Strong Performance Guarantees Research Articles

Runtime Analysis of the (μ + 1) GA: Provable Speed-Ups from Strong Drift towards Diverse Populations

Most evolutionary algorithms used in practice heavily employ crossover. In contrast, the rigorous understanding of how crossover is beneficial is largely lagging behind. In this work, we make a considerable step forward by analyzing the population dynamics of the (µ+1) genetic algorithm when optimizing the Jump benchmark. We observe (and prove via mathematical means) that once the population contains two different individuals on the local optimum, the diversity in the population increases in expectation. From this drift towards more diverse states, we show that a diversity suitable for crossover to be effective is reached quickly and, more importantly, then persists for a time that is at least exponential in the population size µ. This drastically improves over the previously best known guarantee, which is only quadratic in µ. Our new understanding of the population dynamics easily gives stronger performance guarantees. In particular, we derive that population sizes logarithmic in the problem size n suffice to gain an Ω(n)-factor runtime improvement from crossover (previous works achieved comparable bounds only with µ = Θ(n) or a non-standard mutation rate).

Fast deterministic algorithms for non-submodular maximization with strong performance guarantees

Fast deterministic algorithms for non-submodular maximization with strong performance guarantees

Sequential Search with Acquisition Uncertainty

We study a variation of the classical Pandora’s problem in which a decision maker (DM) sequentially explores alternatives from a given set and learns their values while trying to acquire the best alternative. The variations in the model we study are (i) alternatives randomly become unavailable during exploration and (ii) the DM’s ability to acquire a remaining alternative is uncertain and depends on a chosen offer price. Such acquisition uncertainties arise in many applications, including housing search, hiring problems, and e-commerce, but greatly complicate the search problem in that optimal policies retain all previously explored alternative values as part of the problem state, as opposed to only the highest explored value as in Pandora’s rule. Our central insight is that despite the complexity that these acquisition uncertainties create, simple greedy policies based on static sequencing and a single threshold value enjoy strong performance guarantees. We develop such a class of policies and show how to compute them using a greedy algorithm whose worst-case run-time scales linearly (up to logarithmic factors) in the number of alternative types. We show that our policies (a) are asymptotically optimal in high multiplicity regimes with many alternatives and (b) obtain at least [Formula: see text] of the optimal value under a broad set of conditions. Extensive numerical examples support this theory: We illustrate our policies on a variation of Pandora’s problem with disappearing alternatives and housing search on models calibrated on data from the online brokerage Redfin. In these examples, our policies significantly outperform policies based on Pandora’s rule. This paper was accepted by Omar Besbes, revenue management and market analytics. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.00203 .

Learning-Augmented Mechanism Design: Leveraging Predictions for Facility Location

In this work, we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in “learning-augmented algorithms.” Aiming to complement the traditional worst-case analysis approach in computer science, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions. The algorithms can use the predictions as a guide to inform their decisions, aiming to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are inaccurate (robustness). We initiate the design and analysis of strategyproof mechanisms that are augmented with predictions regarding the private information of the participating agents. To exhibit the important benefits of this approach, we revisit the canonical problem of facility location with strategic agents in the two-dimensional Euclidean space. We study both the egalitarian and utilitarian social cost functions, and we propose new strategyproof mechanisms that leverage predictions to guarantee an optimal trade-off between consistency and robustness. Furthermore, we also prove parameterized approximation results as a function of the prediction error, showing that our mechanisms perform well, even when the predictions are not fully accurate. Funding: The work of E. Balkanski was supported in part by the National Science Foundation [Grants CCF-2210501 and IIS-2147361]. The work of V. Gkatzelis and X. Tan was supported in part by the National Science Foundation [Grant CCF-2210502] and [CAREER Award CCF-2047907]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0225 .

PAC-Bayes Bounds for Bandit Problems: A Survey and Experimental Comparison.

