A key problem in statistics and machine learning is the determination of network structure from data. We consider the case where the structure of the graph to be reconstructed is known to be scale-free. We show that in such cases it is natural to formulate structured sparsity inducing priors using submodular functions, and we use their Lovász extension to obtain a convex relaxation. For tractable classes such as Gaussian graphical models, this leads to a convex optimization problem that can be efficiently solved. We show that our method results in an improvement in the accuracy of reconstructed networks for synthetic data. We also show how our prior encourages scale-free reconstructions on a bioinfomatics dataset.
INRIA（法国国家信息与自动化研究所）；Ecole normale supérieure（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；PSL Research University（巴黎文理研究大学）
We consider the minimization of submodular functions subject to ordering constraints. We show that this optimization problem can be cast as a convex optimization problem on a space of uni-dimensional measures, with ordering constraints corresponding to first-order stochastic dominance. We propose new discretization schemes that lead to simple and efficient algorithms based on zero-th, first, or higher order oracles; these algorithms also lead to improvements without isotonic constraints. Finally, our experiments show that non-convex loss functions can be much more robust to outliers for isoton…
Future University Hakodate（函馆未来大学）；Osaka University（大阪大学）
A number of discrete and continuous optimization problems in machine learning are related to convex minimization problems under submodular constraints. In this paper, we deal with a submodular function with a directed graph structure, and we show that a wide range of convex optimization problems under submodular constraints can be solved much more efficiently than general submodular optimization methods by a reduction to a maximum flow problem. Furthermore, we give some applications, including sparse optimization methods, in which the proposed methods are effective. Additionally, we evaluate t…
ETH Zurich（苏黎世联邦理工学院）；Shanghai Jiao Tong University（上海交通大学）
Submodularity is one of the most well-studied properties of problem classes in combinatorial optimization and many applications of machine learning and data mining, with strong implications for guaranteed optimization. In this thesis, we investigate the role of submodularity in provable non-convex optimization and validation of algorithms. A profound understanding which classes of functions can be tractably optimized remains a central challenge for non-convex optimization. By advancing the notion of submodularity to continuous domains (termed "continuous submodularity"), we characterize a clas…
Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the lovasz extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this monograph, we present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, we show how submodular function minimization is equivalent t…
Continuous submodular functions are a category of generally non-convex/non-concave functions with a wide spectrum of applications. The celebrated property of this class of functions - continuous submodularity - enables both exact minimization and approximate maximization in poly. time. Continuous submodularity is obtained by generalizing the notion of submodularity from discrete domains to continuous domains. It intuitively captures a repulsive effect amongst different dimensions of the defined multivariate function. In this paper, we systematically study continuous submodularity and a class o…
In this paper, we study fundamental problems of maximizing DR-submodular continuous functions that have real-world applications in the domain of machine learning, economics, operations research and communication systems. It captures a subclass of non-convex optimization that provides both theoretical and practical guarantees. Here, we focus on minimizing regret for online arriving non-monotone DR-submodular functions over different types of convex sets: hypercube, down-closed and general convex sets. First, we present an online algorithm that achieves a $1/e$-approximation ratio with the regre…
Diminishing-returns (DR) submodular optimization is an important field with many real-world applications in machine learning, economics and communication systems. It captures a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we study the fundamental problem of maximizing non-monotone DR-submodular functions over down-closed and general convex sets in both offline and online settings. First, we show that for offline maximizing non-monotone DR-submodular functions over a general convex set, the Frank-Wolfe algorithm achieves an approxim…
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）；University of Pennsylvania（宾夕法尼亚大学）
Online optimization has been a successful framework for solving large-scale problems under computational constraints and partial information. Current methods for online convex optimization require either a projection or exact gradient computation at each step, both of which can be prohibitively expensive for large-scale applications. At the same time, there is a growing trend of non-convex optimization in machine learning community and a need for online methods. Continuous DR-submodular functions, which exhibit a natural diminishing returns condition, have recently been proposed as a broad cla…
