We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse approximation. We analyze the performance of widely used greedy heuristics, using insights from the maximization of submodular functions and spectral analysis. We introduce the submodularity ratio as a key quantity to help understand why greedy algorithms perform well even when the variables are highly correlated. Using our techniques, we obtain the strongest known approximation guarantees for this problem, both in terms of the submodularity ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further demonstrate the wide applicability of our techniques by analyzing greedy algorithms for the dictionary selection problem, and significantly improve the previously known guarantees. Our theoretical analysis is complemented by experiments on real-world and synthetic data sets; the experiments show that the submodularity ratio is a stronger predictor of the performance of greedy algorithms than other spectral parameters.
University of Tokyo（东京大学）；NTT Communication Science Laboratories（NTT通信科学实验室）
We propose a new concept named adaptive submodularity ratio to study the greedy policy for sequential decision making. While the greedy policy is known to perform well for a wide variety of adaptive stochastic optimization problems in practice, its theoretical properties have been analyzed only for a limited class of problems. We narrow the gap between theory and practice by using adaptive submodularity ratio, which enables us to prove approximation guarantees of the greedy policy for a substantially wider class of problems. Examples of newly analyzed problems include important applications su…
We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm's memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm ST…
Graduate School of Information Sciences and Technology（信息科学技术研究生院）；The University of Tokyo（东京大学）
In dictionary selection, several atoms are selected from finite candidates that successfully approximate given data points in the sparse representation. We propose a novel efficient greedy algorithm for dictionary selection. Not only does our algorithm work much faster than the known methods, but it can also handle more complex sparsity constraints, such as average sparsity. Using numerical experiments, we show that our algorithm outperforms the known methods for dictionary selection, achieving competitive performances with dictionary learning algorithms in a smaller running time.
Yale University（耶鲁大学）；The Open University of Israel（以色列开放大学）
Submodular functions are a broad class of set functions, which naturally arise in diverse areas. Many algorithms have been suggested for the maximization of these functions. Unfortunately, once the function deviates from submodularity, the known algorithms may perform arbitrarily poorly. Amending this issue, by obtaining approximation results for set functions generalizing submodular functions, has been the focus of recent works. One such class, known as weakly submodular functions, has received a lot of attention. A key result proved by Das and Kempe (2011) showed that the approximation ratio…
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 evaluat…
Sharif University（谢里夫大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）
The standard greedy algorithm has been recently shown to enjoy approximation guarantees for constrained non-submodular nondecreasing set function maximization. While these recent results allow to better characterize the empirical success of the greedy algorithm, they are only applicable to simple cardinality constraints. In this paper, we study the problem of maximizing a non-submodular nondecreasing set function subject to a general matroid constraint. We first show that the standard greedy algorithm offers an approximation factor of $\frac{0.4 γ^{2}}{\sqrt{γr} + 1}$, where $γ$ is the submodu…
In this work we propose a new deep learning tool called deep dictionary learning. Multi-level dictionaries are learnt in a greedy fashion, one layer at a time. This requires solving a simple (shallow) dictionary learning problem, the solution to this is well known. We apply the proposed technique on some benchmark deep learning datasets. We compare our results with other deep learning tools like stacked autoencoder and deep belief network; and state of the art supervised dictionary learning tools like discriminative KSVD and label consistent KSVD. Our method yields better results than all.
