A refinement of Bennett's inequality with applications to portfolio optimization

A refinement of Bennett's inequality is introduced which is strictly tighter than the classical bound. The new bound establishes the convergence of the average of independent random variables to its expected value. It also carefully exploits information about the potentially heterogeneous mean, variance, and ceiling of each random variable. The bound is strictly sharper in the homogeneous setting and very often significantly sharper in the heterogeneous setting. The improved convergence rates are obtained by leveraging Lambert's W function. We apply the new bound in a portfolio optimization setting to allocate a budget across investments with heterogeneous returns

Frank-Wolfe Algorithms for Saddle Point Problems

We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained smooth convex-concave saddle point (SP) problems. Remarkably, the method only requires access to linear minimization oracles. Leveraging recent advances in FW optimization, we provide the first proof of convergence of a FW-type saddle point solver over polytopes, thereby partially answering a 30 year-old conjecture. We also survey other convergence results and highlight gaps in the theoretical underpinnings of FW-style algorithms. Motivating applications without known efficient alternatives are explored through structured prediction with combinatorial penalties as well as games over matching polytopes involving an exponential number of constraints.Comment: Appears in: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS 2017). 39 page

Square Root Propagation

We propose a message propagation scheme for numerically stable inference in Gaussian graphical models which can otherwise be susceptible to errors caused by finite numerical precision. We adapt square root algorithms, popular in Kalman filtering, to graphs with arbitrary topologies. The method consists of maintaining potentials and generating messages that involve the square root of precision matrices. Combining this with the machinery of the junction tree algorithm leads to an efficient and numerically stable algorithm. Experiments are presented to demonstrate the robustness of the method to numerical errors that can arise in complex learning and inference problems