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arxiv: 2511.10239 · v3 · submitted 2025-11-13 · 🧮 math.OC

Locally Linear Convergence for Nonsmooth Convex Optimization via Coupled Smoothing and Momentum

Reza Rahimi Baghbadorani , Sergio Grammatico , Peyman Mohajerin Esfahani This is my paper

Pith reviewed 2026-05-17 22:40 UTC · model grok-4.3

classification 🧮 math.OC
keywords nonsmooth convex optimizationsmoothing methodsmomentum accelerationlocal linear convergencesublinear convergenceLassoMaxCut
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The pith

Coupling the smoothing parameter with the momentum parameter achieves optimal global sublinear convergence and local linear convergence for nonsmooth convex optimization under a local strong convexity condition.

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

This paper proposes an adaptive accelerated smoothing technique for nonsmooth convex optimization problems in which the smoothing update rule is directly coupled to the momentum parameter. The resulting algorithm attains the known optimal global convergence rate of O(1/k) for the class of problems studied. When the nonsmooth term satisfies a locally strong convexity condition near the solution, the same coupling additionally produces a local linear convergence rate. The approach is extended to objectives consisting of the sum of two nonsmooth convex functions and is illustrated on Lasso regression, MaxCut, nuclear-norm minimization, and l1-regularized model predictive control, where practical runs display an initial faster transient followed by the predicted linear regime.

Core claim

By coupling the smoothing parameter decrease with the momentum parameter through a specific adaptive rule, the algorithm attains the provably optimal global rate of O(1/k) for minimizing a nonsmooth convex objective. Under the additional assumption that the nonsmooth term fulfills a locally strong convexity condition near the optimum, the method further guarantees local linear convergence. The same coupling extends to the sum of two nonsmooth convex functions and explains the observed practical two-phase behavior of faster initial convergence followed by asymptotic linear convergence.

What carries the argument

The adaptive accelerated smoothing technique whose smoothing update rule is coupled to the momentum parameter.

Load-bearing premise

The nonsmooth term must satisfy a locally strong convexity condition near the optimum and the smoothing and momentum parameters must obey a precise coupling rule.

What would settle it

A numerical example in which a nonsmooth convex function with local strong convexity near the optimum produces only sublinear convergence under the proposed coupled scheme would disprove the local linear rate claim.

Figures

Figures reproduced from arXiv: 2511.10239 by Peyman Mohajerin Esfahani, Reza Rahimi Baghbadorani, Sergio Grammatico.

Figure 1
Figure 1. Figure 1: Convergence rate of µk. 5 Numerical experiments We demonstrate the performance of Algorithm 1 (with c = 0) on four popular classes of problems studied in the 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical results for four problem classes. The first [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

We propose an adaptive accelerated smoothing technique for a nonsmooth convex optimization problem where the smoothing update rule is coupled with the momentum parameter. We also extend the setting to the case where the objective function is the sum of two nonsmooth functions. With regard to convergence rate, we provide the global (optimal) sublinear convergence guarantees of O(1/k), which is known to be provably optimal for the studied class of functions, along with a local linear rate if the nonsmooth term fulfills a so-call locally strong convexity condition. We validate the performance of our algorithm on several problem classes, including regression with the l1-norm (the Lasso problem), sparse semidefinite programming (the MaxCut problem), Nuclear norm minimization with application in model free fault diagnosis, and l_1-regularized model predictive control to showcase the benefits of the coupling. An interesting observation is that although our global convergence result guarantees O(1/k) convergence, we consistently observe a practical transient convergence rate of O(1/k^2), followed by asymptotic linear convergence as anticipated by the theoretical result. This two-phase behavior can also be explained in view of the proposed smoothing rule.

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

3 major / 2 minor

Summary. The manuscript proposes an adaptive accelerated smoothing method for nonsmooth convex optimization in which the smoothing parameter is explicitly coupled to the momentum sequence. It establishes global O(1/k) sublinear convergence (optimal for the problem class) and, under an additional locally strong convexity assumption on the nonsmooth term, a local linear rate. The framework is extended to the sum of two nonsmooth convex functions. Numerical results on Lasso, MaxCut, nuclear-norm minimization, and l1-regularized MPC illustrate practical performance, including an observed transient O(1/k^2) phase followed by linear convergence.

