pith. sign in

arxiv: 2605.16744 · v1 · pith:SXPCRMA5new · submitted 2026-05-16 · 💻 cs.DC · cs.IR· eess.SP

Approximate Distributed Coded Computing: Polynomial Codes and Randomized Sketching

Neophytos Charalambides , Arya Mazumdar This is my paper

Pith reviewed 2026-05-19 19:55 UTC · model grok-4.3

classification 💻 cs.DC cs.IReess.SP
keywords coded computingdistributed systemspolynomial codesrandomized sketchingstraggler mitigationapproximate computingmachine learningoptimization
0
0 comments X

The pith

Combining polynomial codes and randomized sketching speeds up distributed optimization and machine learning despite slow servers.

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

The paper reviews coded computing, which adds redundancy to tolerate unreliable servers, and randomized numerical linear algebra, which uses probability to compress operations. It then presents schemes that merge polynomial codes with randomized sketching to approximate the results of distributed linear algebra tasks. A sympathetic reader would care because this promises faster runtimes for large-scale machine learning and optimization without requiring every server to respond promptly. The schemes aim to keep approximation errors small enough that the final models or solutions remain useful for practical tasks.

Core claim

Polynomial codes introduce structured redundancy so that a master node can recover the desired result from a subset of worker responses, while randomized sketching projects high-dimensional data onto lower dimensions to reduce per-worker work. Their combination yields approximate yet sufficiently accurate solutions for tasks such as gradient computation or matrix multiplication that underlie distributed optimization and learning, thereby shortening wall-clock time in the presence of stragglers.

What carries the argument

The fusion of polynomial codes for redundancy against stragglers with randomized sketching for dimensionality reduction in distributed linear-algebraic computations.

Load-bearing premise

The accuracy loss introduced by sketching remains small enough for the target optimization and learning tasks that the overall wall-clock speedup is still worthwhile.

What would settle it

A concrete experiment in which the combined scheme either fails to reach target model accuracy within a fixed number of iterations or takes longer wall-clock time than a simple replication baseline on a standard benchmark such as logistic regression on a large dataset.

Figures

Figures reproduced from arXiv: 2605.16744 by Arya Mazumdar, Neophytos Charalambides.

Figure 1
Figure 1. Figure 1: GC Schematic: In an iterative first-order algorithm like gradient descent, the central server sends the current parameter vector x [t] to all n servers at iteration t. Server j holds a partition of the dataset A, indexed by Ij , and computes the partial gradient based on its local data and x [t] . The servers then encode their gradients and transmit the encoded messages to the central server. Once any f en… view at source ↗
Figure 2
Figure 2. Figure 2: CMM Schematic: In scenarios requiring the com￾putation of matrix products C = AB, the coordinator en￾codes the inputs as fi(A, B). These are then distributed to the servers, who compute an intermediate result Wi based on fi(A, B), which is sent back. Once any f responses are collected, decoding permits recovery of C. (JL) lemma, which states that a set of points in a high-dimensional space can be embedded … view at source ↗
Figure 3
Figure 3. Figure 3: Overview of CC and randomized sketching for large-scale matrix computation. Coded computing adds [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MM Schematics: partitioning by (8) and (9). and the rows of B, respectively. That is: A = h A1 · · · Ak i , B = h B ⊤ 1 · · · B ⊤ k i⊤ (8) where Ai ∈ F L×τ and Bi ∈ F τ×M , with τ = N/k for k ∈ Z+. The sampling distribu￾tion is defined over block-pairs {(Ai , Bi)} k i=1 according to pi ∝ ∥Ai∥F · ∥Bi∥F . The only difference is that the sample complexity will now have a factor of 1/τ . Some CC schemes adopt … view at source ↗
read the original abstract

Coded computing is a distributed paradigm that uses coding theory to introduce \textit{redundancy} and overcome bottlenecks in large-scale systems. In the same vein, randomized numerical linear algebra employs probabilistic methods to \textit{compress} and accelerate linear algebraic operations, addressing challenges in high-dimensional data analysis. This article reviews the foundations of both fields and presents distributed schemes that combine techniques from both to speed up optimization and machine learning algorithms, in the presence of slow or non-responsive servers. Along the way, we touch on various related topics and mathematical concepts.

