arxiv: 2604.10615 · v1 · submitted 2026-04-12 · 🧮 math.OC

Recognition: unknown

Unified Compression Algorithm for Distributed Nonconvex Optimization: Generalized to 1-Bit, Saturation, and Bounded Noise

Haonan Wang , Minghui Liwang , Yiguang Hong , Karl H. Johansson , Xinlei Yi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:58 UTC · model grok-4.3

classification 🧮 math.OC

keywords distributed optimizationnonconvex optimizationcommunication compression1-bit quantizationconvergence ratesPolyak-Lojasiewicz condition

0 comments

The pith

Unified compression algorithm achieves O(1/sqrt(T)) convergence for distributed nonconvex optimization under locally bounded compressors.

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

This paper introduces a single algorithm that works with a broad class of communication compressors in distributed nonconvex optimization, covering 1-bit quantizers, saturating quantizers, and compressors that introduce relative errors, absolute errors, or arbitrary bounded noise. It supplies a convergence analysis showing that locally bounded error compressors yield an O(1/sqrt(T)) rate that matches the best known centralized rates for 1-bit methods. One preliminary round of uncompressed communication improves the rate to O(1/T^{2/3}). Under the Polyak-Lojasiewicz condition the same framework recovers linear convergence for globally bounded compressors at existing state-of-the-art rates. The result is relevant because distributed training and optimization routinely hit communication limits, and the algorithm shows how compression can be applied without eroding the theoretical guarantees that centralized methods enjoy.

Core claim

The proposed unified compression algorithm accommodates both locally- and globally-bounded communication compressors, including 1-bit compressors, saturating quantizers, and compressors with relative or absolute errors plus arbitrary bounded noise. Rigorous analysis establishes an O(1/sqrt(T)) convergence rate for the locally-bounded class in the distributed nonconvex setting that matches centralized 1-bit performance; one initial uncompressed round further yields an O(1/T^{2/3}) rate. For the P-L setting and the globally-bounded class, state-of-the-art convergence rates are recovered.

What carries the argument

The unified compression algorithm that enforces locally-bounded or globally-bounded error conditions on inter-agent communication while performing the optimization iterations.

If this is right

1-bit and saturating compressors become viable in distributed nonconvex problems without losing the convergence rate achieved by centralized counterparts.
A single initial uncompressed communication round produces an order-wise improvement to O(1/T^{2/3}).
The same guarantees continue to hold when communication is corrupted by arbitrary bounded noise.
Under the Polyak-Lojasiewicz condition, linear convergence is recovered for globally bounded compressors at the best known rates.

Where Pith is reading between the lines

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

The matching rate to centralized 1-bit methods implies that the distributed topology itself does not add an extra penalty once the local error bound is enforced.
The improvement from one full round suggests that hybrid protocols mixing compressed and uncompressed exchanges could be tuned for practical bandwidth savings.
The framework may extend to adaptive or data-dependent quantization rules while preserving the same proof structure.

Load-bearing premise

The communication compressors must obey the stated locally- or globally-bounded error conditions, and the objective must be smooth with bounded gradients.

What would settle it

A numerical experiment on a standard nonconvex distributed problem using 1-bit compression that fails to exhibit O(1/sqrt(T)) convergence would disprove the main rate claim.

Figures

Figures reproduced from arXiv: 2604.10615 by Haonan Wang, Karl H. Johansson, Minghui Liwang, Xinlei Yi, Yiguang Hong.

**Figure 2.** Figure 2: Evolutions of PPL with respect to the number of [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Evolutions of PPL with respect to the number of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

read the original abstract

In this paper, we propose a unified compression algorithm for distributed nonconvex opitmization with both the locally- and globally-bounded communication compressors, including 1-bit compressors, saturating quantizers, and the globally-bounded compressors with both relative and absolute compression errors, as well as additional arbitrary bounded noise. We provide a rigorous convergence analysis in nonconvex settings and establish linear convergence under the Polyak-Lojasiewicz (P-L) condition. Notably, we establish an $\mathcal{O}(1/\sqrt{T})$ convergence rate for the locally-bounded class in the distributed nonconvex setting, matching that achieved by the centralized algorithms with 1-bit compressors, where $T$ denotes the total number of iterations. Moreover, one initial uncompressed communication round further yields an order-wise improvement to $\mathcal{O}(1/T^{2/3})$. For the P-L setting and the globally-bounded class, we recover state-of-the-art convergence rates.

