Georgia Institute of Technology College of Sciences

We will discuss two types of structured multi-objective optimization programs.  In the first, the goal is to minimize a sum of functions described by a graph: each function is associated with a vertex, and there is an edge between vertices if two functions share a subset of their variables.   Problems of this type arise in state estimation problems, including simultaneous localization and mapping (SLAM) in robotics, tracking, and streaming reconstruction problems in signal processing.  We will show that under mild smoothness conditions, these types of problems exhibit a type of locality: if a node is added to the graph (changing the optimization problem), the optimal solution changes only for variables that are ``close’’ to the added node, immediately giving us a quick way to update the solution as the graph grows.

In the second part of the talk, we will consider a multi-task learning problem where the solutions are expected to lie in a low-dimensional subspace.  This corresponds to a low-rank matrix recover problem where the columns of the matrix have been ``sketched’’ independently.  We show that a novel convex relaxation of this problem results in optimal sample complexity bounds.  These bounds demonstrate the statistical leverage we gain by solving the problem jointly over solving each individually.