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Adaptive Rotation for iSOMA: Geometry, Benchmarking, and Noise Robustness in Variational Quantum Objectives

Vojtěch Novák, Ivan Zelinka

arXiv:2609.37193v1cs.NEquant-ph

TL;DR

The paper asks whether iSOMA’s coordinate dependence can be reduced while retaining its leader-directed migration, and introduces iSOMA-AR, which learns a basis from successful displacements for applying the perturbation mask. Adaptive rotation improves iSOMA robustly on BBOB and most clearly on Q2, while its advantage is selective or absent on other objectives and under strong sampling noise.

  • Problem

    The study asks whether iSOMA’s coordinate-dependent perturbation mask can be made more effective on objectives whose useful search directions are oblique combinations of variables.

    Source: §1 · p. 2

  • Method

    iSOMA-AR estimates covariance from successful normalized migration displacements and applies the existing PRT mask in the learned basis when off-axis structure is sufficiently strong.

    Source: §1 · p. 2

  • Results

    Adaptive rotation robustly improves iSOMA on BBOB and gives its clearest VQE gain on Q2, while Q1 shows no benefit and strong-noise performance is not significantly better than baseline iSOMA.

    Source: Statevector results · p. 13, §5 · p. 17

  • Takeaways & Limitations

    Adaptive rotation is most useful for deterministic problems with coordinate-sensitive geometry, whereas the observed high-noise robustness is primarily associated with SOMA’s migration mechanism.

    Source: §5 · p. 17

    The VQE noise model omits state-dependent Pauli variances, measurement grouping, shot allocation, readout error, and hardware drift, so explicit finite-shot evaluation remains a next test.

    Source: §4.4 · p. 17

Abstract

from arXiv · show

We study whether the coordinate dependence of the improved Self-Organizing Migrating Algorithm (iSOMA) can be reduced while retaining its inexpensive leader-directed migration mechanism. We introduce iSOMA-AR, which learns a basis from successful migration displacements and selectively applies the standard perturbation mask in that basis. On the complete noiseless BBOB suite, iSOMA- AR significantly outperformed baseline iSOMA across matched conditions, with the largest gains on geometrically difficult landscapes. A targeted ablation shows that the learned orientation is beneficial on a rotated ill-conditioned landscape and that moderate changes of the gate threshold and rotation cap preserve the qualitative result. On CEC 2011 Real World Optimization Problems, iSOMA-AR outperformed iL-SHADE on most problems, although its advantage over baseline iSOMA was not statistically significant. A canonical-jSO rerun is reported as a post-hoc sensitivity check alongside the original jSO-derived comparator. On frustrated-spin variational quantum objectives, adaptive rotation improved most transverse-field conditions, while gains on the diagonal and anisotropic models were absent or selective. Under strong effective sampling noise, the SOMA variants were the most robust population-based methods in the comparison, but iSOMA-AR was not significantly better than baseline iSOMA. Repairing all-zero PRT masks greatly reduced repeated-point evaluations without changing endpoint quality significantly, making this implementation detail unlikely to explain the noise result. Overall, adaptive rotation is most useful on coordinate-sensitive deterministic problems, while the observed noise robustness appears to arise mainly from the underlying SOMA migration mechanism.

1 Introduction

The study targets coordinate dependence in iSOMA’s perturbation mask, asking whether learning a basis can improve search on geometrically difficult objectives without replacing leader-directed migration. It situates this question across controlled optimization benchmarks, application problems, and noisy variational quantum objectives.