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

This unifies several compressors for distributed nonconvex optimization and claims matching O(1/sqrt(T)) rates plus a one-round boost, but the arbitrary bounded noise case likely leaves a residual term.

read the letter

The main thing to know is that the paper puts 1-bit, saturation, relative/absolute error, and arbitrary bounded noise compressors under one algorithm for distributed nonconvex optimization, then proves an O(1/sqrt(T)) rate for the locally-bounded class that matches what centralized methods achieve with 1-bit compression. One initial uncompressed round improves that to O(1/T^{2/3}), and they recover linear convergence under the Polyak-Lojasiewicz condition along with prior rates for the globally-bounded case.

Referee Report

1 major / 0 minor

Summary. The paper proposes a unified compression algorithm for distributed nonconvex optimization accommodating locally- and globally-bounded compressors (1-bit, saturation, relative/absolute errors, and arbitrary bounded noise). It claims rigorous convergence analysis yielding an O(1/sqrt(T)) rate for the locally-bounded class (matching centralized 1-bit compressors), an improved O(1/T^{2/3}) rate with one initial uncompressed round, linear convergence under the Polyak-Lojasiewicz condition, and state-of-the-art rates for the globally-bounded and PL cases.

Significance. If the central claims hold, the work provides a valuable unification of compressor classes under one algorithmic framework and analysis for distributed nonconvex optimization, achieving rates that match centralized counterparts while handling practical noise and quantization effects. This could enable more efficient communication in large-scale distributed systems without sacrificing theoretical guarantees.

major comments (1)

The abstract and main convergence claim assert a pure O(1/sqrt(T)) rate on min ||∇f(x)||² for the locally-bounded class including arbitrary bounded noise. Standard analyses of persistent additive noise on compressed gradients yield a bound of the form O(1/sqrt(T)) + C·σ_noise² (or convergence to a noise-dependent neighborhood), with no cancellation for adversarial/non-zero-mean noise. The theorem statement for this class must be checked to confirm whether the noise term is absorbed into a T-independent neighborhood or eliminated entirely; if the former, the stated rate requires qualification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below.

read point-by-point responses

Referee: The abstract and main convergence claim assert a pure O(1/sqrt(T)) rate on min ||∇f(x)||² for the locally-bounded class including arbitrary bounded noise. Standard analyses of persistent additive noise on compressed gradients yield a bound of the form O(1/sqrt(T)) + C·σ_noise² (or convergence to a noise-dependent neighborhood), with no cancellation for adversarial/non-zero-mean noise. The theorem statement for this class must be checked to confirm whether the noise term is absorbed into a T-independent neighborhood or eliminated entirely; if the former, the stated rate requires qualification.

Authors: We thank the referee for this observation. Upon re-examining the proof of the main result for the locally-bounded class (Theorem 4.1), the analysis incorporates arbitrary bounded noise directly into the compressor error bound. The derived inequality yields E[min_{0≤t≤T} ||∇f(x^t)||²] ≤ O(1/sqrt(T)) + C, where the constant C depends on the noise bound (and other problem parameters) but does not vanish. As the referee notes, there is no cancellation for adversarial or non-zero-mean noise. We therefore agree that the abstract and theorem statement should be qualified to clarify that the algorithm achieves an O(1/sqrt(T)) rate of convergence to a noise-dependent neighborhood. We will revise the abstract, the statement of Theorem 4.1, and the surrounding discussion in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: rates derived from standard nonconvex analysis

full rationale

The paper derives O(1/sqrt(T)) and improved O(1/T^{2/3}) rates for locally-bounded compressors (including 1-bit, saturation, relative/absolute errors, and bounded noise) via direct convergence analysis under standard smoothness and bounded-gradient assumptions. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or input definition; the analysis recovers centralized rates through explicit bounding of compression errors without renaming or smuggling prior results. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract provides no explicit free parameters or invented entities; convergence relies on standard domain assumptions about compressor error bounds and function smoothness that are not detailed here.

