arXiv subjects

Will Ma

Publications and source records attributed to Will Ma.

54 records · Page 3Linked to original sources

arXiv2023

Order-optimal Correlated Rounding for Fulfilling Multi-item E-commerce Orders

We study the dynamic fulfillment problem in e-commerce, in which incoming (multi-item) customer orders must be immediately dispatched to (a combination of) fulfillment centers that have the required inventory. A prevailing approach to this problem, pioneered by Jasin and Sinha (2015), is to write a ``deterministic'' linear program that dictates, for each item in an incoming multi-item order from a particular region, how frequently it should be dispatched to each fulfillment center (FC). However, dispatching items in a way that satisfies these frequency constraints, without splitting the order across too many FC's, is challenging. Jasin and Sinha identify this as a correlated rounding problem, and propose an intricate rounding scheme that they prove is suboptimal by a factor of at most $\approx q/4$ on a $q$-item order. This paper provides to our knowledge the first substantially improved scheme for this correlated rounding problem, which is suboptimal by a factor of at most $1+\ln(q)$. We provide another scheme for sparse networks, which is suboptimal by a factor of at most $d$ if each item is stored in at most $d$ FC's. We show both of these guarantees to be tight in terms of the dependence on $q$ or $d$. Our schemes are simple and fast, based on an intuitive idea -- items wait for FC's to ``open'' at random times, but observe them on ``dilated'' time scales. This also implies a new randomized rounding method for the classical Set Cover problem, which could be of general interest. We numerically test our new rounding schemes under the same realistic setups as Jasin and Sinha (2015) and find that they improve runtimes, shorten code, and robustly improve performance. Our code is made publicly available.

cs.DS↗

arXiv2022

Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

In the prophet secretary problem, $n$ values are drawn independently from known distributions, and presented in a uniformly random order. A decision-maker must accept or reject each value when it is presented, and may accept at most $k$ values in total. The objective is to maximize the expected sum of accepted values. We analyze the performance of static threshold policies, which accept the first $k$ values exceeding a fixed threshold (or all such values, if fewer than $k$ exist). We show that an appropriate threshold guarantees $γ_k = 1 - e^{-k}k^k/k!$ times the value of the offline optimal solution. Note that $γ_1 = 1-1/e$, and by Stirling's approximation $γ_k \approx 1-1/\sqrt{2 πk}$. This represents the best-known guarantee for the prophet secretary problem for all $k>1$, and is tight for all $k$ for the class of static threshold policies. We provide two simple methods for setting the threshold. Our first method sets a threshold such that $k \cdot γ_k$ values are accepted in expectation, and offers an optimal guarantee for all $k$. Our second sets a threshold such that the expected number of values exceeding the threshold is equal to $k$. This approach gives an optimal guarantee if $k > 4$, but gives sub-optimal guarantees for $k \le 4$. Our proofs use a new result for optimizing sums of independent Bernoulli random variables, which extends a classical result of Hoeffding (1956) and is likely to be of independent interest. Finally, we note that our methods for setting thresholds can be implemented under limited information about agents' values.

cs.GT↗

arXiv2022

When is Assortment Optimization Optimal?

Assortment optimization concerns the problem of selling items with fixed prices to a buyer who will purchase at most one. Typically, retailers select a subset of items, corresponding to an "assortment" of brands to carry, and make each selected item available for purchase at its brand-recommended price. Despite the tremendous importance in practice, the best method for selling these fixed-price items is not well understood, as retailers have begun experimenting with making certain items available only through a lottery. In this paper we analyze the maximum possible revenue that can be earned in this setting, given that the buyer's preference is private but drawn from a known distribution. In particular, we introduce a Bayesian mechanism design problem where the buyer has a random ranking over fixed-price items and an outside option, and the seller optimizes a (randomized) allocation of up to one item. We show that allocations corresponding to assortments are suboptimal in general, but under many commonly-studied Bayesian priors for buyer rankings such as the MNL and Markov Chain choice models, assortments are in fact optimal. Therefore, this large literature on assortment optimization has much greater significance than appreciated before -- it is not only computing optimal assortments; it is computing the economic limit of the seller's revenue for these fixed-price substitute items. We derive several further results -- a more general sufficient condition for assortments being optimal that captures choice models beyond Markov Chain, a proof that Nested Logit choice models cannot be captured by Markov Chain but can be partially captured by our condition, and suboptimality gaps for assortments when our condition does not hold. Finally, we show that our mechanism design problem provides the tightest-known LP relaxation for assortment optimization under the ranking distribution model.

