An Ordinal Minimax Theorem

In the early 1950s Lloyd Shapley proposed an ordinal and set-valued solution concept for zero-sum games called \emph{weak saddle}. We show that all weak saddles of a given zero-sum game are interchangeable and equivalent. As a consequence, every such game possesses a unique set-based value.Comment: 10 pages, 2 figure

When is it Better to Compare than to Score?

When eliciting judgements from humans for an unknown quantity, one often has the choice of making direct-scoring (cardinal) or comparative (ordinal) measurements. In this paper we study the relative merits of either choice, providing empirical and theoretical guidelines for the selection of a measurement scheme. We provide empirical evidence based on experiments on Amazon Mechanical Turk that in a variety of tasks, (pairwise-comparative) ordinal measurements have lower per sample noise and are typically faster to elicit than cardinal ones. Ordinal measurements however typically provide less information. We then consider the popular Thurstone and Bradley-Terry-Luce (BTL) models for ordinal measurements and characterize the minimax error rates for estimating the unknown quantity. We compare these minimax error rates to those under cardinal measurement models and quantify for what noise levels ordinal measurements are better. Finally, we revisit the data collected from our experiments and show that fitting these models confirms this prediction: for tasks where the noise in ordinal measurements is sufficiently low, the ordinal approach results in smaller errors in the estimation

Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence

Data in the form of pairwise comparisons arises in many domains, including preference elicitation, sporting competitions, and peer grading among others. We consider parametric ordinal models for such pairwise comparison data involving a latent vector $w^* \in \mathbb{R}^d$ that represents the "qualities" of the $d$ items being compared; this class of models includes the two most widely used parametric models--the Bradley-Terry-Luce (BTL) and the Thurstone models. Working within a standard minimax framework, we provide tight upper and lower bounds on the optimal error in estimating the quality score vector $w^*$ under this class of models. The bounds depend on the topology of the comparison graph induced by the subset of pairs being compared via its Laplacian spectrum. Thus, in settings where the subset of pairs may be chosen, our results provide principled guidelines for making this choice. Finally, we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre-factors.Comment: 39 pages, 5 figures. Significant extension of arXiv:1406.661

Counting submodules of a module over a noetherian commutative ring

We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple module as subquotient. We deduce in particular a characterization of uniserial modules over commutative noetherian rings.Comment: 34 pages. v2: expanded introduction and preliminarie

Hybrid Shrinkage Estimators Using Penalty Bases For The Ordinal One-Way Layout

This paper constructs improved estimators of the means in the Gaussian saturated one-way layout with an ordinal factor. The least squares estimator for the mean vector in this saturated model is usually inadmissible. The hybrid shrinkage estimators of this paper exploit the possibility of slow variation in the dependence of the means on the ordered factor levels but do not assume it and respond well to faster variation if present. To motivate the development, candidate penalized least squares (PLS) estimators for the mean vector of a one-way layout are represented as shrinkage estimators relative to the penalty basis for the regression space. This canonical representation suggests further classes of candidate estimators for the unknown means: monotone shrinkage (MS) estimators or soft-thresholding (ST) estimators or, most generally, hybrid shrinkage (HS) estimators that combine the preceding two strategies. Adaptation selects the estimator within a candidate class that minimizes estimated risk. Under the Gaussian saturated one-way layout model, such adaptive estimators minimize risk asymptotically over the class of candidate estimators as the number of factor levels tends to infinity. Thereby, adaptive HS estimators asymptotically dominate adaptive MS and adaptive ST estimators as well as the least squares estimator. Local annihilators of polynomials, among them difference operators, generate penalty bases suitable for a range of numerical examples.Comment: Published at http://dx.doi.org/10.1214/009053604000000652 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust Algorithms for the Secretary Problem

In classical secretary problems, a sequence of n elements arrive in a uniformly random order, and we want to choose a single item, or a set of size K. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin’s popular 1/e-secretary algorithm is sensitive to even a single adversarial arrival: if the adversary gives one large bid at the beginning of the stream, the algorithm does not select any element at all. We investigate a robust version of the secretary problem. In the Byzantine Secretary model, we have two kinds of elements: green (good) and red (rogue). The values of all elements are chosen by the adversary. The green elements arrive at times uniformly randomly drawn from [0,1]. The red elements, however, arrive at adversarially chosen times. Naturally, the algorithm does not see these colors: how well can it solve secretary problems? We show that selecting the highest value red set, or the single largest green element is not possible with even a small fraction of red items. However, on the positive side, we show that these are the only bad cases, by giving algorithms which get value comparable to the value of the optimal green set minus the largest green item. (This benchmark reminds us of regret minimization and digital auctions, where we subtract an additive term depending on the "scale" of the problem.) Specifically, we give an algorithm to pick K elements, which gets within (1-ε) factor of the above benchmark, as long as K ≥ poly(ε^{-1} log n). We extend this to the knapsack secretary problem, for large knapsack size K. For the single-item case, an analogous benchmark is the value of the second-largest green item. For value-maximization, we give a poly log^* n-competitive algorithm, using a multi-layered bucketing scheme that adaptively refines our estimates of second-max over time. For probability-maximization, we show the existence of a good randomized algorithm, using the minimax principle. We hope that this work will spur further research on robust algorithms for the secretary problem, and for other problems in sequential decision-making, where the existing algorithms are not robust and often tend to overfit to the model.ISSN:1868-896

Infinite presentability of groups and condensation

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.Comment: 32 pages, no figure. 1->2 Major changes (the 13-page first version, authored by Y.C. and L.G., was entitled "On infinitely presented soluble groups") 2->3 some changes including cuts in Section