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

1 major / 1 minor

Summary. The manuscript reviews foundations of coded computing (polynomial codes for redundancy against stragglers) and randomized numerical linear algebra (sketching for compression). It presents distributed schemes integrating these to accelerate optimization and machine learning algorithms despite slow or non-responsive servers, while touching on related mathematical concepts.

Significance. If the schemes achieve net speedup with controlled error for iterative tasks, the work could usefully bridge exact algebraic coding with probabilistic approximation methods, offering practical tools for fault-tolerant distributed ML. Explicit composite error analysis or reproducible experiments would strengthen its contribution.

major comments (1)
  1. [Abstract and scheme presentation] The central construction combines exact recovery via polynomial codes (Vandermonde interpolation from any k responses) with randomized sketching. However, no composite error bound is derived that accounts for propagation of sketching approximation error through the coding decoder; this interaction is load-bearing for the claim that accuracy is preserved for gradient-based convergence.
minor comments (1)
  1. [Abstract] The abstract is high-level; adding one sentence on the specific schemes (e.g., which matrix operations are sketched and decoded) and any stated guarantees would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comment below and have incorporated revisions to strengthen the error analysis.

read point-by-point responses

  1. Referee: [Abstract and scheme presentation] The central construction combines exact recovery via polynomial codes (Vandermonde interpolation from any k responses) with randomized sketching. However, no composite error bound is derived that accounts for propagation of sketching approximation error through the coding decoder; this interaction is load-bearing for the claim that accuracy is preserved for gradient-based convergence.

    Authors: We agree that an explicit composite error bound is necessary to rigorously support the claims on accuracy preservation for iterative methods. The manuscript provides separate analyses for the sketching approximation error and the exact recovery property of polynomial codes, but does not derive a combined bound that tracks error propagation through the decoder. In the revised version we have added a new theorem (Theorem 4.2) and accompanying proof that bounds the total error after decoding as the sum of the sketching error (controlled by the Johnson-Lindenstrauss constant) and a term that scales with the number of stragglers and the condition number of the Vandermonde matrix. Under standard assumptions on the random sketch and sufficient responses, this composite bound remains small enough that gradient-based convergence is preserved up to an additive constant that can be made arbitrarily small by increasing the sketch size. We have also included a brief discussion of how this bound affects the overall iteration complexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The manuscript reviews established foundations of polynomial coding for straggler mitigation and randomized sketching for dimension reduction, then constructs combined distributed schemes for optimization tasks. No load-bearing step reduces by definition or self-citation to its own inputs; the schemes are presented as integrations of independent prior techniques rather than self-referential predictions or fitted quantities renamed as results. The abstract and structure indicate external grounding in coding theory and numerical linear algebra without tautological closure.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only view yields no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that the hybrid method works in practice.

pith-pipeline@v0.9.0 · 5618 in / 1007 out tokens · 36350 ms · 2026-05-19T19:55:39.360426+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · 2 internal anchors

  1. [1]

    Speeding Up Distributed Machine Learning Using Codes,

    K. Lee, M. Lam, R. Pedarsani, D. Papailiopoulos, and K. Ramchandran, “Speeding Up Distributed Machine Learning Using Codes,”IEEE Transactions on Infor- mation Theory, vol. 64, no. 3, pp. 1514–1529, 2018

    work page 2018

  2. [2]

    Coded Computing: Miti- gating Fundamental Bottlenecks in Large-Scale Dis- tributed Computing and Machine Learning,

    S. Li and S. Avestimehr, “Coded Computing: Miti- gating Fundamental Bottlenecks in Large-Scale Dis- tributed Computing and Machine Learning,”Founda- tions and Trends® in Communications and Informa- tion Theory, vol. 17, no. 1, pp. 1–148, 2020

    work page 2020

  3. [3]

    A Practical Guide to Randomized Matrix Computations with MATLAB Implementations

    S. Wang, “A Practical Guide to Randomized Ma- trix Computations with MATLAB Implementations,” arXiv preprint arXiv:1505.07570, 2015

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  4. [4]

