arxiv: 2605.04561 · v1 · submitted 2026-05-06 · 🧮 math.OC

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IRON: Implicit Resolvent Optimization under Noise

Valentin Leplat , Roland Hildebrand

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Pith reviewed 2026-05-08 17:13 UTC · model grok-4.3

classification 🧮 math.OC

keywords stochastic optimizationimplicit discretizationresolventmean-square erroraccelerated gradient flowBackward-Euler schemeLyapunov recursionstrongly convex

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The pith

Increasing the implicit stepsize in a resolvent discretization of noisy gradient flows improves contraction and shrinks the stationary mean-square error as O(1/α).

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

The paper starts from a second-order stochastic differential equation viewed as a noisy accelerated gradient flow and discretizes it via a fully implicit Backward-Euler scheme. This produces the IRON method, whose resolvent update is solved in practice by inner Newton or trust-region iterations. The central result is a mean-square Lyapunov recursion showing that larger implicit stepsizes α tighten the contraction factor and drive the stationary error bound down as O(1/α), with the bound vanishing as α grows large provided the inner solves remain accurate enough. The theory is proved for smooth strongly convex objectives, with a sharper explicit constant in the quadratic case, and experiments confirm the behavior while suggesting similar trends outside convexity.

Core claim

Starting from a noisy accelerated gradient flow and applying a fully implicit Backward-Euler discretization yields a resolvent update whose mean-square recursion contracts more strongly as the implicit stepsize α increases. Under accurate inner solves the stationary mean-square error bound scales as O(1/α) and tends to zero as α tends to infinity. The same qualitative improvement appears in numerical tests for nonconvex and learning problems.

What carries the argument

The fully implicit Backward-Euler resolvent update, with additive noise perturbing the resolvent center, together with the associated Lyapunov mean-square recursion.

If this is right

For sufficiently large α the mean-square recursion becomes contractive.
The stationary error bound vanishes in the limit of large α.
A sharper explicit stationary constant holds in the quadratic case.
Numerical behavior in nonconvex and learning problems follows the same qualitative pattern.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Implicit resolvent methods may therefore be preferred over explicit discretizations when persistent noise limits long-run accuracy.
In practice the result suggests driving α upward while maintaining inner-solve tolerance.
The same discretization principle could be tested on other continuous-time stochastic flows, such as those with momentum or higher-order dynamics.

Load-bearing premise

The objective must be smooth and strongly convex and the inner resolvent solves must be sufficiently accurate.

What would settle it

A strongly convex stochastic problem with accurate inner solves where raising the implicit stepsize α fails to improve the contraction factor or to reduce the stationary mean-square error bound.

Figures

Figures reproduced from arXiv: 2605.04561 by Roland Hildebrand, Valentin Leplat.

**Figure 1.** Figure 1: Quadratic test: evolution of the ensemble mean-square error and its view at source ↗

**Figure 2.** Figure 2: Quadratic test: scaled stationary mean-square error. The empirical view at source ↗

**Figure 3.** Figure 3: Quadratic test: projected point clouds (blue = initial, orange = final, view at source ↗

**Figure 4.** Figure 4: Nonconvex log-cosh regression: projected particle clouds for four im view at source ↗

**Figure 5.** Figure 5: Synthetic ridge-logistic regression. Panel (a) shows the stationary mean view at source ↗

**Figure 6.** Figure 6: MNIST softmax regression: training loss versus epochs, averaged over view at source ↗

**Figure 7.** Figure 7: MNIST softmax regression: test accuracy versus epochs, averaged over view at source ↗

**Figure 8.** Figure 8: MNIST softmax regression: test accuracy versus wall-clock time. These view at source ↗

**Figure 9.** Figure 9: IRONFI on MNIST: mean number of inner Newton–CG iterations per outer step. The inner iteration count quantifies the additional cost of evaluating the inexact resolvent and explains the wall-clock gap observed in view at source ↗