PAC-Bayes has recently re-emerged as an effective theory with which one can derive principled learning algorithms with tight performance guarantees. However, applications of PAC-Bayes to bandit problems are relatively rare, which is a great misfortune. Many decision-making problems in healthcare, finance and natural sciences can be modelled as bandit problems. In many of these applications, principled algorithms with strong performance guarantees would be very much appreciated. This survey provides an overview of PAC-Bayes bounds for bandit problems and an experimental comparison of these bounds. On the one hand, we found that PAC-Bayes bounds are a useful tool for designing offline bandit algorithms with performance guarantees. In our experiments, a PAC-Bayesian offline contextual bandit algorithm was able to learn randomised neural network polices with competitive expected reward and non-vacuous performance guarantees. On the other hand, the PAC-Bayesian online bandit algorithms that we tested had loose cumulative regret bounds. We conclude by discussing some topics for future work on PAC-Bayesian bandit algorithms.

Multiproduct Dynamic Pricing with Limited Inventories Under a Cascade Click Model

Problem definition: Designing effective operational strategies requires a good understanding of customer behavior. The classic economic theory of customer choice has long been the paradigm in the operations literature. However, the rise of online marketplaces such as e-commerce has triggered considerable efforts in academia and industry to develop alternative models that not only provide a good approximation of customer behavior but also are easily scalable for large-scale implementations. In this paper, we consider a multiproduct dynamic pricing problem with limited inventories under the so-called cascade click model, which is one of the most popular click models used in practice and has been intensively studied in the computer science literature. Methodology/results: We present some fundamental results. First, we derive a sufficiently general characterization of the optimal pricing policy and show that it has a different structure than the optimal policy under the standard pricing model. Second, we show that the optimal expected total revenue under the cascade click model can be upper bounded by the objective value of an approximate deterministic pricing problem. Third, we show that two policies that are known to have strong performance guarantees in the standard revenue management setting can be properly adapted (in a nontrivial way) to the setting with cascade click model while retaining their strong performance. Finally, we also briefly discuss the joint ranking and pricing problem and provide an iterative heuristic to calculate an approximate ranking. Managerial implications: Taking into account customers’ click-and-search behavior leads to different structures of the optimal pricing policy, and some common insights under the standard pricing models may no longer hold. Moreover, our simulation studies show that pricing under a (misspecified) classic choice model that is oblivious to customers click-and-search behavior can severely impact profitability. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.0504 .

Informative core identification in complex networks

AbstractIn a complex network, the core component with interesting structures is usually hidden within noninformative connections. The noises and bias introduced by the noninformative component can obscure the salient structure and limit many network modeling procedures’ effectiveness. This paper introduces a novel core–periphery model for the noninformative periphery structure of networks without imposing a specific form of the core. We propose spectral algorithms for core identification for general downstream network analysis tasks under the model. The algorithms enjoy strong performance guarantees and are scalable for large networks. We evaluate the methods by extensive simulation studies demonstrating advantages over multiple traditional core–periphery methods. The methods are also used to extract the core structure from a citation network, which results in a more interpretable hierarchical community detection.

Online training data acquisition for federated learning in cloud–edge networks

Federated learning (FL) is an effective approach to exploiting different data sources for collaborative model training while maintaining the privacy of users. Adequate data is necessary to improve the accuracy of the federated learning. However, it is difficult for all data sources to acquire sufficient data for each iteration of federated training. In this paper, we study the data acquisition problem in cloud–edge networks, where edges acquire data from users and conduct the local training while the cloud aggregates the local models. Since data acquired by users with uncertainty, it is not easy to make the data acquisition decision for the FL online. Even worse, due to the unknown cost of data, it is difficult to make a macro-timescale decision. We propose a two-timescale online scheduling for federated learning to confront the uncertainty of the data acquisition. By learning empirical state information of the system with a carefully designed Lyapunov virtual queue and coordinating the data acquisition in different timescales in an online manner, the proposed approach minimizes the data acquisition cost of federated learning, reduces the data transmission delay and accelerates the convergence speed of federated learning. Rigorous theoretical analysis shows strong performance guarantees of the proposed two-timescale Lyapunov optimization algorithm and extensive trace-driven experimental results suggests that the algorithm achieves outstanding performance gains over existing benchmarks.