cs.GT↗

arXiv2021

Fairness Maximization among Offline Agents in Online-Matching Markets

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.

cs.GT↗

arXiv2021

Inventory Balancing with Online Learning

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.

cs.AI↗

arXiv2021

Group-level Fairness Maximization in Online Bipartite Matching

We consider the allocation of limited resources to heterogeneous customers who arrive in an online fashion. We would like to allocate the resources "fairly", so that no group of customers is marginalized in terms of their overall service rate. We study whether this is possible to do so in an online fashion, and if so, what a good online allocation policy is. We model this problem using online bipartite matching under stationary arrivals, a fundamental model in the literature typically studied under the objective of maximizing the total number of customers served. We instead study the objective of maximizing the minimum service rate across all groups, and propose two notions of fairness: long-run and short-run. For these fairness objectives, we analyze how competitive online algorithms can be, in comparison to offline algorithms which know the sequence of demands in advance. For long-run fairness, we propose two online heuristics (Sampling and Pooling) which establish asymptotic optimality in different regimes (no specialized supplies, no rare demand types, or imbalanced supply/demand). By contrast, outside all of these regimes, we show that the competitive ratio of online algorithms is between 0.632 and 0.732. For short-run fairness, we show for complete bipartite graphs that the competitive ratio of online algorithms is between 0.863 and 0.942; we also derive a probabilistic rejection algorithm which is asymptotically optimal in the total demand. Depending on the overall scarcity of resources, either our Sampling or Pooling heuristics could be desirable. The most difficult situation for online allocation occurs when the total supply is just enough to serve the total demand. We simulate our algorithms on a public ride-hailing dataset, which both demonstrates the efficacy of our heuristics and validates our managerial insights.

cs.DS↗

arXiv2021

Reaping the Benefits of Bundling under High Production Costs

It is well-known that selling different goods in a single bundle can significantly increase revenue. However, bundling is no longer profitable if the goods have high production costs. To overcome this challenge, we introduce a new mechanism, Pure Bundling with Disposal for Cost (PBDC), where after buying the bundle, the customer is allowed to return any subset of goods for their costs. We provide two types of guarantees on the profit of PBDC mechanisms relative to the optimum in the presence of production costs, under the assumption that customers have valuations which are additive over the items and drawn independently. We first provide a distribution-dependent guarantee which shows that PBDC earns at least 1-6c^{2/3} of the optimal profit, where c denotes the coefficient of variation of the welfare random variable. c approaches 0 if there are a large number of items whose individual valuations have bounded coefficients of variation, and our constants improve upon those from the classical result of Bakos and Brynjolfsson (1999) without costs. We then provide a distribution-free guarantee which shows that either PBDC or individual sales earns at least 1/5.2 times the optimal profit, generalizing and improving the constant of 1/6 from the celebrated result of Babaioff et al. (2014). Conversely, we also provide the best-known upper bound on the performance of any partitioning mechanism (which captures both individual sales and pure bundling), of 1/1.19 times the optimal profit, improving on the previously-known upper bound of 1/1.08. Finally, we conduct simulations under the same playing field as the extensive numerical study of Chu et al. (2011), which confirm that PBDC outperforms other simple pricing schemes overall.

cs.GT↗

arXiv2020

Dynamic Pricing (and Assortment) under a Static Calendar

This work is motivated by our collaboration with a large consumer packaged goods (CPG) company. We have found that while the company appreciates the advantages of dynamic pricing, they deem it operationally much easier to plan out a static price calendar in advance. We investigate the efficacy of static control policies for revenue management problems whose optimal solution is inherently dynamic. In these problems, a firm has limited inventory to sell over a finite time horizon, over which heterogeneous customers stochastically arrive. We consider both pricing and assortment controls, and derive simple static policies in the form of a price calendar or a planned sequence of assortments, respectively. In the assortment planning problem, we also differentiate between the static vs. dynamic substitution models of customer demand. We show that our policies are within 1-1/e (approximately 0.63) of the optimum under stationary (IID) demand, and 1/2 of the optimum under non-stationary demand, with both guarantees approaching 1 if the starting inventories are large. We adapt the technique of prophet inequalities from optimal stopping theory to pricing and assortment problems, and our results are relative to the linear programming relaxation. Under the special case of IID single-item pricing, our results improve the understanding of irregular and discrete demand curves, by showing that a static calendar can be (1-1/e)-approximate if the prices are sorted high-to-low. Finally, we demonstrate on both data from the CPG company and synthetic data from the literature that our simple price and assortment calendars are effective.

