Consider the task of estimating a 3-order $n \times n \times n$ tensor from noisy observations of randomly chosen entries in the sparse regime. We introduce a similarity based collaborative filtering algorithm for estimating a tensor from sparse observations and argue that it achieves sample complexity that nearly matches the conjectured computationally efficient lower bound on the sample complexity for the setting of low-rank tensors. Our algorithm uses the matrix obtained from the flattened tensor to compute similarity, and estimates the tensor entries using a nearest neighbor estimator. We prove that the algorithm recovers a finite rank tensor with maximum entry-wise error (MEE) and mean-squared-error (MSE) decaying to $0$ as long as each entry is observed independently with probability $p = Ω(n^{-3/2 + κ})$ for any arbitrarily small $κ> 0$. More generally, we establish robustness of the estimator, showing that when arbitrary noise bounded by $\varepsilon \geq 0$ is added to each observation, the estimation error with respect to MEE and MSE degrades by $\text{poly}(\varepsilon)$. Consequently, even if the tensor may not have finite rank but can be approximated within $\varepsilon \geq 0$ by a finite rank tensor, then the estimation error converges to $\text{poly}(\varepsilon)$. Our analysis sheds insight into the conjectured sample complexity lower bound, showing that it matches the connectivity threshold of the graph used by our algorithm for estimating similarity between coordinates.
London School of Economics and Political Science（伦敦政治经济学院）；University of Minnesota（明尼苏达大学）
Matrix completion is a class of machine learning methods that concerns the prediction of missing entries in a partially observed matrix. This paper studies matrix completion for mixed data, i.e., data involving mixed types of variables (e.g., continuous, binary, ordinal). We formulate it as a low-rank matrix estimation problem under a general family of non-linear factor models and then propose entrywise consistent estimators for estimating the low-rank matrix. Tight probabilistic error bounds are derived for the proposed estimators. The proposed methods are evaluated by simulation studies and …
This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model --- the Bradley-Terry-Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress towards characterizing the performance (e.g. the m…
This paper introduces a novel message-passing (MP) framework for the collaborative filtering (CF) problem associated with recommender systems. We model the movie-rating prediction problem popularized by the Netflix Prize, using a probabilistic factor graph model and study the model by deriving generalization error bounds in terms of the training error. Based on the model, we develop a new MP algorithm, termed IMP, for learning the model. To show superiority of the IMP algorithm, we compare it with the closely related expectation-maximization (EM) based algorithm and a number of other matrix co…
Item neighbourhood methods for collaborative filtering learn a weighted graph over the set of items, where each item is connected to those it is most similar to. The prediction of a user's rating on an item is then given by that rating of neighbouring items, weighted by their similarity. This paper presents a new neighbourhood approach which we call item fields, whereby an undirected graphical model is formed over the item graph. The resulting prediction rule is a simple generalization of the classical approaches, which takes into account non-local information in the graph, allowing its best r…
Within the task of collaborative filtering two challenges for computing conditional probabilities exist. First, the amount of training data available is typically sparse with respect to the size of the domain. Thus, support for higher-order interactions is generally not present. Second, the variables that we are conditioning upon vary for each query. That is, users label different variables during each query. For this reason, there is no consistent input to output mapping. To address these problems we purpose a maximum entropy approach using a non-standard measure of entropy. This approach can…
We investigate top-$m$ arm identification, a basic problem in bandit theory, in a multi-agent learning model in which agents collaborate to learn an objective function. We are interested in designing collaborative learning algorithms that achieve maximum speedup (compared to single-agent learning algorithms) using minimum communication cost, as communication is frequently the bottleneck in multi-agent learning. We give both algorithmic and impossibility results, and conduct a set of experiments to demonstrate the effectiveness of our algorithms.