Encoder Blind Combinatorial Compressed Sensing

In its most elementary form, compressed sensing studies the design of decoding algorithms to recover a sufficiently sparse vector or code from a lower dimensional linear measurement vector. Typically it is assumed that the decoder has access to the encoder matrix, which in the combinatorial case is sparse and binary. In this paper we consider the problem of designing a decoder to recover a set of sparse codes from their linear measurements alone, that is without access to encoder matrix. To this end we study the matrix factorisation task of recovering both the encoder and sparse coding matrices from the associated linear measurement matrix. The contribution of this paper is a computationally efficient decoding algorithm, Decoder-Expander Based Factorisation, with strong performance guarantees. Under mild assumptions on the sparse coding matrix and by deploying a novel random encoder matrix, we prove that Decoder-Expander Based Factorisation recovers both the encoder and sparse coding matrix at the optimal measurement rate with high probability and from a near optimal number of measurement vectors. In addition, our experiments demonstrate the efficacy and computational efficiency of our algorithm in practice. Beyond compressed sensing, our results may be of interest for researchers working in areas as diverse as linear sketching, coding theory, matrix compression and dictionary learning.

Robustness of Proactive Intensive Care Unit Transfer Policies

Patients whose transfer to the intensive care unit (ICU) is unplanned are prone to higher mortality rates. In “Robustness of Proactive Intensive Care Unit Transfer Policies,” the authors study the problem of finding robust patient transfer policies to the ICU, which account for uncertainty in statistical estimates because of data limitations when optimizing to improve overall patient care. Under general assumptions, it is shown that an optimal transfer policy has a threshold structure. A robust policy also has a threshold structure, and it is more aggressive in transferring patients than the optimal nominal policy, which does not consider parameter uncertainty. The sensitivity of various hospital metrics to small changes in the parameters is highlighted using a data set of close to 300,000 hospitalizations at 21 Kaiser Permanente Northern California hospitals. This work provides useful insights into the impact of parameter uncertainty on deriving simple policies for proactive ICU transfer that have strong empirical performance and theoretical guarantees.

Distributionally Robust Chance Constrained Data-Enabled Predictive Control

In this article we study the problem of finite-time constrained optimal control of unknown stochastic linear time-invariant (LTI) systems, which is the key ingredient of a predictive control algorithm—albeit typically having access to a model. We propose a novel distributionally robust data-enabled predictive control (DeePC) algorithm which uses noise-corrupted input/output data to predict future trajectories and compute optimal control inputs while satisfying output chance constraints. The algorithm is based on 1) a nonparametric representation of the subspace spanning the system behavior, where past trajectories are sorted in Page or Hankel matrices; and 2) a distributionally robust optimization formulation which gives rise to strong probabilistic performance guarantees. We show that for certain objective functions, DeePC exhibits strong out-of-sample performance, and at the same time respects constraints with high probability. The algorithm provides an end-to-end approach to control design for unknown stochastic LTI systems. We illustrate the closed-loop performance of the DeePC in an aerial robotics case study.

Sample Average Approximation for Stochastic Optimization with Dependent Data: Performance Guarantees and Tractability

Sample average approximation (SAA), a popular method for tractably solving stochastic optimization problems, enjoys strong asymptotic performance guarantees in settings with independent training samples. However, these guarantees are not known to hold generally with dependent samples, such as in online learning with time series data or distributed computing with Markovian training samples. In this paper, we show that SAA remains tractable when the distribution of unknown parameters is only observable through dependent instances and still enjoys asymptotic consistency and finite sample guarantees. Specifically, we provide a rigorous probability error analysis to derive 1 - beta confidence bounds for the out-of-sample performance of SAA estimators and show that these estimators are asymptotically consistent. We then, using monotone operator theory, study the performance of a class of stochastic first-order algorithms trained on a dependent source of data. We show that approximation error for these algorithms is bounded and concentrates around zero, and establish deviation bounds for iterates when the underlying stochastic process is phi-mixing. The algorithms presented can be used to handle numerically inconvenient loss functions such as the sum of a smooth and non-smooth function or of non-smooth functions with constraints. To illustrate the usefulness of our results, we present several stochastic versions of popular algorithms such as stochastic proximal gradient descent (S-PGD), stochastic relaxed Peaceman-Rachford splitting algorithms (S-rPRS), and numerical experiment.

