arXiv subjects

Enlu Zhou

Publications and source records attributed to Enlu Zhou.

At least 37 records · Page 2Linked to original sources

arXiv2024

Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes

We consider infinite-horizon Markov Decision Processes where parameters, such as transition probabilities, are unknown and estimated from data. The popular distributionally robust approach to addressing the parameter uncertainty can sometimes be overly conservative. In this paper, we utilize the recently proposed formulation, Bayesian risk Markov Decision Process (BR-MDP), to address parameter (or epistemic) uncertainty in MDPs. To solve the infinite-horizon BR-MDP with a class of convex risk measures, we propose a computationally efficient approach called approximate bilevel difference convex programming (ABDCP). The optimization is performed offline and produces the optimal policy that is represented as a finite state controller with desirable performance guarantees. We also demonstrate the empirical performance of the BR-MDP formulation and the proposed algorithm.

eess.SY↗

arXiv2024

Optimal Computing Budget Allocation for Data-driven Ranking and Selection

In a fixed budget ranking and Selection (R&S) problem, one aims to identify the best design among a finite number of candidates by efficiently allocating the given computing budget to evaluate design performance. Classical methods for R&S usually assume the distribution of the randomness in the system is exactly known. In this paper, we consider the practical scenario where the true distribution is unknown but can be estimated from streaming input data that arrive in batches over time. We formulate the R&S problem in this dynamic setting as a multi-stage problem, where we adopt the Bayesian approach to estimate the distribution and formulate a stage-wise optimization problem to allocate the computing budget. We characterize the optimality conditions for the stage-wise problem by applying the large deviations theory to maximize the decay rate of probability of false selection. Based on the optimality conditions and combined with the updating of distribution estimates, we design two sequential budget allocation procedures for R&S under streaming input data. We theoretically guarantee the consistency and asymptotic optimality of the proposed procedures. We demonstrate the practical efficiency through numerical experiments in comparison with the equal allocation policy and an extension of the Optimal Computing Budget Allocation algorithm.

math.OC↗

arXiv2023

Bayesian Stochastic Gradient Descent for Stochastic Optimization with Streaming Input Data

We consider stochastic optimization under distributional uncertainty, where the unknown distributional parameter is estimated from streaming data that arrive sequentially over time. Moreover, data may depend on the decision of the time when they are generated. For both decision-independent and decision-dependent uncertainties, we propose an approach to jointly estimate the distributional parameter via Bayesian posterior distribution and update the decision by applying stochastic gradient descent on the Bayesian average of the objective function. Our approach converges asymptotically over time and achieves the convergence rates of classical SGD in the decision-independent case. We demonstrate the empirical performance of our approach on both synthetic test problems and a classical newsvendor problem.

math.OC↗

arXiv2023

Bayesian Risk-Averse Q-Learning with Streaming Observations

We consider a robust reinforcement learning problem, where a learning agent learns from a simulated training environment. To account for the model mis-specification between this training environment and the real environment due to lack of data, we adopt a formulation of Bayesian risk MDP (BRMDP) with infinite horizon, which uses Bayesian posterior to estimate the transition model and impose a risk functional to account for the model uncertainty. Observations from the real environment that is out of the agent's control arrive periodically and are utilized by the agent to update the Bayesian posterior to reduce model uncertainty. We theoretically demonstrate that BRMDP balances the trade-off between robustness and conservativeness, and we further develop a multi-stage Bayesian risk-averse Q-learning algorithm to solve BRMDP with streaming observations from real environment. The proposed algorithm learns a risk-averse yet optimal policy that depends on the availability of real-world observations. We provide a theoretical guarantee of strong convergence for the proposed algorithm.