1 Introduction

  • Black-box optimization becomes difficult when objectives are ill-conditioned, coupled, rotated, multimodal, rugged, or noisy, because these properties alter informative directions and ranking reliability.The challenge is to search globally with proposal mechanisms that remain effective across differing landscape geometries.Source: §1 · p. 1
  • Standard SOMA’s coordinate-selective PRT mask restricts migration to coordinate-aligned subspaces, which can be inefficient when useful directions combine variables obliquely.The study seeks to reduce this coordinate dependence while retaining inexpensive leader-directed migration.Source: §1 · p. 2
  • Learning a useful basis may matter as much as scalar-parameter adaptation when an optimizer uses coordinate-selective operators.DE illustrates the distinction: componentwise crossover can underperform on correlated and rotated problems, motivating covariance- and eigenvector-based operators.Source: §1 · p. 2
  • BBOB isolates geometric failure modes under controlled transformations, whereas CEC 2011 tests transfer to heterogeneous application-oriented objectives.CEC 2011 includes 22 problems spanning applications such as parameter estimation, control, power systems, and spacecraft trajectories.Source: §1 · p. 2
  • Variational quantum objectives combine optimization geometry with stochastic finite-shot evaluations, making optimizer performance dependent on the problem and sampling setup.Even gradient-free optimization does not generally evade loss of resolvable cost differences in vanishing-gradient regions.Source: §1 · p. 2
  • iSOMA-AR learns a basis from successful migration displacements and applies the existing PRT mask in that basis when off-axis structure is strong.The study tests it across noiseless BBOB, CEC 2011, and frustrated-spin variational quantum objectives, comparing against CMA-ES, adaptive DE methods, and SPSA where noise is present.Source: §1 · p. 2

2 Methods

The methods preserve SOMA’s leader-directed migration while applying its perturbation mask in a learned basis when successful displacements indicate off-axis structure. Evaluation spans BBOB, CEC 2011, and variational quantum objectives, with matched protocols and targeted controls.

2.1 Canonical SOMA, iSOMA baseline, and adaptive rotation

  • SOMA proposes path points by masking the leader-directed displacement, making the mask-induced covariance diagonal in the current coordinate system.An active coordinate interpolates toward, projects onto, or overshoots the target depending on the path position.Source: §2.1 · p. 3

iSOMA baseline used in this study

  • The baseline samples migrants and leader candidates, evaluates path locations in reverse order, and accepts the first point no worse than the migrant.It increases PRT activation with consumed evaluations and restarts part of the population after prolonged unsuccessful attempts without global-best improvement.Source: iSOMA baseline used in this study · p. 3

iSOMA-AR: Adaptive-rotation extension

  • iSOMA-AR updates a success matrix from improving normalized displacements, derives an orthogonal basis, and conditionally applies the PRT mask in that basis without extra objective evaluations.Rotation is gated by the number of successful displacements and an off-axis score; otherwise ordinary coordinate masking is used.Source: iSOMA-AR: Adaptive-rotation extension · p. 4

2.2 Comparator algorithms and common evaluation protocol

  • The comparison includes iSOMA, iSOMA-AR, jSO-derived, iL-SHADE, and CMA-ES; canonical jSO is rerun on BBOB and CEC 2011 as a post-hoc sensitivity analysis.Methods share an objective-evaluation counter and seed labels, although heterogeneous algorithms consume different random streams.Source: §2.2 · p. 5, §2.7 · p. 8

2.3 BBOB experiment

  • BBOB covers all 24 functions at dimensions 5, 10, and 20, with five instances and budgets of 500D and 2000D evaluations.Final errors use matched-condition medians and ranks; fixed-target runtimes are also summarized with ECDFs at 2000D.Source: §2.3 · p. 5

2.4 CEC 2011 real-world experiment

  • CEC 2011 evaluation covers all 22 real-world problems, with 50,000 evaluations and ten runs per optimizer; four Minion rewrite discrepancies are marked provisional.Comparisons use per-problem median objective values because scales and usable target values differ.Source: §2.4 · pp. 5–6

Variational ansatz and error definitions.

Sampling-noise protocol.

