Unified Theory of Efficient Optimization and Estimation

ended

The goal of this project is to make progress toward a unified theory of efficient optimization and estimation. In many computing applications, especially machine learning, optimization and estimation problems play an increasingly important role. For that reason, a large research effort is devoted to developing and understanding the limitations of efficient algorithms for these problems. For many of these problems, achieving the best known provable guarantees required the use of algorithms that are tailored to problem specifics. In recent years, the PI’s research with collaborators has shown that for many optimization problems, the conceptually simple sum-of-squares meta-algorithm, despite not being tailored to problem specifics, can match and often significantly outperform previous efficient algorithms in terms of provable guarantees. This project aims to better understand the capabilities and limitations of this meta-algorithm, especially for estimation problems, which have only recently begun to be studied in this light. In this way, the project will establish new algorithmic guarantees for basic optimization and estimation problems even in the face of non-convexity and adversarial outliers. In the same way, the project will shed light on the limitations of efficient algorithms for basic average-case problems like planted clique and stochastic block models. The project also aims to transfer the obtained theoretical insights into practical algorithms building on recent works by the PI and collaborators. Toward this goal the project will develop new algorithms with close to linear running times that match the guarantees of the best known polynomial-time algorithms. In order to assess their practicality, the project will perform systematic empirical evaluations of these algorithms.