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arxiv: 2606.06280 · v1 · pith:6HCKP4UNnew · submitted 2026-06-04 · 🧮 math.OC

Second order splitting dynamics for stochastic monotone inclusions with closed loop distribution

Wutao Si , Hamza Ennaji , Jalal Fadili This is my paper

Pith reviewed 2026-06-28 00:11 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic monotone inclusionssecond-order dynamicsforward-backward splittinguniform monotonicityperformative predictionviscous dampingexponential convergence
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The pith

Second-order dynamics governed by a distributionally evaluated forward-backward splitting operator converge strongly to the unique equilibrium of sums of maximal monotone and cocoercive operators under uniform monotonicity.

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

The paper analyzes continuous-time second-order dynamical systems for finding zeros of sums of maximal monotone and cocoercive operators in Hilbert spaces. These systems use a forward-backward splitting operator evaluated at the current distribution, which models stochastic optimization where the distribution depends on the decision variable. Under a uniform monotonicity assumption, the dynamics admit a unique equilibrium to which trajectories converge strongly when using vanishing viscous damping, with fast velocity decay. With constant Polyak damping and strong monotonicity of the regularizer, the system achieves global exponential convergence. This approach provides a theoretical foundation for understanding algorithms in performative prediction settings.

Core claim

We establish the existence and uniqueness of the equilibrium point under a general uniform monotonicity assumption. In this setting, employing a vanishing viscous damping coefficient, we prove the strong convergence of the trajectories to the equilibrium, accompanied by fast asymptotic convergence rates for the velocities. Furthermore, when the regularizing operator is strongly monotone, we consider a constant Polyak-type damping coefficient and we establish global exponential convergence rates for the dynamical system.

What carries the argument

distributionally evaluated forward-backward splitting operator in the second-order inertial dynamics

If this is right

  • Existence and uniqueness of an equilibrium point for the dynamical system.
  • Strong convergence of trajectories to the equilibrium with vanishing viscous damping.
  • Fast asymptotic convergence rates for the velocities under vanishing damping.
  • Global exponential convergence rates when using constant Polyak-type damping and strong monotonicity of the regularizer.
  • The framework models stochastic optimization problems with decision-dependent distributions.

Where Pith is reading between the lines

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

  • Discretizations of these second-order dynamics may yield practical algorithms with inherited convergence guarantees for performative prediction tasks.
  • The uniform monotonicity condition could be checked in concrete applications such as certain supervised learning models with feedback effects.
  • Similar inertial splitting structures might extend to related problems involving time-varying or closed-loop distributions outside the monotone inclusion setting.

Load-bearing premise

The sum of the maximal monotone operator and the cocoercive operator satisfies a uniform monotonicity condition.

What would settle it

A concrete counterexample consisting of a maximal monotone operator A and cocoercive operator Bm whose sum is monotone but not uniformly monotone, for which trajectories of the proposed dynamics fail to converge strongly.

read the original abstract

In this paper, we investigate the problem of finding a zero of the sum of a maximal monotone operator $A$ and a cocoercive operator $\Bm$ in a Hilbert space. This formulation naturally captures stochastic optimization problems with decision-dependent distributions, often referred to as performative prediction. We propose and analyze continuous-time second-order dynamics governed by a distributionally evaluated forward-backward splitting operator. We establish the existence and uniqueness of the equilibrium point under a general uniform monotonicity assumption. In this setting, employing a vanishing viscous damping coefficient, we prove the strong convergence of the trajectories to the equilibrium, accompanied by fast asymptotic convergence rates for the velocities. Furthermore, when the regularizing operator is strongly monotone, we consider a constant Polyak-type damping coefficient and we establish global exponential convergence rates for the dynamical system.

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.

Referee Report

0 major / 3 minor

Summary. The paper studies the zero-finding problem for the sum of a maximal monotone operator A and a cocoercive operator Bm in a Hilbert space, motivated by performative prediction with decision-dependent distributions. It introduces second-order continuous-time dynamics driven by a distributionally evaluated forward-backward splitting operator. Under a uniform monotonicity assumption on A + Bm, the work claims existence and uniqueness of an equilibrium, strong convergence of trajectories to the equilibrium (with fast velocity rates) when using vanishing viscous damping, and global exponential convergence when Bm is strongly monotone and a constant Polyak-type damping is used.

Significance. If the derivations hold, the results supply a continuous-time convergence theory for second-order splitting methods in closed-loop monotone inclusions, extending standard forward-backward analysis to performative settings and furnishing explicit rates under vanishing and constant damping. The explicit conditioning on uniform monotonicity and the separation of vanishing versus constant damping cases are strengths that keep the claims falsifiable.

minor comments (3)
  1. [Abstract] Abstract and §1: the operator is denoted Bm throughout; the manuscript should clarify at first use whether m indexes the distribution or is part of the operator symbol, and whether the cocoercivity constant depends on the distribution.
  2. [Abstract] The abstract states 'fast asymptotic convergence rates for the velocities' but does not indicate the precise rate (e.g., o(1/t) or better); a brief statement of the rate in the abstract or introduction would improve readability.
  3. [§2] Notation section or §2: ensure that the Hilbert-space inner product and norm are fixed once and that the definition of the distributionally evaluated forward-backward operator is given before the dynamics are written.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring point-by-point rebuttal or manuscript changes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims—existence and uniqueness of equilibrium under uniform monotonicity of A + Bm, strong convergence of trajectories under vanishing damping, and exponential convergence under constant Polyak damping when the regularizer is strongly monotone—are standard monotone-operator arguments conditioned explicitly on the stated assumption. No derivation step reduces a claimed rate or equilibrium to a fitted quantity, self-referential definition, or self-citation chain; the performative-prediction setting is presented only as motivation, not as an input that forces the conclusions. The derivation chain is therefore self-contained against the external benchmark of the uniform monotonicity hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper rests on standard background from monotone operator theory in Hilbert spaces. No free parameters or invented entities are introduced in the abstract. The uniform monotonicity assumption is the key domain assumption enabling the convergence claims.

axioms (4)
  • standard math The underlying space is a real Hilbert space.
    Invoked implicitly for the definition of the operators and inner-product structure.
  • domain assumption A is a maximal monotone operator.
    Stated in the problem formulation; required for the forward-backward splitting to be well-defined.
  • domain assumption Bm is a cocoercive operator.
    Part of the sum whose zero is sought; enables the splitting step.
  • domain assumption The sum operator satisfies a uniform monotonicity condition.
    Explicitly required for existence/uniqueness of equilibrium and all convergence statements.

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Reference graph

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