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…
Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation guarantees. Specifically, i) We introduce the weak DR property that gives a unified characterization of submodularity for all set, integer-lattice and continuous functions; ii) for maximizing monotone DR-submodular continuous functions under general down-closed convex constraints, we propose a Frank-Wolfe variant with $(1-1/e)$ approximation guarantee, and su…
INRIA（法国国家信息与自动化研究所）；Département d’Informatique de l’Ecole Normale Supérieure（巴黎高等师范学院计算机系）
Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the submodular set-function to a convex function, which opens up tools from convex optimization. Submodularity goes beyond set-functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains, where it corresponds essentially to cross second-derivatives being nonpositive. In this paper, we show that mo…
INRIA（法国国家信息与自动化研究所）；Laboratoire d’Informatique de l’Ecole Normale Supérieure（巴黎高等师范学院计算机实验室）
Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the L1-norm. In this paper, we investigate more general set-functions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nondecreasing submodular set…
DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints. We start by investigating geometric properties that underlie such objective…
We study the problem of optimizing expensive blackbox functions over combinatorial spaces (e.g., sets, sequences, trees, and graphs). BOCS (Baptista and Poloczek, 2018) is a state-of-the-art Bayesian optimization method for tractable statistical models, which performs semi-definite programming based acquisition function optimization (AFO) to select the next structure for evaluation. Unfortunately, BOCS scales poorly for large number of binary and/or categorical variables. Based on recent advances in submodular relaxation (Ito and Fujimaki, 2016) for solving Binary Quadratic Programs, we study …
University of Pennsylvania（宾夕法尼亚大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Minimax optimization has been central in addressing various applications in machine learning, game theory, and control theory. Prior literature has thus far mainly focused on studying such problems in the continuous domain, e.g., convex-concave minimax optimization is now understood to a significant extent. Nevertheless, minimax problems extend far beyond the continuous domain to mixed continuous-discrete domains or even fully discrete domains. In this paper, we study mixed continuous-discrete minimax problems where the minimization is over a continuous variable belonging to Euclidean space an…
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…
Improved Approximate Regret for Decentralized Online Continuous Submodular Maximization via Reductions
School of Software Technology, Zhejiang University, Ningbo, China（浙江大学宁波校区软件学院）；State Key Laboratory of Blockchain and Data Security, Zhejiang University, Hangzhou, China（浙江大学区块链与数据安全国家重点实验室）；School of Artificial Intelligence, Nanjing University, Nanjing, China（南京大学人工智能学院）；Department of Computer Science, City University of Hong Kong, Hong Kong, China（香港城市大学计算机科学系）
Continuous DR-submodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing works have studied monotone continuous DR-submodular maximization subject to a convex constraint and have proposed efficient algorithms with approximation guarantees. However, in many applications, e.g., computing the stability number of a graph and mean-field inference for probabilistic log-submodular models, the DR-submodular function has the additional property of being \emph{strongly} concave along non-ne…
Submodular function minimization is well studied, and existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, such as structured sparse learning or batch Bayesian optimization, the objective function is not exactly submodular, but close. In this case, no theoretical guarantees exist. Indeed, submodular minimization algorithms rely on intricate connections between submodularity and convexity. We show how these relations can be extended to obtain approximation guarantees for minimizing non-submodular functions, characterized by how close the function is t…
Yale Institute for Network Science（耶鲁大学网络科学研究所）；Department of Electrical Engineering, Yale University（耶鲁大学电气工程系）；Department of Electrical and Systems Engineering, University of Pennsylvania（宾夕法尼亚大学电气与系统工程系）
In this paper, we consider an online optimization process, where the objective functions are not convex (nor concave) but instead belong to a broad class of continuous submodular functions. We first propose a variant of the Frank-Wolfe algorithm that has access to the full gradient of the objective functions. We show that it achieves a regret bound of $O(\sqrt{T})$ (where $T$ is the horizon of the online optimization problem) against a $(1-1/e)$-approximation to the best feasible solution in hindsight. However, in many scenarios, only an unbiased estimate of the gradients are available. For su…
