arXiv subjects

Henry Lam

Publications and source records attributed to Henry Lam.

103 records · Page 6Linked to original sources

arXiv2016

The Empirical Likelihood Approach to Quantifying Uncertainty in Sample Average Approximation

We study the empirical likelihood approach to construct confidence intervals for the optimal value and the optimality gap of a given solution, henceforth quantify the statistical uncertainty of sample average approximation, for optimization problems with expected value objectives and constraints where the underlying probability distributions are observed via limited data. This approach relies on two distributionally robust optimization problems posited over the uncertain distribution, with a divergence-based uncertainty set that is suitably calibrated to provide asymptotic statistical guarantees.

stat.ME↗

arXiv2016

Sensitivity to Serial Dependency of Input Processes: A Robust Approach

Procedures in assessing the impact of serial dependency on performance analysis are usually built on parametrically specified models. In this paper, we propose a robust, nonparametric approach to carry out this assessment, by computing the worst-case deviation of the performance measure due to arbitrary dependence. The approach is based on optimizations, posited on the model space, that have constraints specifying the level of dependency measured by a nonparametric distance to some nominal i.i.d. input model. We study approximation methods for these optimizations via simulation and analysis-of-variance (ANOVA). Numerical experiments demonstrate how the proposed approach can discover the hidden impacts of dependency beyond those revealed by conventional parametric modeling and correlation studies.

stat.ME↗

arXiv2016

Accelerated Evaluation of Automated Vehicles Safety in Lane Change Scenarios Based on Importance Sampling Techniques

Automated vehicles (AVs) must be evaluated thoroughly before their release and deployment. A widely-used evaluation approach is the Naturalistic-Field Operational Test (N-FOT), which tests prototype vehicles directly on the public roads. Due to the low exposure to safety-critical scenarios, N-FOTs are time-consuming and expensive to conduct. In this paper, we propose an accelerated evaluation approach for AVs. The results can be used to generate motions of the primary other vehicles to accelerate the verification of AVs in simulations and controlled experiments. Frontal collision due to unsafe cut-ins is the target crash type of this paper. Human-controlled vehicles making unsafe lane changes are modeled as the primary disturbance to AVs based on data collected by the University of Michigan Safety Pilot Model Deployment Program. The cut-in scenarios are generated based on skewed statistics of collected human driver behaviors, which generate risky testing scenarios while preserving the statistical information so that the safety benefits of AVs in non-accelerated cases can be accurately estimated. The Cross Entropy method is used to recursively search for the optimal skewing parameters. The frequencies of occurrence of conflicts, crashes and injuries are estimated for a modeled automated vehicle, and the achieved accelerated rate is around 2,000 to 20,000. In other words, in the accelerated simulations, driving for 1,000 miles will expose the AV with challenging scenarios that will take about 2 to 20 million miles of real-world driving to encounter. This technique thus has the potential to reduce greatly the development and validation time for AVs.

cs.RO↗

arXiv2016

Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust Optimization

We investigate the use of distributionally robust optimization (DRO) as a tractable tool to recover the asymptotic statistical guarantees provided by the Central Limit Theorem, for maintaining the feasibility of an expected value constraint under ambiguous probability distributions. We show that using empirically defined Burg-entropy divergence balls to construct the DRO can attain such guarantees. These balls, however, are not reasoned from the standard data-driven DRO framework since by themselves they can have low or even zero probability of covering the true distribution. Rather, their superior statistical performances are endowed by linking the resulting DRO with empirical likelihood and empirical processes. We show that the sizes of these balls can be optimally calibrated using chi-square process excursion. We conduct numerical experiments to support our theoretical findings.

math.OC↗

arXiv2015

Robust Sensitivity Analysis for Stochastic Systems

We study a worst-case approach to measure the sensitivity to model misspecification in the performance analysis of stochastic systems. The situation of interest is when only minimal parametric information is available on the form of the true model. Under this setting, we post optimization programs that compute the worst-case performance measures, subject to constraints on the amount of model misspecification measured by Kullback-Leibler (KL) divergence. Our main contribution is the development of infinitesimal approximations for these programs, resulting in asymptotic expansions of their optimal values in terms of the divergence. The coefficients of these expansions can be computed via simulation, and are mathematically derived from the representation of the worst-case models as changes of measure that satisfy a well-defined class of functional fixed point equations.

math.PR↗

arXiv2015

A Bayesian Approach for Online Classifier Ensemble

We propose a Bayesian approach for recursively estimating the classifier weights in online learning of a classifier ensemble. In contrast with past methods, such as stochastic gradient descent or online boosting, our approach estimates the weights by recursively updating its posterior distribution. For a specified class of loss functions, we show that it is possible to formulate a suitably defined likelihood function and hence use the posterior distribution as an approximation to the global empirical loss minimizer. If the stream of training data is sampled from a stationary process, we can also show that our approach admits a superior rate of convergence to the expected loss minimizer than is possible with standard stochastic gradient descent. In experiments with real-world datasets, our formulation often performs better than state-of-the-art stochastic gradient descent and online boosting algorithms.

cs.LG↗

arXiv2014

From Black-Scholes to Online Learning: Dynamic Hedging under Adversarial Environments