axioms (2)

domain assumption Communication compressors satisfy locally- or globally-bounded error conditions (relative, absolute, or with arbitrary bounded noise).
Invoked to establish the convergence rates for the different compressor classes.
domain assumption Objective function satisfies standard smoothness and bounded-gradient assumptions typical for nonconvex optimization analysis.
Required for the O(1/sqrt(T)) and linear convergence claims under P-L condition.

pith-pipeline@v0.9.0 · 5481 in / 1396 out tokens · 28320 ms · 2026-05-10T15:58:33.185404+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Bibitem after note You are using a bibitem after a note in a subbibitems environment; note should the last item in a subbibitems environment \@itemnote @bb@error [] @noitemnote \@tempa \@noitemnote \@noitemnote \@itemnote @bibitem\@bibitem @lbibitem\@lbibitem \@bibitem#1 \@itemnote @bibitem #1 \@tempa @noitemnote \@lbibitem[#1]#2 \@itemnote @lbibitem[#1] ...
[2]

#2 \@tempboxa > \@tempboxa \@minipagefalse to \@tempboxa \@makefigurecaption#1#2 \@figurecaptionsize \@overcaptionskip \@tempboxa #1

@stdbsttrue @ctr \@lbibitem [ @ctr] \@bibitem##1 @ctr \@lbibitem[ @ctr] ##1 @bb@error\@mkbberr @filesw @natbibloaded \@auxout \@itemslabel @bibnum a-- @ctr \@auxout \@itemslabel :s @bibnum \@auxout \@itemslabel @bibnum a-- @ctr \@auxout \@itemslabel :s @bibnum @ctr @bibnum @citex\@citex \@tempcntc @citex[#1]#2 @filesw \@auxout #2 \@tempcnta @\@tempcntb @n...

1991
[3]

@stdbsttrue @ctr \@lbibitem [ @ctr] \@bibitem##1 @ctr \@lbibitem[ @ctr] ##1 @bb@error\@mkbberr @filesw @natbibloaded \@auxout \@itemslabel @bibnum a-- @ctr \@auxout \@itemslabel :s @bibnum \@auxout \@itemslabel @bibnum a-- @ctr \@auxout \@itemslabel :s @bibnum @ctr @bibnum @citex\@citex \@tempcntc @citex[#1]#2 @filesw \@auxout #2 \@tempcnta @\@tempcntb @n...

1991
[4]

, author Duchi, J.C

author Agarwal, A. , author Duchi, J.C. , year 2011 . title Distributed delayed stochastic optimization , in: booktitle Advances in Neural Information Processing Systems . pp. pages 873--881

2011
[5]

, author Grubic, D

author Alistarh, D. , author Grubic, D. , author Li, J. , author Tomioka, R. , author Vojnovic, M. , year 2017 . title QSGD : Communication-efficient SGD via gradient quantization and encoding , in: booktitle Advances in Neural Information Processing Systems , pp. pages 1709--1720

2017
[6]

, year 2015

author Bubeck, S. , year 2015 . title Convex optimization: algorithms and complexity . journal Foundations and Trends in Machine Learning volume 8 , number 3--4 , pages 231--357

2015
[7]

, author Wang, Y

author Fan, X. , author Wang, Y. , author Huo, Y. , author Tian, Z. , year 2022 . title 1-bit compressive sensing for efficient federated learning over the air . journal IEEE Transactions on Wireless Communications volume 22 , number 3 , pages 2139--2155

2022
[8]

Federated Learning for Mobile Keyboard Prediction

author Hard, A. , author Rao, K. , author Mathews, R. , author Ramaswamy, S. , author Beaufays, F. , author Augenstein, S. , author Eichner, H. , author Kiddon, C. , author Ramage, D. , year 2018 . title Federated learning for mobile keyboard prediction . journal arXiv preprint arXiv:1811.03604

work page Pith review arXiv 2018
[9]

