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Scalable High-Dimensional Multivariate Linear Regression for Feature-Distributed Data

Core Concepts
The author proposes a two-stage relaxed greedy algorithm (TSRGA) for scalable multivariate linear regression on feature-distributed data, reducing communication complexity and improving statistical performance.
The paper introduces TSRGA for efficient multivariate linear regression on feature-distributed data. It addresses the challenges of high communication complexity and offers theoretical guarantees on convergence and communication costs. TSRGA is shown to outperform existing algorithms in terms of speed and accuracy, making it a valuable tool for analyzing large datasets. Key Points: Feature-distributed data partitioned by features across nodes. TSRGA reduces communication complexity with a two-stage approach. The algorithm converges fast and provides accurate estimates. Applications include financial analysis using unstructured data from reports.
The proposed TSRGA enjoys a communication complexity of Op(sn(n + dn)) bytes. Hydra algorithm requires O(np log(1/ϵ)) bytes of communication for Lasso problem estimation. DIDRP algorithm needs O(n2 +n log(1/ϵ)) bytes for estimating ridge regression.
"The main advantage of TSRGA is that its communication complexity does not depend on the feature dimension." "TSRGA achieved the smallest estimation error using the least number of iterations." "TSRGA efficiently utilizes information from texts in high-dimensional feature matrices."

Deeper Inquiries

How can TSRGA be adapted for other types of regression models beyond linear

TSRGA can be adapted for other types of regression models beyond linear by modifying the algorithm to accommodate different loss functions and constraints specific to the new model. For example, for logistic regression in classification tasks, the squared loss function in TSRGA can be replaced with a log-likelihood function. This adaptation would involve updating the optimization criteria and stopping criteria accordingly. Similarly, for Poisson regression in count data modeling, adjustments need to be made to suit the specific characteristics of this type of regression.

What are potential drawbacks or limitations of using TSRGA in practical applications

One potential drawback of using TSRGA in practical applications is that it may require strong assumptions or conditions to achieve optimal performance. For instance, TSRGA relies on assumptions such as sparsity and certain properties of the predictors and errors which may not always hold true in real-world datasets. If these assumptions are violated, it could lead to suboptimal results or even algorithm failure. Additionally, while TSRGA reduces communication complexity compared to some existing algorithms, it still requires coordination among multiple computing nodes which can introduce overhead and complexity in implementation.

How does the concept of sparsity impact the performance and efficiency of TSRGA

The concept of sparsity plays a crucial role in determining the performance and efficiency of TSRGA. In TSRGA, sparsity refers to the number of relevant predictors selected by the algorithm relative to the total number of predictors available. A higher degree of sparsity implies that only a small subset of predictors significantly contribute to explaining variability in the response variables. In such cases, TSRGA can effectively screen out irrelevant predictors early on through its two-stage approach, leading to faster convergence and reduced communication costs. On the other hand, if there is low sparsity where many predictors have non-negligible coefficients contributing towards predicting outcomes (low rank assumption), then TSRGA may struggle due to increased computational complexity from handling a larger set of predictors simultaneously during estimation iterations.