We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization. The problem is closely related to decomposable submodular function minimization and arises in many learning on graphs and hypergraphs settings, such as graph-based semi-supervised learning and PageRank. We approach the problem via a new dual strategy and describe an objective that may be optimized via random coordinate descent (RCD) methods and projections onto cones. We also establish the linear convergence rate of the RCD algorithm and develop efficient projection algorithms with p…
University of Pennsylvania（宾夕法尼亚大学）；University of Southern California（南加州大学）；Yale University（耶鲁大学）
In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient a…
In this paper, we propose an unifying view of several recently proposed structured sparsity-inducing norms. We consider the situation of a model simultaneously (a) penalized by a set- function de ned on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) regularized in Lp-norm. We show that the natural combinatorial optimization problems obtained may be relaxed into convex optimization problems and introduce a notion, the lower combinatorial envelope of a set-function, that characterizes the tightness of our relaxations. We moreover establish links…
Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation techniques to find a convex surrogate for the problem. Unfortunately, convex relaxations typically must be found on a problem-by-problem basis. Thus, providing a general-purpose strategy to estimate a convex relaxation would have a wide reaching impact. Here, we introduce Convex Relaxation Regression (CoRR), an approach for learning convex relaxations for a clas…
In this paper, we present an algorithm for minimizing the difference between two submodular functions using a variational framework which is based on (an extension of) the concave-convex procedure [17]. Because several commonly used metrics in machine learning, like mutual information and conditional mutual information, are submodular, the problem of minimizing the difference of two submodular problems arises naturally in many machine learning applications. Two such applications are learning discriminatively structured graphical models and feature selection under computational complexity const…
Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have high running times and are unsuitable for large-scale problems. Recent work have used convex optimization techniques to obtain very practical algorithms for minimizing functions that are sums of ``simple" functions. In this paper, we use random coordinate descent methods to obtain algorithms with faster linear convergence rates and cheaper iteration costs. C…
In this paper, we consider an online optimization problem over $T$ rounds where at each step $t\in[T]$, the algorithm chooses an action $x_t$ from the fixed convex and compact domain set $\mathcal{K}$. A utility function $f_t(\cdot)$ is then revealed and the algorithm receives the payoff $f_t(x_t)$. This problem has been previously studied under the assumption that the utilities are adversarially chosen monotone DR-submodular functions and $\mathcal{O}(\sqrt{T})$ regret bounds have been derived. We first characterize the class of strongly DR-submodular functions and then, we derive regret boun…
Purdue University（普渡大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization (QDSFM), which allows to model a number of learning tasks on graphs and hypergraphs. The problem exhibits close ties to decomposable submodular function minimization (DSFM), yet is much more challenging to solve. We approach the problem via a new dual strategy and formulate an objective that can be optimized through a number of double-loop algorithms. The outer-loop uses either random coordinate descent (RCD) or alternative projection (AP) methods, for both of which we prove linear co…
Georgia Institute of Technology（佐治亚理工学院）；Universidad San Pablo-CEU（圣巴布洛CEU大学）
Distributed automatic speech recognition (ASR) requires to aggregate outputs of distributed deep neural network (DNN)-based models. This work studies the use of submodular functions to design a rank aggregation on score-based permutations, which can be used for distributed ASR systems in both supervised and unsupervised modes. Specifically, we compose an aggregation rank function based on the Lovasz Bregman divergence for setting up linear structured convex and nested structured concave functions. The algorithm is based on stochastic gradient descent (SGD) and can obtain well-trained aggregati…
Learning with non-modular losses is an important problem when sets of predictions are made simultaneously. The main tools for constructing convex surrogate loss functions for set prediction are margin rescaling and slack rescaling. In this work, we show that these strategies lead to tight convex surrogates iff the underlying loss function is increasing in the number of incorrect predictions. However, gradient or cutting-plane computation for these functions is NP-hard for non-supermodular loss functions. We propose instead a novel surrogate loss function for submodular losses, the Lovász hinge…
INRIA（法国国家信息与自动化研究所）；Laboratoire d’Informatique de l’Ecole Normale Supérieure（巴黎高等师范学院计算机实验室）
Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of submodular functions is presented, in a self-contained way, with all results shown from first principles. A good knowledge of convex analysis is assumed.
