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Register a new userWorst-Case Risk Quantification under Distributional Ambiguity using Kernel Mean Embedding in Moment Problem
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Added by
Jia-Jie Zhu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In order to anticipate rare and impactful events, we propose to quantify the worst-case risk under distributional ambiguity using a recent development in kernel methods -- the kernel mean embedding. Specifically, we formulate the generalized moment problem whose ambiguity set (i.e., the moment constraint) is described by constraints in the associated reproducing kernel Hilbert space in a nonparametric manner. We then present the tractable approximation and its theoretical justification. As a concrete application, we numerically test the proposed method in characterizing the worst-case constraint violation probability in the context of a constrained stochastic control system.
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This paper studies distributionally robust optimization (DRO) when the ambiguity set is given by moments for the distributions. The objective and constraints are given by polynomials in decision variables. We reformulate the DRO with equivalent moment conic constraints. Under some general assumptions, we prove the DRO is equivalent to a linear optimization problem with moment and psd polynomial cones. A moment-SOS relaxation method is proposed to solve it. Its asymptotic and finite convergence are shown under certain assumptions. Numerical examples are presented to show how to solve DRO problems.
This paper introduces for the first time a framework to obtain provable worst-case guarantees for neural network performance, using learning for optimal power flow (OPF) problems as a guiding example. Neural networks have the potential to substantially reduce the computing time of OPF solutions. However, the lack of guarantees for their worst-case performance remains a major barrier for their adoption in practice. This work aims to remove this barrier. We formulate mixed-integer linear programs to obtain worst-case guarantees for neural network predictions related to (i) maximum constraint violations, (ii) maximum distances between predicted and optimal decision variables, and (iii) maximum sub-optimality. We demonstrate our methods on a range of PGLib-OPF networks up to 300 buses. We show that the worst-case guarantees can be up to one order of magnitude larger than the empirical lower bounds calculated with conventional methods. More importantly, we show that the worst-case predictions appear at the boundaries of the training input domain, and we demonstrate how we can systematically reduce the worst-case guarantees by training on a larger input domain than the domain they are evaluated on.
We propose to analyse the conditional distributional treatment effect (CoDiTE), which, in contrast to the more common conditional average treatment effect (CATE), is designed to encode a treatments distributional aspects beyond the mean. We first introduce a formal definition of the CoDiTE associated with a distance function between probability measures. Then we discuss the CoDiTE associated with the maximum mean discrepancy via kernel conditional mean embeddings, which, coupled with a hypothesis test, tells us whether there is any conditional distributional effect of the treatment. Finally, we investigate what kind of conditional distributional effect the treatment has, both in an exploratory manner via the conditional witness function, and in a quantitative manner via U-statistic regression, generalising the CATE to higher-order moments. Experiments on synthetic, semi-synthetic and real datasets demonstrate the merits of our approach.
This work provides analysis of a variant of the Risk-Sharing Principal-Agent problem in a single period setting with additional constant lower and upper bounds on the wage paid to the Agent. First the effect of the extra constraints on optimal contract existence is analyzed and leads to conditions on utilities under which an optimum may be attained. Solution characterization is then provided along with the derivation of a Borch rule for Limited Liability. Finally the CARA utility case is considered and a closed form optimal wage and action are obtained. This allows for analysis of the classical CARA utility and gaussian setting.
Flow routing over inter-datacenter networks is a well-known problem where the network assigns a path to a newly arriving flow potentially according to the network conditions and the properties of the new flow. An essential system-wide performance metric for a routing algorithm is the flow completion times, which affect the performance of applications running across multiple datacenters. Current static and dynamic routing approaches do not take advantage of flow size information in routing, which is practical in a controlled environment such as inter-datacenter networks that are managed by the datacenter operators. In this paper, we discuss Best Worst-case Routing (BWR), which aims at optimizing the tail completion times of long-running flows over inter-datacenter networks with non-uniform link capacities. Since finding the path with the best worst-case completion time for a new flow is NP-Hard, we investigate two heuristics, BWRH and BWRHF, which use two different upper bounds on the worst-case completion times for routing. We evaluate BWRH and BWRHF against several real WAN topologies and multiple traffic patterns. Although BWRH better models the BWR problem, BWRH and BWRHF show negligible difference across various system-wide performance metrics, while BWRHF being significantly faster. Furthermore, we show that compared to other popular routing heuristics, BWRHF can reduce the mean and tail flow completion times by over $1.5times$ and $2times$, respectively.
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