Active Ranking using Pairwise Comparisons

This paper examines the problem of ranking a collection of objects using pairwise comparisons (rankings of two objects). In general, the ranking of $n$ objects can be identified by standard sorting methods using $n log_2 n$ pairwise comparisons. We are interested in natural situations in which relationships among the objects may allow for ranking using far fewer pairwise comparisons. Specifically, we assume that the objects can be embedded into a $d$-dimensional Euclidean space and that the rankings reflect their relative distances from a common reference point in $R^d$. We show that under this assumption the number of possible rankings grows like $n^{2d}$ and demonstrate an algorithm that can identify a randomly selected ranking using just slightly more than $d log n$ adaptively selected pairwise comparisons, on average. If instead the comparisons are chosen at random, then almost all pairwise comparisons must be made in order to identify any ranking. In addition, we propose a robust, error-tolerant algorithm that only requires that the pairwise comparisons are probably correct. Experimental studies with synthetic and real datasets support the conclusions of our theoretical analysis.Comment: 17 pages, an extended version of our NIPS 2011 paper. The new version revises the argument of the robust section and slightly modifies the result there to give it more impac

Active classification with comparison queries

We study an extension of active learning in which the learning algorithm may ask the annotator to compare the distances of two examples from the boundary of their label-class. For example, in a recommendation system application (say for restaurants), the annotator may be asked whether she liked or disliked a specific restaurant (a label query); or which one of two restaurants did she like more (a comparison query). We focus on the class of half spaces, and show that under natural assumptions, such as large margin or bounded bit-description of the input examples, it is possible to reveal all the labels of a sample of size $n$ using approximately $O(\log n)$ queries. This implies an exponential improvement over classical active learning, where only label queries are allowed. We complement these results by showing that if any of these assumptions is removed then, in the worst case, $\Omega(n)$ queries are required. Our results follow from a new general framework of active learning with additional queries. We identify a combinatorial dimension, called the \emph{inference dimension}, that captures the query complexity when each additional query is determined by $O(1)$ examples (such as comparison queries, each of which is determined by the two compared examples). Our results for half spaces follow by bounding the inference dimension in the cases discussed above.Comment: 23 pages (not including references), 1 figure. The new version contains a minor fix in the proof of Lemma 4.

Individualized Rank Aggregation using Nuclear Norm Regularization

In recent years rank aggregation has received significant attention from the machine learning community. The goal of such a problem is to combine the (partially revealed) preferences over objects of a large population into a single, relatively consistent ordering of those objects. However, in many cases, we might not want a single ranking and instead opt for individual rankings. We study a version of the problem known as collaborative ranking. In this problem we assume that individual users provide us with pairwise preferences (for example purchasing one item over another). From those preferences we wish to obtain rankings on items that the users have not had an opportunity to explore. The results here have a very interesting connection to the standard matrix completion problem. We provide a theoretical justification for a nuclear norm regularized optimization procedure, and provide high-dimensional scaling results that show how the error in estimating user preferences behaves as the number of observations increase

Stochastic Non-convex Ordinal Embedding with Stabilized Barzilai-Borwein Step Size

Learning representation from relative similarity comparisons, often called ordinal embedding, gains rising attention in recent years. Most of the existing methods are batch methods designed mainly based on the convex optimization, say, the projected gradient descent method. However, they are generally time-consuming due to that the singular value decomposition (SVD) is commonly adopted during the update, especially when the data size is very large. To overcome this challenge, we propose a stochastic algorithm called SVRG-SBB, which has the following features: (a) SVD-free via dropping convexity, with good scalability by the use of stochastic algorithm, i.e., stochastic variance reduced gradient (SVRG), and (b) adaptive step size choice via introducing a new stabilized Barzilai-Borwein (SBB) method as the original version for convex problems might fail for the considered stochastic \textit{non-convex} optimization problem. Moreover, we show that the proposed algorithm converges to a stationary point at a rate $\mathcal{O}(\frac{1}{T})$ in our setting, where $T$ is the number of total iterations. Numerous simulations and real-world data experiments are conducted to show the effectiveness of the proposed algorithm via comparing with the state-of-the-art methods, particularly, much lower computational cost with good prediction performance.Comment: 11 pages, 3 figures, 2 tables, accepted by AAAI201

Query Complexity of Derivative-Free Optimization

This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same