Dynamic Stochastic Matching Under Limited Time

In “Dynamic Stochastic Matching Under Limited Time,” Aouad and Sarıtaç analyze the design of matching policies in dynamic markets such as carpooling platforms and kidney exchange schemes. A crucial distinction with previous literature is that the agents’ arrivals and departures are fully dynamic. The demand and supply side are constantly replenished; each market participant remains available for potential matches during a limited period of time. Specifically, the authors formulate a general dynamic matching model over edge-weighted graphs, where the agents' arrivals and abandonments are stochastic and heterogeneous. The platform controls how long each agent waits and whom s/he is matched with. These decisions are subject to a fundamental tradeoff between increasing market thickness and mitigating the risk of abandonments from certain participants. The authors’ main contribution is to devise simple matching algorithms with strong performance guarantees for a broad class of networks. In contrast, they show that widely used batching algorithms have an arbitrary bad performance on certain graph-theoretic structures. Their analysis involves novel techniques including linear programming benchmarks, value function approximations, and proxies for continuous-time Markov chains, which may be of broader interest. Extensive simulations on real-world taxi demand data demonstrate that the newly developed algorithms can significantly improve cost efficiency against batching algorithms.

Product Ranking on Online Platforms

On online platforms, consumers face an abundance of options that are displayed in the form of a position ranking. Only products placed in the first few positions are readily accessible to the consumer, and she needs to exert effort to access more options. For such platforms, we develop a two-stage sequential search model where, in the first stage, the consumer sequentially screens positions to observe the preference weight of the products placed in them and forms a consideration set. In the second stage, she observes the additional idiosyncratic utility that she can derive from each product and chooses the highest-utility product within her consideration set. For this model, we first characterize the optimal sequential search policy of a welfare-maximizing consumer. We then study how platforms with different objectives should rank products. We focus on two objectives: (i) maximizing the platform’s market share and (ii) maximizing the consumer’s welfare. Somewhat surprisingly, we show that ranking products in decreasing order of their preference weights does not necessarily maximize market share or consumer welfare. Such a ranking may shorten the consumer’s consideration set due to the externality effect of high-positioned products on low-positioned ones, leading to insufficient screening. We then show that both problems—maximizing market share and maximizing consumer welfare—are NP-complete. We develop novel near-optimal polynomial-time ranking algorithms for each objective. Further, we show that, even though ranking products in decreasing order of their preference weights is suboptimal, such a ranking enjoys strong performance guarantees for both objectives. We complement our theoretical developments with numerical studies using synthetic data, in which we show (1) that heuristic versions of our algorithms that do not rely on model primitives perform well and (2) that our model can be effectively estimated using a maximum likelihood estimator. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Pareto frontiers of bicriteria continuous convex problems can be efficiently computed and optimal theoretical performance bounds have been established. In the case of bicriteria mixed-integer problems, the approximation of the Pareto frontier becomes, however, significantly harder. In this paper, we propose a new algorithm for approximating the Pareto frontier of bicriteria mixed-integer programs with convex constraints. Such Pareto frontiers are composed of patches of solutions with shared assignments for the discrete variables. By adaptively creating such a patchwork, our algorithm is able to create approximations that converge quickly to the true Pareto frontier. As a quality measure, we use the difference in hypervolume between the approximation and the true Pareto frontier. At least a certain number of patches is required to obtain an approximation with a given quality. This patch complexity gives a lower bound on the number of required computations. We show that our algorithm performs a number of optimization steps that are of a similar order as this lower bound. We provide an efficient MIP-based implementation of this algorithm. The efficiency of our algorithm is illustrated with numerical results showing that our algorithm has a strong theoretical performance guarantee while being competitive with other state-of-the-art approaches in practice.