  • The noise study corrupts optimizer-returned evaluations with an effective Gaussian surrogate at 8192 or 128 shots, while exact oracle evaluations score queried points offline.The surrogate omits Pauli-term-resolved measurement statistics; targeted controls test gate thresholds, rotation caps, and all-zero-mask repair.Source: Sampling-noise protocol · p. 7, §2.7 · p. 8

3 Results

Across BBOB, iSOMA-AR significantly outperformed baseline iSOMA, with the largest gains on some conditioned landscapes, though stronger comparators remained ahead on others. Results across CEC 2011 and quantum objectives were more dependent on problem family and evaluation noise.

3.1 BBOB benchmark

  • 507 of 720 matched BBOB conditions favored iSOMA-AR over iSOMA; the median condition-wise error ratio was approximately 0.81, with pHolm < 10^-49.Median time per 1000 evaluations increased by about 12% relative to baseline iSOMA.Source: §3.1 · p. 8

Landscape dependence

  • AR/base median error ratios were 0.55 for low/moderate conditioning and 0.45 for high-conditioning unimodal functions, but iL-SHADE and CMA-ES remained stronger on several difficult groups.iSOMA-AR won 137 of 150 high-conditioning conditions against iSOMA, while it lost 105 of 150 to iL-SHADE in that group.Source: Landscape dependence · p. 9, Table 3 · p. 9

Targeted adaptive-rotation ablation

  • On the rotated Ellipsoid, default adaptive rotation outperformed both baseline iSOMA and a random-basis control (p = 1.2 × 10^-4 for both paired comparisons).Threshold and rotation-cap perturbations preserved the qualitative result; canonical jSO was reported separately as a post-hoc sensitivity comparator.Source: Targeted adaptive-rotation ablation · pp. 10–11

Overall results

  • On CEC 2011, iSOMA-AR ranked third (mean rank 3.27), ahead of baseline iSOMA (3.41) and iL-SHADE (4.05), but its advantage over baseline was not significant (pHolm = 0.34).It won 14 problems, tied two, and lost six to iL-SHADE; the median regret ratio was approximately 0.125 (pHolm = 0.052).Source: Overall results · p. 11, Table 4 · p. 11

Application families

  • In power-system problems, iSOMA-AR beat baseline iSOMA on eight of 12 and iL-SHADE on 10 of 12, while it lost all three molecular/material comparisons to baseline.The control/parameter-estimation set was too small for much inference, and the two-problem spacecraft result was provisional because of reference discrepancies.Source: Application families · p. 11, Table 5 · p. 12

Statevector results

  • On exact-objective VQE tasks, iSOMA-AR reduced Q2 median error by 24–48% in all four size–budget conditions, while Q1 showed no benefit and Q3 improved only at larger budgets without significant corrected comparisons.On Q2, three of four direct AR–baseline comparisons remained Holm-significant; the saved Q3 variational reference was unchanged.Source: Statevector results · p. 13, Table 7 · p. 14

High-noise results; Low-noise results

All-zero PRT-mask control.

  • Activating one coordinate whenever a PRT mask was all-zero substantially reduced unchanged-point evaluations, but none of eight paired endpoint comparisons was significant (all Wilcoxon p ≥ 0.252).Thus, the mask repair reduced wasted evaluations without materially changing the high-noise result.Source: All-zero PRT-mask control · p. 15

4 Discussion

Adaptive rotation is most useful when coordinate masking limits iSOMA, while the SOMA family’s high-noise performance is not significantly improved by rotation itself. The discussion also identifies comparator differences, modest optimizer overhead, and limits in the benchmark and noise settings.