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…
Amazon Development Center（亚马逊开发中心）；IBM Research（IBM研究院）
In this paper, we show that if the optimization function is restricted-strongly-convex (RSC) and restricted-smooth (RSM) -- a rich subclass of weakly submodular functions -- then a streaming algorithm with constant factor approximation guarantee is possible. More generally, our results are applicable to any monotone weakly submodular function with submodularity ratio bounded from above. This (positive) result which provides a sufficient condition for having a constant factor streaming guarantee for weakly submodular functions may be of special interest given the recent negative result (Elenber…
Empirical risk minimization frequently employs convex surrogates to underlying discrete loss functions in order to achieve computational tractability during optimization. However, classical convex surrogates can only tightly bound modular loss functions, sub-modular functions or supermodular functions separately while maintaining polynomial time computation. In this work, a novel generic convex surrogate for general non-modular loss functions is introduced, which provides for the first time a tractable solution for loss functions that are neither super-modular nor submodular. This convex surro…
Massachusetts Institute of Technology（麻省理工学院）；University of Pennsylvania（宾夕法尼亚大学）；Yale University（耶鲁大学）
This paper considers stochastic optimization problems for a large class of objective functions, including convex and continuous submodular. Stochastic proximal gradient methods have been widely used to solve such problems; however, their applicability remains limited when the problem dimension is large and the projection onto a convex set is costly. Instead, stochastic conditional gradient methods are proposed as an alternative solution relying on (i) Approximating gradients via a simple averaging technique requiring a single stochastic gradient evaluation per iteration; (ii) Solving a linear …
Imperial College London（帝国理工学院）；INRIA（法国国家信息与自动化研究所）；Ecole normale superieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Graz University of Technology（格拉茨工业大学）
We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lovász extensions and the squared Euclidean norm, leading to algorithms requiring total variation oracles of the summand functions; without further assumptions, these more complex oracles require many calls to the simpler minimization oracles often available in practice. In this paper, we consider a modified convex problem requiring constrained version of the …
Submodular functions are discrete analogs of convex functions, which have applications in various fields, including machine learning and computer vision. However, in large-scale applications, solving Submodular Function Minimization (SFM) problems remains challenging. In this paper, we make the first attempt to extend the emerging technique named screening in large-scale sparse learning to SFM for accelerating its optimization process. We first conduct a careful studying of the relationships between SFM and the corresponding convex proximal problems, as well as the accurate primal optimum esti…
In this paper, we study the problem of monotone (weakly) DR-submodular continuous maximization. While previous methods require the gradient information of the objective function, we propose a derivative-free algorithm LDGM for the first time. We define $β$ and $α$ to characterize how close a function is to continuous DR-submodulr and submodular, respectively. Under a convex polytope constraint, we prove that LDGM can achieve a $(1-e^{-β}-ε)$-approximation guarantee after $O(1/ε)$ iterations, which is the same as the best previous gradient-based algorithm. Moreover, in some special cases, a var…
Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, state-of-the-art algorithms for general submodular minimization are intractable for larger problems. In this paper, we introduce a novel subclass of submodular minimization problems that we call decomposable. Decomposable submodular functions are those that can be represented as sums of concave functions applied to modular functions. We d…
Decentralized learning has been studied intensively in recent years motivated by its wide applications in the context of federated learning. The majority of previous research focuses on the offline setting in which the objective function is static. However, the offline setting becomes unrealistic in numerous machine learning applications that witness the change of massive data. In this paper, we propose \emph{decentralized online} algorithm for convex and continuous DR-submodular optimization, two classes of functions that are present in a variety of machine learning problems. Our algorithms a…
