: ARC-φ1 changes only the step rule and uses the same outer scheme, with rejection halving the horizon and sustained success doubling it toward the Newton limit.The stabilized acceptance guard remains the mechanism deciding whether a trial is accepted.
  • Per-mode behavior: For positive definite G and sufficiently small h, φ1 produces quadratic-model descent, while its per-mode behavior interpolates between gradient and Newton limits.Negative modes are amplified exponentially with h, requiring an explicit bound.
  • Descent guarantees: The frozen-quadratic decrease is strictly positive for every spectrum sign pattern and increases with h, with a Cauchy-type lower bound.The corresponding gradient contracts exactly under the quadratic flow.
  • Complexity and stationarity: ARC-φ1 reaches an ε-first-order point in O(ε^-2) accepted steps, whereas the O(ε^-3/2) cubic-regularization rate does not transfer.Second-order escape is guaranteed only when the gradient has a nonzero component along the bottom eigenvector under the stated clamped mechanism.
  • Free per-trial horizon ranking: Free horizon ranking evaluates candidate horizons through diagonal reweighting and reduced-model evaluations after the Ritz decomposition, without additional HVPs or function evaluations.On the small MLP control benchmark, the ranked variant selected shorter horizons 68% of the time and longer horizons 29%.

6 Experimental Results

The experiments show that stabilized blockwise cubic methods improve scalability and accuracy by preserving curvature-aware steps on large tensors, while adaptive safeguards prevent stalls and over-regularization. Across nano and full-scale INR benchmarks, these methods outperform competing second-order and first-order baselines, especially on coupled or saddle-dominated landscapes.

  • Mechanism ablations: +10.9 dB over the damped-Newton fallback is obtained at 0.66× the grad-equivalent cost on identical routing.Removing fresh phase-B gradients lowers every mechanism variant by 6–8.5 dB, showing the importance of the stabilization design.
  • Nano ablation: 32.8±1.4 dB is achieved by CubicKrylov across 20 images, ahead of gradient regularized Newton on 17 of 20 and lazy cubic Newton on 19 of 20.It uses 59% of their oracle budget and approximately 14× less wall-clock time.
  • Full-scale ablation: 51.65 dB is reached by CubicKrylov in 100 steps, exceeding Chebyshev-ON by 9.90 dB and damped Newton on large blocks by 29.4 dB.The gain comes from large decoder and attention tensors where CubicKrylov retains genuine curvature-aware cubic updates.
  • Stabilization: 64.4 dB is reached with stabilized CubicKrylov and 64.2 dB with the Chebyshev recurrence, whereas a global single-subspace ARC stalls at 4.0 dB.The Chebyshev recurrence uses 7.0–8.7 GB peak memory, compared with approximately 11 GB for stored-basis Lanczos.

7 Comparison with Other ARC-Based Approaches

This section compares the paper’s per-tensor, true-curvature cubic design with recent ARC-based methods across curvature computation, scale, and cubic-step exactness.

  • The comparison covers ARC-based families by curvature computation or approximation, largest demonstrated scale, and exactness of the cubic step on true curvature.These dimensions are collated in Table 10 and related comparisons.
  • The review spans Krylov cubic minimization, blockwise and partition-based cubic methods, quasi-Newton models, Chebyshev recurrences, lazy and stochastic cubic Newton, subspace methods, and trust-region hybrids.Appendix D provides the detailed family-by-family review.

8 Limitations

The method reaches the per-block cubic step at substantial computational cost, while its guarantees and evaluation remain bounded by theoretical, numerical, and experimental scope limitations.

  • On FINER, the per-block cubic step costs ∼33× more gradient-equivalents than converged Adam.Tuned first-order methods can match accuracy faster on axis-aligned ill-conditioning.
  • The φ1 subsolver has no proven optimality or O(ε^-3/2) rate, although its accepted steps satisfy a Cauchy-type decrease rule.The outer scheme retains monotonicity through per-block acceptance, but second-order stationarity transfers only generically.
  • The clamped escape guarantee requires a nonzero gradient component along the bottom eigenvector.This is the stated condition for the second-order stationarity transfer.
  • The full-convergence FINER result of 133.5 dB at 2,824 sweeps and 17,173 s comes from a single image and seed.The authors call for broader experiments across neural networks and loss landscapes.
  • The diagonal φ1 variant fails on FINER’s saddle-rich landscape because diagonal dominance statistics depend on the search trajectory.

