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We give an online algorithm and prove novel mistake and regret bounds for online binary matrix completion with side information. The mistake bounds we prove are of the form $tilde{O}(D/gamma^2)$. The term $1/gamma^2$ is analogous to the usual margin term in SVM (perceptron) bounds. More specifically, if we assume that there is some factorization of the underlying $m times n$ matrix into $P Q^intercal$ where the rows of $P$ are interpreted as classifiers in $mathcal{R}^d$ and the rows of $Q$ as instances in $mathcal{R}^d$, then $gamma$ is the maximum (normalized) margin over all factorizations $P Q^intercal$ consistent with the observed matrix. The quasi-dimension term $D$ measures the quality of side information. In the presence of vacuous side information, $D= m+n$. However, if the side information is predictive of the underlying factorization of the matrix, then in an ideal case, $D in O(k + ell)$ where $k$ is the number of distinct row factors and $ell$ is the number of distinct column factors. We additionally provide a generalization of our algorithm to the inductive setting. In this setting, we provide an example where the side information is not directly specified in advance. For this example, the quasi-dimension $D$ is now bounded by $O(k^2 + ell^2)$.
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Supervised, semi-supervised, and unsupervised learning estimate a function given input/output samples. Generalization of the learned function to unseen data can be improved by incorporating side information into learning. Side information are data that are neither from the input space nor from the output space of the function, but include useful information for learning it. In this paper we show that learning with side information subsumes a variety of related approaches, e.g. multi-task learning, multi-view learning and learning using privileged information. Our main contributions are (i) a new perspective that connects these previously isolated approaches, (ii) insights about how these methods incorporate different types of prior knowledge, and hence implement different patterns, (iii) facilitating the application of these methods in novel tasks, as well as (iv) a systematic experimental evaluation of these patterns in two supervised learning tasks.
Feature missing is a serious problem in many applications, which may lead to low quality of training data and further significantly degrade the learning performance. While feature acquisition usually involves special devices or complex process, it is expensive to acquire all feature values for the whole dataset. On the other hand, features may be correlated with each other, and some values may be recovered from the others. It is thus important to decide which features are most informative for recovering the other features as well as improving the learning performance. In this paper, we try to train an effective classification model with least acquisition cost by jointly performing active feature querying and supervised matrix completion. When completing the feature matrix, a novel target function is proposed to simultaneously minimize the reconstruction error on observed entries and the supervised loss on training data. When querying the feature value, the most uncertain entry is actively selected based on the variance of previous iterations. In addition, a bi-objective optimization method is presented for cost-aware active selection when features bear different acquisition costs. The effectiveness of the proposed approach is well validated by both theoretical analysis and experimental study.
We propose orthogonal inductive matrix completion (OMIC), an interpretable approach to matrix completion based on a sum of multiple orthonormal side information terms, together with nuclear-norm regularization. The approach allows us to inject prior knowledge about the singular vectors of the ground truth matrix. We optimize the approach by a provably converging algorithm, which optimizes all components of the model simultaneously. We study the generalization capabilities of our method in both the distribution-free setting and in the case where the sampling distribution admits uniform marginals, yielding learning guarantees that improve with the quality of the injected knowledge in both cases. As particular cases of our framework, we present models which can incorporate user and item biases or community information in a joint and additive fashion. We analyse the performance of OMIC on several synthetic and real datasets. On synthetic datasets with a sliding scale of user bias relevance, we show that OMIC better adapts to different regimes than other methods. On real-life datasets containing user/items recommendations and relevant side information, we find that OMIC surpasses the state-of-the-art, with the added benefit of greater interpretability.
In this paper, we study Contextual Unsupervised Sequential Selection (USS), a new variant of the stochastic contextual bandits problem where the loss of an arm cannot be inferred from the observed feedback. In our setup, arms are associated with fixed costs and are ordered, forming a cascade. In each round, a context is presented, and the learner selects the arms sequentially till some depth. The total cost incurred by stopping at an arm is the sum of fixed costs of arms selected and the stochastic loss associated with the arm. The learners goal is to learn a decision rule that maps contexts to arms with the goal of minimizing the total expected loss. The problem is challenging as we are faced with an unsupervised setting as the total loss cannot be estimated. Clearly, learning is feasible only if the optimal arm can be inferred (explicitly or implicitly) from the problem structure. We observe that learning is still possible when the problem instance satisfies the so-called Contextual Weak Dominance (CWD) property. Under CWD, we propose an algorithm for the contextual USS problem and demonstrate that it has sub-linear regret. Experiments on synthetic and real datasets validate our algorithm.
Due to the insufficient measurements in the distribution system state estimation (DSSE), full observability and redundant measurements are difficult to achieve without using the pseudo measurements. The matrix completion state estimation (MCSE) combines the matrix completion and power system model to estimate voltage by exploring the low-rank characteristics of the matrix. This paper proposes a robust matrix completion state estimation (RMCSE) to estimate the voltage in a distribution system under a low-observability condition. Tradition state estimation weighted least squares (WLS) method requires full observability to calculate the states and needs redundant measurements to proceed a bad data detection. The proposed method improves the robustness of the MCSE to bad data by minimizing the rank of the matrix and measurements residual with different weights. It can estimate the system state in a low-observability system and has robust estimates without the bad data detection process in the face of multiple bad data. The method is numerically evaluated on the IEEE 33-node radial distribution system. The estimation performance and robustness of RMCSE are compared with the WLS with the largest normalized residual bad data identification (WLS-LNR), and the MCSE.
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