David Wolpert

Brendan Tracey, David Wolpert, Juan J. Alonso

Monte Carlo (MC) techniques are often used to estimate integrals of a multivariate function using randomly generated samples of the function. In light of the increasing interest in uncertainty quantification and robust design applications in aerospace engineering, the calculation of expected values of such functions (e.g. performance measures) becomes important. However, MC techniques often suffer from high variance and slow convergence as the number of samples increases. In this paper we present Stacked Monte Carlo (StackMC), a new method for post-processing an existing set of MC samples to improve the associated integral estimate. StackMC is based on the supervised learning techniques of fitting functions and cross validation. It should reduce the variance of any type of Monte Carlo integral estimate (simple sampling, importance sampling, quasi-Monte Carlo, MCMC, etc.) without adding bias. We report on an extensive set of experiments confirming that the StackMC estimate of an integral is more accurate than both the associated unprocessed Monte Carlo estimate and an estimate based on a functional fit to the MC samples. These experiments run over a wide variety of integration spaces, numbers of sample points, dimensions, and fitting functions. In particular, we apply StackMC in estimating the expected value of the fuel burn metric of future commercial aircraft and in estimating sonic boom loudness measures. We compare the efficiency of StackMC with that of more standard methods and show that for negligible additional computational cost significant increases in accuracy are gained.

Dev Rajnarayan, David Wolpert

In this paper, we examine the CE method in the broad context of Monte Carlo Optimization (MCO) and Parametric Learning (PL), a type of machine learning. A well-known overarching principle used to improve the performance of many PL algorithms is the bias-variance tradeoff. This tradeoff has been used to improve PL algorithms ranging from Monte Carlo estimation of integrals, to linear estimation, to general statistical estimation. Moreover, as described by, MCO is very closely related to PL. Owing to this similarity, the bias-variance tradeoff affects MCO performance, just as it does PL performance. In this article, we exploit the bias-variance tradeoff to enhance the performance of MCO algorithms. We use the technique of cross-validation, a technique based on the bias-variance tradeoff, to significantly improve the performance of the Cross Entropy (CE) method, which is an MCO algorithm. In previous work we have confirmed that other PL techniques improve the perfomance of other MCO algorithms. We conclude that the many techniques pioneered in PL could be investigated as ways to improve MCO algorithms in general, and the CE method in particular.

Kunal Menda, Yi-Chun Chen, Justin Grana et al.

The incorporation of macro-actions (temporally extended actions) into multi-agent decision problems has the potential to address the curse of dimensionality associated with such decision problems. Since macro-actions last for stochastic durations, multiple agents executing decentralized policies in cooperative environments must act asynchronously. We present an algorithm that modifies generalized advantage estimation for temporally extended actions, allowing a state-of-the-art policy optimization algorithm to optimize policies in Dec-POMDPs in which agents act asynchronously. We show that our algorithm is capable of learning optimal policies in two cooperative domains, one involving real-time bus holding control and one involving wildfire fighting with unmanned aircraft. Our algorithm works by framing problems as "event-driven decision processes," which are scenarios in which the sequence and timing of actions and events are random and governed by an underlying stochastic process. In addition to optimizing policies with continuous state and action spaces, our algorithm also facilitates the use of event-driven simulators, which do not require time to be discretized into time-steps. We demonstrate the benefit of using event-driven simulation in the context of multiple agents taking asynchronous actions. We show that fixed time-step simulation risks obfuscating the sequence in which closely separated events occur, adversely affecting the policies learned. In addition, we show that arbitrarily shrinking the time-step scales poorly with the number of agents.

Justin Grana, David Wolpert, Joshua Neil et al.

The rapid detection of attackers within firewalls of enterprise computer net- works is of paramount importance. Anomaly detectors address this problem by quantifying deviations from baseline statistical models of normal network behav- ior and signaling an intrusion when the observed data deviates significantly from the baseline model. However, many anomaly detectors do not take into account plausible attacker behavior. As a result, anomaly detectors are prone to a large number of false positives due to unusual but benign activity. This paper first in- troduces a stochastic model of attacker behavior which is motivated by real world attacker traversal. Then, we develop a likelihood ratio detector that compares the probability of observed network behavior under normal conditions against the case when an attacker has possibly compromised a subset of hosts within the network. Since the likelihood ratio detector requires integrating over the time each host be- comes compromised, we illustrate how to use Monte Carlo methods to compute the requisite integral. We then present Receiver Operating Characteristic (ROC) curves for various network parameterizations that show for any rate of true posi- tives, the rate of false positives for the likelihood ratio detector is no higher than that of a simple anomaly detector and is often lower. We conclude by demon- strating the superiority of the proposed likelihood ratio detector when the network topologies and parameterizations are extracted from real-world networks.

Scott Backhaus, Russell Bent, James Bono et al.

Recent years have seen increased interest in the design and deployment of smart grid devices and control algorithms. Each of these smart communicating devices represents a potential access point for an intruder spurring research into intruder prevention and detection. However, no security measures are complete, and intruding attackers will compromise smart grid devices leading to the attacker and the system operator interacting via the grid and its control systems. The outcome of these machine-mediated human-human interactions will depend on the design of the physical and control systems mediating the interactions. If these outcomes can be predicted via simulation, they can be used as a tool for designing attack-resilient grids and control systems. However, accurate predictions require good models of not just the physical and control systems, but also of the human decision making. In this manuscript, we present an approach to develop such tools, i.e. models of the decisions of the cyber-physical intruder who is attacking the systems and the system operator who is defending it, and demonstrate its usefulness for design.