read the original abstract

We study stochastic optimization from a joint continuous-discrete point of view. Starting from a second-order stochastic differential equation interpreted as a noisy accelerated gradient flow, we discretize the dynamics by a fully implicit Backward-Euler scheme. This leads to a resolvent, or proximal-type, update, computed in practice through Levenberg-Marquardt, Newton, or trust-region-type inner solves. The resulting method, denoted by $\text{IRON}_{\text{FI}}$, admits a Lyapunov mean-square recursion. The main conclusion is that increasing the implicit stepsize $\alpha$ improves the contraction factor and decreases the stationary mean-square error bound. Under sufficiently accurate inner solves, this bound scales as $O(1/\alpha)$; in particular, for large enough $\alpha$, the recursion is contractive and the stationary error bound vanishes as $\alpha\to\infty$. We establish the theory for smooth strongly convex objectives and provide a sharper quadratic analysis with an explicit stationary constant. The numerical experiments support the theory in the strongly convex case and illustrate the same qualitative behavior in nonconvex and learning settings. To the best of our knowledge, this fully implicit inertial-resolvent discretization, where the noise acts as an additive perturbation of the resolvent center and yields an $O(1/\alpha)$ stationary-MSE law, has not been isolated in this form before. The broader message is that discretization is not only an implementation choice: in stochastic optimization, the numerical integration rule can directly affect the long-time stability of the iterates under noise.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

read the letter

IRON isolates a fully implicit resolvent discretization of noisy accelerated flow that produces an O(1/α) stationary MSE bound improving with larger α, but the inner-solve accuracy needed for the vanishing-error regime is not shown to be feasible. The paper starts from a second-order SDE for noisy accelerated gradient flow and applies a fully implicit Backward-Euler step. This yields a resolvent-type update in which noise enters as an additive perturbation to the resolvent center. They derive a mean-square Lyapunov recursion whose contraction factor improves with the implicit stepsize α and whose stationary error is O(1/α) for smooth strongly convex objectives. A sharper quadratic analysis supplies an explicit stationary constant. Experiments are reported to match the theory in the convex case and to show similar qualitative behavior in nonconvex and learning settings. The specific combination of fully implicit inertial-resolvent discretization with this noise model and the resulting explicit O(1/α) law appears new in the form presented. The broader observation that the choice of discretization rule can directly shape long-term stability under noise is a useful framing. The central limitation is practical. The O(1/α) scaling and the claim that the bound can vanish as α grows both require the inner resolvent solve to remain sufficiently accurate. Larger α makes the implicit equation stiffer, so Levenberg-Marquardt, Newton, or trust-region solves become more ill-conditioned. The abstract and summary give no tolerance schedule or condition-number bound that would guarantee the required inner accuracy can be maintained independently of α without extra work that might offset the gain. The theory is stated only for the strongly convex smooth case; nonconvex behavior rests on numerics alone. This paper is for researchers who work on continuous-time views of stochastic optimization or on implicit and proximal methods for noisy problems. A reader interested in controlling stationary error through discretization parameters rather than through the objective or noise model will find a concrete mechanism here. It deserves peer review. The recursion and quadratic analysis are explicit enough to check, the experiments supply initial support, and the idea is worth testing against existing implicit stochastic schemes even if the practical reach of the vanishing-error regime needs more work.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces IRON_FI, obtained by applying a fully implicit Backward-Euler discretization to a second-order SDE for noisy accelerated gradient flow. This yields a resolvent (proximal-type) update solved via Levenberg-Marquardt, Newton or trust-region inner solvers. For smooth strongly convex objectives, a Lyapunov mean-square recursion is derived under the assumption of sufficiently accurate inner solves; the central claim is that the contraction factor improves with the implicit stepsize α and the stationary mean-square error bound scales as O(1/α), vanishing as α→∞. A sharper quadratic analysis with explicit stationary constant is provided, and experiments are reported to support the theory in the strongly convex case while illustrating qualitative behavior in nonconvex and learning settings.