9 Conclusion

The paper introduces a blockwise stabilized adaptive cubic regularization optimizer with interchangeable exact cubic subsolvers for parameter tensors of different sizes. Its φ1 variant substantially improves FINER image-fitting accuracy while retaining the shared blockwise outer scheme.

  • The optimizer partitions parameters by tensor and maintains an independent cubic constant and acceptance test for each block.
  • Large tensors use a Chebyshev-bounded Krylov subspace, while small tensors use lazy exact-Hessian cubic steps.The cubic subproblem solver is treated as an interchangeable component, including a three-term recurrence when solver memory is scarce.
  • 64.4 dB is reached by the stabilized configuration at 91.4M parameters, versus 51.65 dB without stabilization and 4.0 dB for global single-subspace ARC.
  • 133.5 dB is reached by ARC-φ1 at full FINER convergence, versus Adam’s converged 78.2 dB and a +17.4 dB advantage at equal extended budget.ARC-φ1 reaches 95.6 dB within the time in which tuned Adam plateaus at 78.2 dB.

A Feasibility of Second-Order Steps at the 91.4M-Parameter Scale

The feasibility study shows that matrix-free, blockwise cubic steps can scale to a 91.4M-parameter model, whereas dense-Hessian methods remain near the 15k–20k-parameter range. Per-block regularization addresses the failure of a single global regularization weight across heterogeneous tensor curvatures.

  • The 91.4M-parameter ViSIR contains 45 tensors, with the largest decoder block holding 88,473,600 parameters, or 96.8% of the model.
  • Eleven Hessian–vector products compute the largest block’s cubic step, requiring about 3.9 GB for its Lanczos basis.A dense Hessian for this block alone would occupy approximately 3 × 10^7 GB.
  • The matrix-free cubic step charges L+2 gradient-equivalents per step independently of block size, although HVP time and basis memory scale linearly with tensor size.
  • Dense-solve cubic Newton methods remained impractical beyond roughly 15k–20k parameters, while HVP-based variants remained feasible at 91.4M parameters.
  • 4.0 dB is reached by global ARC, versus 51.7 dB for CubicKrylov, while global regularization drives σ to 5×10^11 and rejects 46% of trials.
  • Per-block constants allow larger steps in well-conditioned tensors while regularizing ill-conditioned tensors individually.The controlled block-σ ARC changes the regularization granularity while retaining block-restricted cubic solves and per-block acceptance tests.
  • The deterministic non-convex block-cubic scheme has no O(ε^-3/2) guarantee from the cited Cartis et al. theory.

B The Cost Structure of the Subsolver: stored basis (CubicKrylov) vs Chebyshev recurrence as a subproblem solver

Stored-basis Lanczos and three-term Chebyshev recurrence construct cubic steps in the same HVP-generated Krylov setting but exchange memory for repeated HVP work. The stored basis is preferable when memory permits; the recurrence becomes attractive when basis storage is limiting.

  • Both subsolver families build cubic steps in a Krylov subspace using one Hessian–vector product per degree.
  • The stored-basis solver retains L+1 vectors and reuses the subspace across secular trials, whereas the recurrence keeps three vectors regardless of degree.
  • At 91.4M parameters, replacing the stored basis relieves its approximately 11 GB memory footprint but multiplies HVP work across secular solves.
  • 35.18 dB is reached by the best hybrid with EMA-cached bounds at 14.4 HVPs per block step.
  • The recurrence trades approximately 1.5 dB at equal oracle budget for a degree-independent memory footprint.
  • The recurrence is viable when storing L+1 block vectors exceeds memory, as in the 91.4M-parameter model, while retaining only three vectors.
  • On ESM-20, v2r averages 63.8 ± 4.0 dB with zero failures, above CubicKrylov’s 58.5 dB and below per-block ARC controls at 67.0–67.4 dB.
  • Stale γb estimates can produce incorrect shifts, whereas secular-solving hybrids retain acceptance tests and reject such steps.