We consider the problem of maximizing submodular functions; while this problem is known to be NP-hard, several numerically efficient local search techniques with approximation guarantees are available. In this paper, we propose a novel convex relaxation which is based on the relationship between submodular functions, entropies and probabilistic graphical models. In a graphical model, the entropy of the joint distribution decomposes as a sum of marginal entropies of subsets of variables; moreover, for any distribution, the entropy of the closest distribution factorizing in the graphical model p…
Simons Institute for the Theory of Computing（西蒙斯理论计算研究所）
In this paper, we study the problem of \textit{constrained} and \textit{stochastic} continuous submodular maximization. Even though the objective function is not concave (nor convex) and is defined in terms of an expectation, we develop a variant of the conditional gradient method, called \alg, which achieves a \textit{tight} approximation guarantee. More precisely, for a monotone and continuous DR-submodular function and subject to a \textit{general} convex body constraint, we prove that \alg achieves a $[(1-1/e)\text{OPT} -\eps]$ guarantee (in expectation) with $\mathcal{O}{(1/\eps^3)}$ stoc…
In recent years, maximization of DR-submodular continuous functions became an important research field, with many real-worlds applications in the domains of machine learning, communication systems, operation research and economics. Most of the works in this field study maximization subject to down-closed convex set constraints due to an inapproximability result by Vondrák (2013). However, Durr et al. (2021) showed that one can bypass this inapproximability by proving approximation ratios that are functions of $m$, the minimum $\ell_{\infty}$-norm of any feasible vector. Given this observation,…
Microsoft Corporation（微软公司）；University of Washington（华盛顿大学）
We are motivated by large scale submodular optimization problems, where standard algorithms that treat the submodular functions in the \emph{value oracle model} do not scale. In this paper, we present a model called the \emph{precomputational complexity model}, along with a unifying memoization based framework, which looks at the specific form of the given submodular function. A key ingredient in this framework is the notion of a \emph{precomputed statistic}, which is maintained in the course of the algorithms. We show that we can easily integrate this idea into a large class of submodular opt…
University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of Surrey（萨里大学）
Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of the energy and the objective functions of linear programming (LP) relaxations for the corresponding MAP estimation problem. This allows us to (i) establish the worst-case optimality of the submodular extension for Potts model used in the literature; (ii) identify the worst-case …
We propose convex relaxations for convolutional neural nets with one hidden layer where the output weights are fixed. For convex activation functions such as rectified linear units, the relaxations are convex second order cone programs which can be solved very efficiently. We prove that the relaxation recovers the global minimum under a planted model assumption, given sufficiently many training samples from a Gaussian distribution. We also identify a phase transition phenomenon in recovering the global minimum for the relaxation.
Convex relaxations are effective for training and certifying neural networks against norm-bounded adversarial attacks, but they leave a large gap between certifiable and empirical robustness. In principle, convex relaxation can provide tight bounds if the solution to the relaxed problem is feasible for the original non-convex problem. We propose two regularizers that can be used to train neural networks that yield tighter convex relaxation bounds for robustness. In all of our experiments, the proposed regularizers result in higher certified accuracy than non-regularized baselines.
Google Research（谷歌研究院）；Rice University（莱斯大学）；Columbia University（哥伦比亚大学）；Polytechnique Montréal（蒙特利尔理工学院）
We improve the effectiveness of propagation- and linear-optimization-based neural network verification algorithms with a new tightened convex relaxation for ReLU neurons. Unlike previous single-neuron relaxations which focus only on the univariate input space of the ReLU, our method considers the multivariate input space of the affine pre-activation function preceding the ReLU. Using results from submodularity and convex geometry, we derive an explicit description of the tightest possible convex relaxation when this multivariate input is over a box domain. We show that our convex relaxation is…