4.1 Interpretation across benchmark families

  • iSOMA-AR improved baseline iSOMA in 507 of 720 matched BBOB conditions, with its largest gains in conditioned groups; Q2 median errors fell 24–48% in all four conditions.A controlled rotated-ill-conditioned Ellipsoid ablation found learned rotation significantly outperformed baseline masking and a random-basis control.Source: §4.1 · pp. 15–16

4.2 Noise robustness of the SOMA migration mechanism

  • Under high noise, iSOMA-AR had the best mean rank and ranked first or second in all six model–budget conditions, but none of 12 AR–baseline comparisons was significant.At lower noise, SOMA methods tied for best mean rank; SPSA dominated Q3 while DE/CMA methods became competitive on Q2.Source: §4.2 · pp. 16–17

4.3 Comparator behavior and computational cost

  • Canonical jSO remained competitive, while the jSO-derived implementation was stronger in aggregate BBOB sensitivity comparisons; the two variants are reported separately.The derived comparator was statistically indistinguishable from CMA-ES on aggregate BBOB and significantly stronger on CEC 2011 under this protocol.Source: §4.3 · p. 17
  • On BBOB, iSOMA-AR’s median cost per 1000 evaluations was 1.12 times baseline, versus 1.54 for jSO-derived, 1.84 for iL-SHADE, and 3.72 for CMA-ES.On CEC 2011, adaptive basis cost was 1.75 times baseline; it remained much cheaper per FE than CMA-ES.Source: §4.3 · p. 17

4.4 Limitations and next experiments

  • The CEC 2011 study used one 50,000-evaluation budget and ten runs, and four Minion functions require independent validation against the reference implementation.Its normalized-target ECDF is diagnostic, not an official benchmark metric.Source: §4.4 · p. 17
  • The VQE noise study used additive Gaussian noise and omitted state-dependent Pauli variances, measurement grouping, shot allocation, readout error, and hardware drift.An explicit finite-shot estimator on Q2 and Q3 is identified as a natural next test.Source: §4.4 · p. 17
  • A proposed extension would adaptively choose coordinate masking, rotated masking, and step/pathlength settings using recent successful moves and evaluation efficiency.Source: §4.4 · p. 17

5 Conclusion

iSOMA-AR provides a targeted geometric correction to iSOMA, with its strongest benefits on coordinate-sensitive deterministic problems. Under strong sampling noise, the SOMA variants remain competitive, but the improvement is not significantly attributable to adaptive rotation.

  • Adaptive rotation improves deterministic handling of coupled geometry, while high-noise robustness appears primarily to arise from the underlying SOMA migration mechanism.The conclusion also reports no significant AR–baseline noise advantage and no endpoint-quality change from repairing all-zero PRT masks.Source: §5 · p. 17

A Optimizer configurations and implementation details

The appendix records the optimizer implementations used in the experiments and explains why the primary comparator is labelled jSO-derived. It distinguishes that implementation from published jSO mechanisms and details the choices retained or changed.

A Optimizer configurations and implementation details

Why the comparator is labelled jSO-derived.

  • The reported comparator is labelled jSO-derived because its implementation differs substantially from published jSO, including population initialization, memory settings, parameter restrictions, and bound repair.It retains success-history adaptation, dynamic p, an external archive, and linear population-size reduction.Source: Why the comparator is labelled… · p. 18
  • The derived implementation uses log10 D with a minimum population of 30, initializes MF entries to 0.5, applies the same F to both mutation terms, and clips out-of-bound trial coordinates.Source: Why the comparator is labelled… · p. 18

Canonical-jSO sensitivity implementation.

The post-hoc sensitivity comparator restores principal published jSO mechanisms while making an explicit choice about the ambiguous printed p schedule.

  • The canonical rerun uses the prose convention in which p decreases from 0.25 to 0.125 and is reported as a post-hoc sensitivity comparator.It restores published mechanisms including the weighted current-to-pbest term, stage-dependent F and CR restrictions, and midpoint-to-parent bound repair.Source: Canonical-jSO sensitivity implementation · p. 18

A.1 Runtime overhead

iSOMA-AR limits adaptation overhead by retaining sparse baseline masking and gating costly rotations; its relative overhead is larger on higher-dimensional CEC 2011 problems.

