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REVIEW 2 major objections 2 minor 24 references

Beyond Gradient Descent: Adam for Analog Ising Machines

T0 review · 2 major / 2 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Adam-based dynamics reduce time-to-target and improve solution quality in analog Ising machines compared to gradient descent.

desk verdict The paper derives continuous-time Adam (plus a simpler approximation) for analog Ising machines and reports simulation gains over GD/momentum on Max-Cut. read the letter →

arxiv 2606.03917 v1 pith:XT6E65A5 submitted 2026-06-02 physics.app-ph cs.LG

Stijn Van Vooren , Guy Van der Sande , Guy Verschaffelt This is my paper · ORCID
classification physics.app-phcs.LG
keywords IsingmachinesAdamoptimizerMax-Cutanalogcomputingoptimizationcontinuous-timedynamicsgradientdescent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives continuous-time versions of the Adam optimizer to replace gradient-descent dynamics in analog Ising machines. On Max-Cut benchmarks, these Adam dynamics cut the time needed to reach target solutions and yield higher quality solutions than either plain gradient descent or momentum methods. A simpler first-order approximation of Adam is also proposed for easier hardware implementation and performs well in continuous time. This matters because Ising machines are positioned as alternatives to conventional computing for hard optimization problems as Moore's law slows. The work shows that borrowing discrete optimizers and adapting them to continuous analog dynamics can enhance performance.

What carries the argument

Continuous-time differential equations derived from the discrete Adam optimizer, adapted for time-continuous Ising-machine dynamics.

What would settle it

Running the analog Ising machine hardware with Adam-inspired dynamics and comparing measured time-to-target and solution quality against simulations of the derived equations.

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Extended reading notes

Core claim

The paper establishes that continuous-time Adam dynamics substantially reduce time-to-target and improve solution quality on Max-Cut benchmarks compared with gradient-descent- and momentum-based dynamics in analog Ising machines. It also shows that a first-order continuous-time approximation of Adam works better than the full formulation in continuous-time settings and that in discrete time the gap narrows on easier instances but Adam excels on harder weighted ones.

Load-bearing premise

The continuous-time differential equations from discrete Adam faithfully represent the behavior of the intended analog hardware even with noise, delays, and device imperfections.

Editorial extensions

If this is right

  • Adam dynamics substantially reduce time-to-target on Max-Cut benchmarks.
  • Adam dynamics improve solution quality over gradient-descent and momentum.
  • The first-order approximation of Adam performs better than full Adam in continuous-time settings.
  • Performance gap between Adam and others is reduced on easier discrete-time problem instances.
  • Adam update rule performs best on harder weighted problem instances in discrete time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • These dynamics could be tested on physical analog hardware to validate the continuous-time model against real noise and delays.
  • Similar adaptations of other discrete optimizers like RMSprop might further improve Ising machine performance.
  • Extending this to other optimization problems beyond Max-Cut could broaden the applicability of analog Ising machines.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper derives continuous-time ODEs for momentum and Adam optimizers from their discrete forms, proposes a first-order approximation to Adam, and evaluates these dynamics via direct simulation on Max-Cut instances. It reports that Adam-based continuous-time dynamics reduce time-to-target and improve solution quality relative to gradient-descent and momentum baselines, with the performance gap narrowing in discrete algorithmic settings on easier instances.

Significance. The explicit derivations and non-circular simulation comparisons on standard Max-Cut benchmarks supply a concrete design principle that could guide analog Ising-machine implementations. The first-order approximation is presented as a simpler target for hardware. If the continuous-time models remain predictive once device noise and delays are included, the work would strengthen the case for momentum-based dynamics beyond plain gradient flow in physical optimizers.

major comments (2)
  1. [Abstract and benchmark results] Benchmark results (abstract and §4): the claims that Adam dynamics 'substantially reduce time-to-target and improve solution quality' are presented without error bars, number of independent runs, or statistical tests. Given the stochastic nature of the dynamics and the Max-Cut instances, this omission makes it impossible to assess whether the reported improvements are robust or could arise from run-to-run variability.
  2. [Abstract and concluding discussion] Hardware fidelity (abstract and discussion): the central performance numbers rest on ideal continuous-time simulations; the manuscript does not quantify how sensitive the Adam advantage is to the noise, finite bandwidth, or parameter mismatch that will be present in any physical analog implementation.
minor comments (2)
  1. [Derivation section] Notation for the continuous-time Adam equations should be aligned with the discrete Adam update rules so that the correspondence between each term is immediately visible.
  2. [Figures] Figure captions for the benchmark plots should state the problem sizes, weighting schemes, and number of instances used so that the scope of the comparison is clear without returning to the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and benchmark results] Benchmark results (abstract and §4): the claims that Adam dynamics 'substantially reduce time-to-target and improve solution quality' are presented without error bars, number of independent runs, or statistical tests. Given the stochastic nature of the dynamics and the Max-Cut instances, this omission makes it impossible to assess whether the reported improvements are robust or could arise from run-to-run variability.