cs.LG↗

arXiv2023

Bayesian Distributionally Robust Optimization

We introduce a new framework, Bayesian Distributionally Robust Optimization (Bayesian-DRO), for data-driven stochastic optimization where the underlying distribution is unknown. Bayesian-DRO contrasts with most of the existing DRO approaches in the use of Bayesian estimation of the unknown distribution. To make computation of Bayesian updating tractable, Bayesian-DRO first assumes the underlying distribution takes a parametric form with unknown parameter and then computes the posterior distribution of the parameter. To address the model uncertainty brought by the assumed parametric distribution, Bayesian-DRO constructs an ambiguity set of distributions with the assumed parametric distribution as the reference distribution and then optimizes with respect to the worst case in the ambiguity set. We show the consistency of the Bayesian posterior distribution and subsequently the convergence of objective functions and optimal solutions of Bayesian-DRO. Our consistency result of the Bayesian posterior requires simpler assumptions than the classical literature on Bayesian consistency. We also consider several approaches for selecting the ambiguity set size in Bayesian-DRO and compare them numerically. Our numerical experiments demonstrate the out-of-sample performance of Bayesian-DRO in comparison with Kullback-Leibler-based (KL-) and Wasserstein-based empirical DRO as well as risk-neutral Bayesian Risk Optimization. Our numerical results shed light on how to choose the modeling framework (Bayesian-DRO, KL-DRO, Wasserstein-DRO) for specific problems, but the choice for general problems still remains an important and open question.

math.OC↗

arXiv2022

Bayesian Risk Markov Decision Processes

We consider finite-horizon Markov Decision Processes where parameters, such as transition probabilities, are unknown and estimated from data. The popular distributionally robust approach to addressing the parameter uncertainty can sometimes be overly conservative. In this paper, we propose a new formulation, Bayesian risk Markov Decision Process (BR-MDP), to address parameter uncertainty in MDPs, where a risk functional is applied in nested form to the expected total cost with respect to the Bayesian posterior distribution of the unknown parameters. The proposed formulation provides more flexible risk attitutes towards parameter uncertainty and takes into account the availability of data in future times stages. To solve the proposed formulation with the conditional value-at-risk (CVaR) risk functional, we propose an efficient approximation algorithm by deriving an analytical approximation of the value function and utilizing the convexity of CVaR. We demonstrate the empirical performance of the BR-MDP formulation and proposed algorithms on a gambler's betting problem and an inventory control problem.

eess.SY↗

arXiv2022

Data-driven Ranking and Selection under Input Uncertainty

We consider a simulation-based Ranking and Selection (R&S) problem with input uncertainty, where unknown input distributions can be estimated using input data arriving in batches of varying sizes over time. Each time a batch arrives, additional simulations can be run using updated input distribution estimates. The goal is to confidently identify the best design after collecting as few batches as possible. We first introduce a moving average estimator for aggregating simulation outputs generated under heterogenous input distributions. Then, based on a Sequential Elimination framework, we devise two major R&S procedures by establishing exact and asymptotic confidence bands for the estimator. In deriving the latter confidence bands, we incorporate the result of "Multiple Comparison with Best" and establish an asymptotic normality result which explicitly characterizes the tradeoff between input uncertainty and stochastic uncertainty in an online environment. We also extend our procedures to the indifference zone setting, which helps save simulation effort for practical usage. Numerical results show the effectiveness and necessity of our procedures. Moreover, the efficiency can be further boosted through optimizing the "drop rate" parameter of the estimator.

math.OC↗

arXiv2022

Risk-averse Contextual Multi-armed Bandit Problem with Linear Payoffs

In this paper we consider the contextual multi-armed bandit problem for linear payoffs under a risk-averse criterion. At each round, contexts are revealed for each arm, and the decision maker chooses one arm to pull and receives the corresponding reward. In particular, we consider mean-variance as the risk criterion, and the best arm is the one with the largest mean-variance reward. We apply the Thompson Sampling algorithm for the disjoint model, and provide a comprehensive regret analysis for a variant of the proposed algorithm. For $T$ rounds, $K$ actions, and $d$-dimensional feature vectors, we prove a regret bound of $O((1+ρ+\frac{1}ρ) d\ln T \ln \frac{K}δ\sqrt{d K T^{1+2ε} \ln \frac{K}δ \frac{1}ε})$ that holds with probability $1-δ$ under the mean-variance criterion with risk tolerance $ρ$, for any $0<ε<\frac{1}{2}$, $0<δ<1$. The empirical performance of our proposed algorithms is demonstrated via a portfolio selection problem.