cs.DS↗

arXiv2020

The Convex Relaxation Barrier, Revisited: Tightened Single-Neuron Relaxations for Neural Network Verification

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.

cs.LG↗

arXiv2020

Revenue-Optimal Deterministic Auctions for Multiple Buyers with Ordinal Preferences over Fixed-price Items

In this paper, we introduce a Bayesian revenue-maximizing mechanism design model where the items have fixed, exogenously-given prices. Buyers are unit-demand and have an ordinal ranking over purchasing either one of these items at its given price, or purchasing nothing. This model arises naturally from the assortment optimization problem, in that the single-buyer optimization problem over deterministic mechanisms reduces to deciding on an assortment of items to "show". We study its multi-buyer generalization in the simplest setting of single-winner auctions, or more broadly, any service-constrained environment. Our main result is that if the buyer rankings are drawn independently from Markov Chain ranking models, then the optimal mechanism is computationally tractable, and structurally a virtual welfare maximizer. We also show that for ranking distributions not induced by Markov Chains, the optimal mechanism may not be a virtual welfare maximizer.

cs.GT↗

arXiv2020

Multi-stage and Multi-customer Assortment Optimization with Inventory Constraints

We consider an assortment optimization problem where a customer chooses a single item from a sequence of sets shown to her, while limited inventories constrain the items offered to customers over time. In the special case where all of the assortments have size one, our problem captures the online stochastic matching with timeouts problem. For this problem, we derive a polynomial-time approximation algorithm which earns at least 1-ln(2-1/e), or 0.51, of the optimum. This improves upon the previous-best approximation ratio of 0.46, and furthermore, we show that it is tight. For the general assortment problem, we establish the first constant-factor approximation ratio of 0.09 for the case that different types of customers value items differently, and an approximation ratio of 0.15 for the case that different customers value each item the same. Our algorithms are based on rounding an LP relaxation for multi-stage assortment optimization, and improve upon previous randomized rounding schemes to derive the tight ratio of 1-ln(2-1/e).

cs.DS↗

arXiv2020

Strong mixed-integer programming formulations for trained neural networks

We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks. These formulations can be used for a number of important tasks, such as verifying that an image classification network is robust to adversarial inputs, or solving decision problems where the objective function is a machine learning model. We present a generic framework, which may be of independent interest, that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domains. We apply this result to derive MIP formulations for a number of the most popular nonlinear operations (e.g. ReLU and max pooling) that are strictly stronger than other approaches from the literature. We corroborate this computationally, showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks.

math.OC↗

arXiv2020

On Policies for Single-leg Revenue Management with Limited Demand Information

In this paper we study the single-item revenue management problem, with no information given about the demand trajectory over time. When the item is sold through accepting/rejecting different fare classes, Ball and Queyranne (2009) have established the tight competitive ratio for this problem using booking limit policies, which raise the acceptance threshold as the remaining inventory dwindles. However, when the item is sold through dynamic pricing instead, there is the additional challenge that offering a low price may entice high-paying customers to substitute down. We show that despite this challenge, the same competitive ratio can still be achieved using a randomized dynamic pricing policy. Our policy incorporates the price-skimming technique from Eren and Maglaras (2010), but importantly we show how the randomized price distribution should be stochastically-increased as the remaining inventory dwindles. A key technical ingredient in our policy is a new "valuation tracking" subroutine, which tracks the possible values for the optimum, and follows the most "inventory-conservative" control which maintains the desired competitive ratio. Finally, we demonstrate the empirical effectiveness of our policy in simulations, where its average-case performance surpasses all naive modifications of the existing policies.

