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Register a new userScalable Greedy Feature Selection via Weak Submodularity
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Greedy algorithms are widely used for problems in machine learning such as feature selection and set function optimization. Unfortunately, for large datasets, the running time of even greedy algorithms can be quite high. This is because for each greedy step we need to refit a model or calculate a function using the previously selected choices and the new candidate. Two algorithms that are faster approximations to the greedy forward selection were introduced recently ([Mirzasoleiman et al. 2013, 2015]). They achieve better performance by exploiting distributed computation and stochastic evaluation respectively. Both algorithms have provable performance guarantees for submodular functions. In this paper we show that divergent from previously held opinion, submodularity is not required to obtain approximation guarantees for these two algorithms. Specifically, we show that a generalized concept of weak submodularity suffices to give multiplicative approximation guarantees. Our result extends the applicability of these algorithms to a larger class of functions. Furthermore, we show that a bounded submodularity ratio can be used to provide data dependent bounds that can sometimes be tighter also for submodular functions. We empirically validate our work by showing superior performance of fast greedy approximations versus several established baselines on artificial and real datasets.
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We connect high-dimensional subset selection and submodular maximization. Our results extend the work of Das and Kempe (2011) from the setting of linear regression to arbitrary objective functions. For greedy feature selection, this connection allows us to obtain strong multiplicative performance bounds on several methods without statistical modeling assumptions. We also derive recovery guarantees of this form under standard assumptions. Our work shows that greedy algorithms perform within a constant factor from the best possible subset-selection solution for a broad class of general objective functions. Our methods allow a direct control over the number of obtained features as opposed to regularization parameters that only implicitly control sparsity. Our proof technique uses the concept of weak submodularity initially defined by Das and Kempe. We draw a connection between convex analysis and submodular set function theory which may be of independent interest for other statistical learning applications that have combinatorial structure.
The problem of inferring the direct causal parents of a response variable among a large set of explanatory variables is of high practical importance in many disciplines. Recent work in the field of causal discovery exploits invariance properties of models across different experimental conditions for detecting direct causal links. However, these approaches generally do not scale well with the number of explanatory variables, are difficult to extend to nonlinear relationships, and require data across different experiments. Inspired by {em Debiased} machine learning methods, we study a one-vs.-the-rest feature selection approach to discover the direct causal parent of the response. We propose an algorithm that works for purely observational data, while also offering theoretical guarantees, including the case of partially nonlinear relationships. Requiring only one estimation for each variable, we can apply our approach even to large graphs, demonstrating significant improvements compared to established approaches.
We consider the class of convex minimization problems, composed of a self-concordant function, such as the $logdet$ metric, a convex data fidelity term $h(cdot)$ and, a regularizing -- possibly non-smooth -- function $g(cdot)$. This type of problems have recently attracted a great deal of interest, mainly due to their omnipresence in top-notch applications. Under this emph{locally} Lipschitz continuous gradient setting, we analyze the convergence behavior of proximal Newton schemes with the added twist of a probable presence of inexact evaluations. We prove attractive convergence rate guarantees and enhance state-of-the-art optimization schemes to accommodate such developments. Experimental results on sparse covariance estimation show the merits of our algorithm, both in terms of recovery efficiency and complexity.
We consider the problem of model selection for the general stochastic contextual bandits under the realizability assumption. We propose a successive refinement based algorithm called Adaptive Contextual Bandit ({ttfamily ACB}), that works in phases and successively eliminates model classes that are too simple to fit the given instance. We prove that this algorithm is adaptive, i.e., the regret rate order-wise matches that of {ttfamily FALCON}, the state-of-art contextual bandit algorithm of Levi et. al 20, that needs knowledge of the true model class. The price of not knowing the correct model class is only an additive term contributing to the second order term in the regret bound. This cost possess the intuitive property that it becomes smaller as the model class becomes easier to identify, and vice-versa. We then show that a much simpler explore-then-commit (ETC) style algorithm also obtains a regret rate of matching that of {ttfamily FALCON}, despite not knowing the true model class. However, the cost of model selection is higher in ETC as opposed to in {ttfamily ACB}, as expected. Furthermore, {ttfamily ACB} applied to the linear bandit setting with unknown sparsity, order-wise recovers the model selection guarantees previously established by algorithms tailored to the linear setting.
Modern LiDAR-SLAM (L-SLAM) systems have shown excellent results in large-scale, real-world scenarios. However, they commonly have a high latency due to the expensive data association and nonlinear optimization. This paper demonstrates that actively selecting a subset of features significantly improves both the accuracy and efficiency of an L-SLAM system. We formulate the feature selection as a combinatorial optimization problem under a cardinality constraint to preserve the information matrixs spectral attributes. The stochastic-greedy algorithm is applied to approximate the optimal results in real-time. To avoid ill-conditioned estimation, we also propose a general strategy to evaluate the environments degeneracy and modify the feature number online. The proposed feature selector is integrated into a multi-LiDAR SLAM system. We validate this enhanced system with extensive experiments covering various scenarios on two sensor setups and computation platforms. We show that our approach exhibits low localization error and speedup compared to the state-of-the-art L-SLAM systems. To benefit the community, we have released the source code: https://ram-lab.com/file/site/m-loam.
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