492 research outputs found
Incorporating statistical model error into the calculation of acceptability prices of contingent claims
The determination of acceptability prices of contingent claims requires the
choice of a stochastic model for the underlying asset price dynamics. Given
this model, optimal bid and ask prices can be found by stochastic optimization.
However, the model for the underlying asset price process is typically based on
data and found by a statistical estimation procedure. We define a confidence
set of possible estimated models by a nonparametric neighborhood of a baseline
model. This neighborhood serves as ambiguity set for a multi-stage stochastic
optimization problem under model uncertainty. We obtain distributionally robust
solutions of the acceptability pricing problem and derive the dual problem
formulation. Moreover, we prove a general large deviations result for the
nested distance, which allows to relate the bid and ask prices under model
ambiguity to the quality of the observed data.Comment: 27 pages, 2 figure
Complexity-Free Generalization via Distributionally Robust Optimization
Established approaches to obtain generalization bounds in data-driven
optimization and machine learning mostly build on solutions from empirical risk
minimization (ERM), which depend crucially on the functional complexity of the
hypothesis class. In this paper, we present an alternate route to obtain these
bounds on the solution from distributionally robust optimization (DRO), a
recent data-driven optimization framework based on worst-case analysis and the
notion of ambiguity set to capture statistical uncertainty. In contrast to the
hypothesis class complexity in ERM, our DRO bounds depend on the ambiguity set
geometry and its compatibility with the true loss function. Notably, when using
maximum mean discrepancy as a DRO distance metric, our analysis implies, to the
best of our knowledge, the first generalization bound in the literature that
depends solely on the true loss function, entirely free of any complexity
measures or bounds on the hypothesis class
Predict-then-Calibrate: A New Perspective of Robust Contextual LP
Contextual optimization, also known as predict-then-optimize or prescriptive
analytics, considers an optimization problem with the presence of covariates
(context or side information). The goal is to learn a prediction model (from
the training data) that predicts the objective function from the covariates,
and then in the test phase, solve the optimization problem with the covariates
but without the observation of the objective function. In this paper, we
consider a risk-sensitive version of the problem and propose a generic
algorithm design paradigm called predict-then-calibrate. The idea is to first
develop a prediction model without concern for the downstream risk profile or
robustness guarantee, and then utilize calibration (or recalibration) methods
to quantify the uncertainty of the prediction. While the existing methods
suffer from either a restricted choice of the prediction model or strong
assumptions on the underlying data, we show the disentangling of the prediction
model and the calibration/uncertainty quantification has several advantages.
First, it imposes no restriction on the prediction model and thus fully
unleashes the potential of off-the-shelf machine learning methods. Second, the
derivation of the risk and robustness guarantee can be made independent of the
choice of the prediction model through a data-splitting idea. Third, our
paradigm of predict-then-calibrate applies to both (risk-sensitive) robust and
(risk-neutral) distributionally robust optimization (DRO) formulations.
Theoretically, it gives new generalization bounds for the contextual LP problem
and sheds light on the existing results of DRO for contextual LP. Numerical
experiments further reinforce the advantage of the predict-then-calibrate
paradigm in that an improvement on either the prediction model or the
calibration model will lead to a better final performance.Comment: 30 pages, 8 figure
Distributionally Robust Optimization: A Review
The concepts of risk-aversion, chance-constrained optimization, and robust
optimization have developed significantly over the last decade. Statistical
learning community has also witnessed a rapid theoretical and applied growth by
relying on these concepts. A modeling framework, called distributionally robust
optimization (DRO), has recently received significant attention in both the
operations research and statistical learning communities. This paper surveys
main concepts and contributions to DRO, and its relationships with robust
optimization, risk-aversion, chance-constrained optimization, and function
regularization
A Survey of Contextual Optimization Methods for Decision Making under Uncertainty
Recently there has been a surge of interest in operations research (OR) and
the machine learning (ML) community in combining prediction algorithms and
optimization techniques to solve decision-making problems in the face of
uncertainty. This gave rise to the field of contextual optimization, under
which data-driven procedures are developed to prescribe actions to the
decision-maker that make the best use of the most recently updated information.
A large variety of models and methods have been presented in both OR and ML
literature under a variety of names, including data-driven optimization,
prescriptive optimization, predictive stochastic programming, policy
optimization, (smart) predict/estimate-then-optimize, decision-focused
learning, (task-based) end-to-end learning/forecasting/optimization, etc.
Focusing on single and two-stage stochastic programming problems, this review
article identifies three main frameworks for learning policies from data and
discusses their strengths and limitations. We present the existing models and
methods under a uniform notation and terminology and classify them according to
the three main frameworks identified. Our objective with this survey is to both
strengthen the general understanding of this active field of research and
stimulate further theoretical and algorithmic advancements in integrating ML
and stochastic programming
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