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arxiv: 2604.09789 · v2 · submitted 2026-04-10 · 🧮 math.OC · cs.NA· math.NA

ProxiCBO: A Provably Convergent Consensus-Based Method for Composite Optimization

Haoyu Zhang , Yanting Ma , Ruangrawee Kitichotkul , Joshua Rapp , Petros Boufounos This is my paper

Pith reviewed 2026-05-10 16:38 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords consensus-based optimizationproximal optimizationcomposite optimizationnonconvex optimizationparticle-based methodsglobal convergencesignal processing
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The pith

ProxiCBO combines consensus-based optimization with proximal gradients for provable global convergence in non-convex composite problems.

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

This paper presents ProxiCBO, a method that merges consensus-based optimization with proximal gradient techniques for composite optimization problems that may be non-convex. These problems appear often in signal processing. The work proves global convergence for the continuous-time version with a finite number of particles and offers a practical alternating update scheme. If correct, this provides a reliable way to optimize difficult functions more efficiently than previous approaches, using fewer particles for similar accuracy.

Core claim

The central claim is that ProxiCBO, by integrating CBO with proximal operators, achieves global convergence guarantees for its continuous-time finite-particle dynamics on composite optimization problems, even when non-convex, and supports an efficient discrete implementation via alternating updates.

What carries the argument

The alternating consensus-proximal particle dynamics, which uses particle interactions for exploration and proximal mappings to handle the nonsmooth composite term while maintaining convergence.

If this is right

  • The continuous-time particle system converges globally to an optimizer.
  • The alternating scheme allows practical implementation without losing the convergence properties.
  • ProxiCBO shows superior performance in signal recovery from one-bit measurements and lidar parameter estimation compared to prior methods.
  • It requires fewer particles for effective optimization in tested tasks.

Where Pith is reading between the lines

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

  • This hybrid could be adapted to other structured optimization problems beyond signal processing.
  • The convergence result might encourage combining particle methods with other structure-exploiting techniques like constraints or regularizers.
  • Testing on higher-dimensional or real-world large-scale problems would reveal scalability.
  • Similar methods might emerge for stochastic or distributed optimization settings.

Load-bearing premise

That incorporating proximal operators into the consensus-based particle dynamics preserves the global convergence even in non-convex settings.

What would settle it

Demonstrating a non-convex composite objective where the ProxiCBO particle trajectories diverge or fail to reach the known global minimum, contrary to the established guarantees.

Figures

Figures reproduced from arXiv: 2604.09789 by Haoyu Zhang, Joshua Rapp, Petros Boufounos, Ruangrawee Kitichotkul, Yanting Ma.

Figure 1
Figure 1. Figure 1: Success rate by objective function value defined in (12). achieving global minimum, as this is important for evaluating an optimization algorithm. For each trial, we estimate the global minimum by running PG initialized at the ground truth signal and a trial is said to be successful if the excess error in objective function value above the estimated global minimum is smaller than a threshold. Specifically,… view at source ↗
Figure 2
Figure 2. Figure 2: Reconstruction SNR for sparse signal recovery with one-bit quantized measurements. The number of particles is 1000. Sparse recovery. When x0 is known to be sparse, we use ℓ1 norm as the regularizer and estimate x0 by solving min x∈Rd {E(x) := D(x; A, y, u) + λ∥x∥1} . In our simulations, x0 has dimension d = 200 with sparsity s = 10, and the number of measurements is m = 4d. All initial particles have i.i.d… view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction SNR for image recovery with one-bit quantized mea￾surements. The number of particles is 500 and the regularization parameter is λ = β · 10−4∥y∥ 2 2 . Doppler SPL. We now consider the case where the target is moving with constant velocity v. The photon detection process is still an inhomogeneous Poisson process, but the intensity function has changed due to the Doppler-shift effect. Suppose t… view at source ↗
Figure 4
Figure 4. Figure 4: Estimation of S (Fig. 4a), b (Fig. 4b) and τ (Fig. 4c) for single-photon lidar. The relatively high root mean squared error (RMSE) of ProxiCBO with 60 particles for τ estimation in Fig. 4c is due to a few (10 out of 10,000) outliers as we can see in Fig. 4d, which explains the high success rate reported in Fig. 1b [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Estimation of S (Fig. 5a), b (Fig. 5b), τ0 (Fig. 5c), and v (Fig. 5d) for Doppler single-photon lidar. The results are computed from 10,000 independent trials [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

This paper introduces an interacting-particle optimization method tailored to possibly non-convex composite optimization problems, which arise widely in signal processing. The proposed method, \emph{ProxiCBO}, integrates consensus-based optimization (CBO) with proximal gradient techniques to handle challenging optimization landscapes and exploit the composite structure of the objective function. We establish global convergence guarantees for the continuous-time finite-particle dynamics and develop an alternating update scheme for efficient practical implementation. Simulation results for signal processing tasks, including signal recovery from one-bit quantized measurements and parameter estimation from single-photon lidar data, demonstrate that ProxiCBO outperforms existing proximal gradient methods and CBO methods in terms of both accuracy and particle-efficiency.