B.2 Robustness to a bf16 HVP Oracle

The bf16 HVP study finds that both solvers operate on a noisy curvature oracle, but the recurrence is more sensitive because it applies each noisy HVP once with fixed coefficients. The stored-basis solver therefore loses less PSNR under mixed precision.

  • The robustness evaluation rebuilds loss, gradient, and HVP graphs under bf16 autocast while retaining fp32 master weights and acceptance evaluations.
  • The stored-basis solver relies on consistency across successive HVP columns, whereas the recurrence uses each HVP once with fixed coefficients.
  • Under bf16, stationarity residuals rise to 0.17 for the basis solver and 0.75–0.98 for the recurrence.
  • At 91.4M parameters, the recurrence loses 4.5 dB against 0.9 dB for the stored-basis solver.

C Experiments on INR architectures

The experiments compare second-order and first-order optimizers across INR image and SDF fitting tasks, using convergence, residual-frequency, and wall-clock evidence. FINER shows a large reachability gap, whereas SIREN methods are comparable at convergence but differ in cost.

  • Evaluation setup: The evaluation covers SIREN, FINER, and WIRE image regression, plus Thai-statue signed-distance fitting on SIREN and FINER backbones.Image regression uses 240 × 240 RGB images and approximately 199k-parameter networks; SDF fitting uses IGR supervision.
  • Residual-frequency behavior: Second-order runs reduce DC and Low residual coefficients to approximately 10^-7–10^-6 while matching Adam’s high-band accuracy.These coefficients are 3–6 orders below Adam, making the within-run hl ratio large because its denominator is small.
  • Residual-frequency behavior: Total MSE is dominated by the coefficient-rich High band.
  • FINER results: On FINER, converged Adam plateaus at 65.2 dB in 2D, while the blockwise cubic step continues to 120–129 dB with a band-uniform residual.In 3D, Adam’s chamfer error is 4.6–5.5× above every curvature-aware method, and additional budget does not change the result.
  • SIREN results: On SIREN, methods are comparable at convergence, with CubicKrylov reaching 49.5 dB SDF test accuracy but requiring 8.3× Adam’s wall time.

D Extended Comparison with ARC-Based Approaches

The extended comparison situates the blockwise ARC family among Krylov, trust-region, quasi-Newton, stochastic, and structured second-order methods. The distinguishing scope is per-block adaptive cubic regularization combined with network-scale deterministic training.

  • Krylov and cubic methods: Bellavia et al. minimize cubic models in reusable low-dimensional Krylov subspaces, whereas this work uses per-block Krylov subsolvers with separate regularization.
  • Blockwise methods: ARC-Block updates all blocks per step with individual cubic constants and acceptance tests, unlike greedy single-block cubic updates.
  • Quasi-Newton methods: Related quasi-Newton cubic methods replace exact block curvature with limited-memory symmetric-rank-one or quasi-Newton structure.
  • Recurrence stability: Chebyshev-recurrence stability analyses address recurrence perturbations, while this work combines recurrence control with negative-mode clamping and trust-controlled horizons.
  • Theory and scope: Prior ARC theory covers stale higher-order approximations, stochastic oracle conditions, or Hessian reuse, but does not combine these analyses with per-block adaptive cubic constants at network scale.
  • Theory and scope: The paper considers only full-batch, deterministic training.

E.1 Iso-time results

At equal wall-clock time, CubicKrylov outperforms first-order and diagonal-curvature variants on the reported 91.4M-parameter ViSIR reconstruction. Its advantage reflects cheaper curvature-aware steps rather than merely taking more steps.

