Fundamental Limits of Coded Polynomial Aggregation

TL;DR

This work addresses the problem of recovering a weighted polynomial aggregation in straggler-prone distributed systems by introducing straggler-aware CPA. It develops a framework with pre-specified non-straggler patterns and derives a necessary-and-sufficient orthogonality condition that enables exact recovery with fewer worker responses than traditional, individual-decoding approaches. The paper quantifies the minimum number of responses needed as $N^* = \left\lfloor\frac{K-1}{2}\right\rfloor + 1$ for $d=1$ and $N^* = (d-1)(K-1) + 1$ for $d\ge 2$, and provides explicit CPA constructions achieving this bound, along with proofs and algorithmic guidance. Simulations demonstrate a sharp feasibility transition at the predicted threshold, supporting the tightness and practicality of the proposed limits.

Abstract

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.

Figures (1)

• Figure 1: Feasible regimes of CPA and the baseline CPA scheme based on individual decoding for computation degrees $d=1$ and $d=2$. For a fixed $K$, the feasible regime consists of the values of $N$ for which the corresponding scheme is feasible. The red region corresponds to CPA, which enables feasibility with fewer responses ($N \le d(K-1)$) via the orthogonality conditions in Theorem \ref{['th-condition']}, with the minimum number of responses $N^\star$ characterized in Theorem \ref{['th-bound-case1']}. The gray region corresponds to the baseline scheme based on individual decoding, which is feasible only when $N \ge d(K-1)+1$, as stated in Lemma \ref{['le-local']}.

Theorems & Definitions (12)

• Definition 1: Feasibility of CPA
• Definition 2: Minimum Number of Responses
• Definition 3: CPA Based on Individual Decoding
• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Example 1
• Lemma 2
• Lemma 3