Matrix Factorization Techniques for Recommender Systems

Order Restricted Statistical Inference.

Thurstonian and Bradley-Terry models are the most commonly applied models in the analysis of paired comparison data. Since their introduction, numerous developments have been proposed in different areas. This paper provides an updated overview of these extensions, including how to account for object- and subject-specific covariates and how to deal with ordinal paired comparison data. Special emphasis is given to models for dependent comparisons. Although these models are more realistic, their use is complicated by numerical difficulties. We therefore concentrate on implementation issues. In particular, a pairwise likelihood approach is explored for models for dependent paired comparison data, and a simulation study is carried out to compare the performance of maximum pairwise likelihood with other limited information estimation methods. The methodology is illustrated throughout using a real data set about university paired comparisons performed by students.

Models for Paired Comparison Data: A Review with Emphasis on Dependent Data

We consider data in the form of pairwise comparisons of n items, with the goal of precisely identifying the top k items for some value of k < n, or alternatively, recovering a ranking of all the items. We analyze the Copeland counting algorithm that ranks the items in order of the number of pairwise comparisons won, and show it has three attractive features: (a) its computational efficiency leads to speed-ups of several orders of magnitude in computation time as compared to prior work; (b) it is robust in that theoretical guarantees impose no conditions on the underlying matrix of pairwise-comparison probabilities, in contrast to some prior work that applies only to the BTL parametric model; and (c) it is an optimal method up to constant factors, meaning that it achieves the information-theoretic limits for recovering the top k-subset. We extend our results to obtain sharp guarantees for approximate recovery under the Hamming distortion metric, and more generally, to any arbitrary error requirement that satisfies a simple and natural monotonicity condition.

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Simple, Robust and Optimal Ranking from Pairwise Comparisons

The celebrated Vickrey-Clarke-Grove(VCG) mechanism induces selfish agents to behave truthfully by paying them a premium. In the process, it may end up paying more than the actual cost to the agents. For the minimum spanning tree problem, if the market is competitive, one can show that VCG never pays too much. On the other hand, for the shortest s-t path problem, Archer and Tardos [5] showed that VCG can overpay by a factor of Ω(n). A natural question that arises then is: For what problems does VCG overpay by a lot? We quantify this notion of overpayment, and show that the class of instances for which VCG never overpays is a natural generalization of matroids, that we call frugoids. We then give some sufficient conditions to upper bound and lower bound the overpayment in other cases, and apply these to several important combinatorial problems. We also relate the overpayment in an suitable model to the locality ratio of a natural local search procedure.

The price of truth: Frugality in truthful mechanisms

Author(s): Shah, Nihar Bhadresh | Advisor(s): Wainwright, Martin J; Ramchandran, Kannan | Abstract: Learning from people represents a new and expanding frontier for data science. Crowdsourcing, where data is collected from non-experts online, is now extensively employed in academic research, industry, and also for many societal causes. Two critical challenges in crowdsourcing and learning form people are that of (i) developing algorithms for maximally accurate learning and estimation that operate under minimal modeling assumptions, and (ii) designing incentive mechanisms to elicit high-quality data from people. In this thesis, we addresses these fundamental challenges in the context of several canonical problem settings that arise in learning from people.For the challenge of estimation, there are various algorithms proposed in past literature, but their reliance on strong parameter-based assumptions is severely limiting. In this thesis, we introduce a class of "permutation-based" models that are considerably richer than classical parameter-based models. We present algorithms for estimation which we show are both statistically optimal and significantly more robust than prior state-of-the-art methods. We also prove that our estimators automatically adapt and are simultaneously optimal over the classical parameter-based models as well, thereby enjoying a surprising win-win in the statistical bias-variance tradeoff. As for the second challenge of incentivizing people, we design a class of payment mechanisms that take a "multiplicative" form. For several common interfaces in crowdsourcing, we show that these multiplicative mechanisms are surprisingly the only mechanisms that can guarantee honest responses and satisfy a mild and natural requirement which we call no-free-lunch. We show that our mechanisms have several additional desirable qualities. The simplicity of our mechanisms imparts them with an additional practical appeal.

Nihar Bhadresh Shah

Papers

Learning From People