cs.LG↗

arXiv2022

Robust Multi-Objective Bayesian Optimization Under Input Noise

Bayesian optimization (BO) is a sample-efficient approach for tuning design parameters to optimize expensive-to-evaluate, black-box performance metrics. In many manufacturing processes, the design parameters are subject to random input noise, resulting in a product that is often less performant than expected. Although BO methods have been proposed for optimizing a single objective under input noise, no existing method addresses the practical scenario where there are multiple objectives that are sensitive to input perturbations. In this work, we propose the first multi-objective BO method that is robust to input noise. We formalize our goal as optimizing the multivariate value-at-risk (MVaR), a risk measure of the uncertain objectives. Since directly optimizing MVaR is computationally infeasible in many settings, we propose a scalable, theoretically-grounded approach for optimizing MVaR using random scalarizations. Empirically, we find that our approach significantly outperforms alternative methods and efficiently identifies optimal robust designs that will satisfy specifications across multiple metrics with high probability.

cs.LG↗

arXiv2022

Noise Regularizes Over-parameterized Rank One Matrix Recovery, Provably

We investigate the role of noise in optimization algorithms for learning over-parameterized models. Specifically, we consider the recovery of a rank one matrix $Y^*\in R^{d\times d}$ from a noisy observation $Y$ using an over-parameterization model. We parameterize the rank one matrix $Y^*$ by $XX^\top$, where $X\in R^{d\times d}$. We then show that under mild conditions, the estimator, obtained by the randomly perturbed gradient descent algorithm using the square loss function, attains a mean square error of $O(σ^2/d)$, where $σ^2$ is the variance of the observational noise. In contrast, the estimator obtained by gradient descent without random perturbation only attains a mean square error of $O(σ^2)$. Our result partially justifies the implicit regularization effect of noise when learning over-parameterized models, and provides new understanding of training over-parameterized neural networks.

cs.LG↗

arXiv2022

Contextual Ranking and Selection with Gaussian Processes

In many real world problems, we are faced with the problem of selecting the best among a finite number of alternatives, where the best alternative is determined based on context specific information. In this work, we study the contextual Ranking and Selection problem under a finite-alternative-finite-context setting, where we aim to find the best alternative for each context. We use a separate Gaussian process to model the reward for each alternative, and derive the large deviations rate function for both the expected and worst-case contextual probability of correct selection. We propose the GP-C-OCBA sampling policy, which uses the Gaussian process posterior to iteratively allocate observations to maximize the rate function. We prove its consistency and show that it achieves the optimal convergence rate under the assumption of a non-informative prior. Numerical experiments show that our algorithm is highly competitive in terms of sampling efficiency, while having significantly smaller computational overhead.

math.OC↗

arXiv2021

Asymptotically Optimal Sampling Policy for Selecting Top-m Alternatives

We consider selecting the top-$m$ alternatives from a finite number of alternatives via Monte Carlo simulation. Under a Bayesian framework, we formulate the sampling decision as a stochastic dynamic programming problem, and develop a sequential sampling policy that maximizes a value function approximation one-step look ahead. To show the asymptotic optimality of the proposed procedure, the asymptotically optimal sampling ratios which optimize large deviations rate of the probability of false selection for selecting top-$m$ alternatives has been rigorously defined. The proposed sampling policy is not only proved to be consistent but also achieves the asymptotically optimal sampling ratios. Numerical experiments demonstrate superiority of the proposed allocation procedure over existing ones.

