This paper studies distributed sketching for ordinary least squares (OLS) regression, an approach that distributes small sketches of a large data set over multiple machines to separately construct OLS estimators and average them. Unlike prior studies that consider sketching on the whole data set, we consider sketching on partitioned subsets to further reduce computational cost. Under the fixed design setting, we characterize the exact excess loss of the averaged OLS estimator. Results show that this loss is comparable to the established loss for sketching on the whole data set when the divergence among subset covariances is small.
This paper analyzes bidirectional random projections for ordinary least squares (OLS) regression under the fixed design setting. Let $(X,Y) \in \mathbb{R}^{n \times p} \times \mathbb{R}^n$ be a sample and $R \in \mathbb{R}^{n_1 \times n}, W \in \mathbb{R}^{p \times p_1}$ be two properly distributed random projections. We develop an expected excess loss bound for the OLS estimator built on $(WXR, WY)$. Compared to an established bound for OLS estimator built on $(XR, Y)$, the gap is approximately $O\left( p_1 + C \frac{1}{p_1} \right)$, where $C$ scales with $n_1/n$ and can be negative for small $n_1/n$. Its implications are confirmed by numerical results on real-world data.