Learning and Optimization with Submodular Functions

In many naturally occurring optimization problems one needs to ensure that the definition of the optimization problem lends itself to solutions that are tractable to compute. In cases where exact solutions cannot be computed tractably, it is beneficial to have strong guarantees on the tractable approximate solutions. In order operate under these criterion most optimization problems are cast under the umbrella of convexity or submodularity. In this report we will study design and optimization over a common class of functions called submodular functions. Set functions, and specifically submodular set functions, characterize a wide variety of naturally occurring optimization problems, and the property of submodularity of set functions has deep theoretical consequences with wide ranging applications. Informally, the property of submodularity of set functions concerns the intuitive "principle of diminishing returns. This property states that adding an element to a smaller set has more value than adding it to a larger set. Common examples of submodular monotone functions are entropies, concave functions of cardinality, and matroid rank functions; non-monotone examples include graph cuts, network flows, and mutual information. In this paper we will review the formal definition of submodularity; the optimization of submodular functions, both maximization and minimization; and finally discuss some applications in relation to learning and reasoning using submodular functions.Comment: Tech Report - USC Computer Science CS-599, Convex and Combinatorial Optimizatio

Mixed Robust/Average Submodular Partitioning: Fast Algorithms, Guarantees, and Applications to Parallel Machine Learning and Multi-Label Image Segmentation

We study two mixed robust/average-case submodular partitioning problems that we collectively call Submodular Partitioning. These problems generalize both purely robust instances of the problem (namely max-min submodular fair allocation (SFA) and min-max submodular load balancing (SLB) and also generalize average-case instances (that is the submodular welfare problem (SWP) and submodular multiway partition (SMP). While the robust versions have been studied in the theory community, existing work has focused on tight approximation guarantees, and the resultant algorithms are not, in general, scalable to very large real-world applications. This is in contrast to the average case, where most of the algorithms are scalable. In the present paper, we bridge this gap, by proposing several new algorithms (including those based on greedy, majorization-minimization, minorization-maximization, and relaxation algorithms) that not only scale to large sizes but that also achieve theoretical approximation guarantees close to the state-of-the-art, and in some cases achieve new tight bounds. We also provide new scalable algorithms that apply to additive combinations of the robust and average-case extreme objectives. We show that these problems have many applications in machine learning (ML). This includes: 1) data partitioning and load balancing for distributed machine algorithms on parallel machines; 2) data clustering; and 3) multi-label image segmentation with (only) Boolean submodular functions via pixel partitioning. We empirically demonstrate the efficacy of our algorithms on real-world problems involving data partitioning for distributed optimization of standard machine learning objectives (including both convex and deep neural network objectives), and also on purely unsupervised (i.e., no supervised or semi-supervised learning, and no interactive segmentation) image segmentation

Differentiable Greedy Networks

Optimal selection of a subset of items from a given set is a hard problem that requires combinatorial optimization. In this paper, we propose a subset selection algorithm that is trainable with gradient-based methods yet achieves near-optimal performance via submodular optimization. We focus on the task of identifying a relevant set of sentences for claim verification in the context of the FEVER task. Conventional methods for this task look at sentences on their individual merit and thus do not optimize the informativeness of sentences as a set. We show that our proposed method which builds on the idea of unfolding a greedy algorithm into a computational graph allows both interpretability and gradient-based training. The proposed differentiable greedy network (DGN) outperforms discrete optimization algorithms as well as other baseline methods in terms of precision and recall.Comment: Work in progress and under revie

A Unified Multi-Faceted Video Summarization System

This paper addresses automatic summarization and search in visual data comprising of videos, live streams and image collections in a unified manner. In particular, we propose a framework for multi-faceted summarization which extracts key-frames (image summaries), skims (video summaries) and entity summaries (summarization at the level of entities like objects, scenes, humans and faces in the video). The user can either view these as extractive summarization, or query focused summarization. Our approach first pre-processes the video or image collection once, to extract all important visual features, following which we provide an interactive mechanism to the user to summarize the video based on their choice. We investigate several diversity, coverage and representation models for all these problems, and argue the utility of these different mod- els depending on the application. While most of the prior work on submodular summarization approaches has focused on combining several models and learning weighted mixtures, we focus on the explain-ability of different the diversity, coverage and representation models and their scalability. Most importantly, we also show that we can summarize hours of video data in a few seconds, and our system allows the user to generate summaries of various lengths and types interactively on the fly.Comment: 18 pages, 11 Figure

Class Subset Selection for Transfer Learning using Submodularity

In recent years, it is common practice to extract fully-connected layer (fc) features that were learned while performing image classification on a source dataset, such as ImageNet, and apply them generally to a wide range of other tasks. The general usefulness of some large training datasets for transfer learning is not yet well understood, and raises a number of questions. For example, in the context of transfer learning, what is the role of a specific class in the source dataset, and how is the transferability of fc features affected when they are trained using various subsets of the set of all classes in the source dataset? In this paper, we address the question of how to select an optimal subset of the set of classes, subject to a budget constraint, that will more likely generate good features for other tasks. To accomplish this, we use a submodular set function to model the accuracy achievable on a new task when the features have been learned on a given subset of classes of the source dataset. An optimal subset is identified as the set that maximizes this submodular function. The maximization can be accomplished using an efficient greedy algorithm that comes with guarantees on the optimality of the solution. We empirically validate our submodular model by successfully identifying subsets of classes that produce good features for new tasks

