Linear real-valued computations over distributed datasets are common in many applications, most notably as part of machine learning inference. In particular, linear computations that are quantized, i.e., where the coefficients are restricted to a predetermined set of values (such as $\pm 1$), have gained increasing interest lately due to their role in efficient, robust, or private machine learning models. Given a dataset to store in a distributed system, we wish to encode it so that all such computations could be conducted by accessing a small number of servers, called the access parameter of the system. Doing so relieves the remaining servers to execute other tasks. Minimizing the access parameter gives rise to an access-redundancy tradeoff, where a smaller access parameter requires more redundancy in the system, and vice versa. In this paper, we study this tradeoff and provide several explicit low-access schemes for $\{\pm1\}$ quantized linear computations based on covering codes in a novel way. While the connection to covering codes has been observed in the past, our results strictly outperform the state-of-the-art for two-valued linear computations. We further show that the same storage scheme can be used to retrieve any linear combination with two distinct coefficients -- regardless of what those coefficients are -- with the same access parameter. This universality result is then extended to all possible quantizations with any number of values; while the storage remains identical, the access parameter increases according to a new additive-combinatorics property we call coefficient complexity. We then turn to study the coefficient complexity -- we characterize the complexity of small sets of coefficients, provide bounds, and identify coefficient sets having the highest and lowest complexity.

翻译：在分布式数据集上进行线性实值计算在许多应用中都很常见，尤其是作为机器学习推理的一部分。特别是量化线性计算（即系数被限制在预定义值集，例如$\pm 1$），因其在高效、鲁棒或隐私保护的机器学习模型中的作用，近来受到越来越多的关注。对于要存储在分布式系统中的数据集，我们希望对其进行编码，使得所有此类计算可以通过访问少量服务器（称为系统的访问参数）来完成。这样做可以解放其余服务器以执行其他任务。最小化访问参数会导致一种访问-冗余权衡：更小的访问参数需要系统中更多的冗余，反之亦然。本文研究这种权衡，并基于覆盖码提出了一种新颖的显式低访问方案，用于$\{\pm1\}$量化线性计算。尽管过去已观察到与覆盖码的联系，但我们的结果在二值线性计算方面严格优于现有技术。我们进一步证明，相同的存储方案可用于检索具有两个不同系数的任意线性组合——无论这些系数是什么——且访问参数保持不变。这一普适性结果随后被扩展到所有可能的多值量化：存储方式保持不变，但访问参数根据一种我们称为系数复杂度的新加性组合学性质而增加。接着，我们研究系数复杂度——我们刻画了小系数集的复杂度，提供了界限，并确定了具有最高和最低复杂度的系数集。