Monotone Approximation of Decision Problems

Chehrazi, Naveed; Weber, Thomas A.

doi:10.2139/ssrn.1324024

Cited by 4 publications

(3 citation statements)

References 39 publications

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“…Several methods for modeling ambiguity sets have been proposed relying on discrete distributions [14,15], moment constraints [2,16,17], Kullback-Leibler divergence [18,19], Prohorov metric [20], and the Wasserstein distance [21,22,1], among others. Of particular interest in distributionally-robust optimization is the family of data-driven distributionally-robust optimization, where the ambiguity set is parameterized based on samples of the distribution [23,20,1,13]. These studies have proved to be of extreme importance in machine learning [24,25,3,26,27].…”

Section: Related Workmentioning

confidence: 99%

Online Stochastic Convex Optimization: Wasserstein Distance Variation

Shames,

Farokhi

2020

Preprint

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Distributionally-robust optimization is often studied for a fixed set of distributions rather than timevarying distributions that can drift significantly over time (which is, for instance, the case in finance and sociology due to underlying expansion of economy and evolution of demographics). This motivates understanding conditions on probability distributions, using the Wasserstein distance, that can be used to model time-varying environments. We can then use these conditions in conjunction with online stochastic optimization to adapt the decisions. We considers an online proximal-gradient method to track the minimizers of expectations of smooth convex functions parameterised by a random variable whose probability distributions continuously evolve over time at a rate similar to that of the rate at which the decision maker acts. We revisit the concepts of estimation and tracking error inspired by systems and control literature and provide bounds for them under strong convexity, Lipschitzness of the gradient, and bounds on the probability distribution drift characterised by the Wasserstein distance. Further, noting that computing projections for a general feasible sets might not be amenable to online implementation (due to computational constraints), we propose an exact penalty method. Doing so allows us to relax the uniform boundedness of the gradient and establish dynamic regret bounds for tracking and estimation error. We further introduce a constraint-tightening approach and relate the amount of tightening to the probability of satisfying the constraints.

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Section: Related Workmentioning

confidence: 99%

Online Stochastic Convex Optimization: Wasserstein Distance Variation

Shames,

Farokhi

2020

Preprint

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“…The additional diffusion term tends to augment the differentiability of the value function, which, in turn, simplifies optimality proofs for threshold-type policies. 4 The possibility of early settlement offers is subject to future research; see Chehrazi and Weber (2010) for a (robust) static approach. 5 The chance of recovering an outstanding balance in the long run increases in λ ∞ .…”

Section: Appendix D Implementation Detailsmentioning

confidence: 99%

Dynamic Credit-Collections Optimization

Chehrazi

Weber

2019

Management Science

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Based on a dynamic model of the stochastic repayment behavior exhibited by delinquent credit-card accounts in the form of a self-exciting point process, a bank can control the arrival intensity of repayments using costly account-treatment actions. A semianalytic solution to the corresponding stochastic optimal control problem is obtained using a recursive approach. For a linear cost of treatment effort, the optimal policy in the two-dimensional (intensity, balance) space is described by the frontier of a convex action region. The unique optimal policy significantly reduces a bank's loss given default and concentrates the collection effort onto the best possible actions at the best possible times so as to minimize the sum of the expected discounted outstanding balance and the discounted cost of the collection effort, thus maximizing the net value of any given delinquent credit-card account.

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“…Another common application can be found in disease screening, where the probability of disease is assumed non-decreasing with increasing measurements of a pertinent biomarker [25,2]. Or in economics, the demand and supply curve is in general assumed to be monotone [10]. Motivated by this large variety of applications, a panoply of statistical approaches has been developed for estimating monotone curves, i.e., onedimensional monotone functions.…”

Section: Introductionmentioning

confidence: 99%

A projection approach for multiple monotone regression

Lin,

Thomas,

Piegorsch

et al. 2019

Preprint

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Shape-constrained inference has wide applicability in bioassay, medicine, economics, risk assessment, and many other fields. Although there has been a large amount of work on monotone-constrained univariate curve estimation, multivariate shape-constrained problems are much more challenging, and fewer advances have been made in this direction. With a focus on monotone regression with multiple predictors, this current work proposes a projection approach to estimate a multiple monotone regression function. An initial unconstrained estimator -such as a local polynomial estimator or spline estimator -is first obtained, which is then projected onto the shape-constrained space. A shape-constrained estimate is obtained by sequentially projecting an (adjusted) initial estimator along each univariate direction. Compared to the initial unconstrained estimator, the projection estimate results in a reduction of estimation error in terms of both L p (p ≥ 1) distance and supremum distance. We also derive the asymptotic distribution of the projection estimate. Simple computational algorithms are available for implementing the projection in both the unidimensional and higher dimensional cases. Our work provides a simple recipe for practitioners to use in real applications, and is illustrated with a joint-action example from environmental toxicology.

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Monotone Approximation of Decision Problems

Cited by 4 publications

References 39 publications

Online Stochastic Convex Optimization: Wasserstein Distance Variation

Online Stochastic Convex Optimization: Wasserstein Distance Variation

Dynamic Credit-Collections Optimization

A projection approach for multiple monotone regression

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