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

2 major / 2 minor

Summary. The paper introduces ProxiCBO, an interacting-particle method that augments consensus-based optimization (CBO) with proximal operators to address composite optimization problems that may be non-convex. It establishes global convergence for the continuous-time finite-particle dynamics and proposes an alternating proximal-consensus update scheme for practical use. Numerical experiments on one-bit signal recovery and single-photon lidar parameter estimation are reported to show improved accuracy and particle efficiency relative to standard proximal gradient and CBO baselines.

Significance. The continuous-time global convergence result for the proximal-augmented particle system is a clear theoretical contribution, as it extends CBO-style methods to composite objectives while retaining global guarantees even in non-convex regimes. If the discrete alternating scheme can be rigorously connected to this analysis, the work would supply a useful, structure-exploiting alternative for signal-processing optimization tasks.

major comments (2)
  1. [abstract and alternating-update section] The abstract and the section describing the practical algorithm state that ProxiCBO is a 'provably convergent' method, yet the only convergence theorem supplied applies exclusively to the continuous-time finite-particle dynamics. No separate result, discretization-error bound, or invariance argument is given for the alternating discrete scheme (proximal step followed by consensus update) that is actually implemented, particularly when the objective is non-convex and the proximal map may be set-valued or non-expansive. This gap is load-bearing for the central claim.
  2. [problem formulation and assumptions] The weakest-assumption paragraph asserts that proximal operators can be integrated while preserving global convergence properties of the CBO dynamics in non-convex settings, but the manuscript supplies no explicit conditions on the proximal map (e.g., single-valuedness, non-expansiveness, or subdifferential regularity) that would be required to carry the continuous-time analysis over to the discrete alternating scheme.
minor comments (2)
  1. [numerical experiments] The simulation section would benefit from explicit reporting of quantitative metrics (e.g., mean and standard deviation of recovery error over multiple runs), clear baseline implementations, and statistical significance tests to support the outperformance claims.
  2. [method description] Notation for the particle positions, consensus drift, and proximal step should be introduced with a single consistent table or list of symbols to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the precise scope of our theoretical results. We address each major point below and will revise the manuscript accordingly to avoid any overstatement of the claims.

read point-by-point responses

  1. Referee: [abstract and alternating-update section] The abstract and the section describing the practical algorithm state that ProxiCBO is a 'provably convergent' method, yet the only convergence theorem supplied applies exclusively to the continuous-time finite-particle dynamics. No separate result, discretization-error bound, or invariance argument is given for the alternating discrete scheme (proximal step followed by consensus update) that is actually implemented, particularly when the objective is non-convex and the proximal map may be set-valued or non-expansive. This gap is load-bearing for the central claim.

    Authors: We agree that the global convergence theorem applies exclusively to the continuous-time finite-particle dynamics. The alternating proximal-consensus scheme is presented as a practical discretization motivated by the continuous model. We will revise the abstract and algorithm section to state explicitly that the method is provably convergent in the continuous-time limit, while the discrete implementation is a heuristic approximation whose justification rests on the continuous analysis as the time-step parameter vanishes. A brief remark will be added noting the lack of a full discretization-error bound for the non-convex case, thereby accurately delimiting the central claim. revision: yes

  2. Referee: [problem formulation and assumptions] The weakest-assumption paragraph asserts that proximal operators can be integrated while preserving global convergence properties of the CBO dynamics in non-convex settings, but the manuscript supplies no explicit conditions on the proximal map (e.g., single-valuedness, non-expansiveness, or subdifferential regularity) that would be required to carry the continuous-time analysis over to the discrete alternating scheme.

    Authors: The continuous-time analysis relies on the proximal operator being single-valued, which is guaranteed when the regularizer is convex. For the general non-convex composite setting we will insert an explicit paragraph in the problem formulation section stating the standing assumptions on the proximal map: single-valuedness, upper semicontinuity, and the one-sided Lipschitz-type condition needed to preserve the well-posedness and convergence properties of the particle dynamics. These conditions will be listed alongside the existing CBO assumptions, making the framework transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence analysis is independent of fitted quantities or self-referential definitions.

full rationale

The paper derives global convergence guarantees directly for the continuous-time finite-particle dynamics by incorporating proximal operators into the CBO drift term. The alternating discrete scheme is introduced separately as a practical implementation without any claim that its convergence follows by construction from the continuous analysis or from parameter fitting. No equations reduce a claimed prediction to an input by definition, no load-bearing self-citations close the argument, and no ansatz is smuggled via prior work. The derivation chain remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; full manuscript would be required to enumerate any that support the convergence claim.

pith-pipeline@v0.9.0 · 5430 in / 1169 out tokens · 44494 ms · 2026-05-10T16:38:03.037341+00:00 · methodology

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