The area of data science needs efficient computational methods with provable guarantees that can cope with complex nature and the high nonlinearity of many real-world systems. Practitioners often design heuristic algorithms tailored to specific applications, but the theoretical underpinnings of these methods remain a mystery and this limits their usage in safety-critical systems. In this talk, we aim to address the above issue for some machine learning problems.

First, we study the problem of certifying the robustness of neural networks against adversarial inputs. To diminish the relaxation error caused by popular linear programming and semidefinite programming certification methods, we propose partitioning the input uncertainty set and show that this approach reduces the relaxation error. We then develop a practical partitioning technique for large-scale networks. We also study the problem of robust neural network training and develop convex formulations to train networks that are robust to adversarial inputs, followed by efficient training algorithms with global convergence guarantees. 

In order to accelerate the computation, there is a major effort in the machine learning community to understand when simple local search algorithms could solve nonlinear problems to global optimality. A key proof technique relies on the notion of Restricted Isometry Property, whose conservatism is not well understood and cannot be applied to nonsmooth problems either. We discuss our recent results on addressing these problems. In particular, we introduce the notion of “global functions”, as a major generalization of convex functions, which allows us to study the non-existence of spurious local minima for nonconvex and nonsmooth learning problems. We demonstrate the results on tensor decomposition with outliers, video processing, and online optimization in machine learning.

Somayeh Sojoudi is an Assistant Professor in the Departments of Electrical Engineering & Computer Sciences and Mechanical Engineering at the University of California, Berkeley. She is an Associate Editor for the journals of the IEEE Transactions on Smart Grid, Systems & Control Letters, IEEE Access, and IEEE Open Journal of Control Systems. She is also a member of the conference editorial board of the IEEE Control Systems Society. She has received several awards, including INFORMS Optimization Society Prize for Young Researchers, INFORMS Energy Best Publication Award, INFORMS Data Mining Best Paper Award, NSF CAREER Award, and ONR Young Investigator Award. She has also received several best student conference paper awards (as advisor or co-author) from the Control Systems Society.