  • Iso-time comparison: The iso-time experiment grants every optimizer the same 148-second wall-clock budget.
  • Iso-time comparison: At T = 148 s, CubicKrylov performs better than first-order and diagonal Hutchinson variants by a wide margin.Adam and SOAP remain near 20–21 dB despite running thousands of steps in the same time.
  • Oracle cost: CubicKrylov costs approximately 368 gevals per step, versus approximately 8000 for full-Hessian Chebyshev variants, a roughly 22× reduction.
  • Oracle cost: Within 36,836 gevals, full-Hessian methods complete only approximately five steps and reach ≤19 dB, while CubicKrylov reaches 67 dB.
  • Iso-time comparison: Figure 2 reports PSNR and MSE loss against wall-clock time at the 148-second budget.
  • Oracle cost: The Krylov step’s block-size independence is crucial to reducing curvature-aware cubic-step cost from approximately 8000 to approximately 368 gevals.

F Multi-Image, Multi-Seed ViSIR-Nano Benchmark (ESM-20)

The ESM-20 multi-image, multi-seed benchmark tests whether the single-image ranking persists across 20 images and three seeds. CubicKrylov retains the strongest reported ranking while using less oracle budget and wall-clock time.

  • Benchmark design: The nano configurations are rerun on the first 20 ESM images with three random seeds each, adding spectral-preconditioned gradient descent and DSO-BlockHess references.
  • Results: CubicKrylov reaches 32.8±1.4 dB and beats gradient regularized Newton on 17 of 20 images and lazy cubic Newton on 19 of 20.
  • Results: CubicKrylov consumes 59% of the oracle budget and approximately 14× less wall-clock than the compared Newton methods.
  • Scale effects: At nano scale, dense 15k × 15k Hessians are affordable, making lazy m=n variants genuine competitors.
  • Oracle-cost comparison: At matched oracle budget, CubicKrylov dominates full-Hessian Chebyshev steps by approximately 48 dB because its matrix-free Krylov step is approximately 22× cheaper.
  • Scale effects: Other configurations lose on all 20 images, while SSCN’s τ=64 subspace cannot match full-space curvature.

G Wall-Clock Statistics of the Multi-Image Nano Benchmark

The nano benchmark reports per-run quality, wall-clock, and gradient-equivalent costs across 60 image-seed runs, while an equal-budget control separates computational expense from step quality. CubicKrylov leads the budgeted comparison and has a substantial wall-clock advantage over lazy cubic Newton references, whereas frequent Hessian recomputation is heavily constrained by the budget.

  • Benchmark design: 60 runs cover 20 images and 3 seeds, with Table 18 aggregating best PSNR, mean wall-clock, and total grad-equivalents per method.Table 19 gives the corresponding mean, spread, extremes, and coefficient of variation.
  • Wall-clock cost: CubicKrylov’s worst 60.4 s run is approximately 4.5× faster than the fastest lazy cubic Newton reference run at 274.9 s.CubicKrylov also has an approximately 14× mean wall-clock advantage over those references.
  • Budget effects: Frequent dense Hessian recomputation is strongly affected by the nano oracle budget, stopping several methods after only a handful of steps.The equal-budget control grants 6.53×10^6 grad-equivalents and a 2,400 s wall-clock limit, approximately 185× the benchmark rule.
  • Budget effects: DSO-Adaptive reaches 11–41 steps and 12.6–19.9 dB inside the benchmark budget, while its K=5 plateau reaches 30.0 dB after approximately 2,400 s and 3,000 steps.The K=5 plateau consumes 6.53×10^6 grad-equivalents and 8,867 block rebuilds.
  • Failure modes: The equal-budget control distinguishes expensive-but-potentially-strong methods from algorithmic failures, including unregularized Newton attraction to saddles and a plain cubic Newton eigendecomposition crash.The abs-eig DSO-BlockHess variant reaches 33.1 dB with greater budget, while plain Newton DSO-BlockHess remains at 8.2 dB.
  • Step quality: CubicKrylov leads both tested budget levels at 31.66 dB and 36.64 dB, while global-Krylov ARC reaches 37.21 dB in the equal-budget control.The global-Krylov ARC lead over CubicKrylov is less than 0.6 dB.
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