544 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
Least quantile regression via modern optimization
We address the Least Quantile of Squares (LQS) (and in particular the Least
Median of Squares) regression problem using modern optimization methods. We
propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which
allows us to find a provably global optimal solution for the LQS problem. Our
MIO framework has the appealing characteristic that if we terminate the
algorithm early, we obtain a solution with a guarantee on its sub-optimality.
We also propose continuous optimization methods based on first-order
subdifferential methods, sequential linear optimization and hybrid combinations
of them to obtain near optimal solutions to the LQS problem. The MIO algorithm
is found to benefit significantly from high quality solutions delivered by our
continuous optimization based methods. We further show that the MIO approach
leads to (a) an optimal solution for any dataset, where the data-points
's are not necessarily in general position, (b) a simple
proof of the breakdown point of the LQS objective value that holds for any
dataset and (c) an extension to situations where there are polyhedral
constraints on the regression coefficient vector. We report computational
results with both synthetic and real-world datasets showing that the MIO
algorithm with warm starts from the continuous optimization methods solve small
() and medium () size problems to provable optimality in under
two hours, and outperform all publicly available methods for large-scale
(10,000) LQS problems.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1223 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Voice of Optimization
We introduce the idea that using optimal classification trees (OCTs) and
optimal classification trees with-hyperplanes (OCT-Hs), interpretable machine
learning algorithms developed by Bertsimas and Dunn [2017, 2018], we are able
to obtain insight on the strategy behind the optimal solution in continuous and
mixed-integer convex optimization problem as a function of key parameters that
affect the problem. In this way, optimization is not a black box anymore.
Instead, we redefine optimization as a multiclass classification problem where
the predictor gives insights on the logic behind the optimal solution. In other
words, OCTs and OCT-Hs give optimization a voice. We show on several realistic
examples that the accuracy behind our method is in the 90%-100% range, while
even when the predictions are not correct, the degree of suboptimality or
infeasibility is very low. We compare optimal strategy predictions of OCTs and
OCT-Hs and feedforward neural networks (NNs) and conclude that the performance
of OCT-Hs and NNs is comparable. OCTs are somewhat weaker but often
competitive. Therefore, our approach provides a novel insightful understanding
of optimal strategies to solve a broad class of continuous and mixed-integer
optimization problems
Bootstrap Robust Prescriptive Analytics
We address the problem of prescribing an optimal decision in a framework
where its cost depends on uncertain problem parameters that need to be
learned from data. Earlier work by Bertsimas and Kallus (2014) transforms
classical machine learning methods that merely predict from supervised
training data into prescriptive methods
taking optimal decisions specific to a particular covariate context .
Their prescriptive methods factor in additional observed contextual information
on a potentially large number of covariates to take context specific
actions which are superior to any static decision . Any naive
use of limited training data may, however, lead to gullible decisions
over-calibrated to one particular data set. In this paper, we borrow ideas from
distributionally robust optimization and the statistical bootstrap of Efron
(1982) to propose two novel prescriptive methods based on (nw) Nadaraya-Watson
and (nn) nearest-neighbors learning which safeguard against overfitting and
lead to improved out-of-sample performance. Both resulting robust prescriptive
methods reduce to tractable convex optimization problems and enjoy a limited
disappointment on bootstrap data. We illustrate the data-driven decision-making
framework and our novel robustness notion on a small news vendor problem as
well as a small portfolio allocation problem
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