A Fast Stochastic Riemannian Eigensolver

We propose a fast stochastic Riemannian gradient eigensolver for a real and symmetric matrix, and prove its local, eigengap-dependent and linear convergence. The fast convergence is brought by deploying the variance reduction technique which was originally developed for the Euclidean strongly convex problems. In this paper, this technique is generalized to Riemannian manifolds for solving the geodesically non-convex problem of finding a group of top eigenvectors of such a matrix. We first propose the general variance reduction form of the stochastic Riemannian gradient, giving rise to the stochastic variance reduced Riemannian gradient method (SVRRG). It turns out that the operation of vector transport is necessary in addition to using Riemannian gradients and retraction operations. We then specialize it to the problem in question resulting in our SVRRG-EIGS algorithm. We are among the first to propose and analyze the generalization of the stochastic variance reduced gradient (SVRG) to Riemannian manifolds. As an extension of the linearly convergent VR-PCA, it is significant and nontrivial for the proposed algorithm to theoretically achieve a further speedup and empirically make a difference, due to our respect to the inherent geometry of the problem. ID: 49 link sup A Kernel Conditional Independence Test for Relational Data Sanghack Lee; Vasant Honavar Conditional independence (CI) tests play a central role in statistical inference, machine learning, and causal discovery. Most existing CI tests assume that the samples are independently and identically distributed (i.i.d.). However, this assumption often does not hold in the case of relational data. We define Relational Conditional Independence (RCI), a generalization of CI to the relational setting. We show how, under a set of structural assumptions, we can test for RCI by reducing the task of testing for RCI on non-i.i.d. data to the problem of testing for CI on several data sets each of which consists of i.i.d. samples. We develop Kernel Relational CI test (KRCIT), a nonparametric test as a practical approach to testing for RCI by relaxing the structural assumptions used in our analysis of RCI. We describe results of experiments with synthetic relational data that show the benefits of KRCIT relative to traditional CI tests that don’t account for the non-i.i.d. nature of relational data. ID: 171 link sup A Practical Method for Solving Contextual Bandit Problems Using Decision Trees Adam N. Elmachtoub; Ryan McNellis; Sechan Oh; Marek Petrik Many efficient algorithms with strong theoretical guarantees have been proposed for the contextual multi-armed bandit problem. However, applying these algorithms in practice can be difficult because they require domain expertise to build appropriate features and to tune their parameters. We propose a new method for the contextual bandit problem that is simple, practical, and can be applied with little or no domain expertise. Our algorithm relies on decision trees to model the context-reward relationship. Decision trees are non-parametric, interpretable, and work well without hand-crafted features. To guide the exploration-exploitation trade-off, we use a bootstrapping approach which abstracts Thompson sampling to non-Bayesian settings. We also discuss several computational heuristics and demonstrate the performance of our method on several datasets. ID: 278 link A Probabilistic Framework for Zero-Shot Multi-Label Learning Abhilash Gaure; Aishwarya Gupta; Vinay Kumar Verma; Piyush Rai We present a probabilistic framework for multi-label learning for the setting when the test data may require predicting labels that were not available at training time (i.e., the zero-shot learning setting). We develop a probabilistic model that leverages the co-occurrence statistics of the labels via a joint generative model for the label matrix (which denotes the label presence/absence for each training example) and for the label co-occurrence matrix (which denotes how many times a pair of labels co-occurs with each other). In addition to handling the unseen labels at test time, leveraging the co-occurrence information may also help in the standard multi-label learning setting, especially if the number of training examples is very small and/or the label matrix of training examples has a large fraction of missing entries. Our experimental results demonstrate the efficacy of our model in handling unseen labels. ID: 209 link A Reinforcement Learning Approach to Weaning of Mechanical Ventilation in Intensive Care Units Niranjani Prasad; Li-Fang Cheng; Corey Chivers; Michael Draugelis; Barbara E Engelhardt The management of invasive mechanical ventilation, and the regulation of sedation and analgesia during ventilation, constitutes a major part of the care of patients admitted to intensive care units. Both prolonged dependence on mechanical ventilation and premature extubation are associated with increased risk of complications and higher hospital costs, but clinical opinion on the best protocol for weaning patients off the ventilator varies. This work looks to develop a decision support tool that uses available information to predict time to extubation readiness and recommend a personalized regime of sedation dosage and ventilator support. To this end, we employ off-policy reinforcement learning algorithms to determine the best action at a given patient state from sub-optimal historical ICU data. We compare treatment policies from fitted Q-iteration with extremely randomized trees and with feedforward neural networks, and demonstrate that the policies learnt show promise in recommending weaning protocols with improved outcomes, in terms of minimizing the rate of reintubation and closer regulation of physiological stability. ID: 220 link sup A Tractable Probabilistic Model for Subset Selection Yujia Shen; Arthur Choi; Adnan Darwiche Subset selection tasks, such as top-k ranking, induce datasets where examples have cardinalities that are known a priori. In this paper, we propose a tractable probabilistic model for subset selection and show how it can be learned from data. Our proposed model is interpretable and subsumes a previously introduced model based on logistic regression. We show how the parameters of our model can be estimated in closed form given complete data, and propose an algorithm for learning its structure in an interpretable space. We highlight the intuitive structures that we learn via case studies. We finally show how our proposed model can be viewed as an instance of the recently proposed Probabilistic Sentential Decision Diagram. ID: 306 link sup Adversarial Sets for Regularising Neural Link Predictors Pasquale Minervini; Thomas Demeester; Tim Rockt{ä}schel; Sebastian Riedel In adversarial training, a set of models learn together by pursuing competing goals, usually defined on single data instances. However, in relational learning and other non-i.i.d domains, goals can also be defined over sets of instances. For example, a link predictor for the is-a relation needs to be consistent with the transitivity property: if is-a(x_1, x_2) and is-a(x_2, x_3) hold, is-a(x_1, x_3) needs to hold as well. Here we use such assumptions for deriving an inconsistency loss, measuring the degree to which the model violates the assumptions on an adversarially-generated set of examples. The training objective is defined as a minimax problem, where an adversary finds the most offending adversarial examples by maximising the inconsistency loss, and the model is trained by jointly minimising a supervised loss and the inconsistency loss on the adversarial examples. This yields the first method that can use function-free Horn clauses (as in Datalog) to regularise any neural link predictor, with complexity independent of the domain size. We show that for several link prediction models, the optimisation problem faced by the adversary has efficient closed-form solutions. Experiments on link prediction benchmarks indicate that given suitable prior knowledge, our method can significantly improve neural link predictors on all relevant metrics. ID: 277 link sup Algebraic Equivalence of Linear Structural Equation Models Thijs van Ommen; Joris M. Mooij Despite their popularity, many questions about the algebraic constraints imposed by linear structural equation models remain open problems. For causal discovery, two of these problems are especially important: the enumeration of the constraints imposed by a model, and deciding whether two graphs define the same statistical model. We show how the half-trek criterion can be used to make progress in both of these problems. We apply our theoretical results to a small-scale model selection problem, and find that taking the additional algebraic constraints into account may lead to significant improvements in model selection accuracy. ID: 54 link sup Best Student Paper Honorable Mention