A.1 Runtime overhead

  • Most proposals retain O(D) baseline masking, while dense O(D2) rotations are gated and probabilistic and eigendecomposition follows every five improving updates.These sparse adaptation choices account for iSOMA-AR’s relatively low overhead.Source: §A.1 · p. 18
  • DE variants perform mutation, crossover, population and archive management, and parameter adaptation for each trial, while CMA-ES updates a full covariance-based search distribution.The passage contrasts these operations with iSOMA-AR’s adaptation overhead.Source: §A.1 · p. 18
  • iSOMA-AR’s larger relative overhead on CEC 2011 is consistent with the suite’s higher problem dimensions.For expensive simulation or quantum objectives, optimizer-side differences are expected to be small relative to objective-evaluation cost.Source: §A.1 · p. 18

Ising-backbone selection.

The quantum benchmarks use fixed, matched disorder instances and stored reference energies, with Q3 sharing a classical frustrated backbone at N = 10 and 12.

Ising-backbone selection.

  • For each Ising size, the benchmark retains the seed-1234 candidate disorder instance with the most enumerated one-spin-flip local minima among 50 candidates.The stored arrays, ground energy, and instance seed are reused across optimizers; Q2 loads the matched Q1/Q2 arrays and adds the transverse-field term.Source: Ising-backbone selection · p. 18
  • Table 9 describes optimizer implementations and numerical settings, with B denoting the FE budget and D the decision-space dimension.Table 10 reports median computational overhead per 1000 objective evaluations; its wall-clock values are implementation- and hardware-dependent indicators.Source: Table 9 · p. 19, Table 10 · p. 20

Heisenberg disorder.

  • At N = 10 and 12, Q3 reuses the corresponding Q1/Q2 Ising backbone while drawing independent Jx and Jy couplings at fixed RNG positions from seed 1234.At N = 8, the complete Q3 realization is fixed by the same seed-1234 construction.Source: Heisenberg disorder · p. 20

Q3 variational reference.

  • The shallow 2N-parameter circuit does not generally contain the exact Heisenberg ground state, so N = 10 and 12 use the lowest of 20 polished differential-evolution references.The N = 8 reference is retained after consistency checks, and any benchmark endpoint improving the Q3 reference by more than 10^-12 replaces it; iSOMA-AR changed no final Q3 reference.Source: Q3 variational reference · p. 20

C Adaptive-rotation mechanism ablation

The targeted ablation finds learned orientation most beneficial on rotated ill-conditioned landscapes, while the mask-repair control reduces repeated evaluations without significant endpoint gains.

C Adaptive-rotation mechanism ablation

  • The ablation separates learned orientation from simply injecting a rotated mask by comparing learned AR with a random orthogonal basis under the same gate score and rotation probability.Threshold and rotation-cap variants change one gate parameter at a time; the largest sensitivity-sweep degradation occurs when maximum rotation probability falls to 0.25 on rotated Ellipsoid.Source: C Adaptive-rotation mechanism ablation · p. 20
  • The controlled ablation table compares each control with default learned AR at D = 10 and 20,000 FEs across 25 paired runs, with ratios above one favoring default AR.The gate sensitivity table pairs variants against τ = 0.18 and pmax = 0.45.Source: Table 11 · p. 20, Table 12 · p. 21
  • On the rotated Ellipsoid, learned AR significantly outperforms both baseline iSOMA and random-basis AR; the learned-orientation effect is not significant on axis-aligned Ellipsoid or either Rastrigin comparison.The selected BBOB check also finds learned AR significantly better than baseline and random-basis controls on rotated Rosenbrock f9 and rotated Ellipsoid f10, while random-basis AR is competitive on f2.Source: C Adaptive-rotation mechanism ablation · pp. 20–21

D All-zero PRT-mask control

  • An all-zero baseline mask re-evaluates the unchanged point, producing another noisy observation at that location; the paired repair activates one uniformly random coordinate instead.The paired variants share the optimizer seed and FE-indexed Gaussian-noise stream, with a separate repair RNG.Source: D All-zero PRT-mask control · p. 21

E Canonical-jSO sensitivity analysis

The post-hoc canonical-jSO rerun covers the complete BBOB and CEC 2011 grids, while accompanying controls describe selected BBOB and noisy-mask comparisons.

