Bregman Augmented Lagrangian Method: Convergence, acceleration, and applications in reinforcement learning
In this thesis, the algorithm Bergman proximal point method (BPP), and its application to Bregman augmented Lagrangian method(BALM) is considered. Unlike classical augmented Lagrangian method (ALM ), whose convergence rate and its relation with the proximal point method is well-understood, the convergence rate for BALM has not yet been thoroughly studied in the literature. We analyze, in this thesis, the convergence rates of BALM in terms of the primal objective as well as the feasibility violation. We show that the algorithm can also be applied to variational inequality problems with convex constraints, and fully characterize the iteration complexity of the algorithm derived from the inexact version of BALM. Furthermore, we develop, for the first time, an accelerated Bregman proximal point method, that improves the convergence rate from $\cO(1/\sum_{k=0}^{T-1}\eta_k)$ to $\cO(1/(\sum_{k=0}^{T-1}\sqrt{\eta_k})^2)$, where $\{\eta_k\}_{k=0}^{T-1}$ is the sequence of proximal parameters. When applied to the dual of convex constrained convex programs, this leads to the construction of an accelerated BALM, that achieves the improved rates for both primal and dual convergences. Finally, numerical experiments comparing the performance of different Bregman divergences as well as the acceleration versions, with applications to Markov decision problems/reinforcement learning are presented at the end.
Bregman Augmented Lagrangian Method (BALM), Bregman Proximal Point Method (BPP), Acceleration (acc-BPP, acc-BALM), Reinforcement Learning
University of Illinois - Urbana-Champaign
Networked Digital Library of Theses and Dissertations
University of Illinois at Urbana-Champaign