8 research outputs found
Universal Convexification via Risk-Aversion
We develop a framework for convexifying a fairly general class of
optimization problems. Under additional assumptions, we analyze the
suboptimality of the solution to the convexified problem relative to the
original nonconvex problem and prove additive approximation guarantees. We then
develop algorithms based on stochastic gradient methods to solve the resulting
optimization problems and show bounds on convergence rates. %We show a simple
application of this framework to supervised learning, where one can perform
integration explicitly and can use standard (non-stochastic) optimization
algorithms with better convergence guarantees. We then extend this framework to
apply to a general class of discrete-time dynamical systems. In this context,
our convexification approach falls under the well-studied paradigm of
risk-sensitive Markov Decision Processes. We derive the first known model-based
and model-free policy gradient optimization algorithms with guaranteed
convergence to the optimal solution. Finally, we present numerical results
validating our formulation in different applications
On the Link between Gaussian Homotopy Continuation and Convex Envelopes
Abstract. The continuation method is a popular heuristic in computer vision for nonconvex optimization. The idea is to start from a simpli-fied problem and gradually deform it to the actual task while tracking the solution. It was first used in computer vision under the name of graduated nonconvexity. Since then, it has been utilized explicitly or im-plicitly in various applications. In fact, state-of-the-art optical flow and shape estimation rely on a form of continuation. Despite its empirical success, there is little theoretical understanding of this method. This work provides some novel insights into this technique. Specifically, there are many ways to choose the initial problem and many ways to progres-sively deform it to the original task. However, here we show that when this process is constructed by Gaussian smoothing, it is optimal in a specific sense. In fact, we prove that Gaussian smoothing emerges from the best affine approximation to Vese’s nonlinear PDE. The latter PDE evolves any function to its convex envelope, hence providing the optimal convexification
On the convex formulations of robust Markov decision processes
Robust Markov decision processes (MDPs) are used for applications of dynamic
optimization in uncertain environments and have been studied extensively. Many
of the main properties and algorithms of MDPs, such as value iteration and
policy iteration, extend directly to RMDPs. Surprisingly, there is no known
analog of the MDP convex optimization formulation for solving RMDPs. This work
describes the first convex optimization formulation of RMDPs under the
classical sa-rectangularity and s-rectangularity assumptions. By using entropic
regularization and exponential change of variables, we derive a convex
formulation with a number of variables and constraints polynomial in the number
of states and actions, but with large coefficients in the constraints. We
further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or
entropy-based uncertainty sets, showing that, in these cases, RMDPs can be
reformulated as conic programs based on exponential cones, quadratic cones, and
non-negative orthants. Our work opens a new research direction for RMDPs and
can serve as a first step toward obtaining a tractable convex formulation of
RMDPs