100 research outputs found
Making Robust Decisions in Discrete Optimization Problems as a Game against Nature
In this paper a discrete optimization problem under uncertainty is discussed. Solving such a problem can be seen as a game against nature. In order to choose a solution, the minmax and minmax regret criteria can be applied. In this paper an extension of the known minmax (regret) approach is proposed. It is shown how different types of uncertainty can be simultaneously taken into account. Some exact and approximation algorithms for choosing a best solution are constructed.Discrete optimization, minmax, minmax regret, game against nature
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
Algorithmic Processes And Social Values
In this thesis, we study several problems at the interface of algorithmic decision-making and society, focusing on the tensions that arise between these processes and social values like fairness and privacy. In the first chapter, we examine the design of financial portfolios which adequately serve all segments of the population. In the second, we examine an allocation setting where the allocator wishes to distribute a scarce resource across many groups fairly, but does not know ahead of time which groups have a need for the resource. In the third, we study a game-theoretic model of information aggregation and the effects of individuals acting to preserve the privacy of their personal beliefs on the collective welfare of the population. Finally, we look at some of the issues that arise from the desire to apply automated techniques to problems in redistricting, including fundamental flaws in the definitions and frameworks typically used
Probably Approximately Correct Nash Equilibrium Learning
We consider a multi-agent noncooperative game with agents' objective
functions being affected by uncertainty. Following a data driven paradigm, we
represent uncertainty by means of scenarios and seek a robust Nash equilibrium
solution. We treat the Nash equilibrium computation problem within the realm of
probably approximately correct (PAC) learning. Building upon recent
developments in scenario-based optimization, we accompany the computed Nash
equilibrium with a priori and a posteriori probabilistic robustness
certificates, providing confidence that the computed equilibrium remains
unaffected (in probabilistic terms) when a new uncertainty realization is
encountered. For a wide class of games, we also show that the computation of
the so called compression set - a key concept in scenario-based optimization -
can be directly obtained as a byproduct of the proposed solution methodology.
Finally, we illustrate how to overcome differentiability issues, arising due to
the introduction of scenarios, and compute a Nash equilibrium solution in a
decentralized manner. We demonstrate the efficacy of the proposed approach on
an electric vehicle charging control problem.Comment: Preprint submitted to IEEE Transactions on Automatic Contro
A Unifying Perspective on Multi-Calibration: Game Dynamics for Multi-Objective Learning
We provide a unifying framework for the design and analysis of
multicalibrated predictors. By placing the multicalibration problem in the
general setting of multi-objective learning -- where learning guarantees must
hold simultaneously over a set of distributions and loss functions -- we
exploit connections to game dynamics to achieve state-of-the-art guarantees for
a diverse set of multicalibration learning problems. In addition to shedding
light on existing multicalibration guarantees and greatly simplifying their
analysis, our approach also yields improved guarantees, such as obtaining
stronger multicalibration conditions that scale with the square-root of group
size and improving the complexity of -class multicalibration by an
exponential factor of . Beyond multicalibration, we use these game dynamics
to address emerging considerations in the study of group fairness and
multi-distribution learning.Comment: 45 pages. Authors are ordered alphabeticall
Persuasion, Political Warfare, and Deterrence: Behavioral and Behaviorally Robust Models
This dissertation examines game theory models in the context of persuasion and competition wherein decision-makers are not completely rational by considering two complementary threads of research. The first thread of research pertains to offensive and preemptively defensive behavioral models. Research in this thread makes three notable contributions. First, an offensive modeling framework is created to identify how an entity optimally influences a populace to take a desired course of action. Second, a defensive modeling framework is defined wherein a regulating entity takes action to bound the behavior of multiple adversaries simultaneously attempting to persuade a group of decision-makers. Third, an offensive influence modeling framework under conditions of ambiguity is developed in accordance with historical information limitations, and we demonstrate how it can be used to select a robust course of action on a specific, data-driven use case. The second thread of research pertains to behavioral and behaviorally robust approaches to deterrence. Research in this thread makes two notable contributions. First, we demonstrate the alternative insights behavioral game theory generates for the analysis of classic deterrence games, and explicate the rich analysis generated from its combined use with standard equilibrium models. Second, we define behaviorally robust models for an agent to use in a normal form game under varying forms of uncertainty in order to inform deterrence policy decisions
Robust Restless Bandits: Tackling Interval Uncertainty with Deep Reinforcement Learning
We introduce Robust Restless Bandits, a challenging generalization of
restless multi-arm bandits (RMAB). RMABs have been widely studied for
intervention planning with limited resources. However, most works make the
unrealistic assumption that the transition dynamics are known perfectly,
restricting the applicability of existing methods to real-world scenarios. To
make RMABs more useful in settings with uncertain dynamics: (i) We introduce
the Robust RMAB problem and develop solutions for a minimax regret objective
when transitions are given by interval uncertainties; (ii) We develop a double
oracle algorithm for solving Robust RMABs and demonstrate its effectiveness on
three experimental domains; (iii) To enable our double oracle approach, we
introduce RMABPPO, a novel deep reinforcement learning algorithm for solving
RMABs. RMABPPO hinges on learning an auxiliary "-network" that allows
each arm's learning to decouple, greatly reducing sample complexity required
for training; (iv) Under minimax regret, the adversary in the double oracle
approach is notoriously difficult to implement due to non-stationarity. To
address this, we formulate the adversary oracle as a multi-agent reinforcement
learning problem and solve it with a multi-agent extension of RMABPPO, which
may be of independent interest as the first known algorithm for this setting.
Code is available at https://github.com/killian-34/RobustRMAB.Comment: 18 pages, 3 figure
- …