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We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results., Comment: 21 pages. A shorter version appeared in Advances of NIPS-2013

We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [14, 35] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and an approximation algorithm solving one can be used to obtain an approximation guarantee for the other. We provide hardness results for both problems thus showing that our approximation factors are tight up to log-factors. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms., Comment: 23 pages. A short version of this appeared in Advances of NIPS-2013

We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes, 2012) in several ways. We show that they represent a distortion between a 'score' and an 'ordering', thus providing a new view of rank aggregation and order based clustering with interesting connections to web ranking. We show how the LB divergences have a number of properties akin to many permutation based metrics, and in fact have as special cases forms very similar to the Kendall-$\tau$ metric. We also show how the LB divergences subsume a number of commonly used ranking measures in information retrieval, like the NDCG and AUC. Unlike the traditional permutation based metrics, however, the LB divergence naturally captures a notion of "confidence" in the orderings, thus providing a new representation to applications involving aggregating scores as opposed to just orderings. We show how a number of recently used web ranking models are forms of Lovasz-Bregman rank aggregation and also observe that a natural form of Mallow's model using the LB divergence has been used as conditional ranking models for the 'Learning to Rank' problem., Comment: 18 pages. A shorter version appeared in Proc. Uncertainty in Artificial Intelligence (UAI)-2013, Bellevue, WA

We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (sub- and super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes steps towards providing a unifying paradigm applicable to both submodular min- imization and maximization, problems that historically have been treated quite distinctly. The practicality of our algorithms is important since interest in submodularity, owing to its natural and wide applicability, has recently been in ascendance within machine learning. We analyze theoretical properties of our algorithms for minimization and maximization, and show that many state-of-the-art maximization algorithms are special cases. Lastly, we complement our theoretical analyses with supporting empirical experiments., Comment: This work appeared in Proc. International Conference of Machine Learning (ICML, 2013)

In this work, dynamic Bayesian multinets are introduced where a Markov chain state at time t determines conditional independence patterns between random variables lying within a local time window surrounding t. It is shown how information-theoretic criterion functions can be used to induce sparse, discriminative, and class-conditional network structures that yield an optimal approximation to the class posterior probability, and therefore are useful for the classification task. Using a new structure learning heuristic, the resulting models are tested on a medium-vocabulary isolated-word speech recognition task. It is demonstrated that these discriminatively structured dynamic Bayesian multinets, when trained in a maximum likelihood setting using EM, can outperform both HMMs and other dynamic Bayesian networks with a similar number of parameters., Comment: Appears in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI2000)

This paper introduces new methodology to triangulate dynamic Bayesian networks (DBNs) and dynamic graphical models (DGMs). While most methods to triangulate such networks use some form of constrained elimination scheme based on properties of the underlying directed graph, we find it useful to view triangulation and elimination using properties only of the resulting undirected graph, obtained after the moralization step. We first briefly introduce the Graphical model toolkit (GMTK) and its notion of dynamic graphical models, one that slightly extends the standard notion of a DBN. We next introduce the 'boundary algorithm', a method to find the best boundary between partitions in a dynamic model. We find that using this algorithm, the notions of forward- and backward-interface become moot - namely, the size and fill-in of the best forward- and backward- interface are identical. Moreover, we observe that finding a good partition boundary allows for constrained elimination orders (and therefore graph triangulations) that are not possible using standard slice-by-slice constrained eliminations. More interestingly, with certain boundaries it is possible to obtain constrained elimination schemes that lie outside the space of possible triangulations using only unconstrained elimination. Lastly, we report triangulation results on invented graphs, standard DBNs from the literature, novel DBNs used in speech recognition research systems, and also random graphs. Using a number of different triangulation quality measures (max clique size, state-space, etc.), we find that with our boundary algorithm the triangulation quality can dramatically improve., Comment: Appears in Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence (UAI2003)

Shotgun proteomics is a high-throughput technology used to identify unknown proteins in a complex mixture. At the heart of this process is a prediction task, the spectrum identification problem, in which each fragmentation spectrum produced by a shotgun proteomics experiment must be mapped to the peptide (protein subsequence) which generated the spectrum. We propose a new algorithm for spectrum identification, based on dynamic Bayesian networks, which significantly outperforms the de-facto standard tools for this task: SEQUEST and Mascot., Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI2012)

We introduce a method to learn a mixture of submodular "shells" in a large-margin setting. A submodular shell is an abstract submodular function that can be instantiated with a ground set and a set of parameters to produce a submodular function. A mixture of such shells can then also be so instantiated to produce a more complex submodular function. What our algorithm learns are the mixture weights over such shells. We provide a risk bound guarantee when learning in a large-margin structured-prediction setting using a projected subgradient method when only approximate submodular optimization is possible (such as with submodular function maximization). We apply this method to the problem of multi-document summarization and produce the best results reported so far on the widely used NIST DUC-05 through DUC-07 document summarization corpora., Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI2012)

We show that the class of strongly connected graphical models with treewidth at most k can be properly efficiently PAC-learnt with respect to the Kullback-Leibler Divergence. Previous approaches to this problem, such as those of Chow ([1]), and Ho gen ([7]) have shown that this class is PAC-learnable by reducing it to a combinatorial optimization problem. However, for k > 1, this problem is NP-complete ([15]), and so unless P=NP, these approaches will take exponential amounts of time. Our approach differs significantly from these, in that it first attempts to find approximate conditional independencies by solving (polynomially many) submodular optimization problems, and then using a dynamic programming formulation to combine the approximate conditional independence information to derive a graphical model with underlying graph of the tree-width specified. This gives us an efficient (polynomial time in the number of random variables) PAC-learning algorithm which requires only polynomial number of samples of the true distribution, and only polynomial running time., Comment: Appears in Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI2004)