Vis-DSS: An Open-Source toolkit for Visual Data Selection and Summarization

With increasing amounts of visual data being created in the form of videos and images, visual data selection and summarization are becoming ever increasing problems. We present Vis-DSS, an open-source toolkit for Visual Data Selection and Summarization. Vis-DSS implements a framework of models for summarization and data subset selection using submodular functions, which are becoming increasingly popular today for these problems. We present several classes of models, capturing notions of diversity, coverage, representation and importance, along with optimization/inference and learning algorithms. Vis-DSS is the first open source toolkit for several Data selection and summarization tasks including Image Collection Summarization, Video Summarization, Training Data selection for Classification and Diversified Active Learning. We demonstrate state-of-the art performance on all these tasks, and also show how we can scale to large problems. Vis-DSS allows easy integration for applications to be built on it, also can serve as a general skeleton that can be extended to several use cases, including video and image sharing platforms for creating GIFs, image montage creation, or as a component to surveillance systems and we demonstrate this by providing a graphical user-interface (GUI) desktop app built over Qt framework. Vis-DSS is available at https://github.com/rishabhk108/vis-dssComment: Vis-DSS is available at https://github.com/rishabhk108/vis-ds

Unsupervised Submodular Rank Aggregation on Score-based Permutations

Unsupervised rank aggregation on score-based permutations, which is widely used in many applications, has not been deeply explored yet. This work studies the use of submodular optimization for rank aggregation on score-based permutations in an unsupervised way. Specifically, we propose an unsupervised approach based on the Lovasz Bregman divergence for setting up linear structured convex and nested structured concave objective functions. In addition, stochastic optimization methods are applied in the training process and efficient algorithms for inference can be guaranteed. The experimental results from Information Retrieval, Combining Distributed Neural Networks, Influencers in Social Networks, and Distributed Automatic Speech Recognition tasks demonstrate the effectiveness of the proposed methods

Learning From Less Data: A Unified Data Subset Selection and Active Learning Framework for Computer Vision

Supervised machine learning based state-of-the-art computer vision techniques are in general data hungry. Their data curation poses the challenges of expensive human labeling, inadequate computing resources and larger experiment turn around times. Training data subset selection and active learning techniques have been proposed as possible solutions to these challenges. A special class of subset selection functions naturally model notions of diversity, coverage and representation and can be used to eliminate redundancy thus lending themselves well for training data subset selection. They can also help improve the efficiency of active learning in further reducing human labeling efforts by selecting a subset of the examples obtained using the conventional uncertainty sampling based techniques. In this work, we empirically demonstrate the effectiveness of two diversity models, namely the Facility-Location and Dispersion models for training-data subset selection and reducing labeling effort. We demonstrate this across the board for a variety of computer vision tasks including Gender Recognition, Face Recognition, Scene Recognition, Object Detection and Object Recognition. Our results show that diversity based subset selection done in the right way can increase the accuracy by upto 5 - 10% over existing baselines, particularly in settings in which less training data is available. This allows the training of complex machine learning models like Convolutional Neural Networks with much less training data and labeling costs while incurring minimal performance loss.Comment: Accepted to WACV 2019. arXiv admin note: substantial text overlap with arXiv:1805.1119

Adaptive Sequence Submodularity

In many machine learning applications, one needs to interactively select a sequence of items (e.g., recommending movies based on a user's feedback) or make sequential decisions in a certain order (e.g., guiding an agent through a series of states). Not only do sequences already pose a dauntingly large search space, but we must also take into account past observations, as well as the uncertainty of future outcomes. Without further structure, finding an optimal sequence is notoriously challenging, if not completely intractable. In this paper, we view the problem of adaptive and sequential decision making through the lens of submodularity and propose an adaptive greedy policy with strong theoretical guarantees. Additionally, to demonstrate the practical utility of our results, we run experiments on Amazon product recommendation and Wikipedia link prediction tasks

A Memoization Framework for Scaling Submodular Optimization to Large Scale Problems

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 optimization problems including constrained and unconstrained submodular maximization, minimization, difference of submodular optimization, optimization with submodular constraints and several other related optimization problems. Moreover, memoization can be integrated in both discrete and continuous relaxation flavors of algorithms for these problems. We demonstrate this idea for several commonly occurring submodular functions, and show how the precomputational model provides significant speedups compared to the value oracle model. Finally, we empirically demonstrate this for large scale machine learning problems of data subset selection and summarization.Comment: To Appear in Proc. AISTATS 201