Regret (decision theory)

In decision theory, on making decisions under uncertainty—should information about the best course of action arrive after taking a fixed decision—the human emotional response of regret is often experienced, and can be measured as the value of difference between a made decision and the optimal decision. The theory of regret aversion or anticipated regret proposes that when facing a decision, individuals might anticipate regret and thus incorporate in their choice their desire to eliminate or reduce this possibility. Regret is a negative emotion with a powerful social and reputational component, and is central to how humans learn from experience and to the human psychology of risk aversion. Conscious anticipation of regret creates a feedback loop that transcends regret from the emotional realm—often modeled as mere human behavior—into the realm of the rational choice behavior that is modeled in decision theory. Description Regret theory is a model in theoretical economics si

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This paper develops a methodology for regret minimization with stochastic first-order oracle feedback in online, constrained, non-smooth, non-convex problems. In this setting, the minimization of external regret is beyond reach for first-order methods, and there are no gradient-based algorithmic frameworks capable of providing a solution. On that account, we focus on a local regret measure defined via a proximal-gradient mapping, that also encompasses the original notion proposed by Hazan et al, (2017). To achieve no local regret in this setting, we develop a proximal-gradient method based on stochastic first-order feedback, and a simpler method for when access to a perfect first-order oracle is possible. Both methods are order-optimal (in the min-max sense), and we also establish a bound on the number of proximal-gradient queries these methods require. As an important application of our results, we also obtain a link between online and offline non-convex stochastic optimization manifested as a new proximal-gradient scheme with complexity guarantees matching those obtained via variance reduction techniques.

JMLR-JOURNAL MACHINE LEARNING RESEARCH2021

Inference about the number of contributors to a DNA mixture: Comparative analyses of a Bayesian network approach and the maximum allele count method

Kjell Konis

In the forensic examination of DNA mixtures, the question of how to set the total number of contributors (N) presents a topic of ongoing interest. Part of the discussion gravitates around issues of bias, in particular when assessments of the number of contributors are not made prior to considering the genotypic configuration of potential donors. Further complication may stem from the observation that, in some cases, there may be numbers of contributors that are incompatible with the set of alleles seen in the profile of a mixed crime stain, given the genotype of a potential contributor. In such situations, procedures that take a single and fixed number contributors as their output can lead to inferential impasses. Assessing the number of contributors within a probabilistic framework can help avoiding such complication. Using elements of decision theory, this paper analyses two strategies for inference on the number of contributors. One procedure is deterministic and focuses on the minimum number of contributors required to 'explain' an observed set of alleles. The other procedure is probabilistic using Bayes' theorem and provides a probability distribution for a set of numbers of contributors, based on the set of observed alleles as well as their respective rates of occurrence. The discussion concentrates on mixed stains of varying quality (i.e., different numbers of loci for which genotyping information is available). A so-called qualitative interpretation is pursued since quantitative information such as peak area and height data are not taken into account. The competing procedures are compared using a standard scoring rule that penalizes the degree of divergence between a given agreed value for N, that is the number of contributors, and the actual value taken by N. Using only modest assumptions and a discussion with reference to a casework example, this paper reports on analyses using simulation techniques and graphical models (i.e., Bayesian networks) to point out that setting the number of contributors to a mixed crime stain in probabilistic terms is, for the conditions assumed in this study, preferable to a decision policy that uses categoric assumptions about N. (C) 2012 Elsevier Ireland Ltd. All rights reserved.

Elsevier2012

Regret Minimization and Separation in Multi-Bidder Multi-Item Auctions

Cagil Kocyigit, Daniel Kuhn, Napat Rujeerapaiboon

We study a robust auction design problem with a minimax regret objective, where a seller seeks a mechanism for selling multiple items to multiple anonymous bidders with additive values. The seller knows that the bidders' values range over a box uncertainty set but has no information about their probability distribution. This auction design problem can be viewed as a zero-sum game between the seller, who chooses a mechanism, and a fictitious adversary or `nature,' who chooses the bidders' values from within the uncertainty set with the aim to maximize the seller's regret. We characterize the Nash equilibrium of this game analytically. The Nash strategy of the seller is a mechanism that sells each item via a separate auction akin to a second price auction with a random reserve price. The Nash strategy of nature is mixed and constitutes a probability distribution on the uncertainty set under which each bidder's values for the items are comonotonic.

2020

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