We consider interactive learning and covering problems, in a setting where actions may incur different costs, depending on the response to the action. We propose a natural greedy algorithm for response-dependent costs. We bound the approximation factor of this greedy algorithm in active learning settings as well as in the general setting. We show that a different property of the cost function controls the approximation factor in each of these scenarios. We further show that in both settings, the approximation factor of this greedy algorithm is near-optimal among all greedy algorithms. Experime…
In this paper, we propose a unified framework and an algorithm for the problem of group recommendation where a fixed number of items or alternatives can be recommended to a group of users. The problem of group recommendation arises naturally in many real world contexts, and is closely related to the budgeted social choice problem studied in economics. We frame the group recommendation problem as choosing a subgraph with the largest group consensus score in a completely connected graph defined over the item affinity matrix. We propose a fast greedy algorithm with strong theoretical guarantees, …
University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Rochester（罗切斯特大学）
We study the problem of sampling a bandlimited graph signal in the presence of noise, where the objective is to select a node subset of prescribed cardinality that minimizes the signal reconstruction mean squared error (MSE). To that end, we formulate the task at hand as the minimization of MSE subject to binary constraints, and approximate the resulting NP-hard problem via semidefinite programming (SDP) relaxation. Moreover, we provide an alternative formulation based on maximizing a monotone weak submodular function and propose a randomized-greedy algorithm to find a sub-optimal subset. We t…
We consider a class of discrete optimization problems that aim to maximize a submodular objective function subject to a distributed partition matroid constraint. More precisely, we consider a networked scenario in which multiple agents choose actions from local strategy sets with the goal of maximizing a submodular objective function defined over the set of all possible actions. Given this distributed setting, we develop Constraint-Distributed Continuous Greedy (CDCG), a message passing algorithm that converges to the tight $(1-1/e)$ approximation factor of the optimum global solution using on…
Naveen Jindal School of Management, The University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）
In this paper, we study the non-monotone adaptive submodular maximization problem subject to a cardinality constraint. We first revisit the adaptive random greedy algorithm proposed in \citep{gotovos2015non}, where they show that this algorithm achieves a $1/e$ approximation ratio if the objective function is adaptive submodular and pointwise submodular. It is not clear whether the same guarantee holds under adaptive submodularity (without resorting to pointwise submodularity) or not. Our first contribution is to show that the adaptive random greedy algorithm achieves a $1/e$ approximation rat…
University of Cambridge（剑桥大学）；Georgia Institute of Technology（佐治亚理工学院）
We study the worst-case adaptive optimization problem with budget constraint that is useful for modeling various practical applications in artificial intelligence and machine learning. We investigate the near-optimality of greedy algorithms for this problem with both modular and non-modular cost functions. In both cases, we prove that two simple greedy algorithms are not near-optimal but the best between them is near-optimal if the utility function satisfies pointwise submodularity and pointwise cost-sensitive submodularity respectively. This implies a combined algorithm that is near-optimal w…
Often times, in many design problems, there is a need to select a small set of informative or representative elements from a large ground set of entities in an optimal fashion. Submodular optimization that provides for a formal way to solve such problems, has recently received significant attention from the controls community where such subset selection problems are abound. However, scaling these approaches to large systems can be challenging because of the high computational complexity of the overall flow, in-part due to the high-complexity compute-oracles used to determine the objective func…
This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a self-dictionary multiple measurement vector (SD-MMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is special in that it allows simultaneous identification of the endmember spectral signatures and the number of endmembers. Previous SD-MMV studies mainly focus on convex relaxations. In this study, we explore the alternative of greedy pursuit, which generally provides efficient and simple algorit…
University of Washington（华盛顿大学）；Yahoo Research（雅虎研究院）
We propose a new random pruning method (called "submodular sparsification (SS)") to reduce the cost of submodular maximization. The pruning is applied via a "submodularity graph" over the $n$ ground elements, where each directed edge is associated with a pairwise dependency defined by the submodular function. In each step, SS prunes a $1-1/\sqrt{c}$ (for $c>1$) fraction of the nodes using weights on edges computed based on only a small number ($O(\log n)$) of randomly sampled nodes. The algorithm requires $\log_{\sqrt{c}}n$ steps with a small and highly parallelizable per-step computation. An …
