Randomized Constraints Consensus for Distributed Robust Linear Programming

In this paper we consider a network of processors aiming at cooperatively solving linear programming problems subject to uncertainty. Each node only knows a common cost function and its local uncertain constraint set. We propose a randomized, distributed algorithm working under time-varying, asynchronous and directed communication topology. The algorithm is based on a local computation and communication paradigm. At each communication round, nodes perform two updates: (i) a verification in which they check-in a randomized setup-the robust feasibility (and hence optimality) of the candidate optimal point, and (ii) an optimization step in which they exchange their candidate bases (minimal sets of active constraints) with neighbors and locally solve an optimization problem whose constraint set includes: a sampled constraint violating the candidate optimal point (if it exists), agent's current basis and the collection of neighbor's basis. As main result, we show that if a processor successfully performs the verification step for a sufficient number of communication rounds, it can stop the algorithm since a consensus has been reached. The common solution is-with high confidence-feasible (and hence optimal) for the entire set of uncertainty except a subset having arbitrary small probability measure. We show the effectiveness of the proposed distributed algorithm on a multi-core platform in which the nodes communicate asynchronously.Comment: Accepted for publication in the 20th World Congress of the International Federation of Automatic Control (IFAC

Sequential Randomized Algorithms for Convex Optimization in the Presence of Uncertainty

In this paper, we propose new sequential randomized algorithms for convex optimization problems in the presence of uncertainty. A rigorous analysis of the theoretical properties of the solutions obtained by these algorithms, for full constraint satisfaction and partial constraint satisfaction, respectively, is given. The proposed methods allow to enlarge the applicability of the existing randomized methods to real-world applications involving a large number of design variables. Since the proposed approach does not provide a priori bounds on the sample complexity, extensive numerical simulations, dealing with an application to hard-disk drive servo design, are provided. These simulations testify the goodness of the proposed solution.Comment: 18 pages, Submitted for publication to IEEE Transactions on Automatic Contro

From Cutting Planes Algorithms to Compression Schemes and Active Learning

Cutting-plane methods are well-studied localization(and optimization) algorithms. We show that they provide a natural framework to perform machinelearning ---and not just to solve optimization problems posed by machinelearning--- in addition to their intended optimization use. In particular, theyallow one to learn sparse classifiers and provide good compression schemes.Moreover, we show that very little effort is required to turn them intoeffective active learning methods. This last property provides a generic way todesign a whole family of active learning algorithms from existing passivemethods. We present numerical simulations testifying of the relevance ofcutting-plane methods for passive and active learning tasks.Comment: IJCNN 2015, Jul 2015, Killarney, Ireland. 2015, \<http://www.ijcnn.org/\&g

Robustness and Randomness

Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discussed approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust

Stochastic Optimization Problems with Incomplete Information on Distribution Functions

The main purpose of this paper is to discus: numerical optimization procedures, based on duality theory, for problems in which the distribution function is only partially known. The dual problem is formulated as a minimax-type problem in which the "inner" problem of maximization is not concave. Numerical procedures that avoid the difficulties associated with solving the "inner" problem are proposed

Blending Learning and Inference in Structured Prediction

In this paper we derive an efficient algorithm to learn the parameters of structured predictors in general graphical models. This algorithm blends the learning and inference tasks, which results in a significant speedup over traditional approaches, such as conditional random fields and structured support vector machines. For this purpose we utilize the structures of the predictors to describe a low dimensional structured prediction task which encourages local consistencies within the different structures while learning the parameters of the model. Convexity of the learning task provides the means to enforce the consistencies between the different parts. The inference-learning blending algorithm that we propose is guaranteed to converge to the optimum of the low dimensional primal and dual programs. Unlike many of the existing approaches, the inference-learning blending allows us to learn efficiently high-order graphical models, over regions of any size, and very large number of parameters. We demonstrate the effectiveness of our approach, while presenting state-of-the-art results in stereo estimation, semantic segmentation, shape reconstruction, and indoor scene understanding