arXiv subjects

Enlu Zhou

Publications and source records attributed to Enlu Zhou.

51 records · Page 3Linked to original sources

arXiv2020

Online Quantification of Input Model Uncertainty by Two-Layer Importance Sampling

Stochastic simulation has been widely used to analyze the performance of complex stochastic systems and facilitate decision making in those systems. Stochastic simulation is driven by the input model, which is a collection of probability distributions that model the stochasticity in the system. The input model is usually estimated using a finite amount of data, which introduces the so-called input model uncertainty to the simulation output. How to quantify input uncertainty has been studied extensively, and many methods have been proposed for the batch data setting, i.e., when all the data are available at once. However, methods for "streaming data" arriving sequentially in time are still in demand, despite that streaming data have become increasingly prevalent in modern applications. To fill this gap, we propose a two-layer importance sampling framework that incorporates streaming data for online input uncertainty quantification. Under this framework, we develop two algorithms that suit different application scenarios: the first scenario is when data come at a fast speed and there is no time for any new simulation in between updates; the second is when data come at a moderate speed and a few but limited simulations are allowed at each time stage. We prove the consistency and asymptotic convergence rate results, which theoretically show the efficiency of our proposed approach. We further demonstrate the proposed algorithms on a numerical example of the news vendor problem.

q-fin.RM↗

arXiv2019

Towards Understanding the Importance of Shortcut Connections in Residual Networks

Residual Network (ResNet) is undoubtedly a milestone in deep learning. ResNet is equipped with shortcut connections between layers, and exhibits efficient training using simple first order algorithms. Despite of the great empirical success, the reason behind is far from being well understood. In this paper, we study a two-layer non-overlapping convolutional ResNet. Training such a network requires solving a non-convex optimization problem with a spurious local optimum. We show, however, that gradient descent combined with proper normalization, avoids being trapped by the spurious local optimum, and converges to a global optimum in polynomial time, when the weight of the first layer is initialized at 0, and that of the second layer is initialized arbitrarily in a ball. Numerical experiments are provided to support our theory.

cs.LG↗

arXiv2019

Towards Understanding the Importance of Noise in Training Neural Networks

Numerous empirical evidence has corroborated that the noise plays a crucial rule in effective and efficient training of neural networks. The theory behind, however, is still largely unknown. This paper studies this fundamental problem through training a simple two-layer convolutional neural network model. Although training such a network requires solving a nonconvex optimization problem with a spurious local optimum and a global optimum, we prove that perturbed gradient descent and perturbed mini-batch stochastic gradient algorithms in conjunction with noise annealing is guaranteed to converge to a global optimum in polynomial time with arbitrary initialization. This implies that the noise enables the algorithm to efficiently escape from the spurious local optimum. Numerical experiments are provided to support our theory.

cs.LG↗

arXiv2018

Analyzing and provably improving fixed budget ranking and selection algorithms

This paper studies the fixed budget formulation of the Ranking and Selection (R&S) problem with independent normal samples, where the goal is to investigate different algorithms' convergence rate in terms of their resulting probability of false selection (PFS). First, we reveal that for the well-known Optimal Computing Budget Allocation (OCBA) algorithm and its two variants, a constant initial sample size (independent of the total budget) only amounts to a sub-exponential (or even polynomial) convergence rate. After that, a modification is proposed to achieve an exponential convergence rate, where the improvement is shown by a finite-sample bound on the PFS as well as numerical results. Finally, we focus on a more tractable two-design case and explicitly characterize the large deviations rate of PFS for some simplified algorithms. Our analysis not only develops insights into the algorithms' properties, but also highlights several useful techniques for analyzing the convergence rate of fixed budget R\&S algorithms.

math.OC↗

arXiv2017

Space-filling design for nonlinear models

Performing a computer experiment can be viewed as observing a mapping between the model parameters and the corresponding model outputs predicted by the computer model. In view of this, experimental design for computer experiments can be thought of as devising a reliable procedure for finding configurations of design points in the parameter space so that their images represent the manifold parametrized by such a mapping (i.e., computer experiments). Traditional space-filling designs aim to achieve this goal by filling the parameter space with design points that are as "uniform" as possible in the parameter space. However, the resulting design points may be non-uniform in the model output space and hence fail to provide a reliable representation of the manifold, becoming highly inefficient or even misleading in case the computer experiments are non-linear. In this paper, we propose an iterative algorithm that fills in the model output manifold uniformly---rather than the parameter space uniformly---so that one could obtain a reliable understanding of the model behaviors with the minimal number of design points.