math.OC↗

arXiv2021

A Diffusion Approximation Theory of Momentum SGD in Nonconvex Optimization

Momentum Stochastic Gradient Descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning, e.g., training deep neural networks, variational Bayesian inference, and etc. Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points, but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks.

cs.LG↗

arXiv2021

Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization

Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that is uniquely determined by the injected noise. Our result implies that the noise imposes implicit bias towards certain optima. Numerical experiments are provided to support our theory.

cs.LG↗

arXiv2021

Towards Understanding Acceleration Tradeoff between Momentum and Asynchrony in Nonconvex Stochastic Optimization

Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) is one of the most popular algorithms in distributed machine learning. However, its convergence properties for these complicated nonconvex problems is still largely unknown, because of the current technical limit. Therefore, in this paper, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problem - streaming PCA, which helps us to understand Aync-MSGD better even for more general problems. Specifically, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA by diffusion approximation. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.

cs.LG↗

arXiv2020

Group Testing Enables Asymptomatic Screening for COVID-19 Mitigation: Feasibility and Optimal Pool Size Selection with Dilution Effects

Repeated asymptomatic screening for SARS-CoV-2 promises to control spread of the virus but would require too many resources to implement at scale. Group testing is promising for screening more people with fewer test resources: multiple samples tested together in one pool can be excluded with one negative test result. Existing approaches to group testing design for SARS-CoV-2 asymptomatic screening, however, do not consider dilution effects: that false negatives become more common with larger pools. As a consequence, they may recommend pool sizes that are too large or misestimate the benefits of screening. Modeling dilution effects, we derive closed-form expressions for the expected number of tests and false negative/positives per person screened under two popular group testing methods: the linear and square array methods. We find that test error correlation induced by a common viral load across an individual's samples results in many fewer false negatives than would be expected from less realistic but more widely assumed independent errors. This insight also suggests that false positives can be controlled through repeated tests without significantly increasing false negatives. Using these closed-form expressions to trace a Pareto frontier over error rates and tests, we design testing protocols for repeated asymptomatic screening of a large population. We minimize disease prevalence by optimizing a time-varying pool sizes and screening frequency constrained by daily test capacity and a false positive limit. This provides a testing protocol practitioners can use for mitigating COVID-19. In a case study, we demonstrate the effectiveness of this methodology in controlling spread.

q-bio.PE↗

arXiv2020

Bayesian Optimization of Risk Measures

We consider Bayesian optimization of objective functions of the form $ρ[ F(x, W) ]$, where $F$ is a black-box expensive-to-evaluate function and $ρ$ denotes either the VaR or CVaR risk measure, computed with respect to the randomness induced by the environmental random variable $W$. Such problems arise in decision making under uncertainty, such as in portfolio optimization and robust systems design. We propose a family of novel Bayesian optimization algorithms that exploit the structure of the objective function to substantially improve sampling efficiency. Instead of modeling the objective function directly as is typical in Bayesian optimization, these algorithms model $F$ as a Gaussian process, and use the implied posterior on the objective function to decide which points to evaluate. We demonstrate the effectiveness of our approach in a variety of numerical experiments.

stat.ML↗

arXiv2020

Solving Bayesian Risk Optimization via Nested Stochastic Gradient Estimation

In this paper, we aim to solve Bayesian Risk Optimization (BRO), which is a recently proposed framework that formulates simulation optimization under input uncertainty. In order to efficiently solve the BRO problem, we derive nested stochastic gradient estimators and propose corresponding stochastic approximation algorithms. We show that our gradient estimators are asymptotically unbiased and consistent, and that the algorithms converge asymptotically. We demonstrate the empirical performance of the algorithms on a two-sided market model. Our estimators are of independent interest in extending the literature of stochastic gradient estimation to the case of nested risk functions.

math.OC↗