Algorithms with predictions is a recent framework for decision-making under uncertainty that leverages the power of machine-learned predictions without making any assumption about their quality. The goal in this framework is for algorithms to achieve an improved performance when the predictions are accurate while maintaining acceptable guarantees when the predictions are erroneous. A serious concern with algorithms that use predictions is that these predictions can be biased and, as a result, cause the algorithm to make decisions that are deemed unfair. We show that this concern manifests itself in the classical secretary problem in the learning-augmented setting -- the state-of-the-art algorithm can have zero probability of accepting the best candidate, which we deem unfair, despite promising to accept a candidate whose expected value is at least $\max\{\Omega (1) , 1 - O(\epsilon)\}$ times the optimal value, where $\epsilon$ is the prediction error. We show how to preserve this promise while also guaranteeing to accept the best candidate with probability $\Omega(1)$. Our algorithm and analysis are based on a new "pegging" idea that diverges from existing works and simplifies/unifies some of their results. Finally, we extend to the $k$-secretary problem and complement our theoretical analysis with experiments.

Advances in computational power and AI have increased interest in reinforcement learning approaches to inventory management. This paper provides a theoretical foundation for these approaches and investigates the benefits of restricting to policy structures that are well-established by decades of inventory theory. In particular, we prove generalization guarantees for learning several well-known classes of inventory policies, including base-stock and (s, S) policies, by leveraging the celebrated Vapnik-Chervonenkis (VC) theory. We apply the concepts of the Pseudo-dimension and Fat-shattering dimension from VC theory to determine the generalizability of inventory policies, that is, the difference between an inventory policy's performance on training data and its expected performance on unseen data. We focus on a classical setting without contexts, but allow for an arbitrary distribution over demand sequences and do not make any assumptions such as independence over time. We corroborate our supervised learning results using numerical simulations. Managerially, our theory and simulations translate to the following insights. First, there is a principle of "learning less is more" in inventory management: depending on the amount of data available, it may be beneficial to restrict oneself to a simpler, albeit suboptimal, class of inventory policies to minimize overfitting errors. Second, the number of parameters in a policy class may not be the correct measure of overfitting error: in fact, the class of policies defined by T time-varying base-stock levels exhibits a generalization error comparable to that of the two-parameter (s, S) policy class. Finally, our research suggests situations in which it could be beneficial to incorporate the concepts of base-stock and inventory position into black-box learning machines, instead of having these machines directly learn the order quantity actions.

In this work, we explore a framework for contextual decision-making to study how the relevance and quantity of past data affects the performance of a data-driven policy. We analyze a contextual Newsvendor problem in which a decision-maker needs to trade-off between an underage and an overage cost in the face of uncertain demand. We consider a setting in which past demands observed under ``close by'' contexts come from close by distributions and analyze the performance of data-driven algorithms through a notion of context-dependent worst-case expected regret. We analyze the broad class of Weighted Empirical Risk Minimization (WERM) policies which weigh past data according to their similarity in the contextual space. This class includes classical policies such as ERM, k-Nearest Neighbors and kernel-based policies. Our main methodological contribution is to characterize exactly the worst-case regret of any WERM policy on any given configuration of contexts. To the best of our knowledge, this provides the first understanding of tight performance guarantees in any contextual decision-making problem, with past literature focusing on upper bounds via concentration inequalities. We instead take an optimization approach, and isolate a structure in the Newsvendor loss function that allows to reduce the infinite-dimensional optimization problem over worst-case distributions to a simple line search. This in turn allows us to unveil fundamental insights that were obfuscated by previous general-purpose bounds. We characterize actual guaranteed performance as a function of the contexts, as well as granular insights on the learning curve of algorithms.

We study generalizations of online bipartite matching in which each arriving vertex (customer) views a ranked list of offline vertices (products) and matches to (purchases) the first one they deem acceptable. The number of products that the customer has patience to view can be stochastic and dependent on the products seen. We develop a framework that views the interaction with each customer as an abstract resource consumption process, and derive new results for these online matching problems under the adversarial, non-stationary, and IID arrival models, assuming we can (approximately) solve the product ranking problem for each single customer. To that end, we show new results for product ranking under two cascade-click models: an optimal algorithm when each item has its own hazard rate for making the customer depart, and a 1/2-approximate algorithm when the customer has a general item-independent patience distribution. We also present a constant-factor 0.027-approximate algorithm in a new model where items are not initially available and arrive over time. We complement these positive results by presenting three additional negative results relating to these problems.