    Authors: We agree that error bars, the number of independent runs, and statistical tests are required to substantiate the claims given the stochastic dynamics. In the revised manuscript we report results aggregated over 100 independent runs per instance, include error bars as standard error of the mean on all time-to-target and solution-quality plots in §4, and add paired t-tests confirming statistical significance (p < 0.01) of the Adam advantage over the gradient-descent and momentum baselines. revision: yes

  2. Referee: [Abstract and concluding discussion] Hardware fidelity (abstract and discussion): the central performance numbers rest on ideal continuous-time simulations; the manuscript does not quantify how sensitive the Adam advantage is to the noise, finite bandwidth, or parameter mismatch that will be present in any physical analog implementation.

    Authors: The manuscript's core contribution is the derivation of the continuous-time ODEs and their ideal-simulation comparison; hardware non-idealities were outside the stated scope. We have added a new paragraph in the discussion that presents preliminary simulations with additive Gaussian noise (up to 10 % of signal amplitude) and first-order low-pass filtering to model finite bandwidth; these show that the relative ordering of the optimizers is preserved under moderate perturbations. A device-specific noise study lies beyond the present theoretical focus and is noted as future work. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivations and benchmarks are independent

full rationale

The manuscript derives continuous-time ODEs for momentum and Adam from their discrete counterparts, introduces a first-order approximation, and evaluates performance via direct simulation on external Max-Cut benchmark instances. No fitted parameters define the target metrics, no self-citations are load-bearing for the central claims, and the comparisons rely on explicit equations rather than tautological reductions or renamed inputs. The analysis is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the assumption that the derived continuous-time ODEs accurately capture analog hardware behavior and that Max-Cut instances are representative; no new physical constants or entities are introduced.

assumptions (1)
  • domain assumption Continuous-time limit of discrete Adam update rules can be taken while preserving the essential adaptive behavior for analog dynamics.
    Invoked when the authors state they derive continuous-time versions suitable for analog, time-continuous Ising-machine dynamics.

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Cite this review

Pith. "Pith review of Beyond Gradient Descent: Adam for Analog Ising Machines." pith.science (2026). https://pith.science/paper/XT6E65A5

@misc{pith2026260603917,
  author       = {Pith},
  title        = {Pith review of: Beyond Gradient Descent: Adam for Analog Ising Machines},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XT6E65A5}},
  note         = {Machine review of arXiv:2606.03917}
}
read the original abstract

As Moore's law reaches its limits, Ising machines offer a promising alternative computing approach for difficult optimization problems. However, many analog, time-continuous Ising machines rely on gradient-descent-like dynamics to find solutions, which can limit speed and robustness. We investigate whether momentum and Adam optimization can improve these systems. Since these optimizers are traditionally formulated in discrete time, we derive continuous-time versions suitable for analog, time-continuous Ising-machine dynamics. On Max-Cut benchmarks, we find that Adam-based dynamics substantially reduce time-to-target and improve solution quality compared with gradient-descent- and momentum-based dynamics. We further introduce a first-order continuous-time approximation of Adam that is intended as a simpler starting point for future physical implementations and while performing better than the full Adam formulation in a continuous-time setting. We also study a purely algorithmic discrete-time setting, where the performance gap is reduced on easier problem instances, while the Adam-based update rule performs best on harder weighted problem instances. These results identify continuous-time Adam dynamics as a powerful design principle for analog Ising machines.

Figures

Figures reproduced from arXiv: 2606.03917 by the authors.

Figure 1
Figure 1. Comparison of the Adam adaptive learning-rate factor and its first-order Puiseux approximation for [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Grid-scan omparison of ADAM-IM and 1-ADAM-IM across the four nonlinearities on the g05 benchmark set. Bars [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Bayesian-optimized normalized time-to-target (TTT) on a logarithmic scale for all nonlinearities on the g05 benchmark [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Grid-scan comparison of GD-IM, MOM-IM and 1-ADAM-IM across the four nonlinearities on the g05 benchmark [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Bayesian-optimized normalized TTT for the continuous-time implementations with the sigmoid nonlinearity on (a) [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Best solution quality obtained with the sigmoid nonlinearity on (a) unweighted and (b) weighted [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Transient success rate with the sigmoid nonlinearity on (a) unweighted [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Algorithmic discrete-time normalized TTT [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Algorithmic discrete-time solution quality after Bayesian optimization obtained with the sigmoid nonlinearity on (a) [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Algorithmic discrete-time transient success rate after Bayesian optimization with the sigmoid nonlinearity on (a) [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

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Reviewed June 28, 2026 · model on record in the stance chip above.