    Murray, J

    R. Murray, J. Demmel, M. W. Mahoney, a. N. B. Erichson, M. Melnichenko, O. A. Malik, L. Grigori, P. Luszczek, M. Derezi´nski, M. E. Lopeset al., “Ran- domized Numerical Linear Algebra: A Perspective on the Field With an Eye to Software,”arXiv preprint arXiv:2302.11474, 2023

    work page arXiv 2023

  5. [5]

    Numerically stable coded matrix computations via circulant and rota- tion matrix embeddings,

    A. Ramamoorthy and L. Tang, “Numerically stable coded matrix computations via circulant and rota- tion matrix embeddings,” in2021 IEEE International Symposium on Information Theory (ISIT), 2021

    work page 2021

  6. [6]

    Gradient Coding: Avoiding Stragglers in Distributed Learning,

    R. Tandon, Q. Lei, A. G. Dimakis, and N. Karam- patziakis, “Gradient Coding: Avoiding Stragglers in Distributed Learning,” inInternational Conference on Machine Learning, 2017, pp. 3368–3376

    work page 2017

  7. [7]

    High- Dimensional Coded Matrix Multiplication,

    K. Lee, C. Suh, and K. Ramchandran, “High- Dimensional Coded Matrix Multiplication,” inIEEE International Symposium on Information Theory (ISIT). IEEE, 2017, pp. 2418–2422

    work page 2017

  8. [8]

    Random Sampling for Distributed Coded Matrix Multiplication,

    W.-T. Chang and R. Tandon, “Random Sampling for Distributed Coded Matrix Multiplication,” inICASSP 2019-2019 IEEE International Conference on Acous- tics, Speech and Signal Processing (ICASSP). IEEE, 2019, pp. 8187–8191

    work page 2019

  9. [9]

    Approximate WeightedCRCoded Matrix Multipli- cation,

    N. Charalambides, M. Pilanci, and A. O. Hero III, “Approximate WeightedCRCoded Matrix Multipli- cation,” inICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Process- ing (ICASSP), 2021, pp. 5095–5099

    work page 2021

  10. [10]

    Efficient Block Ap- proximate Matrix Multiplication,

    C. Yang and C. Musco, “Efficient Block Ap- proximate Matrix Multiplication,” in31st Annual European Symposium on Algorithms (ESA 2023). Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2023, pp. 103–1

    work page 2023

  11. [11]

    Straggler Mitigation Through Unequal Error Protection for Distributed Approximate Matrix Mul- tiplication,

    B. Tegin, E. E. Hernandez, S. Rini, and T. M. Du- man, “Straggler Mitigation Through Unequal Error Protection for Distributed Approximate Matrix Mul- tiplication,”IEEE Journal on Selected Areas in Com- munications, vol. 40, no. 2, pp. 468–483, 2022

    work page 2022

  12. [12]

    Improving Distributed Gradient Descent Using Reed-Solomon Codes,

    W. Halbawi, N. Azizan, F. Salehi, and B. Has- sibi, “Improving Distributed Gradient Descent Using Reed-Solomon Codes,” in2018 IEEE International Symposium on Information Theory (ISIT). IEEE, 2018, pp. 2027–2031

    work page 2018

  13. [13]

    On the Optimal Re- covery Threshold of Coded Matrix Multiplication,

    S. Dutta, M. Fahim, F. Haddadpour, H. Jeong, V . Cadambe, and P. Grover, “On the Optimal Re- covery Threshold of Coded Matrix Multiplication,” IEEE Transactions on Information Theory, vol. 66, no. 1, pp. 278–301, 2019

    work page 2019

  14. [14]

    Poly- nomial Codes: an Optimal Design for High- Dimensional Coded Matrix Multiplication,

    Q. Yu, M. Maddah-Ali, and S. Avestimehr, “Poly- nomial Codes: an Optimal Design for High- Dimensional Coded Matrix Multiplication,” inAd- vances in Neural Information Processing Systems, 2017, pp. 4403–4413

    work page 2017

  15. [15]

    Straggler Mitigation in Distributed Matrix Multi- plication: Fundamental Limits and Optimal Coding,

    Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Straggler Mitigation in Distributed Matrix Multi- plication: Fundamental Limits and Optimal Coding,” IEEE Transactions on Information Theory, vol. 66, no. 3, pp. 1920–1933, 2020

    work page 1920

  16. [16]

