Analysis of Crowdsourced Sampling Strategies for HodgeRank with Sparse Random Graphs

Crowdsourcing platforms are now extensively used for conducting subjective pairwise comparison studies. In this setting, a pairwise comparison dataset is typically gathered via random sampling, either \emph{with} or \emph{without} replacement. In this paper, we use tools from random graph theory to analyze these two random sampling methods for the HodgeRank estimator. Using the Fiedler value of the graph as a measurement for estimator stability (informativeness), we provide a new estimate of the Fiedler value for these two random graph models. In the asymptotic limit as the number of vertices tends to infinity, we prove the validity of the estimate. Based on our findings, for a small number of items to be compared, we recommend a two-stage sampling strategy where a greedy sampling method is used initially and random sampling \emph{without} replacement is used in the second stage. When a large number of items is to be compared, we recommend random sampling with replacement as this is computationally inexpensive and trivially parallelizable. Experiments on synthetic and real-world datasets support our analysis

HodgeRank with Information Maximization for Crowdsourced Pairwise Ranking Aggregation

Recently, crowdsourcing has emerged as an effective paradigm for human-powered large scale problem solving in various domains. However, task requester usually has a limited amount of budget, thus it is desirable to have a policy to wisely allocate the budget to achieve better quality. In this paper, we study the principle of information maximization for active sampling strategies in the framework of HodgeRank, an approach based on Hodge Decomposition of pairwise ranking data with multiple workers. The principle exhibits two scenarios of active sampling: Fisher information maximization that leads to unsupervised sampling based on a sequential maximization of graph algebraic connectivity without considering labels; and Bayesian information maximization that selects samples with the largest information gain from prior to posterior, which gives a supervised sampling involving the labels collected. Experiments show that the proposed methods boost the sampling efficiency as compared to traditional sampling schemes and are thus valuable to practical crowdsourcing experiments.Comment: Accepted by AAAI201

Sparse Recovery via Differential Inclusions

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, \emph{i.e.} finding the oracle estimator which is unbiased and sign-consistent using dynamics. We call our dynamics \emph{Bregman ISS} and \emph{Linearized Bregman ISS}. A well-known shortcoming of LASSO and any convex regularization approaches lies in the bias of estimators. However, we show that under proper conditions, there exists a bias-free and sign-consistent point on the solution paths of such dynamics, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the same signs as those of the true signs, \emph{i.e.} the oracle estimator. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased when sign-consistency is reached. We also show how to efficiently compute their solution paths in both continuous and discretized settings: the full solution paths can be exactly computed piece by piece, and a discretization leads to \emph{Linearized Bregman iteration}, which is a simple iterative thresholding rule and easy to parallelize. Theoretical guarantees such as sign-consistency and minimax optimal $l_2$-error bounds are established in both continuous and discrete settings for specific points on the paths. Early-stopping rules for identifying these points are given. The key treatment relies on the development of differential inequalities for differential inclusions and their discretizations, which extends the previous results and leads to exponentially fast recovering of sparse signals before selecting wrong ones.Comment: In Applied and Computational Harmonic Analysis, 201

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