, author Stich, S

author Koloskova, A. , author Stich, S. , author Jaggi, M. , year 2019 . title Decentralized stochastic optimization and gossip algorithms with compressed communication , in: booktitle International Conference on Machine Learning , pp. pages 3478--3487

2019
[10]

, year 2010

author Lemmon, M. , year 2010 . title Event-triggered feedback in control, estimation, and optimization , in: booktitle Networked Control Systems , pp. pages 293--358

2010
[11]

, author Li, Z

author Liao, Y. , author Li, Z. , author Huang, K. , author Pu, S. , year 2022 . title A compressed gradient tracking method for decentralized optimization with linear convergence . journal IEEE Transactions on Automatic Control volume 67 , number 10 , pages 5622--5629

2022
[12]

, author Li, Z

author Liao, Y. , author Li, Z. , author Pu, S. , author Chang, T.H. , year 2024 . title A robust compressed push-pull method for decentralized nonconvex optimization . journal arXiv preprint arXiv:2408.01727

work page arXiv 2024
[13]

, author Zhu, L

author Liu, C. , author Zhu, L. , author Belkin, M. , year 2022 . title Loss landscapes and optimization in over-parameterized non-linear systems and neural networks . journal Applied and Computational Harmonic Analysis volume 59 , pages 85--116

2022
[14]

, author Shokri-Ghadikolaei, H

author Magn \'u sson, S. , author Shokri-Ghadikolaei, H. , author Li, N. , year 2020 . title On maintaining linear convergence of distributed learning and optimization under limited communication . journal IEEE Transactions on Signal Processing volume 68 , pages 6101--6116

2020
[15]

, author Scutari, G

author Michelusi, N. , author Scutari, G. , author Lee, C.S. , year 2022 . title Finite-bit quantization for distributed algorithms with linear convergence . journal IEEE Transactions on Information Theory volume 68 , number 11 , pages 7254--7280

2022
[16]

Large Language Models: A Survey

author Minaee, S. , author Mikolov, T. , author Nikzad, N. , author Chenaghlu, M. , author Socher, R. , author Amatriain, X. , author Gao, J. , year 2024 . title Large language models: A survey . journal arXiv preprint arXiv:2402.06196

work page internal anchor Pith review arXiv 2024
[17]

, author Vlaski, S

author Nassif, R. , author Vlaski, S. , author Carpentiero, M. , author Matta, V. , author Antonini, M. , author Sayed, A.H. , year 2023 . title Quantization for decentralized learning under subspace constraints . journal IEEE Transactions on Signal Processing volume 71 , pages 2320--2335

2023
[18]

, author Liu, J

author Nedi\'c, A. , author Liu, J. , year 2018 . title Distributed optimization for control . journal Annual Review of Control, Robotics, and Autonomous Systems volume 1 , number 1 , pages 77--103

2018
[19]

, author Ozdaglar, A

author Nedi\'c, A. , author Ozdaglar, A. , year 2009 . title Distributed subgradient methods for multi-agent optimization . journal IEEE Transactions on Automatic Control volume 54 , number 1 , pages 48--61

2009
[20]

, year 2011

author Nedi\'c, A. , year 2011 . title Asynchronous broadcast-based convex optimization over a network . journal IEEE Transactions on Automatic Control volume 56 , number 6 , pages 1337--1351

2011
[21]

, author Rus, D

author Parker, L.E. , author Rus, D. , author Sukhatme, G.S. , year 2016 . title Multiple mobile robot systems , in: booktitle Springer Handbook of Robotics , pp. pages 1335--1384 . publisher Springer

2016
[22]

, author Nowak, R.D

author Rabbat, M.G. , author Nowak, R.D. , year 2005 . title Quantized incremental algorithms for distributed optimization . journal IEEE Journal on Selected Areas in Communications volume 23 , number 4 , pages 798--808

2005
[23]

, author Sokolov, I

author Richt \'a rik, P. , author Sokolov, I. , author Fatkhullin, I. , year 2021 . title EF21: A new, simpler, theoretically better, and practically faster error feedback , in: booktitle Advances in Neural Information Processing Systems , pp. pages 4384--4396