We present apricot, an open source Python package for selecting representative subsets from large data sets using submodular optimization. The package implements an efficient greedy selection algorithm that offers strong theoretical guarantees on the quality of the selected set. Two submodular set functions are implemented in apricot: facility location, which is broadly applicable but requires memory quadratic in the number of examples in the data set, and a feature-based function that is less broadly applicable but can scale to millions of examples. Apricot is extremely efficient, using both …
We propose a streaming submodular maximization algorithm "stream clipper" that performs as well as the offline greedy algorithm on document/video summarization in practice. It adds elements from a stream either to a solution set $S$ or to an extra buffer $B$ based on two adaptive thresholds, and improves $S$ by a final greedy step that starts from $S$ adding elements from $B$. During this process, swapping elements out of $S$ can occur if doing so yields improvements. The thresholds adapt based on if current memory utilization exceeds a budget, e.g., it increases the lower threshold, and remov…
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…
In this paper, we propose a new framework for designing fast parallel algorithms for fundamental statistical subset selection tasks that include feature selection and experimental design. Such tasks are known to be weakly submodular and are amenable to optimization via the standard greedy algorithm. Despite its desirable approximation guarantees, the greedy algorithm is inherently sequential and in the worst case, its parallel runtime is linear in the size of the data. Recently, there has been a surge of interest in a parallel optimization technique called adaptive sampling which produces solu…
UT Austin（德克萨斯大学奥斯汀分校）；UC Berkeley（加州大学伯克利分校）；Virginia Tech（弗吉尼亚理工大学）
To cope with the high level of ambiguity faced in domains such as Computer Vision or Natural Language processing, robust prediction methods often search for a diverse set of high-quality candidate solutions or proposals. In structured prediction problems, this becomes a daunting task, as the solution space (image labelings, sentence parses, etc.) is exponentially large. We study greedy algorithms for finding a diverse subset of solutions in structured-output spaces by drawing new connections between submodular functions over combinatorial item sets and High-Order Potentials (HOPs) studied for …
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
Mini-batch gradient descent based methods are the de facto algorithms for training neural network architectures today. We introduce a mini-batch selection strategy based on submodular function maximization. Our novel submodular formulation captures the informativeness of each sample and diversity of the whole subset. We design an efficient, greedy algorithm which can give high-quality solutions to this NP-hard combinatorial optimization problem. Our extensive experiments on standard datasets show that the deep models trained using the proposed batch selection strategy provide better generaliza…
Mean field inference in probabilistic models is generally a highly nonconvex problem. Existing optimization methods, e.g., coordinate ascent algorithms, can only generate local optima. In this work we propose provable mean filed methods for probabilistic log-submodular models and its posterior agreement (PA) with strong approximation guarantees. The main algorithmic technique is a new Double Greedy scheme, termed DR-DoubleGreedy, for continuous DR-submodular maximization with box-constraints. It is a one-pass algorithm with linear time complexity, reaching the optimal 1/2 approximation ratio, …
Chebyshev Greedy Algorithm is a generalization of the well known Orthogonal Matching Pursuit defined in a Hilbert space to the case of Banach spaces. We apply this algorithm for constructing sparse approximate solutions (with respect to a given dictionary) to convex optimization problems. Rate of convergence results in a style of the Lebesgue-type inequalities are proved.
Oak Ridge National Laboratory（橡树岭国家实验室）；University of Tennessee at Knoxville（田纳西大学诺克斯维尔分校）
We present a greedy-based approach to construct an efficient single hidden layer neural network with the ReLU activation that approximates a target function. In our approach we obtain a shallow network by utilizing a greedy algorithm with the prescribed dictionary provided by the available training data and a set of possible inner weights. To facilitate the greedy selection process we employ an integral representation of the network, based on the ridgelet transform, that significantly reduces the cardinality of the dictionary and hence promotes feasibility of the greedy selection. Our approach…
We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $τ$ from the returned set may be removed. We focus on the worst-case setting considered in (Orlin et al., 2016), in which a constant-factor approximation guarantee was given for $τ= o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $τ= o(k)$. Our algorithm constructs partitions consisting of buckets…
We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank estimation and combinatorial optimization, so much so that our bounds are reminiscent of corresponding approximation bounds in submodular maximization. Additionally, we also provide statistical recovery guarantees. Finally, we present empirical comparison of greedy estimation with established baselines on two important real-world problems.