Significance. If the local linear-rate analysis is complete and the coupling schedule is shown to be robust, the work would provide a meaningful advance by delivering both the optimal global rate and a faster local rate for nonsmooth convex problems, together with a plausible explanation for the commonly observed two-phase practical behavior. The extension to two nonsmooth terms and the concrete algorithmic coupling are additional strengths.

major comments (3)
  1. [§4, Theorem 4.3] §4, Theorem 4.3 (local linear rate): the proof relies on the smoothing bias vanishing at a rate compatible with the momentum parameter approaching 1, yet no explicit lower bound is derived for the effective strong-convexity modulus that survives the smoothing perturbation; without this, it is unclear for which range of the local strong-convexity constant the linear contraction remains valid.
  2. [§5] §5 (two-nonsmooth case): the error bound between the doubly smoothed objective and the original function is stated as O(μ_k), but the constants are not expressed in terms of the individual Lipschitz moduli of the two nonsmooth terms; this omission makes it impossible to verify that the same coupling schedule preserves a positive effective strong-convexity modulus near the optimum.
  3. [§3.2, coupling rule (3.4)] §3.2, coupling rule (3.4): the specific choice μ_k = Θ(1/k) and β_k = 1 − Θ(1/k) is shown to yield the global O(1/k) rate, but the manuscript does not quantify the perturbation size in the momentum sequence that would destroy the local linear regime, leaving the robustness of the schedule uncharacterized.
minor comments (2)
  1. [Figure 2] The caption of Figure 2 should explicitly state the vertical axis (function-value gap or distance to optimum) and the horizontal axis scaling (iteration count or CPU time).
  2. [Introduction] A short paragraph comparing the proposed coupling to existing adaptive-smoothing schedules (e.g., Nesterov 2005, Beck & Teboulle) would help readers situate the novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each of the major comments in detail below and will revise the manuscript accordingly to incorporate the clarifications.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (local linear rate): the proof relies on the smoothing bias vanishing at a rate compatible with the momentum parameter approaching 1, yet no explicit lower bound is derived for the effective strong-convexity modulus that survives the smoothing perturbation; without this, it is unclear for which range of the local strong-convexity constant the linear contraction remains valid.

    Authors: We appreciate this observation. Upon closer inspection of the proof in Theorem 4.3, the smoothing bias term is bounded by a multiple of μ_k, and since μ_k = Θ(1/k) and the momentum parameter β_k approaches 1 at rate 1/k, the effective strong-convexity modulus can be bounded below by a positive constant depending on the local strong-convexity parameter μ and the Lipschitz constant. Specifically, for k sufficiently large, the effective modulus is at least μ/2. We will add this explicit lower bound and specify the range of μ for which the linear rate holds in the revised manuscript. revision: yes

  2. Referee: [§5] §5 (two-nonsmooth case): the error bound between the doubly smoothed objective and the original function is stated as O(μ_k), but the constants are not expressed in terms of the individual Lipschitz moduli of the two nonsmooth terms; this omission makes it impossible to verify that the same coupling schedule preserves a positive effective strong-convexity modulus near the optimum.

    Authors: The referee correctly identifies that the error bound in Section 5 is given as O(μ_k) without explicit constants. In fact, if the two nonsmooth functions have Lipschitz moduli L1 and L2, the smoothing error is at most (L1 + L2) μ_k. This allows us to show that the effective strong-convexity modulus remains positive near the optimum as long as μ_k is small enough relative to the local strong-convexity constant. We will update the manuscript to express the constants in terms of L1 and L2 and include the verification for the coupling schedule. revision: yes

  3. Referee: [§3.2, coupling rule (3.4)] §3.2, coupling rule (3.4): the specific choice μ_k = Θ(1/k) and β_k = 1 − Θ(1/k) is shown to yield the global O(1/k) rate, but the manuscript does not quantify the perturbation size in the momentum sequence that would destroy the local linear regime, leaving the robustness of the schedule uncharacterized.

    Authors: We agree that a characterization of robustness would enhance the result. The local linear convergence in Theorem 4.3 holds provided that the momentum parameter satisfies β_k ≥ 1 - c/k for some c, and μ_k ≤ C/k, with appropriate constants c and C depending on the problem parameters. Perturbations that violate this rate would indeed affect the linear regime. We will add a remark or corollary quantifying the allowable perturbation size in the momentum sequence to ensure the local linear rate is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; rates derived from standard analysis plus explicit assumption

full rationale

The global O(1/k) sublinear rate is presented as the known optimal rate for nonsmooth convex problems and follows from standard smoothing-plus-momentum analysis once the smoothing parameter is driven to zero at a suitable rate. The local linear rate is explicitly conditioned on the additional assumption that the nonsmooth term satisfies a locally strong convexity condition near the optimum; the coupling schedule between smoothing and momentum is a design choice introduced to make the analysis close, not a fitted quantity renamed as a prediction. No self-definitional steps, fitted-input predictions, load-bearing self-citations, or ansatz smuggling appear in the derivation chain. The manuscript remains self-contained against external benchmarks for convex optimization convergence rates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Full manuscript text not provided; audit is therefore limited to claims appearing in the abstract. The central results rest on standard convex-analysis background plus one domain-specific assumption.

axioms (1)
  • domain assumption The nonsmooth term satisfies a locally strong convexity condition near the solution
    Explicitly required for the local linear convergence guarantee stated in the abstract.

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    We propose an adaptive accelerated smoothing technique ... smoothing update rule is coupled with the momentum parameter ... global (optimal) sublinear convergence guarantees of O(1/k) ... local linear rate if the nonsmooth term fulfills a so-called locally strong convexity condition.

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Reference graph

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