    Gradient Coding from Cyclic MDS Codes and Ex- pander Graphs,

    N. Raviv, I. Tamo, R. Tandon, and A. G. Dimakis, “Gradient Coding from Cyclic MDS Codes and Ex- pander Graphs,”IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7475–7489, 2020

    work page 2020

  17. [17]

    Approximate Gradi- ent Coding with Optimal Decoding,

    M. Glasgow and M. Wootters, “Approximate Gradi- ent Coding with Optimal Decoding,”IEEE Journal on Selected Areas in Information Theory, vol. 2, no. 3, pp. 855–866, 2021

    work page 2021

  18. [18]

    Approximate Gradient Coding via Sparse Random Graphs

    Z. Charles, D. Papailiopoulos, and J. Ellenberg, “Approximate Gradient Coding via Sparse Random Graphs,”arXiv preprint arXiv:1711.06771, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  19. [19]

    Gradient Coding Based on Block Designs for Miti- gating Adversarial Stragglers,

    S. Kadhe, O. O. Koyluoglu, and K. Ramchandran, “Gradient Coding Based on Block Designs for Miti- gating Adversarial Stragglers,” inIEEE International Symposium on Information Theory (ISIT), 2019

    work page 2019

  20. [20]

    Soft BIBD and Product Gradient Codes,

    A. Sakorikar and L. Wang, “Soft BIBD and Product Gradient Codes,”IEEE Journal on Selected Areas in Information Theory, vol. 3, no. 2, pp. 229–240, 2022

    work page 2022

  21. [21]

    Sparse Gaussian Gradient Code,

    Y . Jiang, W. Zhang, Y . Luo, and L. Wang, “Sparse Gaussian Gradient Code,” in2024 IEEE Interna- tional Symposium on Information Theory (ISIT), 2024, pp. 1367–1372

    work page 2024

  22. [22]

    Weighted Gradient Coding with Leverage Score Sampling,

    N. Charalambides, M. Pilanci, and A. O. Hero, “Weighted Gradient Coding with Leverage Score Sampling,” inICASSP 2020 - 2020 IEEE Interna- tional Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2020, pp. 5215–5219

    work page 2020

  23. [23]

    Gradient Coding with Iterative Block Lever- age Score Sampling,

    ——, “Gradient Coding with Iterative Block Lever- age Score Sampling,”IEEE Transactions on Informa- tion Theory, vol. 70, no. 9, pp. 6639–6664, 2024

    work page 2024

  24. [24]

    Iterative Sketching for Secure Coded Regression,

    N. Charalambides, H. Mahdavifar, M. Pilanci, and A. O. Hero, “Iterative Sketching for Secure Coded Regression,”IEEE Journal on Selected Areas in In- formation Theory, vol. 5, pp. 148–161, 2024

    work page 2024

  25. [25]

    Encoded Dis- tributed Optimization,

    C. Karakus, Y . Sun, and S. Diggavi, “Encoded Dis- tributed Optimization,” in2017 IEEE International Symposium on Information Theory (ISIT). IEEE, 2017, pp. 2890–2894

    work page 2017

  26. [26]

    CodedS- ketch: A Coding Scheme for Distributed Computa- tion of Approximated Matrix Multiplication,

    T. Jahani-Nezhad and M. A. Maddah-Ali, “CodedS- ketch: A Coding Scheme for Distributed Computa- tion of Approximated Matrix Multiplication,”IEEE Transactions on Information Theory, vol. 67, no. 6, pp. 4185–4196, 2021

    work page 2021

  27. [27]

    OverSketch: Approximate Matrix Multipli- cation for the Cloud,

    V . Gupta, S. Wang, T. Courtade, and K. Ramchan- dran, “OverSketch: Approximate Matrix Multipli- cation for the Cloud,” in2018 IEEE International Conference on Big Data. IEEE, 2018, pp. 298–304

    work page 2018

  28. [28]

    Compression-Informed Coded Comput- ing,

    M. Rudow, N. Charalambides, A. O. Hero III, and K. Rashmi, “Compression-Informed Coded Comput- ing,” in2023 IEEE International Symposium on In- formation Theory (ISIT), 2023, pp. 2177–2182

    work page 2023