We consider a non-stochastic online learning approach to price financial options by modeling the market dynamic as a repeated game between the nature (adversary) and the investor. We demonstrate that such framework yields analogous structure as the Black-Scholes model, the widely popular option pricing model in stochastic finance, for both European and American options with convex payoffs. In the case of non-convex options, we construct approximate pricing algorithms, and demonstrate that their efficiency can be analyzed through the introduction of an artificial probability measure, in parallel to the so-called risk-neutral measure in the finance literature, even though our framework is completely adversarial. Continuous-time convergence results and extensions to incorporate price jumps are also presented.

cs.DS↗

arXiv2013

Learning about social learning in MOOCs: From statistical analysis to generative model

We study user behavior in the courses offered by a major Massive Online Open Course (MOOC) provider during the summer of 2013. Since social learning is a key element of scalable education in MOOCs and is done via online discussion forums, our main focus is in understanding forum activities. Two salient features of MOOC forum activities drive our research: 1. High decline rate: for all courses studied, the volume of discussions in the forum declines continuously throughout the duration of the course. 2. High-volume, noisy discussions: at least 30% of the courses produce new discussion threads at rates that are infeasible for students or teaching staff to read through. Furthermore, a substantial portion of the discussions are not directly course-related. We investigate factors that correlate with the decline of activity in the online discussion forums and find effective strategies to classify threads and rank their relevance. Specifically, we use linear regression models to analyze the time series of the count data for the forum activities and make a number of observations, e.g., the teaching staff's active participation in the discussion increases the discussion volume but does not slow down the decline rate. We then propose a unified generative model for the discussion threads, which allows us both to choose efficient thread classifiers and design an effective algorithm for ranking thread relevance. Our ranking algorithm is further compared against two baseline algorithms, using human evaluation from Amazon Mechanical Turk. The authors on this paper are listed in alphabetical order. For media and press coverage, please refer to us collectively, as "researchers from the EDGE Lab at Princeton University, together with collaborators at Boston University and Microsoft Corporation."

cs.SI↗

arXiv2012

Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified

We prove the first Chernoff-Hoeffding bounds for general nonreversible finite-state Markov chains based on the standard L_1 (variation distance) mixing-time of the chain. Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step random walk (v_1,...,v_t) on M starting from the stationary distribution pi has expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the L_1 mixing-time of M. We show that the probability of X deviating from its mean by a multiplicative factor of delta, i.e., Pr [ |X - mu t| >= delta mu t ], is at most exp(-Omega(delta^2 mu t / T)) for 0 <= delta <= 1, and exp(-Omega(delta mu t / T)) for delta > 1. In fact, the bounds hold even if the weight functions f_i's for i in [t] are distinct, provided that all of them have the same mean mu. We also obtain a simplified proof for the Chernoff-Hoeffding bounds based on the spectral expansion lambda of M, which is the square root of the second largest eigenvalue (in absolute value) of M tilde{M}, where tilde{M} is the time-reversal Markov chain of M. We show that the probability Pr [ |X - mu t| >= delta mu t ] is at most exp(-Omega(delta^2 (1-lambda) mu t)) for 0 <= delta <= 1, and exp(-Omega(delta (1-lambda) mu t)) for delta > 1. Both of our results extend to continuous-time Markov chains, and to the case where the walk starts from an arbitrary distribution x, at a price of a multiplicative factor depending on the distribution x in the concentration bounds.

math.PR↗

arXiv2012

Efficient Rare-event Simulation for Perpetuities

We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) + ... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the Y_j's satisfy the so-called Cramer condition with associated root theta_{ast} in (0,infty) and that the tails of the B_j's are appropriately behaved so that D is regularly varying with index theta_{ast}. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the Y_j's according to theta_{ast} fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.

math.PR↗

arXiv2012

Rare-Event Simulation for Many-Server Queues

We develop rare-event simulation methodology for the analysis of loss events in a many-server loss system under quality-driven regime, focusing on the steady-state loss probability (i.e. fraction of lost customers over arrivals) and the behavior of the whole system leading to loss events. The analysis of these events requires working with the full measure-valued process describing the system. This is the first algorithm that is shown to be asymptotically optimal, in the rare-event simulation context, under the setting of many-server queues involving a full measure-valued descriptor.

math.PR↗

arXiv2011

Information Dissemination via Random Walks in d-Dimensional Space

We study a natural information dissemination problem for multiple mobile agents in a bounded Euclidean space. Agents are placed uniformly at random in the $d$-dimensional space $\{-n, ..., n\}^d$ at time zero, and one of the agents holds a piece of information to be disseminated. All the agents then perform independent random walks over the space, and the information is transmitted from one agent to another if the two agents are sufficiently close. We wish to bound the total time before all agents receive the information (with high probability). Our work extends Pettarin et al.'s work (Infectious random walks, arXiv:1007.1604v2, 2011), which solved the problem for $d \leq 2$. We present tight bounds up to polylogarithmic factors for the case $d = 3$. (While our results extend to higher dimensions, for space and readability considerations we provide only the case $d=3$ here.) Our results show the behavior when $d \geq 3$ is qualitatively different from the case $d \leq 2$. In particular, as the ratio between the volume of the space and the number of agents varies, we show an interesting phase transition for three dimensions that does not occur in one or two dimensions.

math.PR↗

arXiv2011

Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson's integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.

math.PR↗