We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

翻译：本研究探讨了结构化随机数据加权和的集中不等式，包括（i）张量内积与（ii）序列矩阵和。我们关注在可交换性条件下这些结构化加权和的尾概率界与集中不等式，其适用范围超越了经典的独立项框架。我们建立了具有结构依赖可交换性的霍夫丁界与伯恩斯坦界。在此过程中，我们以最优常数恢复了可交换随机变量加权和以及独立同分布随机矩阵和中的已知结果。特别地，我们为矩阵阵列的组合和建立了比以往基于Chatterjee可交换对方法所得结果更尖锐的集中界。在应用方面，更丰富的结构为我们提供了新的分析工具，可用于估计多因子响应模型的平均效应以及研究联邦平均中的固定设计草图方法。我们将所得结果应用于这些问题，并发现理论预测得到了数值证据的证实。