Inferring Rankings Using Constrained Sensing

We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over $n$ elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size $\ll$ domain size). Our recovery method is based on finding the sparsest solution (through $\ell_0$ optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through $\ell_0$ optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as $n \to \infty$. $\ell_0$ optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, $\ell_1$ optimization, to find the sparsest solution. However, we show that $\ell_1$ optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as $n \to \infty$. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible.Comment: 19 page

Fair Scheduling in Networks Through Packet Election

We consider the problem of designing a fair scheduling algorithm for discrete-time constrained queuing networks. Each queue has dedicated exogenous packet arrivals. There are constraints on which queues can be served simultaneously. This model effectively describes important special instances like network switches, interference in wireless networks, bandwidth sharing for congestion control and traffic scheduling in road roundabouts. Fair scheduling is required because it provides isolation to different traffic flows; isolation makes the system more robust and enables providing quality of service. Existing work on fairness for constrained networks concentrates on flow based fairness. As a main result, we describe a notion of packet based fairness by establishing an analogy with the ranked election problem: packets are voters, schedules are candidates and each packet ranks the schedules based on its priorities. We then obtain a scheduling algorithm that achieves the described notion of fairness by drawing upon the seminal work of Goodman and Markowitz (1952). This yields the familiar Maximum Weight (MW) style algorithm. As another important result we prove that algorithm obtained is throughput optimal. There is no reason a priori why this should be true, and the proof requires non-traditional methods.Comment: 14 pages (double column), submitted to IEEE Transactions on Information Theor

Inferring rankings using constrained sensing

We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over n elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size <;<; domain size). Our recovery method is based on finding the sparsest solution (through l[subscript 0] optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through l[subscript 0] optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as n → ∞. ℓ0 optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, ℓ[subscript 1] optimization, to find the sparsest solution. However, we show that ℓ[subscript 1] optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as n → ∞. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible

Nonparametric choice modeling : applications to operations management

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 257-263).With the recent explosion of choices available to us in every walk of our life, capturing the choice behavior exhibited by individuals has become increasingly important to many businesses. At the core, capturing choice behavior boils down to being able to predict the probability of choosing a particular alternative from an offer set, given historical choice data about an individual or a group of "similar" individuals. For such predictions, one uses what is called a choice model, which models each choice occasion as follows: given an offer set, a preference list over alternatives is sampled according to a certain distribution, and the individual chooses the most preferred alternative according to the sampled preference list. Most existing literature, which dates back to at least the 1920s, considers parametric approaches to choice modeling. The goal of this thesis is to deviate from the existing approaches to propose a nonparametric approach to modeling choice. Apart from the usual advantages, the primary strength of a nonparametric model is its ability to scale with the data - certainly crucial to applications of our interest where choice behavior is highly dynamic. Given this, the main contribution of the thesis is to operationalize the nonparametric approach and demonstrate its success in several important applications. Specifically, we consider two broad setups: (1) solving decision problems using choice models, and (2) learning the choice models. In both setups, data available corresponds to marginal information about the underlying distribution over rankings. So the problems essentially boil down to designing the 'right' criterion to pick a model from one of the (several) distributions that are consistent with the available marginal information. First, we consider a central decision problem in operations management (OM): find an assortment of products that maximizes the revenues subject to a capacity constraint on the size of the assortment. Solving this problem requires two components: (a) predicting revenues for assortments and (b) searching over all subsets of a certain size for the optimal assortment. In order to predict revenues for an assortment, of all models consistent with the data, we use the choice model that results in the 'worst-case' revenue. We derive theoretical guarantees for the predictions, and show that the accuracy of predictions is good for the cases when the choice data comes from several different parametric models. Finally, by applying our approach to real-world sales transaction data from a major US automaker, we demonstrate an improvement in accuracy of around 20% over state-of-the-art parametric approaches. Once we have revenue predictions, we consider the problem of finding the optimal assortment. It has been shown that this problem is provably hard for most of the important families of parametric of choice models, except the multinomial logit (MNL) model. In addition, most of the approximation schemes proposed in the literature are tailored to a specific parametric structure. We deviate from this and propose a general algorithm to find the optimal assortment assuming access to only a subroutine that gives revenue predictions; this means that the algorithm can be applied with any choice model. We prove that when the underlying choice model is the MNL model, our algorithm can find the optimal assortment efficiently. Next, we consider the problem of learning the underlying distribution from the given marginal information. For that, of all the models consistent with the data, we propose to select the sparsest or simplest model, where we measure sparsity as the support size of the distribution. Finding the sparsest distribution is hard in general, so we restrict our search to what we call the 'signature family' to obtain an algorithm that is computationally efficient compared to the brute-force approach. We show that the price one pays for restricting the search to the signature family is minimal by establishing that for a large class of models, there exists a "sparse enough" model in the signature family that fits the given marginal information well. We demonstrate the efficacy of learning sparse models on the well-known American Psychological Association (APA) dataset by showing that our sparse approximation manages to capture useful structural properties of the underlying model. Finally, our results suggest that signature condition can be considered an alternative to the recently popularized Restricted Null Space condition for efficient recovery of sparse models.by Srikanth Jagabathula.Ph.D