cs.DS↗

arXiv2019

Prophet Inequalities on the Intersection of a Matroid and a Graph

We consider prophet inequalities in a setting where agents correspond to both elements in a matroid and vertices in a graph. A set of agents is feasible if they form both an independent set in the matroid and an independent set in the graph. Our main result is an ex-ante 1/(2d+2)-prophet inequality, where d is a graph parameter upper-bounded by the maximum size of an independent set in the neighborhood of any vertex. We establish this result through a framework that sets both dynamic prices for elements in the matroid (using the method of balanced thresholds), and static but discriminatory prices for vertices in the graph (motivated by recent developments in approximate dynamic programming). The threshold for accepting an agent is then the sum of these two prices. We show that for graphs induced by a certain family of interval-scheduling constraints, the value of d is 1. Our framework thus provides the first constant-factor prophet inequality when there are both matroid-independence constraints and interval-scheduling constraints. It also unifies and improves several results from the literature, leading to a 1/2-prophet inequality when agents have XOS valuation functions over a set of items and use them for a finite interval duration, and more generally, a 1/(d+1)-prophet inequality when these items each require a bundle of d resources to procure.

cs.DS↗

arXiv2019

Separation between Second Price Auctions with Personalized Reserves and the Revenue Optimal Auction

What fraction of the single item $n$ buyers setting's expected optimal revenue MyeRev can the second price auction with reserves achieve? In the special case where the buyers' valuation distributions are all drawn i.i.d. and the distributions satisfy the regularity condition, the second price auction with an anonymous reserve (ASP) is the optimal auction itself. As the setting gets more complex, there are established upper bounds on the fraction of MyeRev that ASP can achieve. On the contrary, no such upper bounds are known for the fraction of MyeRev achievable by the second price auction with eager personalized reserves (ESP). In particular, no separation was earlier known between ESP's revenue and MyeRev even in the most general setting of non-identical product distributions that don't satisfy the regularity condition. In this paper we establish the first separation results for ESP: we show that even in the case of distributions drawn i.i.d., but not necessarily satisfying the regularity condition, the ESP cannot achieve more than a $0.778$ fraction of MyeRev in general. Combined with Correa et al.'s result (EC 2017) that ESP can achieve at least a $0.745$ fraction of MyeRev, this nearly bridges the gap between upper and lower bounds on ESP's approximation factor.

cs.GT↗

arXiv2019

Algorithms for Online Matching, Assortment, and Pricing with Tight Weight-dependent Competitive Ratios

Motivated by the dynamic assortment offerings and item pricings occurring in e-commerce, we study a general problem of allocating finite inventories to heterogeneous customers arriving sequentially. We analyze this problem under the framework of competitive analysis, where the sequence of customers is unknown and does not necessarily follow any pattern. Previous work in this area, studying online matching, advertising, and assortment problems, has focused on the case where each item can only be sold at a single price, resulting in algorithms which achieve the best-possible competitive ratio of 1-1/e. In this paper, we extend all of these results to allow for items having multiple feasible prices. Our algorithms achieve the best-possible weight-dependent competitive ratios, which depend on the sets of feasible prices given in advance. Our algorithms are also simple and intuitive; they are based on constructing a class of universal ``value functions'' which integrate the selection of items and prices offered. Finally, we test our algorithms on the publicly-available hotel data set of Bodea et al. (2009), where there are multiple items (hotel rooms) each with multiple prices (fares at which the room could be sold). We find that applying our algorithms, as a ``hybrid'' with algorithms which attempt to forecast and learn the future transactions, results in the best performance.

cs.DS↗

arXiv2016

Improvements and Generalizations of Stochastic Knapsack and Multi-Armed Bandit Approximation Algorithms: Full Version

We study the multi-armed bandit problem with arms which are Markov chains with rewards. In the finite-horizon setting, the celebrated Gittins indices do not apply, and the exact solution is intractable. We provide approximation algorithms for a more general model which includes Markov decision processes and non-unit transition times. When preemption is allowed, we provide a (1/2-eps)-approximation, along with an example showing this is tight. When preemption isn't allowed, we provide a 1/12-approximation, which improves to a 4/27-approximation when transition times are unity. Our model encompasses the Markovian Bandits model of Gupta et al, the Stochastic Knapsack model of Dean, Goemans, and Vondrak, and the Budgeted Learning model of Guha and Munagala, and our algorithms improve existing results in all three areas. In our analysis, we encounter and overcome to our knowledge a novel obstacle - an algorithm that provably exists via polyhedral arguments, but cannot be found in polynomial time.

cs.DS↗

arXiv2013

A Geometric Approach to Combinatorial Fixed-Point Theorems

We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological fixed-point theorems. Our work naturally suggests several interesting open questions for future research.

math.CO↗