Estimating the Number of Induced Subgraphs from Incomplete Data and Neighborhood Queries

We consider a natural setting where network parameters are estimated from noisy and incomplete information about the network. More specifically, we investigate how we can efficiently estimate the number of small subgraphs (e.g., edges, triangles, etc.) based on full access to one or two noisy and incomplete samples of a large underlying network and on few queries revealing the neighborhood of carefully selected vertices. After specifying a random generator which removes edges from the underlying graph, we present estimators with strong provable performance guarantees, which exploit information from the noisy network samples and query a constant number of the most important vertices for the estimation. Our experimental evaluation shows that, in practice, a single noisy network sample and a couple of hundreds neighborhood queries suffice for accurately estimating the number of triangles in networks with millions of vertices and edges.

Minimum vertex augmentation

This paper introduces a class of graph problems named minimum vertex augmentation (MVA). Given an input graph G where each vertex carries a binary color 0 or 1, we want to flip the colors of the fewest 0-vertices such that the subgraph induced by all the (original and new) 1-vertices satisfies a user-defined predicate π. In other words, the goal is to minimally augment the subset of 1-vertices to uphold the property π. Different formulations of π instantiate the framework into concrete problems at the core of numerous applications. We first describe a suite of techniques for solving MVA problems with strong performance guarantees, and then present a generic algorithmic paradigm that a user can instantiate to deal with ad-hoc MVA problems. The effectiveness and efficiency of our solutions are verified with an extensive experimental evaluation.

Low-rank factorization for rank minimization with nonconvex regularizers

Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective technique to solve the problem with strong performance guarantees. However, nonconvex relaxations have less estimation bias than the nuclear norm and can more accurately reduce the effect of noise on the measurements. We develop efficient algorithms based on iteratively reweighted nuclear norm schemes, while also utilizing the low rank factorization for semidefinite programs put forth by Burer and Monteiro. We prove convergence and computationally show the advantages over convex relaxations and alternating minimization methods. Additionally, the computational complexity of each iteration of our algorithm is on par with other state of the art algorithms, allowing us to quickly find solutions to the rank minimization problem for large matrices.

Online Allocation and Pricing: Constant Regret via Bellman Inequalities

We develop a framework for designing simple and efficient policies for a family of online allocation and pricing problems that includes online packing, budget-constrained probing, dynamic pricing, and online contextual bandits with knapsacks. In each case, we evaluate the performance of our policies in terms of their regret (i.e., additive gap) relative to an offline controller that is endowed with more information than the online controller. Our framework is based on Bellman inequalities, which decompose the loss of an algorithm into two distinct sources of error: (1) arising from computational tractability issues, and (2) arising from estimation/prediction of random trajectories. Balancing these errors guides the choice of benchmarks, and leads to policies that are both tractable and have strong performance guarantees. In particular, in all our examples, we demonstrate constant-regret policies that only require resolving a linear program in each period, followed by a simple greedy action-selection rule; thus, our policies are practical as well as provably near optimal.

Online Ordinal Optimization under Model Misspecification

We consider an ordinal optimization problem, where a decision maker learns the statistical characteristics of a number of systems using sequential sampling in order to ultimately determine the best one (with high probability). In so doing, the decision maker postulates a parametric model which may not precisely represent the true underlying system structure. We show that this misspecification, if not managed properly, can lead to suboptimal performance in the ordinal optimization problem due to a phenomenon identified as sample-selection endogeneity. To address that, we propose online sampling strategies that judiciously learn the unknown model parameters on the fly, and at the same time eliminate the adverse effects of misspecification as the number of samples grows large. The proposed sampling strategies are oblivious to the model misspecification; are shown to have strong performance guarantees regardless of that knowledge; and are computationally tractable.