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Register a new userNatural gradient via optimal transport
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Added by
Wuchen Li
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient flow there. In contrast to the natural Fisher-Rao gradient, the natural Wasserstein gradient incorporates a ground metric on sample space. We illustrate the analysis of elementary exponential family examples and demonstrate an application of the Wasserstein natural gradient to maximum likelihood estimation.
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We elaborate the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott-Strum-Villani, we define this notion based on the geodesic convexity of the Kullback-Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.
In this paper, we consider Strassens version of optimal transport (OT) problem. That is, we minimize the excess-cost probability (i.e., the probability that the cost is larger than a given value) over all couplings of two given distributions. We derive large deviation, moderate deviation, and central limit theorems for this problem. Our proof is based on Strassens dual formulation of the OT problem, Sanovs theorem on the large deviation principle (LDP) of empirical measures, as well as the moderate deviation principle (MDP) and central limit theorems (CLT) of empirical measures. In order to apply the LDP, MDP, and CLT to Strassens OT problem, two nested optimal transport formulas for Strassens OT problem are derived. Based on these nested formulas and using a splitting technique, we carefully design asymptotically optimal solutions to Strassens OT problem and its dual formulation.
The problem of enhancing Quality-of-Service (QoS) in power constrained, mobile relay beamforming networks, by optimally and dynamically controlling the motion of the relaying nodes, is considered, in a dynamic channel environment. We assume a time slotted system, where the relays update their positions before the beginning of each time slot. Modeling the wireless channel as a Gaussian spatiotemporal stochastic field, we propose a novel $2$-stage stochastic programming problem formulation for optimally specifying the positions of the relays at each time slot, such that the expected QoS of the network is maximized, based on causal Channel State Information (CSI) and under a total relay transmit power budget. This results in a schema where, at each time slot, the relays, apart from optimally beamforming to the destination, also optimally, predictively decide their positions at the next time slot, based on causally accumulated experience. Exploiting either the Method of Statistical Differentials, or the multidimensional Gauss-Hermite Quadrature Rule, the stochastic program considered is shown to be approximately equivalent to a set of simple subproblems, which are solved in a distributed fashion, one at each relay. Optimality and performance of the proposed spatially controlled system are also effectively assessed, under a rigorous technical framework; strict optimality is rigorously demonstrated via the development of a version of the Fundamental Lemma of Stochastic Control, and, performance-wise, it is shown that, quite interestingly, the optimal average network QoS exhibits an increasing trend across time slots, despite our myopic problem formulation. Numerical simulations are presented, experimentally corroborating the success of the proposed approach and the validity of our theoretical predictions.
In this paper, we consider a discrete-time stochastic control problem with uncertain initial and target states. We first discuss the connection between optimal transport and stochastic control problems of this form. Next, we formulate a linear-quadratic regulator problem where the initial and terminal states are distributed according to specified probability densities. A closed-form solution for the optimal transport map in the case of linear-time varying systems is derived, along with an algorithm for computing the optimal map. Two numerical examples pertaining to swarm deployment demonstrate the practical applicability of the model, and performance of the numerical method.
This paper develops computable metrics to assign priorities for information collection on network systems made up by binary components. Components are worth inspecting because their condition state is uncertain and the system functioning depends on it. The Value of Information (VoI) allows assessing the impact of information in decision making under uncertainty, including the precision of the observation, the available actions and the expected economic loss. Some VoI-based metrics for system-level and component-level maintenance actions, defined as global and local metrics, respectively, are introduced, analyzed and applied to series and parallel systems. Their computationally complexity of applications to general networks is discussed and, to tame the complexity for the local metric assessment, a heuristic is presented and its performance is compared on some case studies.
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