Abstract:The cross-entropy method is a recent versatile Monte Carlo technique. This article provides a brief introduction to the cross-entropy method and discusses how it can be used for rare-event probability estimation and for solving combinatorial, continuous, constrained and noisy optimization problems. A comprehensive list of references on cross-entropy methods and applications is included.Keywords: cross-entropy, Kullback-Leibler divergence, rare events, importance sampling, stochastic search.The cross-entropy (CE) method is a recent generic Monte Carlo technique for solving complicated simulation and optimization problems. The approach was introduced by R.Y. Rubinstein in [41,42], extending his earlier work on variance minimization methods for rare-event probability estimation [40].The CE method can be applied to two types of problem:, where X is a random variable or vector taking values in some set X and H is function on X . An important special case is the estimation of a probability = P(S(X) γ), where S is another function on X .2. Optimization: Optimize (that is, maximize or minimize) S(x) over all x ∈ X , where S is some objective function on X . S can be either a known or a noisy function. In the latter case the objective function needs to be estimated, e.g., via simulation.In the estimation setting, the CE method can be viewed as an adaptive importance sampling procedure that uses the cross-entropy or Kullback-Leibler divergence as a measure of closeness between two sampling distributions, as is explained further in Section 1. In the optimization setting, the optimization problem is first translated into a rare-event estimation problem, and then the CE method for estimation is used as an adaptive algorithm to locate the optimum, as is explained further in Section 2.An easy tutorial on the CE method is given in [15]. A more comprehensive treatment can be found in [45]; see also [46, Chapter 8]. The CE method homepage can be found at www.cemethod.org .The CE method has been successfully applied to a diverse range of estimation and optimization problems, including buffer allocation [1], queueing models of telecommunication systems [14,16], optimal control of HIV/AIDS spread [48,49], signal detection [30], combinatorial auctions [9], DNA sequence alignment [24,38], scheduling and vehicle routing [3,8,11,20,23,53], neural and reinforcement learning [31,32,34,52,54], project management [12], rare-event simulation with light-and heavy-tail distributions [2,10,21,28], clustering analysis [4,5,29]. Applications to classical combinatorial optimization problems including the max-cut, traveling salesman, and Hamiltonian cycle 1

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We derive generalization bounds for learning algorithms based on their robustness: the property that if a testing sample is "similar" to a training sample, then the testing error is close to the training error. This provides a novel approach, different from complexity or stability arguments, to study generalization of learning algorithms. One advantage of the robustness approach, compared to previous methods, is the geometric intuition it conveys. Consequently, robustness-based analysis is easy to extend to learning in non-standard setups such as Markovian samples or quantile loss. We further show that a weak notion of robustness is both sufficient and necessary for generalizability, which implies that robustness is a fundamental property that is required for learning algorithms to work.

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We consider support vector machines for binary classification. As opposed to most approaches we use the number of support vectors (the "L 0 norm") as a regularizing term instead of the L 1 or L 2 norms. In order to solve the optimization problem we use the cross entropy method to search over the possible sets of support vectors. The algorithm consists of solving a sequence of efficient linear programs. We report experiments where our method produces generalization errors that are similar to support vector machines, while using a considerably smaller number of support vectors.

BackgroundRegular physical activity is known to be beneficial for people with type 2 diabetes. Nevertheless, most of the people who have diabetes lead a sedentary lifestyle. Smartphones create new possibilities for helping people to adhere to their physical activity goals through continuous monitoring and communication, coupled with personalized feedback.ObjectiveThe aim of this study was to help type 2 diabetes patients increase the level of their physical activity.MethodsWe provided 27 sedentary type 2 diabetes patients with a smartphone-based pedometer and a personal plan for physical activity. Patients were sent short message service messages to encourage physical activity between once a day and once per week. Messages were personalized through a Reinforcement Learning algorithm so as to improve each participant’s compliance with the activity regimen. The algorithm was compared with a static policy for sending messages and weekly reminders.ResultsOur results show that participants who received messages generated by the learning algorithm increased the amount of activity and pace of walking, whereas the control group patients did not. Patients assigned to the learning algorithm group experienced a superior reduction in blood glucose levels (glycated hemoglobin [HbA1c]) compared with control policies, and longer participation caused greater reductions in blood glucose levels. The learning algorithm improved gradually in predicting which messages would lead participants to exercise.ConclusionsMobile phone apps coupled with a learning algorithm can improve adherence to exercise in diabetic patients. This algorithm can be used in large populations of diabetic patients to improve health and glycemic control. Our results can be expanded to other areas where computer-led health coaching of humans may have a positive impact. Summary of a part of this manuscript has been previously published as a letter in Diabetes Care, 2016.