stat.CO↗

arXiv2017

Risk Quantification in Stochastic Simulation under Input Uncertainty

When simulating a complex stochastic system, the behavior of output response depends on input parameters estimated from finite real-world data, and the finiteness of data brings input uncertainty into the system. The quantification of the impact of input uncertainty on output response has been extensively studied. Most of the existing literature focuses on providing inferences on the mean response at the true but unknown input parameter, including point estimation and confidence interval construction. Risk quantification of mean response under input uncertainty often plays an important role in system evaluation and control, because it provides inferences on extreme scenarios of mean response in all possible input models. To the best of our knowledge, it has rarely been systematically studied in the literature. In this paper, first we introduce risk measures of mean response under input uncertainty, and propose a nested Monte Carlo simulation approach to estimate them. Then we develop asymptotical properties such as consistency and asymptotic normality for the proposed nested risk estimators. We further study the associated budget allocation problem for efficient nested risk simulation, and finally use a sharing economy example to illustrate the importance of accessing and controlling risk due to input uncertainty.

q-fin.RM↗

arXiv2017

A Bayesian Risk Approach to Data-driven Stochastic Optimization: Formulations and Asymptotics

A large class of stochastic programs involve optimizing an expectation taken with respect to an underlying distribution that is unknown in practice. One popular approach to addressing the distributional uncertainty, known as the distributionally robust optimization (DRO), is to hedge against the worst case over an uncertainty set of candidate distributions. However, it has been observed that inappropriate construction of the uncertainty set can sometimes result in over-conservative solutions. To explore the middle ground between optimistically ignoring the distributional uncertainty and pessimistically fixating on the worst-case scenario, we propose a Bayesian risk optimization (BRO) framework for parametric underlying distributions, which is to optimize a risk functional applied to the posterior distribution of an unknown distribution parameter. Of our particular interest are four risk functionals: mean, mean-variance, value-at-risk, and conditional value-at-risk. To unravel the implication of BRO, we establish the consistency of objective functions and optimal solutions, as well as the asymptotic normality of objective functions and optimal values. More importantly, our analysis reveals a hidden interpretation: the objectives of BRO can be approximately viewed as a weighted sum of posterior mean objective and the (squared) half-width of the true objective's confidence interval.

math.OC↗

arXiv2016

Solving Multi-Objective Optimization via Adaptive Stochastic Search with Domination Measure

For general multi-objective optimization problems, we propose a novel performance metric called domination measure to measure the quality of a solution, which can be intuitively interpreted as the size of the portion of the solution space that dominates that solution. As a result, we reformulate the original multi-objective problem into a stochastic single-objective one and propose a model-based approach to solve it. We show that an ideal version algorithm of the proposed approach converges to a set representation of the global optima of the reformulated problem. We also investigate the numerical performance of an implementable version algorithm by comparing it with numerous existing multi-objective optimization methods on popular benchmark test functions. The numerical results show that the proposed approach is effective in generating a finite and uniformly spread approximation of the Pareto optimal set of the original multi-objective problem, and is competitive to the tested existing methods.

math.OC↗

arXiv2016

Solving the Dual Problems of Dynamic Programs via Regression

In recent years, information relaxation and duality in dynamic programs have been studied extensively, and the resulted primal-dual approach has become a powerful procedure in solving dynamic programs by providing lower-upper bounds on the optimal value function. Theoretically, with the so called value-based optimal dual penalty, the optimal value function could be recovered exactly via strong duality. However, in practice, obtaining tight dual bounds usually requires good approximations of the optimal dual penalty, which could be time-consuming due to the conditional expectations that need to be estimated via nested simulation. In this paper, we will develop a framework of regression approach to approximating the optimal dual penalty in a non-nested manner, by exploring the structure of the function space consisting of all feasible dual penalties. The resulted approximations maintain to be feasible dual penalties, and thus yield valid dual bounds on the optimal value function. We show that the proposed framework is computationally efficient, and the resulted dual penalties lead to numerically tractable dual problems. Finally, we apply the framework to a high-dimensional dynamic trading problem to demonstrate its effectiveness in solving the dual problems of complex dynamic programs.

