16,575 research outputs found
Data-Driven Estimation in Equilibrium Using Inverse Optimization
Equilibrium modeling is common in a variety of fields such as game theory and
transportation science. The inputs for these models, however, are often
difficult to estimate, while their outputs, i.e., the equilibria they are meant
to describe, are often directly observable. By combining ideas from inverse
optimization with the theory of variational inequalities, we develop an
efficient, data-driven technique for estimating the parameters of these models
from observed equilibria. We use this technique to estimate the utility
functions of players in a game from their observed actions and to estimate the
congestion function on a road network from traffic count data. A distinguishing
feature of our approach is that it supports both parametric and
\emph{nonparametric} estimation by leveraging ideas from statistical learning
(kernel methods and regularization operators). In computational experiments
involving Nash and Wardrop equilibria in a nonparametric setting, we find that
a) we effectively estimate the unknown demand or congestion function,
respectively, and b) our proposed regularization technique substantially
improves the out-of-sample performance of our estimators.Comment: 36 pages, 5 figures Additional theorems for generalization guarantees
and statistical analysis adde
Democratic Representations
Minimization of the (or maximum) norm subject to a constraint
that imposes consistency to an underdetermined system of linear equations finds
use in a large number of practical applications, including vector quantization,
approximate nearest neighbor search, peak-to-average power ratio (or "crest
factor") reduction in communication systems, and peak force minimization in
robotics and control. This paper analyzes the fundamental properties of signal
representations obtained by solving such a convex optimization problem. We
develop bounds on the maximum magnitude of such representations using the
uncertainty principle (UP) introduced by Lyubarskii and Vershynin, and study
the efficacy of -norm-based dynamic range reduction. Our
analysis shows that matrices satisfying the UP, such as randomly subsampled
Fourier or i.i.d. Gaussian matrices, enable the computation of what we call
democratic representations, whose entries all have small and similar magnitude,
as well as low dynamic range. To compute democratic representations at low
computational complexity, we present two new, efficient convex optimization
algorithms. We finally demonstrate the efficacy of democratic representations
for dynamic range reduction in a DVB-T2-based broadcast system.Comment: Submitted to a Journa
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
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