Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead

Stackelberg equilibria have become increasingly important as a solution concept in computational game theory, largely inspired by practical problems such as security settings. In practice, however, there is typically uncertainty regarding the model about the opponent. This paper is, to our knowledge, the first to investigate Stackelberg equilibria under uncertainty in extensive-form games, one of the broadest classes of game. We introduce robust Stackelberg equilibria, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new mixed-integer program for the deterministic limited-lookahead setting. We then extend the program to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a mixed-integer program that is of the same asymptotic size as that for the deterministic setting.Comment: Published at AAAI1

Truncated Variance Reduction: A Unified Approach to Bayesian Optimization and Level-Set Estimation

We present a new algorithm, truncated variance reduction (TruVaR), that treats Bayesian optimization (BO) and level-set estimation (LSE) with Gaussian processes in a unified fashion. The algorithm greedily shrinks a sum of truncated variances within a set of potential maximizers (BO) or unclassified points (LSE), which is updated based on confidence bounds. TruVaR is effective in several important settings that are typically non-trivial to incorporate into myopic algorithms, including pointwise costs and heteroscedastic noise. We provide a general theoretical guarantee for TruVaR covering these aspects, and use it to recover and strengthen existing results on BO and LSE. Moreover, we provide a new result for a setting where one can select from a number of noise levels having associated costs. We demonstrate the effectiveness of the algorithm on both synthetic and real-world data sets.Comment: Accepted to NIPS 201

Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead

Stackelberg equilibria have become increasingly important as a solution concept in computational game theory, largely inspired by practical problems such as security settings. In practice, however, there is typically uncertainty regarding the model about the opponent. This paper is, to our knowledge, the first to investigate Stackelberg equilibria under uncertainty in extensive-form games, one of the broadest classes of game. We introduce robust Stackelberg equilibria, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new mixed-integer program for the deterministic limited-lookahead setting. We then extend the program to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a mixed-integer program that is of the same asymptotic size as that for the deterministic setting.Comment: Published at AAAI1

Sequential design of computer experiments for the estimation of a probability of failure

This paper deals with the problem of estimating the volume of the excursion set of a function $f:\mathbb{R}^d \to \mathbb{R}$ above a given threshold, under a probability measure on $\mathbb{R}^d$ that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is available, the budget for simulations is usually severely limited and therefore classical Monte Carlo methods ought to be avoided. One of the main contributions of this article is to derive SUR (stepwise uncertainty reduction) strategies from a Bayesian-theoretic formulation of the problem of estimating a probability of failure. These sequential strategies use a Gaussian process model of $f$ and aim at performing evaluations of $f$ as efficiently as possible to infer the value of the probability of failure. We compare these strategies to other strategies also based on a Gaussian process model for estimating a probability of failure.Comment: This is an author-generated postprint version. The published version is available at http://www.springerlink.co

Replication or exploration? Sequential design for stochastic simulation experiments

We investigate the merits of replication, and provide methods for optimal design (including replicates), with the goal of obtaining globally accurate emulation of noisy computer simulation experiments. We first show that replication can be beneficial from both design and computational perspectives, in the context of Gaussian process surrogate modeling. We then develop a lookahead based sequential design scheme that can determine if a new run should be at an existing input location (i.e., replicate) or at a new one (explore). When paired with a newly developed heteroskedastic Gaussian process model, our dynamic design scheme facilitates learning of signal and noise relationships which can vary throughout the input space. We show that it does so efficiently, on both computational and statistical grounds. In addition to illustrative synthetic examples, we demonstrate performance on two challenging real-data simulation experiments, from inventory management and epidemiology.Comment: 34 pages, 9 figure

Non-Myopic Multifidelity Bayesian Optimization

Bayesian optimization is a popular framework for the optimization of black box functions. Multifidelity methods allows to accelerate Bayesian optimization by exploiting low-fidelity representations of expensive objective functions. Popular multifidelity Bayesian strategies rely on sampling policies that account for the immediate reward obtained evaluating the objective function at a specific input, precluding greater informative gains that might be obtained looking ahead more steps. This paper proposes a non-myopic multifidelity Bayesian framework to grasp the long-term reward from future steps of the optimization. Our computational strategy comes with a two-step lookahead multifidelity acquisition function that maximizes the cumulative reward obtained measuring the improvement in the solution over two steps ahead. We demonstrate that the proposed algorithm outperforms a standard multifidelity Bayesian framework on popular benchmark optimization problems