Near-Optimal Non-Convex Stochastic Optimization under Generalized Smoothness

The generalized smooth condition, $(L_{0},L_{1})$-smoothness, has triggered people's interest since it is more realistic in many optimization problems shown by both empirical and theoretical evidence. Two recent works established the $O(\epsilon^{-3})$ sample complexity to obtain an $O(\epsilon)$-stationary point. However, both require a large batch size on the order of $\mathrm{ploy}(\epsilon^{-1})$, which is not only computationally burdensome but also unsuitable for streaming applications. Additionally, these existing convergence bounds are established only for the expected rate, which is inadequate as they do not supply a useful performance guarantee on a single run. In this work, we solve the prior two problems simultaneously by revisiting a simple variant of the STORM algorithm. Specifically, under the $(L_{0},L_{1})$-smoothness and affine-type noises, we establish the first near-optimal $O(\log(1/(\delta\epsilon))\epsilon^{-3})$ high-probability sample complexity where $\delta\in(0,1)$ is the failure probability. Besides, for the same algorithm, we also recover the optimal $O(\epsilon^{-3})$ sample complexity for the expected convergence with improved dependence on the problem-dependent parameter. More importantly, our convergence results only require a constant batch size in contrast to the previous works.Comment: The whole paper is rewritten with new results in V

Demand Estimation under Uncertain Consideration Sets

To estimate customer demand, choice models rely both on what the individuals do and do not purchase. A customer may not purchase a product because it was not offered but also because it was not considered. To account for this behavior, existing literature has proposed the so-called consider-then-choose (CTC) models, which posit that customers sample a consideration set and then choose the most preferred product from the intersection of the offer set and the consideration set. CTC models have been studied quite extensively in the marketing literature. More recently, they have gained popularity within the operations management (OM) literature to make assortment and pricing decisions. Despite their richness, CTC models are difficult to estimate in practice because firms typically do not observe customers' consideration sets. Therefore, the common assumption in OM has been that customers consider everything on offer, so the consideration set is the same as the offer set. This raises the following question: When firms only collect transaction data, do CTC models provide any predictive advantage over classic choice models? More precisely, under what conditions do CTC models outperform (if ever) classic choice models in terms of prediction accuracy? In this work, we study a general class of CTC models. We propose techniques to estimate these models efficiently from sales transaction data. We then compare their performance against the classic approach. We find that CTC models outperform standard choice models when there is noise in the offer set information and the noise is asymmetric across the training and test offer sets but otherwise lead to no particular predictive advantage over the classic approach. We also demonstrate the benefits of using CTC models in real-world retail settings. In particular, we show that CTC models calibrated on retail transaction data are better at long-term and warehouse level sales forecasts. We also evaluate their performance in the context of an online platform setting: a peer-to-peer car sharing company. In this context, offer sets are even difficult to define. We observe a remarkable performance of CTC models over standard choice models therein.Este documento es la versión aceptada del artículo publicado en Operations Research (ISSN 0030-364X) en Septiembre de 202

A Data-Driven Approach to Modeling Choice

We visit the following fundamental problem: For a 'generic' model of consumer choice (namely, distributions over preference lists) and a limited amount of data on how consumers actually make decisions (such as marginal preference information), how may one predict revenues from offering a particular assortment of choices? This problem is central to areas within operations research, marketing and econometrics. We present a framework to answer such questions and design a number of tractable algorithms (from a data and computational standpoint) for the same.National Science Foundation (U.S.) (CAREER CNS 0546590

Optimal scheduling algorithms for arbitrary networks

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 89-91).We consider the problem of designing scheduling schemes for networks with arbitrary topology and scheduling constraints. We address the optimality of scheduling schemes for packet networks in terms of throughput, delay and fairness. Specifically, we design two scheduling schemes. The first one achieves simultaneous throughput and delay optimization. The second scheme provides fairness. We design a scheduling scheme that guarantees a per-flow average delay bound of O(number of hops), with a constant factor loss of throughput. We derive the constants for a network operating under primary interference constraints. Our scheme guarantees an average delay bound of ... is the number of hops and pj is the effective loading along flow j. This delay guarantee comes at a factor 5 loss of throughput. We also provide a counter-example to prove the essential optimality of our result. For the fair scheduling scheme, we define a packet-based notion of fairness by establishing a novel analogy with the ranked election problem. The election scheme of Goodman and Markowitz (1952) [14] yields a maximum weight style scheduling algorithm. We then prove the throughput optimality of the scheme for a single-hop network. Standard methods for proving the stability of queuing systems fail us and hence we introduce a non-standard proof technique with potential wider applications.by Srikanth Jagabathula.S.M