[Paper] Multivariate Polynomial Codes for Efficient Matrix Chain Multiplication in Distributed Systems

Source: arXiv - 2601.08708v1

Overview

The paper tackles a classic bottleneck in large‑scale data processing: stragglers—slow or failed workers that hold up distributed matrix chain multiplication. While existing coded‑computing tricks work well for a single matrix product, they become costly when the computation involves long chains of multiplications. Gómez‑Vilardebò introduces multivariate polynomial codes that dramatically cut the storage burden on each worker, offering a new trade‑off between computation and memory that is especially relevant for modern AI pipelines and scientific simulations.

Key Contributions

• Two novel multivariate coding schemes tailored for matrix‑chain multiplication in distributed clusters.
• Formal analysis showing that multivariate codes reduce per‑worker storage overhead compared with naïve extensions of univariate polynomial codes.
• Quantitative trade‑off characterization between extra local computation and saved memory, providing designers with a tunable knob.
• Experimental validation on a simulated cluster that demonstrates up to 70 % storage reduction with modest (≤ 30 %) increase in local compute time.
• Generalizable framework that can be extended to other linear‑algebra workloads (e.g., tensor contractions, block‑wise solvers).

Methodology

1. Problem Formalization – The authors model a chain of (L) matrices ({A_1,\dots,A_L}) to be multiplied across (N) worker nodes, each of which may become a straggler.
2. Coding Construction –
   • Univariate baseline: Encode each matrix as coefficients of a single‑variable polynomial; workers evaluate the polynomial at distinct points, then reconstruct the product via interpolation.
   • Multivariate extension: Split the chain into sub‑segments and embed each segment’s data into a different variable of a multivariate polynomial. Workers evaluate the multivariate polynomial at a grid of points, allowing the same worker to store fewer coefficients per variable.
3. Decoding Strategy – The master node collects enough distinct evaluations (from the fastest workers) to solve a system of linear equations and recover the final product, using standard multivariate interpolation techniques.
4. Complexity Analysis – Derive closed‑form expressions for:
   • Storage per worker (number of matrix blocks stored).
   • Local computation (operations needed to evaluate the multivariate polynomial).
   • Recovery threshold (minimum number of non‑straggling workers required).
5. Simulation Setup – Implement the schemes in Python/NumPy, emulate straggler behavior with exponential latency distributions, and compare against the univariate baseline across varying chain lengths, worker counts, and straggler fractions.

Results & Findings

• Storage savings grow with the chain length: a chain of 64 matrices sees a ≈ 70 % reduction in per‑worker memory when using a 2‑variate code.
• Computation overhead remains modest; the extra work is mostly cheap element‑wise multiplications that parallelize well on modern CPUs/GPUs.
• Straggler resilience is unchanged: the master still needs only the fastest (K) workers (where (K) is the recovery threshold) to finish, preserving the latency benefits of coded computing.

Practical Implications

• Deep‑learning training pipelines often involve long sequences of linear layers (e.g., transformer blocks). Deploying multivariate codes can let each GPU/TPU node hold fewer intermediate activations, freeing memory for larger batch sizes or model widths.
• Large‑scale scientific simulations (e.g., finite‑element solvers) that repeatedly multiply many sparse/dense matrices can now scale to more nodes without hitting memory caps, reducing overall wall‑clock time.
• Edge‑cloud hybrid inference: Edge devices with limited RAM can participate in distributed inference by storing only a fraction of the matrix chain, while the cloud handles the heavier compute—thanks to the lower storage footprint.
• Framework integration: The coding scheme can be wrapped as a thin layer over existing distributed linear‑algebra libraries (e.g., MPI, Ray, Dask), requiring only a change in data partitioning and a small pre‑processing step before launch.

In short, developers can trade a modest increase in per‑node compute for a significant memory win, enabling larger problems to run on the same hardware budget and mitigating the impact of stragglers without extra redundancy.

Limitations & Future Work

• Higher‑dimensional polynomials further reduce storage but increase both encoding/decoding complexity and numerical stability concerns; the paper only explores up to three variables.
• Assumes homogeneous workers (same compute power and memory). Heterogeneous clusters may need adaptive variable assignments, which is left for future research.
• Experimental validation is limited to simulated latency models; real‑world cluster experiments (e.g., on AWS or Azure) are needed to confirm robustness under network variability and hardware faults.
• Extension to non‑square or sparse matrices is not fully addressed; tailoring the coding to exploit sparsity patterns could yield additional gains.

The authors suggest exploring adaptive multivariate designs that dynamically choose the number of variables based on observed straggler rates and memory constraints, as well as integrating the approach with gradient‑coding techniques for end‑to‑end distributed training.

Authors

• Jesús Gómez-Vilardebò

Paper Information

• arXiv ID: 2601.08708v1
• Categories: cs.IT, cs.DC
• Published: January 13, 2026
• PDF: Download PDF