E Canonical-jSO sensitivity analysis

  • Paper-faithful canonical jSO was rerun on 7,200 complete-grid BBOB runs and 220 CEC 2011 runs using the same budgets and seed labels as jSO-derived.The rerun is presented as a post-hoc sensitivity analysis; the excerpt states that the original five-method ranks remain the primary analysis.Source: E Canonical-jSO sensitivity analysis · p. 21
  • The selected BBOB mechanism check pools two instances and reports learned-AR ratios against baseline or random-basis controls; learned AR is significant on rotated Rosenbrock f9 and rotated Ellipsoid f10, but random-basis AR is competitive on f2.Ratios above one favor default learned AR.Source: Table 13 · p. 22
  • The all-zero-mask figure reports that repairing masks substantially reduces repeated-point evaluations, especially for Q1, without a corresponding significant endpoint improvement.The endpoint-quality figure states that none of eight paired comparisons is statistically significant.Source: Figure 12 · p. 22, Figure 13 · p. 23
  • The canonical-jSO table reports six-method sensitivity-panel ranks and Top-1 shares, with tight targets of Δf = 10^-8 for BBOB and 10^-6 relative gap for CEC 2011.Its tight-runtime measures are FE/D for BBOB and fraction of the 50,000-FE budget for CEC; direct rows compare jSO-derived with canonical jSO.Source: Table 15 · p. 23
  • The VQE experiment uses jSO-derived only, and CEC target-metric fallback reference values remain fixed to the original comparator panel.Source: E Canonical-jSO sensitivity analysis · p. 23

F Additional BBOB, CEC 2011 and VQE results

The additional results organize BBOB outcomes by diagnostic landscape groups, fixed-target performance by dimension, and function-level matched comparisons. They also present CEC 2011 convergence and problem-level results alongside exact and noisy VQE comparisons; direct noisy-objective tests show no significant advantage for adaptive rotation over baseline iSOMA after Holm correction.

VQE results

  • For noisy-objective selection, Table 21’s direct iSOMA-AR versus baseline iSOMA tests, scoring endpoints by exact energy after noisy selection, found no comparison significant after Holm correction.Table 19 gives direct exact-objective iSOMA–iSOMA-AR tests for all 16 original Q1–Q3 conditions.Source: F Additional BBOB, CEC 2011 and VQE… · p. 23, Table 21 · p. 26

Additional BBOB results

  • The BBOB results include overlapping diagnostic groups, function-level W/T/L outcomes across 30 matched dimension–instance–budget conditions, and fixed-target ECDFs split by dimension.The W/T/L comparisons are for iSOMA-AR against baseline iSOMA and iL-SHADE.Source: Table 16 · p. 23, Table 17 · p. 24, Figure 14 · p. 24

CEC 2011 results

  • CEC 2011 results include mean-rank convergence trajectories by diagnostic application family and problem-level outcomes at 50,000 evaluations.The problem-level table marks documented discrepancies between the Minion C++ rewrite and the Octave/MATLAB reference with a dagger.Source: Figure 15 · p. 25, Table 18 · p. 25
  • CEC 2011 problem-level log10 regret ratios compare iSOMA-AR with baseline iSOMA, with values below zero favoring iSOMA-AR.Table 19 also reports Holm-adjusted two-sided Mann–Whitney U results and effects oriented so positive values favor iSOMA-AR.Source: Figure 16 · p. 25, Table 19 · p. 26

VQE results

  • The VQE extension reports condition-wise ranks for N = 12 under low and high sampling noise, and a noisy-selection penalty comparing selected-point error with the oracle-best queried-point error.Lower values are better for both Table 20 metrics and for the penalty in Figure 17.Source: Table 20 · p. 26, Figure 17 · p. 27
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