Distributed Sketching for Randomized Optimization: Exact Characterization, Concentration and Lower Bounds
This work addresses optimization problems in distributed systems where communication and computation are critical bottlenecks, offering incremental improvements in sketching methods with practical applications in serverless cloud computing.
The paper tackles the challenge of computationally expensive Hessian formation and communication bottlenecks in distributed optimization by using randomized sketches to reduce dimensions and improve privacy and resilience. It provides novel approximation guarantees, tight concentration results, and unbiased parameter averaging methods, with large-scale experiments demonstrating the effectiveness of these approaches.
We consider distributed optimization methods for problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We leverage randomized sketches for reducing the problem dimensions as well as preserving privacy and improving straggler resilience in asynchronous distributed systems. We derive novel approximation guarantees for classical sketching methods and establish tight concentration results that serve as both upper and lower bounds on the error. We then extend our analysis to the accuracy of parameter averaging for distributed sketches. Furthermore, we develop unbiased parameter averaging methods for randomized second order optimization for regularized problems that employ sketching of the Hessian. Existing works do not take the bias of the estimators into consideration, which limits their application to massively parallel computation. We provide closed-form formulas for regularization parameters and step sizes that provably minimize the bias for sketched Newton directions. Additionally, we demonstrate the implications of our theoretical findings via large scale experiments on a serverless cloud computing platform.