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 constraints. A commonly used metric for measuring discriminative capacity is the EAR measure which is the difference between two conditional mutual information terms. Feature selection taking complexity considerations into account also fall into this framework because both the information that a set of features provide and the cost of computing and using the features can be modeled as submodular functions. This problem is NP-hard, and we give a polynomial time heuristic for it. We also present results on synthetic data to show that classifiers based on discriminative graphical models using this algorithm can significantly outperform classifiers based on generative graphical models., Comment: Appears in Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (UAI2005)

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a difference between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the mul- tiplicative inapproximability of minimizing the difference between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the difference between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features., Comment: 17 pages, 8 figures. A shorter version of this appeared in Proc. Uncertainty in Artificial Intelligence (UAI), Catalina Islands, 2012

We introduce a new dynamic model with the capability of recognizing both activities that an individual is performing as well as where that ndividual is located. Our model is novel in that it utilizes a dynamic graphical model to jointly estimate both activity and spatial context over time based on the simultaneous use of asynchronous observations consisting of GPS measurements, and measurements from a small mountable sensor board. Joint inference is quite desirable as it has the ability to improve accuracy of the model. A key goal, however, in designing our overall system is to be able to perform accurate inference decisions while minimizing the amount of hardware an individual must wear. This minimization leads to greater comfort and flexibility, decreased power requirements and therefore increased battery life, and reduced cost. We show results indicating that our joint measurement model outperforms measurements from either the sensor board or GPS alone, using two types of probabilistic inference procedures, namely particle filtering and pruned exact inference., Comment: Appears in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence (UAI2006)

We observe that certain large-clique graph triangulations can be useful to reduce both computational and space requirements when making queries on mixed stochastic/deterministic graphical models. We demonstrate that many of these large-clique triangulations are non-minimal and are thus unattainable via the variable elimination algorithm. We introduce ancestral pairs as the basis for novel triangulation heuristics and prove that no more than the addition of edges between ancestral pairs need be considered when searching for state space optimal triangulations in such graphs. Empirical results on random and real world graphs show that the resulting triangulations that yield significant speedups are almost always non-minimal. We also give an algorithm and correctness proof for determining if a triangulation can be obtained via elimination, and we show that the decision problem associated with finding optimal state space triangulations in this mixed stochastic/deterministic setting is NP-complete., Comment: Appears in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence (UAI2006)

We analyze the generalized Mallows model, a popular exponential model over rankings. Estimating the central (or consensus) ranking from data is NP-hard. We obtain the following new results: (1) We show that search methods can estimate both the central ranking pi0 and the model parameters theta exactly. The search is n! in the worst case, but is tractable when the true distribution is concentrated around its mode; (2) We show that the generalized Mallows model is jointly exponential in (pi0; theta), and introduce the conjugate prior for this model class; (3) The sufficient statistics are the pairwise marginal probabilities that item i is preferred to item j. Preliminary experiments confirm the theoretical predictions and compare the new algorithm and existing heuristics., Comment: Appears in Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007)

This is the Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, which was held in Montreal, QC, Canada, June 18 - 21 2009.

Water in plant xylem is often superheated, and therefore in a meta-stable state. Under certain conditions, it may suddenly turn from the liquid to the vapor state. This cavitation process produces acoustic emissions. We report the measurement of ultrasonic acoustic emissions (UAE) produced by natural and induced cavitation in corn stems. We induced cavitation and UAE in vivo, in well controlled and reproducible experiments, by irradiating the bare stem of the plants with a continuous-wave laser beam. By tracing the source of UAE, we were able to detect absorption and frequency filtering of the UAE propagating through the stem. This technique allows the unique possibility of studying localized embolism of plant conduits, and thus to test hypotheses on the hydraulic architecture of plants. Based on our results, we postulate that the source of UAE is a transient "cavity oscillation" triggered by the disruptive effect of cavitation inception., Comment: 8 pages, 5 figures

We consider active, semi-supervised learning in an offline transductive setting. We show that a previously proposed error bound for active learning on undirected weighted graphs can be generalized by replacing graph cut with an arbitrary symmetric submodular function. Arbitrary non-symmetric submodular functions can be used via symmetrization. Different choices of submodular functions give different versions of the error bound that are appropriate for different kinds of problems. Moreover, the bound is deterministic and holds for adversarially chosen labels. We show exactly minimizing this error bound is NP-complete. However, we also introduce for any submodular function an associated active semi-supervised learning method that approximately minimizes the corresponding error bound. We show that the error bound is tight in the sense that there is no other bound of the same form which is better. Our theoretical results are supported by experiments on real data.

We introduce a natural generalization of submodular set cover and exact active learning with a finite hypothesis class (query learning). We call this new problem interactive submodular set cover. Applications include advertising in social networks with hidden information. We give an approximation guarantee for a novel greedy algorithm and give a hardness of approximation result which matches up to constant factors. We also discuss negative results for simpler approaches and present encouraging early experimental results., Comment: 15 pages, 1 figure

We analyze the expected cost of a greedy active learning algorithm. Our analysis extends previous work to a more general setting in which different queries have different costs. Moreover, queries may have more than two possible responses and the distribution over hypotheses may be non uniform. Specific applications include active learning with label costs, active learning for multiclass and partial label queries, and batch mode active learning. We also discuss an approximate version of interest when there are very many queries., Comment: 14 pages, 2 figures