math.OC↗

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

Simulation Optimization of Risk Measures with Adaptive Risk Levels

Optimizing risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of a general loss distribution is usually difficult, because 1) the loss function might lack structural properties such as convexity or differentiability since it is often generated via black-box simulation of a stochastic system; 2) evaluation of risk measures often requires rare-event simulation, which is computationally expensive. In this paper, we study the extension of the recently proposed gradient-based adaptive stochastic search (GASS) to the optimization of risk measures VaR and CVaR. Instead of optimizing VaR or CVaR at the target risk level directly, we incorporate an adaptive updating scheme on the risk level, by initializing the algorithm at a small risk level and adaptively increasing it until the target risk level is achieved while the algorithm converges at the same time. This enables us to adaptively reduce the number of samples required to estimate the risk measure at each iteration, and thus improving the overall efficiency of the algorithm.

math.OC↗

arXiv2014

Information Relaxation and Dual Formulation of Controlled Markov Diffusions

Information relaxation and duality in Markov decision processes have been studied recently by several researchers with the goal to derive dual bounds on the value function. In this paper we extend this dual formulation to controlled Markov diffusions: in a similar way we relax the constraint that the decision should be made based on the current information and impose penalty to punish the access to the information in advance. We establish the weak duality, strong duality and complementary slackness results in a parallel way as those in Markov decision processes. We explore the structure of the optimal penalties and expose the connection between Markov decision processes and controlled Markov diffusions. We demonstrate the use of the dual representation for controlled Markov diffusions in a classic dynamic portfolio choice problem. We evaluate the lower bounds on the expected utility by Monte Carlo simulation under a sub-optimal policy, and we propose a new class of penalties to derive upper bounds with little extra computation. The small gaps between the lower bounds and upper bounds indicate that the available policy is near optimal as well as the effectiveness of our proposed penalty in the dual method.

math.OC↗

arXiv2014

Weakly Coupled Dynamic Program: Information and Lagrangian Relaxations

"Weakly coupled dynamic program" describes a broad class of stochastic optimization problems in which multiple controlled stochastic processes evolve independently but subject to a set of linking constraints imposed on the controls. One feature of the weakly coupled dynamic program is that it decouples into lower-dimensional dynamic programs by dualizing the linking constraint via the Lagrangian relaxation, which also yields a bound on the optimal value of the original dynamic program. Together with the Lagrangian bound, we utilize the information relaxation approach that relaxes the non-anticipative constraint on the controls to obtain a tighter dual bound. We also investigate other combinations of the relaxations and place the resulting bounds in order. To tackle large-scale problems, we further propose a computationally tractable method based on information relaxation, and provide insightful interpretation and performance guarantee. We implement our method and demonstrate its use through two numerical examples.

math.OC↗

arXiv2013

Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes

Fast pricing of American-style options has been a difficult problem since it was first introduced to financial markets in 1970s, especially when the underlying stocks' prices follow some jump-diffusion processes. In this paper, we propose a new algorithm to generate tight upper bounds on the Bermudan option price without nested simulation, under the jump-diffusion setting. By exploiting the martingale representation theorem for jump processes on the dual martingale, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal-dual algorithm, therefore significantly improves the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our proposed algorithm.

q-fin.CP↗

arXiv2013

Gradient-Based Adaptive Stochastic Search for Non-Differentiable Optimization

In this paper, we propose a stochastic search algorithm for solving general optimization problems with little structure. The algorithm iteratively finds high quality solutions by randomly sampling candidate solutions from a parameterized distribution model over the solution space. The basic idea is to convert the original (possibly non-differentiable) problem into a differentiable optimization problem on the parameter space of the parameterized sampling distribution, and then use a direct gradient search method to find improved sampling distributions. Thus, the algorithm combines the robustness feature of stochastic search from considering a population of candidate solutions with the relative fast convergence speed of classical gradient methods by exploiting local differentiable structures. We analyze the convergence and converge rate properties of the proposed algorithm, and carry out numerical study to illustrate its performance.

math.OC↗