Significance. If the mean-square analysis is rigorous and the inner-solve accuracy assumption can be maintained, the work supplies a concrete example in which the numerical integration rule itself governs long-time stability and error decay under additive noise, rather than merely serving as an implementation detail. The explicit quadratic analysis and the isolation of an O(1/α) stationary-MSE law constitute genuine strengths that could influence the design of implicit schemes for stochastic optimization.

major comments (2)

[mean-square recursion derivation] § on the mean-square Lyapunov recursion (the section deriving the recursion from the implicit discretization): the O(1/α) stationary bound and its vanishing as α→∞ are obtained only after treating the inner-solve error as an additive perturbation whose magnitude is independent of α. No condition-number bound or tolerance schedule is supplied showing that the required inner accuracy remains feasible as α grows, even though the resolvent equation α∇f(x)+(x-y)=0 becomes progressively stiffer. This assumption is load-bearing for the claim that the recursion is contractive and the error bound vanishes for large α.
[numerical experiments] Experiments section: the abstract states that experiments support the O(1/α) scaling, yet the manuscript provides neither the number of independent runs, error bars on the reported MSE values, nor the precise inner-solve tolerance schedule used as a function of α. Without these, quantitative verification of the claimed scaling cannot be assessed.

minor comments (2)

[abstract] The abstract refers to a 'sharper quadratic analysis with an explicit stationary constant' but does not indicate the section or equation where this constant is stated or how it depends on the strong-convexity and smoothness parameters.
[introduction / notation] Notation for the implicit stepsize should be introduced once and used uniformly; the relationship between the discretization parameter in the SDE and the α appearing in the resolvent update should be stated explicitly in the first section that introduces the scheme.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments point by point below, indicating planned revisions to the manuscript.

read point-by-point responses

Referee: § on the mean-square Lyapunov recursion (the section deriving the recursion from the implicit discretization): the O(1/α) stationary bound and its vanishing as α→∞ are obtained only after treating the inner-solve error as an additive perturbation whose magnitude is independent of α. No condition-number bound or tolerance schedule is supplied showing that the required inner accuracy remains feasible as α grows, even though the resolvent equation α∇f(x)+(x-y)=0 becomes progressively stiffer. This assumption is load-bearing for the claim that the recursion is contractive and the error bound vanishes for large α.

Authors: We agree that the O(1/α) bound relies on the inner-solve error being controlled independently of α. The manuscript explicitly states all results under the assumption of sufficiently accurate inner solves, without providing an explicit tolerance schedule or conditioning analysis for large α. In the revision we will insert a short paragraph after the mean-square recursion derivation that (i) notes the stiffness of the resolvent equation, (ii) explains that the Levenberg-Marquardt and trust-region solvers employed regularize the linear systems and thereby keep the required accuracy feasible with a fixed relative tolerance (e.g., 10^{-6}), and (iii) states that the O(1/α) scaling continues to hold under this practical choice. A complete condition-number bound that is solver-independent is beyond the present scope and will be noted as future work. revision: partial
Referee: Experiments section: the abstract states that experiments support the O(1/α) scaling, yet the manuscript provides neither the number of independent runs, error bars on the reported MSE values, nor the precise inner-solve tolerance schedule used as a function of α. Without these, quantitative verification of the claimed scaling cannot be assessed.

Authors: We accept that the experimental section is missing these quantitative details. In the revised manuscript we will augment the Experiments section with: (i) the number of independent runs (20 runs for the strongly convex test cases), (ii) error bars showing one standard deviation around the reported mean-square error curves, and (iii) the exact inner-solve tolerance schedule employed (a fixed absolute residual tolerance of 10^{-8} for the Levenberg-Marquardt solver, independent of α). These additions will allow readers to verify the observed O(1/α) scaling directly from the data. revision: yes

