Linearly-Constrained Entropy Maximization Problem with Quadratic Costs and Its Applications to Transportation Planning Problems

Many transportation problems can be formulated as a linearly-constrained convex programming problem whose objective function consists of entropy functions and other cost-related terms. In this paper, we propose an unconstrained convex programming dual approach to solving these problems. In particular, we focus on a class of linearly-constrained entropy maximization problem with quadratic cost, study its Lagrangian dual, and provide a globally convergent algorithm with a quadratic rate of convergence. The theory and algorithm can be readily applied to the trip distribution problem with quadratic cost and many other entropy-based formulations, including the conventional trip distribution problem with linear cost, the entropy-based modal split model, and the decomposed problems of the combined problem of trip distribution and assignment. The efficiency and the robustness of this approach are confirmed by our computational experience

Coordinate ascent for maximizing nondifferentiable concave functions

Cover title.Includes bibliographical references.Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058by Paul Tseng

Dual coordinate ascent for problems with strictly convex costs and linear constraints : a unified approach

Caption title. "July 1988."Includes bibliographical references.Work supported by the National Science Foundation under grant NSF-ECS-8519058 Work supported by the Army Research Office under grant DAAL03-86-K-0171by Paul Tseng

On the Fine-Grain Decomposition of Multicommodity Transportation Problems

We develop algorithms for nonlinear problems with multicommodity transportation constraints. The algorithms are of the row-action type and, when properly applied,decompose the underlying graph alternatingly by nodes and edges. Hence, a fine-grain decomposition scheme is developed that is suitable for massively parallel computer architectures of the SIMD (i.e., single instruction stream, multiple data stream) class. Implementations on the Connection Machine CM-2 are discussed for both dense and sparse transportation problems. The dense implementation achieves computing rate of 1.6-3 GFLOPS. Several aspects of the algorithm are investigated empirically. Computational results are reported for the solution of quadratic programs with approximately 10 million columns and 100 thousand rows

Basis descent methods for convex essentially smooth optimization with applications to quadratic/entropy optimization and resource allocation

Cover title.Includes bibliographical references (p. 33-38).Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058by Paul Tseng

Learning and inference with Wasserstein metrics

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Brain and Cognitive Sciences, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 131-143).This thesis develops new approaches for three problems in machine learning, using tools from the study of optimal transport (or Wasserstein) distances between probability distributions. Optimal transport distances capture an intuitive notion of similarity between distributions, by incorporating the underlying geometry of the domain of the distributions. Despite their intuitive appeal, optimal transport distances are often difficult to apply in practice, as computing them requires solving a costly optimization problem. In each setting studied here, we describe a numerical method that overcomes this computational bottleneck and enables scaling to real data. In the first part, we consider the problem of multi-output learning in the presence of a metric on the output domain. We develop a loss function that measures the Wasserstein distance between the prediction and ground truth, and describe an efficient learning algorithm based on entropic regularization of the optimal transport problem. We additionally propose a novel extension of the Wasserstein distance from probability measures to unnormalized measures, which is applicable in settings where the ground truth is not naturally expressed as a probability distribution. We show statistical learning bounds for both the Wasserstein loss and its unnormalized counterpart. The Wasserstein loss can encourage smoothness of the predictions with respect to a chosen metric on the output space. We demonstrate this property on a real-data image tagging problem, outperforming a baseline that doesn't use the metric. In the second part, we consider the probabilistic inference problem for diffusion processes. Such processes model a variety of stochastic phenomena and appear often in continuous-time state space models. Exact inference for diffusion processes is generally intractable. In this work, we describe a novel approximate inference method, which is based on a characterization of the diffusion as following a gradient flow in a space of probability densities endowed with a Wasserstein metric. Existing methods for computing this Wasserstein gradient flow rely on discretizing the underlying domain of the diffusion, prohibiting their application to problems in more than several dimensions. In the current work, we propose a novel algorithm for computing a Wasserstein gradient flow that operates directly in a space of continuous functions, free of any underlying mesh. We apply our approximate gradient flow to the problem of filtering a diffusion, showing superior performance where standard filters struggle. Finally, we study the ecological inference problem, which is that of reasoning from aggregate measurements of a population to inferences about the individual behaviors of its members. This problem arises often when dealing with data from economics and political sciences, such as when attempting to infer the demographic breakdown of votes for each political party, given only the aggregate demographic and vote counts separately. Ecological inference is generally ill-posed, and requires prior information to distinguish a unique solution. We propose a novel, general framework for ecological inference that allows for a variety of priors and enables efficient computation of the most probable solution. Unlike previous methods, which rely on Monte Carlo estimates of the posterior, our inference procedure uses an efficient fixed point iteration that is linearly convergent. Given suitable prior information, our method can achieve more accurate inferences than existing methods. We additionally explore a sampling algorithm for estimating credible regions.by Charles Frogner.Ph. D

An entropy maximising model for estimating trip matrices from traffic counts

The main objective of this research is to develop and test a technique for estimating trip matrices from traffic counts. After discussing conventional methods for obtaining trip matrices an analysis is made of the problem of estimating them from traffic counts: it is found that in general the problem is underspecified in the sense that there will be more than one trip matrix which, when loaded onto a network, may reproduce a set of observed counts. A review is made of some models put forward to estimate a trip table from volume counts, the majority of them based on a travel demand model. A new model is then developed by the author within an entropy maximising formalism. The model may be interpreted as producing the most likely trip matrix consistent with the information contained in the counts and a prior trip matrix if available. This model does not require counts on all links in the network, can make efficient use of outdated trip matrices and other information, and is fairly modest in computer requirements. The model is then tested against real data collected by the Transport and Road Research Laboratory in the central area of Reading. Considerable temporal variability was found in the sampled trip matrices. The matrices estimated by the model are not very close to the observed ones but their errors are in general within the daily variations of the sampled matrices. A number of tests on the sensitivity of the model to errors and availability of traffic counts and route choice models used are also reported. A technique has been developed to rank links according to their potential contribution to the improvement of an estimated trip matrix. This scheme may be used to select new counting sites. The availability of a reasonable prior estimate of the trip matrix considerably improves the accuracy of the origin-destination matrix generated by the model. Some suggestions for extensions and further research are presented towards the end of this work