Fundamental Limits of Coded Polynomial Aggregation

Coded polynomial aggregation (CPA) enables the master to directly recover a weighted aggregation of polynomial evaluations without individually decoding each term, thereby reducing the number of required worker responses. In this paper, we extend CPA to straggler-aware distributed computing systems and introduce a straggler-aware CPA framework with pre-specified non-straggler patterns, where exact recovery is required only for a given collection of admissible non-straggler sets. Our main result shows that exact recovery of the desired aggregation is achievable with fewer worker responses than required by polynomial coded computing based on individual decoding, and that feasibility is fundamentally characterized by the intersection structure of the non-straggler patterns. In particular, we establish necessary and sufficient conditions for exact recovery in straggler-aware CPA and identify an intersection-size threshold that is sufficient to guarantee exact recovery. We further prove that this threshold becomes both necessary and sufficient when the number of admissible non-straggler sets is sufficiently large. We also provide an explicit construction of feasible CPA schemes whenever the intersection size exceeds the derived threshold. Finally, simulations reveal a sharp feasibility transition at the predicted threshold, providing empirical evidence that the bound is tight in practice.

💡 Research Summary

This paper investigates a fundamental problem in distributed computing: how to recover a weighted aggregation of polynomial evaluations from a set of worker responses while minimizing the number of required responses. The authors introduce the concept of Coded Polynomial Aggregation (CPA), which differs from the traditional “individual decoding” approach used in most polynomial coded computing schemes. In the individual‑decoding paradigm, the master first reconstructs every individual sub‑computation (i.e., each F(Xₖ)) and then forms the weighted sum Y = ∑ₖ wₖ F(Xₖ). This requires at least d(K − 1)+1 responses for a degree‑d polynomial evaluated on K datasets, because the master must interpolate a polynomial of degree d(K − 1).

The authors ask whether, when the final goal is only the weighted sum, one can exploit the linear aggregation structure to reduce the required number of responses, especially in the regime N ≤ d(K − 1) where individual decoding is impossible. They propose a CPA framework consisting of three phases: (1) encoding a set of K distinct data points αₖ into an encoder polynomial E(z) such that E(αₖ)=Xₖ; (2) sending coded matrices E(βₙ) to N workers, each of which computes F(E(βₙ)) and returns the result; (3) decoding by interpolating a decoder polynomial D(z) that matches the workers’ outputs at the evaluation points βₙ, then evaluating D(αₖ) and forming the weighted sum Ŷ = ∑ₖ wₖ D(αₖ).

The central technical contribution is a set of orthogonality conditions that are both necessary and sufficient for exact recovery of Y under CPA. Define P(z)=∏ₙ(z−βₙ) and let C = d(K − 1) − N + 1. The recovery error can be expressed as Ŷ−Y = ∑ₖ wₖ Δ(αₖ) where Δ(z)=D(z)−F(E(z)). Because Δ(βₙ)=0 for all workers, Δ(z) must be divisible by P(z), i.e., Δ(z)=P(z)·R(z) with deg R ≤ C−1. Substituting this into the error expression yields a linear combination of terms ∑ₖ wₖ P(αₖ) αₖʲ for j=0,…,C−1. Hence, exact recovery is achieved iff all these terms vanish, which is precisely the orthogonality condition:

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