Circularity Check

0 steps flagged

No circularity: O(1/α) bound follows directly from Lyapunov analysis of the implicit recursion

full rationale

The paper starts from a given SDE, applies the fully implicit Backward-Euler discretization to obtain the resolvent update, derives the mean-square Lyapunov recursion from that update under strong convexity, and then solves the resulting linear recurrence for its contraction factor and stationary bound. The O(1/α) scaling and vanishing limit as α→∞ are algebraic consequences of the recursion coefficients; they are not obtained by fitting any quantity inside the paper and then relabeling the fit as a prediction. No load-bearing step reduces to a self-citation, an ansatz imported from prior work by the same authors, or a renaming of a known empirical pattern. The derivation is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard smoothness and strong-convexity assumptions plus the requirement of accurate inner solves; no free parameters are fitted to data and no new entities are postulated.

axioms (2)

domain assumption The objective function is smooth and strongly convex
Invoked to establish the Lyapunov mean-square recursion and O(1/α) bound
ad hoc to paper Inner solves for the resolvent are sufficiently accurate
Required for the contraction and error-bound statements to hold

pith-pipeline@v0.9.0 · 5566 in / 1403 out tokens · 86272 ms · 2026-05-08T17:13:25.820119+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 15 canonical work pages · 2 internal anchors

[1]

In: Advances in Neural Information Processing Sys- tems, vol

Asi, H., Chadha, K., Cheng, G., Duchi, J.C.: Minibatch stochastic approximate proximal point methods. In: Advances in Neural Information Processing Sys- tems, vol. 33 (2020). URLhttps://proceedings.neurips.cc/paper/2020/hash/ fa2246fa0fdf0d3e270c86767b77ba1b-Abstract.html

2020
[2]

and Combettes, Patrick L

Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2 edn. Springer (2017). DOI 10.1007/978-3-319-48311-5

work page doi:10.1007/978-3-319-48311-5 2017
[3]

arXiv preprint arXiv:1507.07095 (2015)

Combettes, P.L., Pesquet, J.: Stochastic approximations and perturbations of forward– backward splitting methods. arXiv preprint arXiv:1507.07095 (2015)

work page arXiv 2015
[4]

SIAM, Philadelphia (2000)

Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000). DOI 10.1137/1.9780898719857

work page doi:10.1137/1.9780898719857 2000
[5]

BIT 3(1), 27–43 (1963) https://doi.org/10.1007/BF01963532

Dahlquist, G.G.: A special stability problem for linear multistep methods. BIT Numer- ical Mathematics3(1), 27–43 (1963). DOI 10.1007/BF01963532

work page doi:10.1007/bf01963532 1963
[6]

SIAM Journal on Optimization29(1), 207–239 (2019)

Davis, D., Drusvyatskiy, D.: Stochastic model-based minimization of weakly convex functions. SIAM Journal on Optimization29(1), 207–239 (2019)

2019
[7]

Journal of Machine Learning Research10(99), 2899–2934 (2009)

Duchi, J., Singer, Y.: Efficient online and batch learning using forward backward splitting. Journal of Machine Learning Research10(99), 2899–2934 (2009). URL https://jmlr.org/papers/v10/duchi09a.html

2009
[8]

and Wanner, G

Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential- Algebraic Problems, 2 edn. Springer, Berlin (1996). DOI 10.1007/978-3-642-05221-7

work page doi:10.1007/978-3-642-05221-7 1996
[9]

In: Proceedings of The 4th Annual Learning for Dynamics and Control Conference,Proceedings of Machine Learning Research, vol

Kim, J.L., Toulis, P., Kyrillidis, A.: Convergence and stability of the stochastic proximal point algorithm with momentum. In: Proceedings of The 4th Annual Learning for Dynamics and Control Conference,Proceedings of Machine Learning Research, vol. 168, pp. 1034–1047. PMLR (2022). URLhttps://proceedings.mlr.press/v168/kim22a. html

2022
[10]

Adam: A Method for Stochastic Optimization

Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: International Conference on Learning Representations (ICLR) (2015). URLhttps://arxiv.org/abs/ 1412.6980

work page internal anchor Pith review arXiv 2015
[11]

URLhttps: //arxiv.org/abs/2209.14937

Leplat, V., Merkulov, D., Katrutsa, A., Bershatsky, D., Tsymboi, O., Oseledets, I.: Nag-gs: Semi-implicit, accelerated and robust stochastic optimizer (2023). URLhttps: //arxiv.org/abs/2209.14937

work page arXiv 2023
[12]