We study the classical Network Revenue Management (NRM) problem with accept/reject decisions and $T$ IID arrivals. We consider a distributional form where each arrival must fall under a finite number of possible categories, each with a deterministic resource consumption vector, but a random value distributed continuously over an interval. We develop an online algorithm that achieves $O(\log^2 T)$ regret under this model, with no further assumptions. We develop another online algorithm that achieves an improved $O(\log T)$ regret, with only a second-order growth assumption. To our knowledge, these are the first results achieving logarithmic-level regret in a continuous-distribution NRM model without further "non-degeneracy" assumptions. Our results are achieved via new techniques including: a new method of bounding myopic regret, a "semi-fluid" relaxation of the offline allocation, and an improved bound on the "dual convergence".

Matching markets involve heterogeneous agents (typically from two parties) who are paired for mutual benefit. During the last decade, matching markets have emerged and grown rapidly through the medium of the Internet. They have evolved into a new format, called Online Matching Markets (OMMs), with examples ranging from crowdsourcing to online recommendations to ridesharing. There are two features distinguishing OMMs from traditional matching markets. One is the dynamic arrival of one side of the market: we refer to these as online agents while the rest are offline agents. Examples of online and offline agents include keywords (online) and sponsors (offline) in Google Advertising; workers (online) and tasks (offline) in Amazon Mechanical Turk (AMT); riders (online) and drivers (offline when restricted to a short time window) in ridesharing. The second distinguishing feature of OMMs is the real-time decision-making element. However, studies have shown that the algorithms making decisions in these OMMs leave disparities in the match rates of offline agents. For example, tasks in neighborhoods of low socioeconomic status rarely get matched to gig workers, and drivers of certain races/genders get discriminated against in matchmaking. In this paper, we propose online matching algorithms which optimize for either individual or group-level fairness among offline agents in OMMs. We present two linear-programming (LP) based sampling algorithms, which achieve online competitive ratios at least 0.725 for individual fairness maximization (IFM) and 0.719 for group fairness maximization (GFM), respectively. We conduct extensive numerical experiments and results show that our boosted version of sampling algorithms are not only conceptually easy to implement but also highly effective in practical instances of fairness-maximization-related models.

We improve the effectiveness of propagation- and linear-optimization-based neural network verification algorithms with a new tightened convex relaxation for ReLU neurons. Unlike previous single-neuron relaxations which focus only on the univariate input space of the ReLU, our method considers the multivariate input space of the affine pre-activation function preceding the ReLU. Using results from submodularity and convex geometry, we derive an explicit description of the tightest possible convex relaxation when this multivariate input is over a box domain. We show that our convex relaxation is significantly stronger than the commonly used univariate-input relaxation which has been proposed as a natural convex relaxation barrier for verification. While our description of the relaxation may require an exponential number of inequalities, we show that they can be separated in linear time and hence can be efficiently incorporated into optimization algorithms on an as-needed basis. Based on this novel relaxation, we design two polynomial-time algorithms for neural network verification: a linear-programming-based algorithm that leverages the full power of our relaxation, and a fast propagation algorithm that generalizes existing approaches. In both cases, we show that for a modest increase in computational effort, our strengthened relaxation enables us to verify a significantly larger number of instances compared to similar algorithms.

We study a general problem of allocating limited resources to heterogeneous customers over time under model uncertainty. Each type of customer can be serviced using different actions, each of which stochastically consumes some combination of resources, and returns different rewards for the resources consumed. We consider a general model where the resource consumption distribution associated with each (customer type, action)-combination is not known, but is consistent and can be learned over time. In addition, the sequence of customer types to arrive over time is arbitrary and completely unknown. We overcome both the challenges of model uncertainty and customer heterogeneity by judiciously synthesizing two algorithmic frameworks from the literature: inventory balancing, which "reserves" a portion of each resource for high-reward customer types which could later arrive, and online learning, which shows how to "explore" the resource consumption distributions of each customer type under different actions. We define an auxiliary problem, which allows for existing competitive ratio and regret bounds to be seamlessly integrated. Furthermore, we show that the performance guarantee generated by our framework is tight, that is, we provide an information-theoretic lower bound which shows that both the loss from competitive ratio and the loss for regret are relevant in the combined problem. Finally, we demonstrate the efficacy of our algorithms on a publicly available hotel data set. Our framework is highly practical in that it requires no historical data (no fitted customer choice models, nor forecasting of customer arrival patterns) and can be used to initialize allocation strategies in fast-changing environments.