A Covariance Matrix Adaptation Evolution Strategy for Direct Policy Search in Reproducing Kernel Hilbert Space

The covariance matrix adaptation evolution strategy (CMA-ES) is an efficient derivative-free optimization algorithm. It optimizes a black-box objective function over a well defined parameter space. In some problems, such parameter spaces are defined using function approximation in which feature functions are manually defined. Therefore, the performance of those techniques strongly depends on the quality of chosen features. Hence, enabling CMA-ES to optimize on a more complex and general function class of the objective has long been desired. Specifically, we consider modeling the input space for black-box optimization in reproducing kernel Hilbert spaces (RKHS). This modeling leads to a functional optimization problem whose domain is a function space that enables us to optimize in a very rich function class. In addition, we propose CMA-ES-RKHS, a generalized CMA-ES framework, that performs black-box functional optimization in the RKHS. A search distribution, represented as a Gaussian process, is adapted by updating both its mean function and covariance operator. Adaptive representation of the function and covariance operator is achieved with sparsification techniques. We evaluate CMA-ES-RKHS on a simple functional optimization problem and bench-mark reinforcement learning (RL) domains. For an application in RL, we model policies for MDPs in RKHS and transform a cumulative return objective as a functional of RKHS policies, which can be optimized via CMA-ES-RKHS. This formulation results in a black-box functional policy search framework

Accelerating Motion Planning via Optimal Transport

Motion planning is still an open problem for many disciplines, e.g., robotics, autonomous driving, due to their need for high computational resources that hinder real-time, efficient decision-making. A class of methods striving to provide smooth solutions is gradient-based trajectory optimization. However, those methods usually suffer from bad local minima, while for many settings, they may be inapplicable due to the absence of easy-to-access gradients of the optimization objectives. In response to these issues, we introduce Motion Planning via Optimal Transport (MPOT) -- a \textit{gradient-free} method that optimizes a batch of smooth trajectories over highly nonlinear costs, even for high-dimensional tasks, while imposing smoothness through a Gaussian Process dynamics prior via the planning-as-inference perspective. To facilitate batch trajectory optimization, we introduce an original zero-order and highly-parallelizable update rule: the Sinkhorn Step, which uses the regular polytope family for its search directions. Each regular polytope, centered on trajectory waypoints, serves as a local cost-probing neighborhood, acting as a \textit{trust region} where the Sinkhorn Step "transports" local waypoints toward low-cost regions. We theoretically show that Sinkhorn Step guides the optimizing parameters toward local minima regions of non-convex objective functions. We then show the efficiency of MPOT in a range of problems from low-dimensional point-mass navigation to high-dimensional whole-body robot motion planning, evincing its superiority compared to popular motion planners, paving the way for new applications of optimal transport in motion planning.Comment: Published as a conference paper at NeurIPS 2023. Project website: https://sites.google.com/view/sinkhorn-step