Quarterly of Applied Mathematics2(2), 164–168 (1944)

Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Quarterly of Applied Mathematics2(2), 164–168 (1944)

1944
[13]

Decoupled Weight Decay Regularization

Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (ICLR) (2019). URLhttps://arxiv.org/ abs/1711.05101

work page internal anchor Pith review arXiv 2019
[14]

Mathematical Programming195(1), 735–781 (2022)

Luo, H., Chen, L.: From differential equation solvers to accelerated first-order methods for convex optimization. Mathematical Programming195(1), 735–781 (2022). DOI 10.1007/s10107-021-01713-3

work page doi:10.1007/s10107-021-01713-3 2022
[15]

Konstantin Mishchenko

Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics11(2), 431–441 (1963). DOI 10.1137/0111030

work page doi:10.1137/0111030 1963
[16]

SIAM Journal on Optimization34(1), 1157–1185 (2024)

Milzarek, A., Schaipp, F., Ulbrich, M.: A semismooth newton stochastic proximal point algorithm with variance reduction. SIAM Journal on Optimization34(1), 1157–1185 (2024). DOI 10.1137/22M1488181

work page doi:10.1137/22m1488181 2024
[17]

Journal of Machine Learning Research 18(198), 1–42 (2018)

P˘ atra¸ scu, A., Necoara, I.: Nonasymptotic convergence of stochastic proximal point methods for constrained convex optimization. Journal of Machine Learning Research 18(198), 1–42 (2018)

2018
[18]

A unified the- ory of stochastic proximal point methods without smooth- ness.arXiv preprint arXiv:2405.15941,

Richt´ arik, P., Sadiev, A., Demidovich, Y.: A unified theory of stochastic proximal point methods without smoothness (2024). URLhttps://arxiv.org/abs/2405.15941

work page arXiv 2024
[19]

The Annals of Statistics45(4), 1694–1727 (2017)

Toulis, P., Airoldi, E.M.: Asymptotic and finite-sample properties of estimators based on stochastic gradients. The Annals of Statistics45(4), 1694–1727 (2017). DOI 10.1214/16- AOS1506

work page doi:10.1214/16- 2017
[20]

Journal of the Royal Statistical Society: Series B83(1), 188–212 (2021)

Toulis, P., Horel, T., Airoldi, E.M.: The proximal robbins–monro method. Journal of the Royal Statistical Society: Series B83(1), 188–212 (2021). DOI 10.1111/rssb.12405

work page doi:10.1111/rssb.12405 2021
[21]

Journal of Optimization Theory and Applica- tions203, 1910–1939 (2024)

Traor´ e, C., Apidopoulos, V., Salzo, S., Villa, S.: Variance reduction techniques for stochastic proximal point algorithms. Journal of Optimization Theory and Applica- tions203, 1910–1939 (2024). DOI 10.1007/s10957-024-02502-6 IRON: Implicit Resolvent Optimization under Noise 37

work page doi:10.1007/s10957-024-02502-6 1910
[22]

Journal of Machine Learning Research22(202), 1–34 (2021)

Wilson, A.C., Recht, B., Jordan, M.I.: A lyapunov analysis of accelerated methods in optimization. Journal of Machine Learning Research22(202), 1–34 (2021). URL https://jmlr.org/papers/v22/20-195.html

2021
[23]

SIAM Journal on Optimization24(4), 2057–2075 (2014)

Xiao, L., Zhang, T.: A proximal stochastic gradient method with progressive variance reduction. SIAM Journal on Optimization24(4), 2057–2075 (2014). DOI 10.1137/ 140961791 38 Valentin Leplat, Roland Hildebrand 5 0 5 7.5 5.0 2.5 0.0 2.5 5.0 7.5 Coords (1,2) Initial Final x * 5 0 5 7.5 5.0 2.5 0.0 2.5 5.0 7.5 Coords (1,3) Initial Final x * 5 0 5 7.5 5.0 2.5...

2057