There is much empirical evidence that item-item collaborative filtering works well in practice. Motivated to understand this, we provide a framework to design and analyze various recommendation algorithms. The setup amounts to online binary matrix completion, where at each time a random user requests a recommendation and the algorithm chooses an entry to reveal in the user's row. The goal is to minimize regret, or equivalently to maximize the number of +1 entries revealed at any time. We analyze an item-item collaborative filtering algorithm that can achieve fundamentally better performance co…
Microsoft Research, Redmond（微软研究院，雷德蒙德）；Duke University（杜克大学）；Brown University（布朗大学）
Collaborative filtering is widely used in modern recommender systems. Recent research shows that variational autoencoders (VAEs) yield state-of-the-art performance by integrating flexible representations from deep neural networks into latent variable models, mitigating limitations of traditional linear factor models. VAEs are typically trained by maximizing the likelihood (MLE) of users interacting with ground-truth items. While simple and often effective, MLE-based training does not directly maximize the recommendation-quality metrics one typically cares about, such as top-N ranking. In this …
University of California, Davis（加利福尼亚大学戴维斯分校）；Department of Computer Science, University of California, Davis（加利福尼亚大学戴维斯分校计算机科学系）；Department of Statistics, University of California, Davis（加利福尼亚大学戴维斯分校统计系）
In this paper, we propose a listwise approach for constructing user-specific rankings in recommendation systems in a collaborative fashion. We contrast the listwise approach to previous pointwise and pairwise approaches, which are based on treating either each rating or each pairwise comparison as an independent instance respectively. By extending the work of (Cao et al. 2007), we cast listwise collaborative ranking as maximum likelihood under a permutation model which applies probability mass to permutations based on a low rank latent score matrix. We present a novel algorithm called SQL-Rank…
We extend variational autoencoders (VAEs) to collaborative filtering for implicit feedback. This non-linear probabilistic model enables us to go beyond the limited modeling capacity of linear factor models which still largely dominate collaborative filtering research.We introduce a generative model with multinomial likelihood and use Bayesian inference for parameter estimation. Despite widespread use in language modeling and economics, the multinomial likelihood receives less attention in the recommender systems literature. We introduce a different regularization parameter for the learning obj…
University of Pennsylvania（宾夕法尼亚大学）；NEC Research Institute（NEC研究院）
Recommender systems leverage product and community information to target products to consumers. Researchers have developed collaborative recommenders, content-based recommenders, and (largely ad-hoc) hybrid systems. We propose a unified probabilistic framework for merging collaborative and content-based recommendations. We extend Hofmann's [1999] aspect model to incorporate three-way co-occurrence data among users, items, and item content. The relative influence of collaboration data versus content data is not imposed as an exogenous parameter, but rather emerges naturally from the given data …
Recommendation and collaborative filtering systems are important in modern information and e-commerce applications. As these systems are becoming increasingly popular in the industry, their outputs could affect business decision making, introducing incentives for an adversarial party to compromise the availability or integrity of such systems. We introduce a data poisoning attack on collaborative filtering systems. We demonstrate how a powerful attacker with full knowledge of the learner can generate malicious data so as to maximize his/her malicious objectives, while at the same time mimickin…
Despite the prevalence of collaborative filtering in recommendation systems, there has been little theoretical development on why and how well it works, especially in the "online" setting, where items are recommended to users over time. We address this theoretical gap by introducing a model for online recommendation systems, cast item recommendation under the model as a learning problem, and analyze the performance of a cosine-similarity collaborative filtering method. In our model, each of $n$ users either likes or dislikes each of $m$ items. We assume there to be $k$ types of users, and all …
University of Hyderabad（海得拉巴大学）；School of Computer & Information Sciences（计算机与信息科学学院）
Machine learning techniques for Recommendation System (RS) and Classification has become a prime focus of research to tackle the problem of information overload. RS are software tools that aim at making informed decisions about the services that a user may like. On the other hand, classification technique deals with the categorization of a data object into one of the several predefined classes. In the multi-label classification problem, unlike the traditional multi-class classification setting, each instance can be simultaneously associated with a subset of labels. The focus of thesis is on th…
HKUST（香港科技大学）；Noah’s Ark Lab of Huawei Co. Ltd.（华为有限公司诺亚方舟实验室）
The challenge of object categorization in images is largely due to arbitrary translations and scales of the foreground objects. To attack this difficulty, we propose a new approach called collaborative receptive field learning to extract specific receptive fields (RF's) or regions from multiple images, and the selected RF's are supposed to focus on the foreground objects of a common category. To this end, we solve the problem by maximizing a submodular function over a similarity graph constructed by a pool of RF candidates. However, measuring pairwise distance of RF's for building the similari…
Content moderation is often performed by a collaboration between humans and machine learning models. However, it is not well understood how to design the collaborative process so as to maximize the combined moderator-model system performance. This work presents a rigorous study of this problem, focusing on an approach that incorporates model uncertainty into the collaborative process. First, we introduce principled metrics to describe the performance of the collaborative system under capacity constraints on the human moderator, quantifying how efficiently the combined system utilizes human dec…
