Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty for Separable Convex Programs in Machine Learning

Many problems in machine learning and other fields can be (re)for-mulated as linearly constrained separable convex programs. In most of the cases, there are multiple blocks of variables. However, the traditional alternating direction method (ADM) and its linearized version (LADM, obtained by linearizing the quadratic penalty term) are for the two-block case and cannot be naively generalized to solve the multi-block case. So there is great demand on extending the ADM based methods for the multi-block case. In this paper, we propose LADM with parallel splitting and adaptive penalty (LADMPSAP) to solve multi-block separable convex programs efficiently. When all the component objective functions have bounded subgradients, we obtain convergence results that are stronger than those of ADM and LADM, e.g., allowing the penalty parameter to be unbounded and proving the sufficient and necessary conditions} for global convergence. We further propose a simple optimality measure and reveal the convergence rate of LADMPSAP in an ergodic sense. For programs with extra convex set constraints, with refined parameter estimation we devise a practical version of LADMPSAP for faster convergence. Finally, we generalize LADMPSAP to handle programs with more difficult objective functions by linearizing part of the objective function as well. LADMPSAP is particularly suitable for sparse representation and low-rank recovery problems because its subproblems have closed form solutions and the sparsity and low-rankness of the iterates can be preserved during the iteration. It is also highly parallelizable and hence fits for parallel or distributed computing. Numerical experiments testify to the advantages of LADMPSAP in speed and numerical accuracy.Comment: Preliminary version published on Asian Conference on Machine Learning 201

Optimal grasping of soft objects with two robotic fingers

Robot grasping of deformable objects is an under-researched area. The difficulty comes from both mechanics and computation. First, deformation caused by the grasp operations changes object\u27s global geometry. Second, under deformation, an object\u27s contacts with the fingers grow from points into areas. Inside such a contact area, points that stick to the finger may later slide while points that slide may later stick. The torques exerted by the grasping fingers vary, in contrast with rigid body grasping whose torques are invariant under forces. In this thesis the object\u27s deformation and configuration of contact with fingers and the plane are tracked with finite element method(FEM) in an event-driven manner based on the contact displacements induced by the finger movements. The first part of the thesis analyzes two-finger squeeze grasping of deformable objects with a focus on two special classes: stable squeezes, which minimize the potential energy of the object among squeezes of the same depth, and pure squeezes, which eliminate all euclidean motions from the resulting deformations. Based on them an algorithm to characterize the best resistance by a grasp to an adversary finger is proposed which minimizes the work done by the grasping fingers. An optimization scheme is offered to handle the general case of frictional segment contact. Simulations and multiple experiments with a Barrett Hand on a rubber foam object are presented. The second part of this thesis describes a strategy for a two-finger robot hand to grasp and lift a 3D deformable object resting on the plane. Inspired by the human hand grasping, the strategy employs two rounded fingers to squeeze the object until a secure grasp is achieved under contact friction. And then lift it by translating upward to pick up the object. During the squeeze, a lift test is repeatedly conducted until it is successful based on the metrics and then trigger the upward translation. The gravitational force acting on the object is accounted for. Simulation is presented and shows some good promise for the sensorless grasping approach for deformable objects