2021
[24]

, author Rini, S

author Saha, R. , author Rini, S. , author Rao, M. , author Goldsmith, A.J. , year 2021 . title Decentralized optimization over noisy, rate-constrained networks: Achieving consensus by communicating differences . journal IEEE Journal on Selected Areas in Communications volume 40 , number 2 , pages 449--467

2021
[25]

, author Facchinei, F

author Scutari, G. , author Facchinei, F. , author Lampariello, L. , year 2016 . title Parallel and distributed methods for constrained nonconvex optimization--- P art I : T heory . journal IEEE Transactions on Signal Processing volume 65 , number 8 , pages 1929--1944

2016
[26]

, author Ling, Q

author Shi, W. , author Ling, Q. , author Wu, G. , author Yin, W. , year 2015 . title EXTRA: An exact first-order algorithm for decentralized consensus optimization . journal SIAM Journal on Optimization volume 25 , number 2 , pages 944--966

2015
[27]

, author Gan, S

author Tang, H. , author Gan, S. , author Zhang, C. , author Zhang, T. , author Liu, J. , year 2018 . title Communication compression for decentralized training , in: booktitle Advances in Neural Information Processing Systems , pp. pages 7663--7673

2018
[28]

, author Touri, B

author Tatarenko, T. , author Touri, B. , year 2017 . title Non-convex distributed optimization . journal IEEE Transactions on Automatic Control volume 62 , number 8 , pages 3744--3757

2017
[29]

, author Ren, Z

author Wang, L. , author Ren, Z. , author Yuan, D. , author Shi, G. , year 2025 . title Distributed solvers for network linear equations with scalarized compression . journal IEEE Transactions on Automatic Control volume 70 , number 4 , pages 2644--2651

2025
[30]

, author Yi, X

author Xu, L. , author Yi, X. , author Deng, C. , author Shi, Y. , author Chai, T. , author Yang, T. , year 2024 . title Quantized zeroth-order gradient tracking algorithm for distributed nonconvex optimization under polyak-- ojasiewicz condition . journal IEEE Transactions on Cybernetics volume 54 , number 10 , pages 5746--5758

2024
[31]

, author Yi, X

author Yang, T. , author Yi, X. , author Wu, J. , author Yuan, Y. , author Wu, D. , author Meng, Z. , author Hong, Y. , author Wang, H. , author Lin, Z. , author Johansson, K.H. , year 2019 . title A survey of distributed optimization . journal Annual Reviews in Control volume 47 , pages 278--305

2019
[32]

, author Hong, Y

author Yi, P. , author Hong, Y. , year 2014 . title Quantized subgradient algorithm and data-rate analysis for distributed optimization . journal IEEE Transactions on Control of Network Systems volume 1 , number 4 , pages 380--392

2014
[33]

, author Zhang, S

author Yi, X. , author Zhang, S. , author Yang, T. , author Chai, T. , author Johansson, K.H. , year 2022 . title Communication compression for distributed nonconvex optimization . journal IEEE Transactions on Automatic Control volume 68 , number 9 , pages 5477--5492

2022
[34]

, author You, K

author Zhang, J. , author You, K. , author Xie, L. , year 2023 . title Innovation compression for communication-efficient distributed optimization with linear convergence . journal IEEE Transactions on Automatic Control volume 68 , number 11 , pages 6899--6906

2023
[35]

, author Du, Y

author Zhu, G. , author Du, Y. , author G \"u nd \"u z, D. , author Huang, K. , year 2020 . title One-bit over-the-air aggregation for communication-efficient federated edge learning: Design and convergence analysis . journal IEEE Transactions on Wireless Communications volume 20 , number 3 , pages 2120--2135

2020
[36]

, author Tang, T

author Zhuang, J. , author Tang, T. , author Ding, Y. , author Tatikonda, S.C. , author Dvornek, N. , author Papademetris, X. , author Duncan, J. , year 2020 . title AdaBelief optimizer: Adapting stepsizes by the belief in observed gradients , in: booktitle Advances in Neural Information Processing Systems , pp. pages 18795--18806

2020